THIS BOOK IS WITH TEARED PAGES THE TEXT IS FLY WITHIN THE BOOK ONLY '-c c ^ ^yfr/rt/'e-w/fra'Cyt 'w//. PBINTBD BY 0. J. CLAY, M.JL AT THE UNIVERSITY PRESS. THE MATHEMATICAL AND OTHER WRITINGS ROBERT LESLIE ELLIS, M.A. LATE FELLOW OF TRINITY COLLEGE, CAMBRIDGE. EDITED BY WILLIAM WALTON, M.A. TRINITY 'COLLEGE ; MATHEMATICAL LECTURER AT MAGDALENE COLLEGE, CAMBRIDGE. a i3fograpi)fcal JWemott BY THE VERY REVEREND HARVEY GOODWIN, D.D. BEAK OF ELY. CAMBBIDGKE : DEIGHTON, BELL, AND CO. LONDON: BELL AND DALDY. 1863. DEDICATORY LETTEE. DEAR LADY AFFLECK, Having undertaken with your sanction the publication of the writings of your lamented brother in a collected form, I may be permitted to address to you a few observations on the principles by which I have been guided in the fulfilment of an act of piety to the memory of a friend of many years, whom I shall ever remember with affection and veneration. The greater part of the writings contained in this volume had previously appeared in Scientific Journals, and Keports, some few in Educational Treatises. I have had no scruple whatever in reprinting these works, since their publication had already taken place by the author's own act or permission. The question of the publication of manuscripts presented to my mind greater difficulty. During his long years of suffering he was in the habit of dictating to his friends various speculations in adaptation to their different tastes and pursuits, evidently for the most part not intended for the press. A great many mathe- matical investigations, which may be described as interesting problems, communicated to me on numerous occasions either by dictation or by letter, are in my possession. Out of these ma- nuscripts I have ventured to publish only two, the one on the Betardation of Sunrise, and the other a new ^ Solution of a Problem in the First Book of Newton's Principia. All the manuscripts entrusted to me by you, classical, philological, botanical, and mathematical, works of a more elaborate cha- racter, I have not hesitated to include in this volume, from a conviction that they will be interesting to many readers, and vi DEDICATORY LETTER, from an impression, grounded on internal evidence, that the pro- bability of their ultimate publication may have been contem- plated by their author. The difficulty under which an amanuensis labours, in trans- ferring accurately to paper the words of one afflicted by severe illness, is at all times considerable. In the manuscripts placed in my hands the errors were necessarily, by reason of the pecu- liar nature of the subjects, very numerous. The obstacles, arising from this source, in the way of preparing for the press some portions of the work with proper emendations, I should have regarded as not a little formidable, had it not been for the zealous assistance of Mr Munro, Fellow of Trinity College, to whom I am indebted for the corrections of the text in all the philological and classical writings which had not previously been published. I may mention also that to the Dean of Ely, who at my request and with your entire approbation undertook most heartily the composition of the Biographical Memoir, I am under obligation for occasional advice, and for much kindly interest in the progress of the work through the press. The engraving has been taken from an admirable portrait by Samuel Lawrence, in the possession of Professor Grote, by his kind permission. Hoping that I have adequately discharged my duties as Editor of this Collection of your brother's writings, and at any rate conscious that I have done my best, I beg to dedicate to you this volume, and to subscribe myself, Tour faithful servant, WILLIAM WALTON. CHBSTIOBTOK, Oct. 26, 1863. CONTENTS. PAGE BlOGBAPHIOAL MEMOIB ix On the Foundations of the Theory of Probabilities .... i On the Method of Least Squares . . . . . . . . 12 Some Remarks on the Theory of Matter 38 Remarks on the Fundamental Principle of the Theory of Probabilities 49 Remarks on an Alleged Proof of the Method of Least Squares , . 53 Note to a Former Paper on an Alleged Proof of the Method of Least Squares "62 On some Properties of the Parabola 63 On the Existence of a Relation among the Co-efficients of the Equation of the Squares of the Differences of the Roots of an Equation . 68 On the Achromatism of Eye- Pieces of Telescopes and Microscopes . 71 On the Condition of Equilibrium of a System of Mutually Attractive Fluid Particles .......... 75 Mathematical Notes . . . .81, 92, 130, 142, 149, 157, 197, 223 "Variation of Node and Inclination . ...... 82 Investigation of the Aberration in Right Ascension and Declination . 84 On the Lines of Curvature on an Ellipsoid ..... 86 On the Tautochrone in a Resisting Medium ...... 94 On the Integration of Certain Differential Equations. No. I. . . 97 No. II. . . 108 Analytical Demonstrations of Dr Matthew Stewart's Theorems . 118 Note on a Definite Integral , . 124 Remark on the Distinction between Algebraical and Functional Equa- tions ,..,... 126 On the Solution of Functional Differential Equations , . . 132 Evaluation of Certain Definite Integrals . , . . . . 143 On the Evaluation of Definite Multiple Integrals . , . . 150 Note on a Definite Multiple Integral ...... 160 Notes on Magnetism. No. I .163 No. II 186 On a Multiple Definite Integral 169 On a Question in the Theory of Probabilities 173 On the Balance of the Chronometer . . , . . . . 180 viii CONTENTS. PAGE Memoir of the late D. F. GBEGORY,, M.A., Follow of Trinity College, Cambridge 193 On the Solution of Equations in Finite Differences .... 202 General Theorems on Multiple Integrals 212 On the Area of the Cycloid 224 Sur leg Inte'grales aux Differences Finies , * . . , , 326 Keport on the Kecent Progress of Analysis (Theory of the Comparison of Transcendentals) 238 Solution of a Dynamical Problem 324 On the Tautochronism of the Cycloid 326 On Napier's Bules 3^8 On the Retardation of Sunrise 33$ A Solution of Problem IX. of the First Book of Newton's Prindpia . 337 On Boman Aqueducts 339 On the Form of Bees' Cells ...,...,, 353 On the Theory of Vegetable Spirals * 358 Some Thoughts on Comparative Metrology 37$ Notes on Boole's Laws of Thought ....... 391 [Remarks on Certain Words in Diez's Efcymologisches Worfcerbuch dor Bomanischen Sprachen .*....., 395 Some Thoughts on the Formation of a Chinese Dictionary , , 400 Value of Boman Money 4115 The Course of Mathematical Studies , . . . . . 417 EBBATUM. p. i, 4, 5. For Bernouilli, read Bernoulli, BIOGRAPHICAL MEMOIR ROBERT LESLIE ELLIS, M.A. LATE FELLOW OF TEINITY COLLEGE, OAMBEIDGE, BY HAEVET GOODWIN, D.D. BBAJT 01 ELI, "II avait une promptitude infinie h tout saisir, uno m&noire prodigious ct uno faculty me'thoclique et rectifiante pour tirer, comme par une chiniio natnrollo, quelque chose de pre*cieux do tout ce qui a'otfrait h lui, soit dans h conversation, soit dans la lecture. Tout sujet d'ontretien lui 4tait bon ; il accoptnit volontiors celui qu'on mettait sur le tapis, et il <^tonnait les indiffdrents par les trdsors qu'il tirait ^ Pinstaiit do la roine qu'ils lui avaient offerte sans y soiigor. Son esprit ^tait comme une bibliotlieque encyclop<5dique Uen ordonn^e, qu'il suffisait d'ou- vrir a la lettre qu'on voulait, pour en faire sorfcir des ricliesses," BIOGRAPHICAL MEMOIR ROBERT LESLIE ELLIS. TITE publication in a collected form of the papers which this volume contains is to be regarded, to a certain extent, as a tribute of affection. From this point of view the work would hardly seem to Tbe complete without some notice of the life and character of the author. That he was a man of no ordi- nary attainments, in at least one field of knowledge, will be sufficiently evident to those who know no more of him than they can gather from this portion of his writings ; that he had powers distinct from those of mathematical research, is evident from what he has done as the editor of the philosophical works of Bacon ; but even these published records of his intellect will perhaps fail to convey to readers that impression of remarkable and various ability which was made, I believe, upon all those who were brought into personal contact with him. It may therefore be interesting to the general reader, besides completing the memorial character of the volume, if an attempt be made by one of his contemporaries to give some account of what he was. The mere facts of the life are few and simple. The ex- ternal picture of it may be very easily drawn. It was short, xii BIOGRAPHICAL MEMOIR quiet, uneventful, but -very full of suffering. The plan which I shall adopt in the following memoir will "be this: I shall first give the story of the life in as compact a form as may be possible, and then endeavour to lay before the reader some estimate of the mind and character. EGBERT LESLIE ELLIS was born at Bath, August 25, 1817, being the youngest of a family consisting of three sons and three daughters. His mother's health was not good, and from her he appears to have inherited that highly nervous constitution, which became, during a considerable portion of his life, as we shall see hereafter, the medium of great suffering. Ills father was a man of cheerful disposition, of active and well cultivated intellect, fond of speculative inquiry, and in worldly circum- stances independent. His character and his mode of dealing with Robert, as a child, had a great influence upon him through- out his life : he became his father's companion from a very early age, and the affection with which he referred in later life to his father's care and to the happy days of his boyhood, could not fail to strike those who had the pleasure of know- ing him intimately, I do not find that as a child he exhibited any extraordinary symptoms of precocity 1 , though it is manifest, from records of his boyish doings made by himself, that he was very forward in his studies, and that he took an interest in his work, and 1 With reference to what is said in the text, and possibly the reader may think in contradiction to it, 1 insert here a memorandum, which I find, amongst the papers intrusted to me, and which appears to be in his father's hand, "The following numerical theorem, if not curious in itself, may perhaps bo esteemed so, as coming from a boy of eight years old, who was not far advanced in the ordinary rules of arithmetic. " If any number be added to its equal,, subtracted from its equal, multiplied by its equal, and divided by its equal, then the sum, the difference, the ffrodttct, and the quotient of these equal numbers, added together, will equal the square of the next higher number." That is to say, if n be the number, (n+n) + (n-n) OF ROBERT LESLIE ELLIS. xiii exerted his mind upon the subjects to which it was directed in a manner by no means usual. He was never at school, but had the advantage of two tutors at Bath, one in classics, the other in mathematics 1 . He worked for them with great earnestness, and I find from his own memoranda, that in the year 1827, when he was about ten years old, he was doing equations, and reading Xenophon and Virgil, besides giving some attention to French and drawing. These same memoranda shew that at this time, in addition to his ordinary work with his tutors, he was reading books not usually read by boys at such an age, Cuvier's Theory of the Earth, The Edinburgh Journal of Science, The Edinburgh, Review, &c. One remark which is suggested by the boyish records left behind him is, that it is clear that from an early age Ellis had an extreme delight in knowledge for its own sake : he had not the ordinary stimulus of school emulation, indeed he was singu- larly free from the influence of competition until his college days : but it is manifest, from his own account, that his pro- gress in knowledge, and perhaps especially in mathematical knowledge, was a source of very keen delight. In 1829, that is, when twelve years old, he began to read Mechanics. In the early part of 1830 he commenced the Dif- ferential Calculus; from which he rapidly proceeded to the Integral Calculus; and towards the middle of the year lie speaks of being engaged with his tutor in finding the lengths and areas of curves. Meanwhile his general reading, for which he was dependent upon his father's library and upon that of the Bath Institution, was most multifarious; but each particular subject seems to have been carefully studied, and an opinion formed upon it. Thus his education proceeded quietly and also rapidly under his father and private tutors for several years. 1 His mathematical tutor was Mr T. S. Davies, afterwards of Woolwich ; his classical, Mr H, A. S. Johnstone. xiv BIOGRAPHICAL MEMOIR This home education had, I think, a perceptible effect upon his future character. The effect was not bad in the sense in which that epithet is generally believed to be applicable to home education ; but there might be observed in him a kind of elderly sobriety of manner, not amounting to stiffness, but con- veying the impression that he had been accustomed to converse with those older than himself, and standing out in marked contrast with that lively boyish freedom and gaiety which is especially the characteristic of young men educated at the great public schools. In October, 1834, he became the pupil of the Kcv. James Challis, then Eector of Papworth St Everarcl, in Cambridge- shire, who soon after was appointed and still remains Plumian Professor of Astronomy in the University of Cambridge. His residence at Papworth was, however, very short; his health gave way, and at the end of six weeks he was compelled to return home. Here he remained for about two years, not coming up to the University in 1835, as originally intended, but postponing the event, on the ground of health, to the follow- ing year. He came into residence as a Pensioner of Trinity College, in October 1836, being entered as a pupil of the Kcv. G< Peacock, afterwards Lowndean Professor and Dean of Ely* During his undergraduate career his health was not strong, but I think he was never compelled by illness to desist from his course of study. He was very much in advance of the men of his year in mathematical acquirement, and had already read most of the subjects which usually occupy an undergraduate'** time. He was himself much amused at the surprise expressed by his tutor, Mr Peacock, when at an early stage of his College life in answer to the question, " What are you chiefly reading now?" he replied, " Woodhousc's Isoporimctrical Problems." He read mathematics chiefly without the aid of a private tutor, but in his third year and his last term had the advantage of OF ROBERT LESLIE ELLIS. xv Mr Hopkins' s direction 1 . I was myself a pupil of Mr Hopkins' at the time ; "but Ellis never read with the class of which I was one; in fact, he did not need the kind of lecture which was adapted to myself and others ; he required only that his reading should "be arranged, and put in a form suitable for the Cam- bridge examinations. The only occasion upon which I was "brought into contact with him as a fellow-student was in attending Professor Peacock's lectures on Plane Astronomy. I remember well the astonish- ment with which I witnessed his demeanour during the lectures: he made no note, he asked no question ; but lie quietly remarked as we left the lecture-room together one day, "It saves one the trouble of reading these things up." It was in fact a great advantage to him to be able to sub- stitute the use of his ears for that of his eyes. His sight was very tender, and during the latter part of his undergraduate career he regularly employed a person to read to him high mathematical subjects. He once mentioned to me incidentally that the theory of the Earth's Figure, as given in Pratt's Mechanical Philosophy, was in this manner read to him ; which instance I herp record as an indication of a power of mental effort possible to very few, and the magnitude of which ma- thematicians will appreciate. During his undergraduate career I was not intimately ac- quainted with him : probably he had no desire to increase his circle of friends beyond that which was naturally brought round him in his own college: and his manner was not such as to encourage rapid intimacy, I do not think that at this period the number of his intimate friends was large even within Trinity College, and sometimes a feeling of desolation and want of 1 In a note to me Mr Hopkins says, with, reference to his recollections of Ellis as a pupil, " On one point lie always seemed to puzzle me. The extent and deftniteness of his acquirement, and his maturity of thought, were so great, so entirely pertaining to the man, that I could hardly conceive when he could have been a boy." 52 xvi BIOGRAPHICAL MEMOIR sympathy oppressed him painfully. Pie once described to me the forcible manner in which he was affected in College Chapel by those words of the Psalm, " I had no place to flee unto, and no man cared for my soul." This melancholy feeling, which sometimes assumed a very painful intensity, was no doubt con- nected with the weak state of his bodily health, and the highly nervous temperament which naturally belonged to him : it was the source of much suffering, I fear, even in that period which preceded the most distressing portion of his life. Ellis was t6 be seen sometimes at the debates of the Union Society, but he seldom took any part in them. On one or two occasions, however, when domestic troubles had arisen, and squabbles of a somewhat personal kind ran high, he stood up as a pacificator; he was heard with marked respect, and his suggestions were readily adopted. In January, 1840, he passed his examination for B.A. degree. In consequence of complaints which had been made of the coldness ,of the Schools, in which the Candidates for Mathematical Honours were then examined, the examination took place for two or three years in the Lecture Booms of Trinity College. The difficulty was afterwards solved by tho proper warming of the Senate-House. When we visited the rooms on the day before the Examination to inspect our places, I found that the alphabetical arrangement of names combined with the conditions imposed by the length of the tables had brought Ellis and myself almost immediately opposite to each other, and I was rather pleased with the thought of seeing him in actual work. He however made a special request that his seat might be changed, (I do not exactly know why,) and was .allowed to be placed in a different room ; so that I saw nothing of him during the examination. Those who knew anything of the relative powers of the men of the year had no doubt as to which place Ellis must occupy, if only his health should enable him to do himself OF ROBERT LESLIE ELLIS. xvii justice In the examination. His health of course introduced an element of uncertainty; but when the examination was con- cluded, and it was found that he had been able to take every paper, the result was quite sure. He was Senior Wrangler; and I can truly say, that for myself I had almost as much satisfaction in seeing his name at the top of the list, as in seeing my own next to It, for I felt convinced that it was his rightful position, and that nothing but the accident of ill health could have put any of his competitors above or even near him, His appearance in the Senate-House when he took his degree was very striking. He looked very pale and ill, but this perhaps enhanced the intellectual beauty of his counte- nance. A person who was present remarked to me very pithily, "If I had seen him before, I could have told you you could not beat him." In October, 1840, he was elected Fellow of Trinity College. He retained his fellowship until the year 1849, that is, for seven years after the degree of M.A., when as a layman he ceased to be a fellow in due course. His intention after taking his degree was to read for the bar, and at one time there was a notion of his entering upon political life by becoming a candidate for his native city Bath. His name was publicly discussed with reference to the election, but the design was given up on the ground of the weakness of his health. Had he been a candidate, it would have been on Whig principles; he. was not a very earnest politician, but always professed himself a Whig 1 , a profession which was probably strengthened by his intimacy with Sir William Napier, to whom he always expressed himself as much at- tached. With regard to the bar, he was duly called, but did not study long with the intention of practising. The fact is that hia worldly position was unexpectedly altered. Both of his xviii BIOaRAPHICAL MEMOIR elder brothers died, and lie thus "became heir in expectation of considerable property, and soon by the death of his father heir in possession. He was thus deprived of the chief induce- ment to labour as a lawyer; and had it been otherwise, it is clear that his health would never have enabled him to undergo the necessary drudgery. Nor indeed would the actual practice of law-courts have been very congenial to his feelings and tastes: law in the abstract he loved exceedingly, as we shall see presently, but law as it is concerned with the actual strifes and quarrels of mankind would have been eminently distasteful to him. As a Fellow of Trinity he made his College his home, except for a short period after his election. Here he continued his mathematical reading, but not with any very definite pur- pose. He became very intimate with the late D. If. Gregory, who was then Fellow of Trinity College, and who did good service to mathematics by the establishment of the CamlridyG Mathematical Journal; when Gregory resigned the editorship shortly before his death, Ellis took the office, and edited part of the third and fourth volumes of the journal, in the klter of which he inserted a short biographical memoir of his friend 1 . In January, 1844, Ellis was Moderator, On this occasion, being myself one of the Examiners, I was thrown into closer relations with him than before, and commenced that real inti- macy which lasted as long as his life. His problem paper on this occasion was singularly elegant, but perhaps too refined for its purpose. His fellow moderator, O'Brion of Oaius Col- 1 Elite's name appears as editor on tho titlo-pa^o of the fourth volume of tho Journal. I may take this opportunity of observing that tho parollul drawn be- tween Gregory and Ellis in a very kind and warmhearted obituary notice of tho latter, which was inserted in the Atfi&mm of Pel), n, 1860, mwitut to wo not justified. They resembled each other, no doubt, iu tho fact that both wre good mathematicians and both real philosophers and lovers of truth; but beyond this very general resemblance the parallel does not hold. They wwo much attached, and Ellia felt the loss of his Mend keenly ; but neither in mind new in manner was there much likeness between them, OF ROBERT LESLIE ELLIS. xix lege, and he, published their problems with their own. solutions, soon after the Examination. He bore the labour of the Exami- nation better than could have been expected; he was a very pleasant workfellow, being always ready to fill up the intervals of work with that rich and varied conversation which his friends remember so well. Shall I be pardoned if I mention that his fees as Moderator were transferred to Addenbrooke's Hospital? , He worked for the Senate-House again in 1845. It is cus- tomary for the Moderator of one year to act as Examiner the next 1 , but he was desirous of escaping the labour, and had declined to serve. There was, however, a difficulty in finding a substitute ; I was myself one of the Moderators and felt anxious that he should serve, which at my earnest entreaty he at length consented to do. It was in this year that Professor W. Thorn- sou took his degree ; great expectations had been excited con- cerning him, and I remember Ellis remarking to me with a smile, "You and I are just about fit to mend his pens." He again got through the Examination much better than could have been expected ; in fact, the effort seemed to do him good; when he had consented to act he said, " I feel all the better for having done something plucky," It has already been mentioned that the study of the prin- ciples of Law was very agreeable to him. At this period of his life he devotedjmuch time to the study of the Civil Law, and he Las left behind him several volumes of notes made in the course of his reading. The only appointment concerning which I ever heard him express any strong wish was that of the Professor- * It may be mentioned, for the benefit of readers not acquainted with Cam- bridge customs, that the distinction between Moderator and Examiner is prac- tically merely this, that the papers of original problems are set wholly by the Moderators. Constitutionally the difference IB, that the office of Moderator is an old statutable office, whereas the Examiners were added by Grace of the Senate, in consequence of the increase of the number of candidates for mathema- tical honours, xx BIOGRAPHICAL MEMOIR ship of Civil Law. He acknowledged to me that lie would have felt gratified Tby the tenure of this office, which is the more remarkable when taken in connection with the fact that on the occasion of one of the Mathematical Professorships "being vacant he expressed no desire to Tbe appointed, but, on the contrary, declared that he would not consent to be nominated as a candi- date. Indeed it is a mistake to suppose that Ellis was in any exclusive or even preponderating degree devoted to mathematics: his mathematical power was no doubt very great, but I think not greater than several other powers, and certainly his taste by no means exclusively leaned in this direction, as his intimate friends very well knew. But of this more hereafter* He did not give himself in any degree to tuition during his Cambridge residence. So far as I know he never had a private pupil ; he gave a few College Lectures upon high mathematical subjects, but he did this only as locum-tenens for friends upon whom the task devolved. Probably his health would have interfered with any regular occupation of this kind ; but besides this, he had not,. I think, any taste or any special fitness for imparting knowledge to average minds; his remarks were always suggestive, and he could throw light upon almost any subject which could be brought forward, but he usually assumed a considerable amount of knowledge on the part of those with whom he conversed, and sometimes (as it seemed to me) he was obscure, in consequence perhaps of the neatness and conciseness which were so remarkable in his conversation. At the request of the British Association, which held its annual meeting at Cambridge in 1845, ho undertook a Iteport upon the progress of certain branches of pure mathematics. This Eeport is reprinted in the present volume. It represents a great amount of labour and research, and I have no doubt that the preparation of it was a source of pleasure to him, as it refers to a department of mathematics which was with Ellis a special favourite. OF ROBERT LESLIE ELLIS. xxi It was during his residence as a Fellow in Trinity College that lie undertook, in conjunction witli Mr James Spedding and Mr Douglas Denon Heath, to edit the works of Bacon. The philosophical section of the works was the share allotted to Ellis. No literary occupation could have been more congenial to his taste, and the prefaces to the several treatises which he was able more or less to complete, especially the " General Preface to the Philosophical Works," are perhaps the most valuable thing which he has left behind him. He was engaged upon the preface to the Novum Organum, when he was stopped by illness ; and so complete and sudden was the break in his health that he ;never completed it. The last sentence that he wrote will be found on page 100 of the first volume of his edition of Bacon's Works. It is an affecting monument, and as such I here produce it "Again he affirms that he does not inculcate, as some might suppose, a ;" to which Mr Spedding has appended a note, " Mr Ellis had written thus far when the fever seized him." The mention of Bacon has led me to anticipate the course of events. In the years 1847 and 1848 he visited Malvern for the benefit of his health, still making Trinity College his head- quarters, and he certainly appeared to be strengthened by the course of treatment to which he was submitted. He had, I believe, always intended, at the expiration of the tenure of his fellowship, to go abroad ; partly perhaps for the general advan- tages of travel, and partly with the belief that his health would be improved by residence in a warmer climate. He desired to settle himself in some place possessing a good library, where he might complete his work for the edition of Bacon. Accordingly in the autumn of 1849 he went to Nice. After remaining there some little time he started, not well in health, for the journey by post along the Riviera. The first night he slept at Mentone, and, as he believed, in a damp Tbed. The next day he arrived early at S. Eemo, but feeling indis- posed determined to proceed no further that day. In the xxii BIOGRAPHICAL MEMOIR evening he took the last walk which lie was ever aTble to take, otherwise than as a cripple ; he described afterwards to one of his friends the profound effect produced upon his mind, possibly rendered more sensitive by approaching illness, by the loveli- ness of the scene. That night, which was one of horrors to him, he was seized with a rheumatic fever, which for several days put his life in great danger. A physician, who was called in, seems to have exerted himself with great kindness and to the utmost of his skill, to do all that could be done. Ellis always retained an affectionate remembrance of him. He ordered his patient to be bled extensively, and after a few days the imminent danger was passed. Rheumatism however re- mained fixed hopelessly upon him; he was ever after in con- stant pain, with very little use of any part of his body ; and the rest of his life, ten years, may be described as a long^ process of gradual dissolution. After a residence of nearly three months at S. Homo, he was brought home by easy stages. He visited several places after his return, London, Brighton, Bath, Malvern, Tunbridge Wells, consulting various physicians, with no apparent result. At length giving up all hope of amendment, he fixed himself in 1853 at Trumpington, a village two miles from Cambridge, of which his friend Professor Grote was Vicar, for the sake of being near the University and his old friends. When he arrived he was unable to walk, but could drive out in a carriage, and in the house he could move from one place to another on the same floor by means of a chair set upon wheels: after some time however he became entirely confined to the house ; then to his bed, where he remained in a sadly suffering condition till the day of his death. He wrote me a letter shortly before his arrival at Trumping- ton, telling me that he had taken Anstey Hall (the name of his house 1 at Trumpington), and adding that he was coming 1 When. Ellis first came into residence at Cambridge "kin father had some OF ROBERT LESLIE ELLIS. xxiii to leave liis bones amongst us, and lie trusted we should " give him a little earth for charity." During his lingering illness I saw him not unfrequently, and am able to record, from personal recollection, some few things which may be pleasant to those who knew him, and not unprofitable perhaps to the general readers of this volume. In the early [part of his residence in Anstey Hall, he was well enough to enjoy"" the society of his friends to a very considerable extent ; he sat in his invalid chair, with sometimes two or three persons present, pouring forth his varied stores of knowledge as in olden days ; in fact, it should be stated here once for all, that during the whole of his lingering sickness his mental powers never ap- peared to be in the smallest degree impaired ; it was a wonder to note the perfect action of the mind, at a time when the body was a mere distorted and attenuated heap of skin and bones. But this brightness of intellect doubtless made the suffering more acute. His life for several years was a constant looking of death in the face, with scarcely an interval of ease or ob- liviousness. By degrees > the resource of the society of his friends began to be diminished ; frequently we called and found him unable to see us ; and even relatives staying in the house could not be admitted into his chamber for days together* In the earlier part of his illness he was able to give some attention, but not much, to Bacon; some of the notes which have been since printed were dictated at this time. The thought of leaving his work imperfect could not fail to be painful to him, and the pain would be increased by the very high standard of excellence which he set up for himself in all matters which he undertook. Latterly he could not bear the subject of Bacon to be alluded to: if it happened to be introduced, he would intention of engaging Anstey Hall, in order that lie might be near his son, whose health even then was, as we have seen, not strong* This circumstance had made Ellis always feel an interest in the house. xxiv BIOGRAPHICAL MEMOIR say, "We don't talk about It in this room." He amused timself also with mathematical investigations, which he was able to carry on in a remarkable manner, without paper or figure, "in his head," as the common phrase is. It was in this way that he discovered for himself what he believed to be a new view of Napier's rules for the solution of right- angled spherical triangles. His discovery involves so curious a piece of history that I shall venture for a moment to dwell upon it. Ellis found out in his illness, that Napier's rules, instead of being, as they have been stated to be in Cam- bridge books, from Professor Woodhouse downwards, ' a mere memorm tec7inica, were all capable of being deduced from one geometrical construction. He sent me a paper which he dic- tated on the subject, and which I requested him to allow me to communicate to the Cambridge Philosophical Society, Hav- ing gained his permission, I thought it well to examine the literature of the subject, and above all to see what Napier had himself said. On turning to Napier's famous tract, Miri- fid Logaritlimorum Canonis Descriptio, I found that Ellis had in fact rediscovered Napier's own original conception of the problem. It was during his , illness that he dictated his remarks con- cerning the construction of bees' cells ; I believe also that he thought out at this time his demonstration of the tautochronous quality of the cycloid. Indeed he had usually some mathe- matical question running in his head, which served him for recreation during his easier moments. Nor were other subjects excluded; it was in this season of extreme bodily weakness that he corresponded with the late Dr Gilly on the Romaunce language, discussed the date of the " Noble Lesson," and criti- cized Dr Gilly's edition of the Vaudois Gospel of S. John. I have also before me a considerable number of letters dictated to friends, dealing with subjects so different, according to the tastes of the persons to whom they were sent, that it seems difficult OF ROBERT LESLIE ELLIS. xxv to believe them to be the production of the same person. He dictated also papers on Vegetable spirals, on Comparative Me- trology, and on various points of Etymology. But the most remarkable effort of his illness was the dic- tation of a pamphlet (reprinted in this volume) on the subject of a Chinese Dictionary, and the best mode of constructing such a work. This pamphlet was in the form of a letter to the Rev, J. Power, the Librarian of the University, who had kindly as- sisted him in his literary researches, and had supplied him with the most recent literature bearing upon the Chinese language. I can give no opinion of its value, but can hardly be wrong in regarding it as a marvellous exertion of mental vigour under very depressing conditions. In truth his taste for language was as marked as that for mathematics. I have just now remarked incidentally upon his study of the Eomaunce language ; Gothic also appears from his letters as having received considerable attention; he has left a paper on Sanscrit; and amongst the modern languages, besides the usual acquirements of French and German, I per- ceive that he was well versed in Italian, and that he had given attention to Banish and Spanish. He thoroughly enjoyed the study of a language, and I remember very well the pleasure which he expressed at having had the courage to communicate a French memoir to Liouville's Journal: it seemed to me that the writing mathematics in a foreign language gave him al- most as much satisfaction as the mathematical results them- selves *. I ought, perhaps, to mention that I found him one day reading the New Testament in Swedish, which he told me he had "picked up" since he had been ill 2 . I have said that through his long illness Ellis retained his 1 He was very fond of translating. Amongst his papers are some translations of Danish ballads, Spanish ballads, Andersen's Tales, &c. 2 During his illness his notes to his physician were usually written in Latin. xxvi BIOG-RAPHICAL MEMOIR vigour of mind. It is wonderful that in such suffering, and in the consciousness of approaching dissolution the mind should have "been capable of dwelling calmly upon subjects of abstract interest, such as investigations in pure mathematics ; he himself felt that there was something strange iu the occupation. In a note accompanying a mathematical paper he writes to me ; " I have been very miserable all this week. God will mend it, when His will is. It seems strange that my mind still runs at all upon triangles, and I am not at all sure that it is right it should. I need not tell you to think charitably of me in this as in other respects." His mind by no means however dwelt upon triangles to the exclusion of more solemn subjects, as I shall have occasion to shew more fully presently ; but it ap- peared to have a vigour of action and a fulness of matter which no external circumstances could affect, and so far as my obser- vation went he conversed with the same facility and command of his subject during his illness, as in earlier days. It would be only painful to draw as vivid a picture as might be easily drawn of the protracted sufferings which he had to endure. Medicine could do nothing more for him than mitigate the severity of the disease, which seemed to claim as its own one muscle after another, as it slowly approached the heart. " One twitch i/wre" said he to me one day, speaking of his heart, "and then I shall know the great secret." For a con- siderable period, reading (as might be supposed) was a relief; the weary hours of night, sometimes rendered horrible by the fear of dreams if sleep should come upon him 1 , were beguiled with books, a lamp suspended over his head giving him the necessary light. By and bye however this resource also failed : the eyes began to be affected by the complaint, and for about 1 He one day represented Ins condition to mo in words ctmoiiHly resembling those of Job; "When I say, My bed shall comfort nio, any couch shall cuiso my complaint ; then Thou scarest me with dreams, and torrifieat mo through visional* Job vii. 13, 14. OF ROBERT LESLIE ELLIS. xxvii two years before his death he was almost entirely "blind 1 . Books were read to him, and he dictated occasionally to an amanuensis, but the loss of sight was a very severe addition to his sufferings. The following lines, which may be taken as a sample of the love of epigram 2 which belonged to him, were sent to Dr Paget, his physician, when the blindness was gaming upon him, Contortos artus nunc culcita celat, at olim Terra teget melius : sit modo et ilia levis. Et quam vix posstmt oculi tolerare dolentes Lux fugit, ac tenebris mox adopertus ero. Mar. 9, 1857. In the earlier part of his illness, I made the remark one day that he appeared to me to be a little better; he at once said 1 His eyes were first attacked in April, 1856 ; he was unable to read in July, 1857. 2 This love of epigram was very striking. Here is an instance. During Ms illness an old friend wrote to him. asking 1 him. for some new conundrums. It so happened that on the day of receiving this request he fancied that he had disco- vered from I)r Paget, that ho was labouring under Blight's disease ; he sent the following answer : Si petis Line oonigroa novum, si ludicra poscis, Quod nuper didici scribere cur dubitem ? Morbus, qui clarurn fecit qui nomine clams Semper erat, solvot vincula queis teneor. On the same day he wrote on the same subject in a different style : DEAB PAQBST, I think it well to thank you for your most kind note. It came a few minutes before my dinner. That over, I told W * its purport, and desired him to take notice how little it disturbed me. Of course, no such communication can ever be matter of indifference, and least of all to a person like me, in whom the power of suffering and of being anxious has been but little impaired by years of suffering and of -anxiety. But, to use John Bradford's words, "He who has helped me till now will not leave mo when I have most need, for His truth and mercy sake ;" and of neces- sity 1 am less anxious about many things than I have long been. Yours, E. L, ELLIS. m. 19, (1857). * His servant. xxviii BIOGRAPHICAL MEMOIR very earnestly, " pray do not say so !" He recognized from the first the sure character of his disease, and he desired that his sufferings might not be protracted ; at the same time he was perfectly patient, even cheerful, and reverently acknowledged in all his afflictions the governing hand of God. But it is time that I should leave the story of my friend's sickness, upon which affectionate remembrance tempts me to linger, in order that I may endeavour to give such estimate as I can of his mind and character. Speaking generally I should say that his intellect was the most remarkable that I have known. It was made up of a combination of powers so delicately balanced, and working together in such perfect harmony, that it would be difficult to say that any one predominated over the rest. Popularly in Cambridge he might be regarded as specially a mathematician, because he was Senior Wrangler, but those who knew anything. of him were fully aware that mathematics was only one of his acquirements, and that in conversation mathematics by no means presented itself as the chief or even the favourite subject of his thoughts. Indeed, I think, it would be difficult to say that there was any one subject in which his mental powers were manifested more decidedly than in others ; and the only marked deficiency which I ever detected was in respect of music, for which he had no special taste. I do not mean that he had no sense of melody : this was far from being the case : but his con- versation never ran upon the great masters of music and their works, as it did upon almost all other subjects. Doubtless, however, mathematical power belonged to him in a very large degree. With the present volume in the reader's hands it would be superfluous to say much concerning the special departments of mathematical investigation in which his taste impelled him; but I may remark that he seemed most naturally to associate his mathematics with the past ; he OF ROBERT LESLIE ELLIS. xxix delighted to discuss the principles of investigations already known, to trace the history of processes, to examine the phi- losophy of a subject, to hunt up its literature, or to simplify its treatment. His memoir on the Foundations of the Theory of Probabilities and that on the Method of Least Squares, which stand at the opening of this volume, appear to me to represent as well as possible his special taste, so far as he had a special taste, with regard to mathematics. He always seemed to talk on the subject of Probabilities with great plea- sure, and as one in which he was thoroughly at home. The remarkable little essay on the Theory of Matter was also one which I think gave him much satisfaction. His taste did not seem to lead him much in the direction of elaborate phy- sical experiment, nor do I remember that on any occasion he worked in this path of investigation. His mind in fact was rather that of the philosopher than the physicist ; his impulse was rather to contemplate existing knowledge, than to take up a particular line of physical investigation and press forward knowledge upon that one line. This characteristic of mind gave a great charm to his con- versation : his thoughts were, so to speak, set in a rich historical framework, and they were produced with an ease and readiness which I have never seen equalled. In referring to his conver- sational powers generally, I may record the singular accuracy of his speech ; this was perhaps partly a natural gift, and partly the result of early education ; certainly it was very wonderful ; his sentences were not only full of thought and of references to literature of all kinds, but they were so remarkably correct in their construction and elegant in diction. He was one of the few men who could have borne a Boswell with great advantage to their reputation 1 , 1 This remark was made to me by one very intimate with him, and I cordially adopt it. In fact, with obviously wide differences, there was a good deal of curious xxx BIOGRAPHICAL MEMOIR He was a good scholar, and very fond of the Greek and Roman literature. I believe I am justified in saying, that his knowledge of that literature was really more extensive and thorough than that of many whose reputation as scholars has "been much greater. He could enjoy a discussion of a point of classical philology as keenly as one on scientific subjects, and when so engaged no one would have supposed that mathematics was his favourite study. Indeed, as 1 have already intimated, mathematics could not in any proper sense be so described: Civil Law was certainly as much a favourite : and he seemed to be most happy in conversation, when the subject was one of a philosophical character. I have often felt disposed to compare his mental constitution in many respects with that of Leibnitz. Each was the philosopher quite as emphatically as the mathematician. Leibnitz, I may observe, was one of his favourites, and he mentioned to me one day with sonic feeling of amusement that a Fellow of Trinity had spoken to him of Leibnitz, under the title " your Leibnitz," as though the old feeling of jealousy were still lurking in the College. His love of philosophy fitted him especially to be the editor of Bacon. It was a work, I believe, which he undertook with all his heart, and relinquished with extreme pain and only under a sense of imperious necessity. He was assisted too by his familiarity with the philosophical speculations of the middle ages: there was something congenial to his own cast of mind in the discussions of the schoolmen, and I think he appreciated Bacon all the more in virtue of his appreciation of those, whoso processes of thought and methods of argument it was Bacon's task to supersede. similarity between Ellis and Dr Johnson, Ellis was an excellent conversationalist : he not only expressed himself with singular precision, but he was a patient Bntenor, and readily caught up and retained in his memory the remarks of those with whom he conversed. OF ROBERT LESLIE ELLIS. xxxi In producing his knowledge he was much assisted "by the strength and clearness of his memory. All his knowledge seemed to be as it were in hand; it was not merely that he knew where to look for information, although that is a great thing and as much as many clever men are satisfied to accom- plish ; but information upon the most various subjects, the most trivial and the most important, seemed to be at call upon all occasions. His long painful illness had no apparent effect upon this faculty. On one occasion, some years after he had been confined to the house, he wished to describe a certain picture in a room which he had only once entered, and that merely for a morning call ; to my astonishment he mentioned the pictures one after another as they hung on the wall, and so identified that to which he desired to refer. Ellis was very fond of expressing his thoughts in verse. Some of his notebooks are full of poetical scraps, generally somewhat melancholy in their tone, but expressing (I have no doubt) the feelings of his mind at the time of writing. I will produce two or three of these scraps in this place: it will be seen that they all belong to the period preceding his last long illness. i. E'en in. the days when life is dear And we would fain live on for aye, Then, let it not forgotten be That death draws near. <2. And when we fall on sadder hours, And gladly would lie down and die, Remember that we live to do God's will; not oura. BIN ANT, 1846, Written April 9. The two following were written, I presume, in Trinity College. xxxiv BIOGRAPHICAL MEMOIR ter ; I say unjust, because the remarkable keenness of his mine! concerning mathematical and other questions, during an illness in which his life was hanging constantly by a thread, may give those who are disposed to do so the occasion of remarking that his mind might very well have been occupied with more solemn thoughts. Let such persons then be satisfied by knowing that more solemn thoughts did occupy his mind. I think that as his sickness advanced and his bodily powers were diminished, his mind gradually found more settled peace and rested more surely upon the love of God and the merits of the Saviour, Certainly there was much to tempt him to murmur, but I never noticed any murmuring propensity or any tendency to do otherwise than bow to God's will and accept with patience a mysterious and painful dispensation. In the early part of his illness, he asked me whether I had ever thought much or hoard a sermon upon Habakkuk iii. 17, 18: " Although the fig-tree shall not blossom, neither shall fruit be in the vines ; the labour of the olive shall fail, and the fields shall yield no meat ; the flock shall be cut off from the fold, and there shall be no herd in the stalls : yet I will rejoice in the Lord, I will joy in the God of my salvation." He made no application of the verses to his own circumstances, only remarking how .striking the language was ; but it was evident to me that his own case was in his mind. He always begged me to read prayers with him. 1 usually introduced the Collect from the Visitation of the Sick ; on one occasion I omitted it ; he noticed the omission at my next visit, and begged me to use it. Anyone who remembers the substance of that Collect will see the value of this simple anecdote. His own bodily weakness and utter abstraction from all works of active piety intensified his desire of doing something for the benefit of his fellow-creatures, and made him grieve over his forced indolence. I do not mean that his charitable feelings first germinated in his sick room : this was very far from being OF ROBERT LESLIE ELLIS. xxxv the case : but his sense of his own inability to discharge any active duty made him more keenly sensible of the privilege of being permitted to exert ourselves for God and for our brethren. On reading the account of the death of Captain Gardiner in his noble but not wisely-arranged effort to found a mission in Pata- gonia, he expressed a wish that his own life might have had a similar termination. Indeed a sense of the spiritual needs of mankind appeared to grow upon him as his own bodily weak- ness brought him nearer to the great realities of existence: I can never forget the earnestness with which he said to me at a late period of his illness, " The thing above all others which strikes me, as I lie here on my bed, is the intense wickedness of mankind." "I feel," he continued, "as if I should be con- strained, did God ever raise me up again, to rush in amongst them, as Barnabas and Paul did amongst the people of Lys- tra, and rend my clothes and say, Sirs! why do ye these things 1 ?" His speculative mind, acting under the peculiar conditions to which it was subjected by his diseased body, could hardly fail to look sometimes anxiously into the future, and guess what might be the nature of the life prepared for him in the world to which lie was brought so near : as time went on however the keen discipline of affliction seemed to have taught him that he must "stand and wait," and simply look forward with calm hope. 1 I ventured to introduce this reminiscence into a sermon preached before the University, March 22, 1863, and since published. I find a similar anecdote amongst some recollections, which have been kindly put in my hands by the Rev. J. P. Norris. Speaking of children, and his strong disapproval of giving them prizes for mere cleverness, he added, "There is another point connected with children that I feel with an intensity which I would give much to have felt years ago, the sacred duty of keeping them pure. Wounded Arthur, speaking to Sir Bedivere, threatens to arise and slay him with his hands if he fail in his behest ; and I feel sometimes as if I could arise from this bed and tear to shreds some of the books that are left in children's way." i BIOGRAPHICAL MEMOIR OF R. L, ELLIS. In fact it was impossible not to observe that throughout his illness lie perceived, that it was to "be regarded in the light of a divine discipline, however mysterious such discipline might be. "Aches and pains," he said to a friend, "have been my teachers of late." "Fiat voluntas Tua," he observed to the same friend, taking up his De Imitation OUnsti^ " is after all the only prayer. Domine, niodo sura, in tribulationc, et non eat cordi meo bene, sed multum vexor a prsosenti ptis&ione......Et nunc inter hsec quid dicam ? Domine, fiat voluntas Tua ; ego bene merui tribal ari et gravari." His last days were days of peace ; and his last words were so striking that I think it right to put them here upon record- It will be remembered that for a considerable period before MB death ho had been quite blind: just before his departure, which was in a certain sense sudden though so long expected, he ex claimed, " I see a light!" and so expired, Possibly some phy- sical explanation of this exclamation may be given : for myself I would rather look upon it as indicative of something spiritual, and as announcing the arrival of a glorious change for which his imprisoned soul had long waited and earnestly prayed \ He was released from his sufferings May 12, 1859, and was buried at Trumpington. His grave is at the South-East corner of the churchyard, and bears the simple inscription EGBERT LESLIE ELLIS BORN 25 AUGUST 1817 D1B1) n MAY 1859. BLESSED is raus MAN THAT HATH saa? HIS HOHS IN MIS LOUD, l l \ 40, V. 5, 1 He once observed that he wondered no on had over chosen, for an epitaph the words of Psalm cxvi, 14, "Thou hast broken my bonds in sunder,*' ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES*. THE Theory of Probabilities is at once a metaphysical and a mathematical science. The mathematical part of it has been fully developed, while, generally speaking, its metaphysical tendencies have not received much attention. This is the more remarkable, as they are in direct opposition to the views of the nature of knowledge, generally adopted at present, 2. The theory received its present form during the ascendancy of the school of Condillac. It rejects all reference to h priori truths as such, and attempts to establish them as mathematical deductions from the simple notion of probability. Are we prepared to admit, that our confidence in the regularity of nature is merely a corollary from Bernouilli's theorem? That until this theorem was published, mankind could give no account of convictions they had always held, and on which they had always acted? If we are not, what refutation have we to give? For these views are entitled to refutation, from the general reception they have met with, from the authority of the great writers by whom they were propounded, and even from the imposing form of the mathematical demonstration in which they are invested. * Transactions of the Cambridge Philosophical "Society, Yol. vnr. [Bead Feb. 14, i%2.] 1 2 ON THE FO UNDA TIONS OF I shall be satisfied if tie present essay does no more than call attention to the inconsistency of the theory of probabilities with any other than a sensational philosophy. 3. As the first principles of the mathematical theory are familiar to every one, I shall merely recapitulate them. If on a given trial, there is no reason to expect one event rather than another, they are said to be equally possible. The probability of an event is the number of equally pos- sible ways in which it may take place, divided by the total num- ber of such ways which may occur on the given trial. If a v b^ m v denote equally possible cases which may occur on one trial, a/^..'..../^ those which may occur on a second trial, a 8 Z> 8 .jp 8 those belonging to a third, &c. : then a^a z a/s.Aj &e. &c. are all equally possible complex results. Hence it follows that on the repetition of the same trial k times, the probability that an event whose simple probability is m will occur p times is this follows merely by the doctrine of combinations. These are all the propositions to which I shall have occasion to refer. 4. If the probability of a given event be correctly deter- mined, the event will, on a long run of trials, tend to recur with frequency proportional to this probability. This is generally proved mathematically. It seems to me to be true & priori. When on a single trial we expect one event rather than another, we necessarily believe that on a series of similar trials the former event will occur more frequently than the latter. The connection between these two things seems to mo to be an ultimate fact, or rather, for I would not be understood to deny the possibility of farther analysis to be a fact, the evidence of which must rest upon an appeal to consciousness. Let any one endeavour to frame a case in which he may expect one event on a single trial, and yet believe that on a scries of trials another will occur more frequently 5 or a case in which lie may be able THE THEOR Y OF PROBABILITIES. 3 to divest lilmself of the belief that tlie expected event will occur more frequently than any other. For myself, after giving a painful degree of attention to the point, I have been unable to sever the judgment that one event is more likely to happen than another, or that it is to be expected in preference to it, from the belief that on the long ' run it will occur more frequently. 5. It follows as a limiting case, that when we expect two events equally, we believe they will recur equally on the long run. In this belief we may of course be mistaken : if we are, we are wrong in expecting the two events equally, and in think- ing them equally possible. Conversely, if the events are truly equally possible, they really will tend to recur equally on a series of trials. But this proves the proposition placed at the head of the section : for if any event can occur in a out of 5 equally possible ways, its probability is j: and if all these I cases tend to recur equally on the long run, the event must tend to occur a times out of I) ; or in the ratio of its probability. Which was to be proved. 6. Let us now examine the mathematical demonstration of this proposition. In entering upon it, we are supposed to have no reason whatever to believe that equally possible events tend to occur with equal frequency. It is well known that what is called Bernonilli's theorem, relates to the comparative magnitudes of the several terms of the binomial expansion. The general term of s which is the probability that an event whose simple probability is m will recur p times on Jc trials ; and hence the connexion between the binomial expansion and the theory of probabilities. 7. A particular example will suffice to illustrate what seems to me to be the essential defect of the mathematical proof of the proposition in question. 12 4 ON THE FO UNDA TIONS OF A coin Is to be thrown 100 times: there arc 2 IO definite sequences of heads and reverses, all equally possible if the coin is fair. One only of these gives an unbroken scries of 100 heads. A very large number give 50 heads and 50 reverses; and Bernouilli's theorem shows that an absolute majority of the 2 100 possible sequences give the difference between the number of heads and reverses less than 5. If we took 1000 throws, the absolute majority of the 2 1000 possible sequences give the difference less than 7, which is pro- portionally smaller than 5. And so on. Now all this is not only true, but important. But it is not what we want. We want a reason for believing that on a series of trials, an event tends to occur with frequency proportional to its probability ; or in other words, that generally speaking, a group of 100 or 1000 will afford an approximate estimate of this probability. But, although a series of 100 heads can occur in one way only, and one of 50 heads and 50 reverses in a great many, there is not the shadow of a reason for saying that therefore the former series is a rare and remarkable event, and the latter, comparatively at least, an ordinary one. Non constat) but the single case producing 100 heads may occur so much oftener than any case which produces 50 only, that a series of 100 heads may be a very common occurrence, and one of 50 heads and 50 reverses may be a curious anomaly* Increase the number of trials to 1000, or to 10,000. Pre- cisely the same objection applies : namely, that in Bernouilli's theorem, it is merely proved that one event is more probable than another, i.e. by the definition can occur in more equally possible ways, and that there is no ground whatever for saying, it will therefore occur oftener, or that it is a more natural occur- rence. On the contrary, the event shown to be improbable may occur 10,000 times for once that the probable one is met with, To deny this, is to admit that if an event can take place in more equally possible ways, it will take place more frequently* But if this is admitted, Bernouilli's theorem is unnecessary* It leaves the matter just where it was before, and introduces no new element into the question. 8. Thus, both by an appeal to consciousness, and by the THE THEOR Y OF PROBABILITIES. 5 impossibility of dispensing with such an admission, we are led to recognize the principle, that when an event is expected rather than another, we "believe it will occur more frequently on the long run. And thus we perceive that we are in the habit of forming judgments as to the comparative frequency of recur- rence of different possible results of similar trials. These judgments are founded, not on the fortuitous and varying cir- cumstances of each trial, but on those which are permanent on what is called the nature of the case. They involve the funda- mental axiom, that on the long run, the action of fortuitous causes disappears. Associated with this axiom is the idea of an average among discordant results, &c. I conceive this axiom to be an a priori truth, supplied by the mind itself, which is ever endeavouring to introduce order and regularity among the objects of its perceptions. 9. With a view to conciseness, I omit several interesting- points which here present themselves namely, the connection between the axiom just stated, and the inductive principle ; the real utility of Bernoulli's theorem; and what seems to me to be the true definition of probability, founded on a reference to the ratios developed on the long run. I proceed to illustrate what has been said by a few passages from Laplace's " Essai Philosophique sur les Probability." 10. It seems obvious that no mathematical deduction from premises which do not relate to laws of nature, can establish such laws. Yet it is beyond doubt that Laplace thought Ber- nouilli's theorem afforded a demonstration of a general law of nature, extending even to the moral world. At p. xlii. of the Essay, prefixed as an Introduction to the third edition of the Th^orie des Probability, after giving some account of the theorem of James Bernouilli, Laplace proceeds: " On peut tirer du th^or&nie pr6cdent cette consequence qui doit fetre regarded comme une loi g&i&ale, savoir que les rap- ports des effets de la nature, sont & fort peu pr&s constants, quand ces effets sont consid&^s en grand nombre....Je n'excepte pas de la loi pr<3c<$dente, les effets dus aux causes morales." It appears not to have occurred to Laplace, that this theorem is founded on the mental phenomenon of expectation. But it is 6 ON THE FO UNDA TIONS OF clear that expectation never could exist, if we did not "believe iu the general similarity of the past to the future, i.e. in the regularity of nature, which is here deduced from It. A little further on,... "II suit encore dc ce thcorbme quo dans une scrie d'dvfcnemens incl6fmiment prolongdc, 1' action des causes r<5guli feres et constantes doit 1'cmportcr it la longue, sur cello, des causes irrdgulifcrcs Ainsi dcs chances favorablos et nombreuses dtant constammcnt attach ees a 1' observation des principes dtemels de raison de justice et d'lumianit<5, - : the ratio of these two expressions is ultimately i ocfdx _ I x p dx J This is the expression applied to determine the probability of a common cause among similar phenomena, as in the case already mentioned of the planets. But this application is founded on a petitio principii: we assume that all the phenomena are allied ; that they are the results of repetitions of the same trial, that they have the same simple probability ; all that, setting other objections aside, we really determine, is the probability, that this simple probability common to all these allied phenomena is > - . 2i But how does this determine the force of the presumption that the phenomena are allied, or, to use Condorcet's illustration, that they all come out of the same infinite lottery ? 19. The object of this little essay being to call attention to the subject rather than fully to discuss it, I have omitted several questions which entered into my original design. The principle on which the whole depends, is the necessity of recognizing the tendency of a series of trials towards regu- larity, as the basis of the theory of probabilities. I have also attempted to show that the estimates furnished by what is called the theory cfc posteriori of the force of inductive results are illusory. If these two positions were satisfactorily established, the theory would cease to be, what I cannot avoid thinking it now is, in opposition to a philosophy of science which recognizes ideal elements of knowledge, and which makes the process of induction depend on them. ON THE METHOD OF LEAST SQUARES*. THE Importance attached to the method of least squares is evident from the attention it has received from some of the most distinguished mathematicians of the present century, and from the variety of ways in which it has been discussed. Something, however, remains to be done namely, to bring the different modes in which the subject has been presented into juxta-position, so that the relations which they bear to one another may be clearly apprehended. For there is an essential difference between the way in which the rule of least squares has been demonstrated by Gauss, and that which was pursued by Laplace. The former of these mathematicians has in fact given two different demonstrations of the method, founded on quite distinct principles. The first of these demonstrations is contained in the Theoria Mottis, and is that which is followed by Encke in a paper of which a translation appeared in the Scientific Memoirs. At a later period Gauss returned to the subject, and subsequently to the publication of Laplace's investi- gation gave his second demonstration in the Theoria Combina* tionis Observationum, The subject has been also discussed by Poisson in the Connaissance des Terns for 1827, and by several other French writers. Poisson's analysis is founded on the same principle as Laplace's : it is more general, and perhaps simpler. It ia not, however, my intention to dwell upon mere differences in the mathematical part of the enquiry. The consequence of the variety of principles which have been made use of by different writers has naturally been to pro- duce some perplexity as to the true foundation of the method. As the results of all the investigations coincided, it wan natural to suppose that the principles on which they were founded were * Transactions of the Cambridge Philosophical JSoc/*'* 1 ', VoL vm [Bead March 4, 1844,] ON THE METHOD OF LEAST SQ UARES. 13 essentially the same. Thus Mr Ivory conceived that if Laplace arrived at the same result as Gauss, it was because in the process of approximation he had introduced an assumption which re- duced his hypothesis to that on which Gauss proceeded. In this I think Mr Ivory was certainly mistaken ; it is at any rate not difficult to show that he had misunderstood some part at least of Laplace's reasoning: "but that so good a mathematician could have come to the conclusion to, which he was led, shows at once Tboth the difficulty of the analytical part of the inquiry, and also the obscurity of the principles on" which it rests. Again, a recent writer on the Theory of Probabilities has adopted Poisson' s investigation, which, as I have said, is the development of La- place's, and which proves in the most general manner the supe- riority of the rule of least squares, whatever be the law of pro- bability of error, provided equal positive and negative errors are equally probable. But in a subsequent chapter we find that he coincides in Mr Ivory's conclusion, that the method of least squares is not established by the theory of probabilities, unless we assume one particular law of probability of error. These two results are irreconcilable; either Poisson or Mr Ivory must be wrong. The latter indeed expressed his dissent from all that had been done by the French mathematicians on the subject, and in a series of papers in the Philosophical Magazine gave several demonstrations of the method of least squares, which he conceived ought not to be derived from the theory of probabilities. In this conclusion I cannot coincide ; nor do I think Mr Ivory's reasoning at all satisfactory. From this imperfect sketch of the history of the subject, we perceive that the methods which have been pursued may be thus classified. 1. Gauss's method in the Tkeoria Mottis, which is followed and developed by Encke and other German writers, 2. That of Laplace and Poisson. 3. Gauss's second method. 4. Those of Mr Ivory. I proceed tor consider these separately, and in detail. For the analysis of Laplace and Poisson, I have substituted another, founded on what is generally known as Fourier's theo- 14 ON THE METHOD OF LEAST SQUARES, rem, Laving teen first given "by him in the TMorie de la Ohalcur. It will be seen that tlie mathematical difficulty is greatly dimin- ished by the change. GAUSS'S FIKST METHOD. This method is founded on the assumption that in a series of direct observations, of the same quantity or magnitude, the arithmetical mean gives the most jprolable result. This seems so natural a postulate that no one would at first refuse to assent to It. For it has been the universal practice of mankind to take the arithmetical mean of any series of equally good direct ob- servations, and to employ the result as the approximately true value of the magnitude observed. The principle of the arithmetical mean seems therefore to be true ct, priori. Undoubtedly the conviction that the effect of fortuitous causes will disappear on a long scries of trials, is an immediate consequence of our confidence in the permanence of nature. And this conviction leads to the rule of the arithmetical mean, as giving a result which as the number of observations increases sine limite, tends to coincide with the true value of the magnitude observed. For let a be this value, x the observed value, e the error, then we have &c. = &c* And as on the long run the action of fortuitous causes disappears, and there is no permanent cause tending to make the sum of the positive differ from that of the negative errors, 20 = 0, and therefore 2(X~a)=0; or, asr-Sa^; which expresses the rule of the arithmetical mean, and which is thus seen to be absolutely true ultimately when n increases sine limite. In this sense therefore the rule in question is deducible from & priori considerations. But it is to be remarked, that it is not the only rale to which these considerations might lead us* For ON THE METHOD OF LEAST JSQ UARE8. 15 not only is 2e = ultimately, but 2j& = 0, where jte is any func- tion such that/6 = -/ ( - e) ; and therefore we should have as an equation which ultimately would give the true value of # when the number of observations increases sine limite, and which therefore for a finite number of observations may be looked on in precisely the same way as the equation which expresses the rule of the arithmetical mean. There is no discrepancy between these two results. At the limit they coincide : short of the limit both are approximations to the truth. Indeed, we might form some idea how far the action of fortuitous causes had disappeared from a given series of observations by assigning different forms to /, and comparing the different values thus found for a. No satisfactory reason can be assigned why, setting aside mere convenience, the rule of the arithmetical mean should be singled out from the other rules which are included in the general equation 2/ (x a) = 0. Let us enquire, therefore, whether there is any sufficient reason for saying that the rule of the arithmetical mean gives the most probable value of the unknown magnitude. In the first place, it is only one rule out of many among which it has no prerogative but that of being in practice more convenient than any other : in the second place, if this were not so, it would not follow that in the accurate sense of the words it gave the most probable result. This objection I shall defer for a moment, and proceed to consider the manner in which Gauss makes use of the postulate on which his method is founded. From the first principles of what is called the theory of probabilities & posteriori, it appears that the most probable value which can be assigned to the magnitude which our observations are intended to determine, is that which shall make the & priori probability of the observed phenomena a maximum. That is to say, if a be the true value sought, x being the value observed at the first observation, x 2 the corresponding quantity for the second, and so on, the errors at the first, second, &c. observation must be x v a, # a a, &c., respectively ; and if (a?! - a) $ (a? a - a)... $ (#- a). Equating to zero the differential of this with respect to a, we find c . = , (-<*) J ' as the equation for determining a in a?. Let 2- ^ 7 then it be- comes 2; ^ (a? - ) o. Now we have assumed that the most probable value of a is given by the equation Sr(*-a) = 0: and it is impossible to make these equations generally coincident, without assuming that e ** ISTow as the error e is necessarily included in the limits oo + oo , we must have or if we adopt the usual notation, and replace m by ~, V 7T V W Consequently, we are thus led to adopt one particular law of probability of error as alone congruent with the rule of the arithmetical mean. But, in fact, we are perfectly sure that in different classes of ON THE METHOD .OF LEAST SQUARES. 17 observations the law of probability of error must vary, and we have no direct proof that in any class it coincides with the form assigned to it. Therefore one of two things must be true, either the rule of the arithmetical mean rests on a mere illusory prejudice, or, if it has a valid foundation, the reasoning now stated must be incorrect. Either alternative is opposed to Gauss's investigation. For th% reasons already given, we are, I think, led to adopt the latter, and then the question arises, wherein does the incorrectness of the reasoning reside ? It resides in the ambiguity of the words most probable. For let us consider what they imply in the theory of probabilities & posteriori. Suppose there were m different magnitudes a 1 a< 3 ,*..a m , and that each of these were observed n times in succession. Let this process be repeated p times, p being a large number which increases sine limite. Thus we shall have pm sets of observa- tions each containing n observations. Of these a certain number K will coincide with the set of observations supposed to be actually under discussion ; and we shall have the equation where Jc is that portion of J{ which is derived from observations of %. Then, ultimately, the most probable value which the given series of observations leads us to assign to a, is (supposing a is susceptible only of the values ^^...aj equal to a ry r being such that the corresponding quantity k r is the maximum value of k. To make the case now stated entirely coincident with the one which we are in the habit of considering, we have only to suppose (making m infinite) that the series of magnitudes a 1 ...a m includes all possible magnitudes from oo to + co . Now from this statement, it is clear there is no reason for supposing that because the arithmetical mean would give the true result if the number of observations were increased sine timite, it must give the most probable result the number of observations being finite. 2 18 ON THE METHOD OF LEAST SQUARED The two notions are heterogeneous : tlic conditions Implied Tby tlie one may "be fulfilled without introducing those required by the other : and we have already seen that "by losing sight of this distinction, we are led to the inadmissible conclusion, that a principle recognised as true & priori necessarily implies a result, viz. the universal existence of a special law of error, not only not true a priori^ but not true at f alL Having stated what seem to me to be the objections in point of logical accuracy to this mode of considering the subject, I will briefly point out the manner in which, from the law of error already obtained, the method of least squares is to bo deduced, Let ^ = aj& + 1$ + &c. - V e 2 = a 2 x + l^j + &c. &c. = &c. be the system of equations of condition, which are to be com- bined together so as to give the values of x, y, &c. The error committed at the first observation is e v at the second e a , and so on ; each observation corresponding to an equation of condition. The probability of the concurrence of all these errors is, (according to the law of error already arrived at) propor- tional to c. - Fi) s 4- (a s and it is to be made a maximum by the most probable values of x, y, &c. These values will therefore make a minimum: that is to say, they will make the sum of the squares of errors a minimum. Hence the method of least squares. The conditions of tlio minimum give the linear equations: tal + &c. a 2a F] 2iF| ............ 08), a &C. ^ ON THE METHOD OF LEAST SQUARES. 19 in which system there are always the same number of equations as there are unknown quantities to be determined. The next investigation of the principle of the method of least squares which I shall attempt to analyze is that of Laplace. LAPLACE'S DEMONSTBATIOK If, in order to determine x from the equations of condition stated in the last paragraph, we multiply the first by /& x , the second by p# &c., and add : (p^ &c. fulfilling the conditions = 0, &c. = 0) we find x lZp V 2/*e; and if we assume that 2/te is equal to zero, then the resulting value of x is 2/^F: the error of this determination being the quantity 2/^e, which we have assumed to be equal to zero, without knowing whether it really is so or not. Now supposing there are n equations of condition, and $ quantities to be determined, and that n is greater than p, then we see that there are n factors ^ /v*-/^ an(i P conditions for them to fulfil. They may therefore be subjected to n -p addi- tional conditions. This being premised, let us consider the probability that the quantity 2/*e will not be less than a, or greater than & a and ft being any quantities whatever. The law of probability of error at each observation being given, the question is evidently analo- gous to the common problem of finding the chance, that with a given set of dice the number of points thrown shall not be less than one given number or greater than another, "We may therefore suppose that the probability in question has been determined: call it P. Suppose also that we have taken a = - I and /3 = I, I being any positive quantity. Then P is a function of 1 9 and of /v . ./v Let us now so determine /v/^> (subject to the conditions already specified,) that P may be a maximum. When this is done it follows that there is a greater probability that the error 22 20 ON THE METHOD OF LEAST SQUARED in our determination of a?, viz. 2^e, lies within tlio limits ?, than if we had made use of any other set of factors whatever. On this principle Laplace determines what he calls the most advantageous system of factors. It does not follow that the value thus obtained for & is the most probable value that could be assigned for it, But if we consider a large number of sets of observations, (the quantities a, 5, &c. being the same for all) then the error which we commit by using Laplace's factors will in a greater proportion of cases lie between 1 9 than if we had used any other system of factors, The investigation has reference merely to the different ways in which by the method of factors a given set of linear equations may be solved. We now enter on the analysis requisite to determine P Let the probability that S/*e will be precisely ecpal to w, be pdu. Then manifestly and we have therefore only to determine p. Let e l a ..,e n be the errors which occur at the first second &c. observation ; fae^d e 15 ^ n e w d e n be the probabilities of their occurrence: the form of the function $ determining the law of probability of error, which, for greater generality, we suppose different at each observation. The probability of the concurrence of these errors is of course and the first principles of the theory of probabilities show that the value of pdu will be obtained by integrating (1), e.,.* n "being subjected to the condition 2^e = u. Thus with the relation ON THE METHOD OF LEAST SQUARES, 21 Consequently Now by Fourier's theorem 1 f" 7 f** / U &* .., = r' da I e w cos a ^ 7T J J .-co V which, replacing by a, becomes rn . oo /*H-oo ^ & <6e- cos a (w ^Jo J.. " ^ Therefore . +00 / + 6 sin otfce(?6 = 0. J -oo Hence (6) will become =^ I"'** fcosawfo f "" 7rJ J, J- 22 ON THE METHOD OF LEAST SQUARES. The next step Is to find an approximate value of this expression. "When -too * + = I e cos a^etle = I ec?e = 1, 1 SO ^ 00 as the error e must have some value lying between cc . It is clear this is the greatest value the integral in question can have, and therefore as n increases sine Umite, the continued product - +00 +OQ $& cos ^ae/e, ... I fe n cos pj*ejle n J -00 * -"00 decreases me Zwwte, ("being the product of w factors each less than unity) except for values of a differing infinitcsimally from zero. Let If = f -*> cos a w - It is certain that S/^e lies "between the limits co . There- fore when I = oo , P should be equal to unity. I proceed to show that this is the case. - - + w /.co . + / + - i 6 -ww du\ da I & r . . / de n fa. . . fa cos a (u - 2/ze) 7T J -00 Jo J-oo */-, when m = 0. Effecting the integration for u, i r a2 r-- w r^" 06 - _. e -ssr (fe ^e x ... & w ^ x ... ^> n 6 w cos aS/t ...... (10) 7^ YTT Jo J -* J - when m = 0, since I" 1 "* 6 ~ m J-a, and A/7T ^ 2 COS aw & ss _ 0"ii5 Integrating for a, we see that when m P n = f + " A, ... f + " e?6,^ ... $. ^-W ... (11). J -00 ^"00 Or, . = r$A ^ - r J00 ^-3 And as each of these integrals is separately equal to unity, P 1, which was to "be proved, ON THE METHOD OF LEAST SQUARES. 27 I proceed to show that in a particular case in which .the value of P can be accurately determined, Laplace's approxima- tion is correct. It has sometimes been thought that the intro- duction of the negative exponential involves a petitio principii, and is equivalent to assuming a particular law of error. It is therefore desirable, and I am not aware that it has hitherto been done, to verify his result in an individual case. Let the law of error be the same in all the observations, and such that $e = J 6+ e , the upper sign to be taken when e is positive. Let fa = p 2 = &c. = 1 , then 1 / ao / + co / +00 =~-l da I (i 6 ^ 1 )^ ...... I ( WJo J^ J-oo or The value of p is thus given by a known definite integral, which has been discussed by M. Catalan in the fifth volume of Liouvillds Journal. It may be developed in a series of powers of u. Up to u* (n ~ 1} no odd power of u can appear in this development, for o? p rrrrr dot, is finite while f) is less than n. and therefore the integral may be developed by Maclaurin's theorem. For higher powers the method ceases to be applicable, and we must com- plete the development by other means. But as we suppose n to increase $. I. the integral tends to become developable in a series of even powers only of u. Thus COBUOL -j [ da , o( TT?j 5 * a Let /; da ,. N T^v- =/(") Then 28 ON THE METHOD OF LEAST SQUARES. and generally, Now /,, . 7T 1 . 3...2W-3 TT and generally, Thus /"coawtda _ , f _ 1 _ 1_ Jo (1 + a 8 )" ~ P 1 2 ' 2n- ___ ..... tt 3 2 . 3 . 4 2n-5.2n-3 The coefficient of w 2l> is . .... za', &c. whatever be the form of . As it thus would serve to establish, at least apparently, an infinity of contradictory results, the inference is that in no case has it any validity, I have now completed, though in an imperfect manner, the design indicated at the outset of this paper, namely, to give an account of the different modes in which the subject has been treated, and to simplify the analytical investigations. If I have succeeded in doing this, the present communication may tend to make a very curious subject more accessible than it has hitherto been. SOME REMARKS ON THE THEORY OF MATTER.* IN tlie present state of Science, there are few subjects of greater interest than the enquiry whether all the phenomena of the uni- verse are to "be explained "by the agency of mechanical force, and if not whether the new principles of causation, such as chemical affinity, and vital action, are to he conceived of as wholly independent of mechanical force, or in some way not hitherto explained cognate and connected with it. One reason among many which makes this enquiry interesting is the cir- cumstance that the application of mathematics to natural philo- sophy has, up to the present time, either Tbeen confined to phenomena, which were supposed to be explicable without assuming any other principle of causation than ordinary " push and pull" forces, or as in Fourier's theory of heat and Ohm's theory of the galvanic circuit, has been based on proximate empiric:;! principles. 2. The intention of the remarks which I have the honour to offer to the Society is to suggest reasons for believing that while on the one hand it is impossible not merely from the short-comings of our analysis but from the nature of the case to reduce, as it appears that Laplace wished to do, all the pheno- mena of the universe to one great dynamical problem, we cannot recognise the existence of any principle of causation wholly disconnected with ordinary mechanical force, or of which the nature could be explained without a reference to local motion : in other words, that the idea of " qualitative action" in the sense which the phrase naturally suggests must be rejected. It will be seen from the explanations I am about to attempt that the objection which Leibnitz has opposed to the atomic, and in effect to any mechanical philosophy, namely, that on such prin- * Transactions of the Cambridge JPhilosopUcal Society, Vol. vm. p, 600. [Bead May 22, 1848.] SOME REMARKS, dc. 39 ciples a finite intelligence might be conceived to exist by which all the phenomena of the universe would Ibe fully comprehended, does not (whatever may be thought of its validity) appear to apply to the views which I have been led to entertain. For these views essentially depend on the conception of what may be called a hierarchy of causes, to which we have no reason for assigning any finite limit. Of this series of principles of causa- tion, ordinary mechanical force is the first term. 3. With respect to the first point, namely, the impossibility of explaining all phenomena mechanically, it may be remarked, that we are met, in the attempt to discuss it, by the difficulty which always attends the establishment of a negative proposition. It is clear that as in the present state of our knowledge we are far from being able to enumerate and classify the phenomena which are or which might be produced by the combined agency of con- ceivable mechanical forces, we are not in a position to decide & priori that any given phenomenon might not be thus produced. Non constat, but that the impossibility we find in the attempt to explain the causes of its existence may have no higher origin than the imperfect command which we have as yet obtained of the principles of mechanical causation. We meet, it may be said, with a multitude of ordinary dynamical problems which have as yet received no adequate solution why then should we have recourse to new kinds of causes, while we have not as yet exhausted the resources, if the expression may thus be used, of those which we already recognise? To this enquiry no con- clusive answer can be given, but the following considerations will I think naturally suggest themselves. 4. In the first place, no even moderately successful attempt has, I think, yet been made to explain any chemical phenomenon on mechanical principles. It is quite true that we are unable, to take a particular instance, fully to comprehend the mechanical constitution of the luminiferous ether ; the determinations which have as yet been attempted of the law of attraction between its molecules cannot, I apprehend, be accepted as any thing more than hypothetical or provisional results, and there are other points involved in yet greater obscurity. Nevertheless the un- dulatory theory of light has, as we all know, given consistent and satisfactory explanations of a great variety of phenomena. 40 SOME REMARKS ON THE Thus it appears, and the same remark might be educed from other though similar considerations, that we are by no means absolutely estopped by the imperfection of our mechanical phi- losophy, from explaining phenomena really due to mechanical forces, even when these phenomena are connected with subjects not as yet fully comprehended : why then cannot some progress be made in the mechanical explanation of chemical phenomena, or of those, to mention no other class, which we are in the habit of referring to vital action ? In these cases, we see or seem to see that the action of mechanical laws is modified or suspended ; and though it is not demonstrably impossible that this is not really the case, and that no other causes are at work beside the "push and pull" forces of ordinary mechanics, yet we are at least much tempted to believe, that the difficulties we meet with do not arise from what may be called the disguised action of mechanical forces but from the presence of an agency of a dis- tinct nature. And to this view we find that most of those in- cline who have made themselves familiar with the science of chemistry or with that which has been called biology; and further that, (with reference to the latter science) the insuffi- ciency not only of a mechanical but even of a chemical phy- siology has been generally admitted. Secondly, it is to be observed that even if it be considered doubtful whether a mechanical philosophy be not after all sufficient for the explanation of all phenomena, it is at least certain that it has not been proved to be so: and that by reject- ing other conceivable modes of action than those which are recognised by it, we unnecessarily and arbitrarily limit the problem which the universe presents to us ; falling thereby into an error similar to that of the atomists, who starting from the assumption that the ap^ai^ or first principles of all things, are atoms and a vacuum proceeded to construct an imaginary world, in accordance with this arbitrary hypothesis. At the same time it must be granted that a purely mechanical* system such as that of Boscovich is more self-consistent and contains, so to * The word mechanical is of course not used in antithesis to dynamical) in the sense in. which the latter is commonly employed by the philosophical writers of Germany. The antithesis in question is foreign to the scope of the present essay, and I have accordingly elsewhere used the word dynamical in its ordinary ac- ceptation. THEORY OF MATTER. 41 speak, less that is discontinuous, than any which should recog- nise other principles, for instance chemical affinity, distinct from force without enquiring into the relation which subsists between them. 5. It may however be asserted that this enquiry is alto- gether superfluous that the power of exerting attractive or repulsive force is one property of matter, that chemical affinity (and so -in other cases) is another that the two are not merely distinct, but absolutely independent and heterogeneous. But to this view the arguments which seem to have led to the adoption of a purely mechanical system, appear to prevent our assenting. I shall therefore attempt to state what I conceive these argu- ments to have been. 6. It is a fundamental principle of the secondary mechanical sciences, for instance of the theory of light, that the secondary qualities of bodies are to be explained by means of the primary. Every substance, to use for a moment the language of Leibnitz, is essentially active ; in other words it is to be conceived of as the formal cause of the sensible qualities which are referred to it. If we ask why gold is yellow and silver white, the answer at once presents itself that the difference of colour corresponds and is due to a difference between the essential constitution of the two substances. Now the^ essential constitution here spoken of, and consequently the differences which individuate it in different cases, may conceivably be something altogether incognisable to the human intellect. The notion that it is so was expressed scholastically by saying that substantial forms are not cog- noscible. But if, setting aside this opinion, we affirm that the essential constitution of each substance is a matter of which the mind can take cognisance, we are led at once to the distinction between primary and secondary qualities. The first are ascribed to each substance as its essential attributes, in virtue of which it is that which it is the second result from the primary*, (by which as we have said the essential or formal constitution of the substance in question is determined,) and have reference to the mind by which they are perceived, while the primary are ascribed to it independently of any reference to a percipient * Or that wliich, in its formation it was to be, rb rl TJV etvai. 42 SOME REMARKS ON THE mind: and a distinction, analogous or identical with that be- tween primary and secondary qualities, has accordingly been expressed by the antithesis between that which is a parte hominis and that which is a parte universi. That the dis- tinction between primary and secondary qualities is necessary on the hypothesis on which we are proceeding^ appears at once from the consideration that if we affirm that all the qualities of bodies of which we can form any conception are equally subjective and phenomenal, nothing will remain of which the mind can take cognisance, and by means of which our con- ception of the nature of any one substance can be discriminated from that of any other*. Let it be granted therefore that the distinction of primary and secondary qualities is a necessary element of physical science. It follows from this that the secondary qualities in a manner disappear when we look at the universe from the scientific point of view. Instead of colours we have vibrations of the luminiferous ether instead of sounds vibrations of the ambient air, and so on. Now from hence it follows that all the phenomena which we see produced, of whatever nature they may be, are all in reality dependent on the primary qualities of matter. Furthermore, these primary qualities themselves all involve the idea of motion or of a ten- dency to motion. A body changes its form in virtue of the local motion (absolute or relative) of some of its parts; and when I press a stone between my hands, I find that I can produce no sensible change of form, while contrariwise the stone reacts against my hands, tending to make them move in opposite directions. I then say that the stone is hard as a mode of expressing this, viz. that when an attempt is made to produce relative local motion of its parts, it resists it in virtue of its reactive tendency to produce motion in that which acts upon it. Again, a body whose parts are readily susceptible of relative local motion is said to be soft or fluid, and when a sensible change of form is accompanied by a tendency to such motion as shall restore the original form, it is said to be elastic, and so on. We thus arrive at a, point of view at which all * The doctrine of the cognoscibility of substantial forms, which is intimately connected with this distinction, is, as Leibnitz in effect remarks, as it were the common character of those who with more or less success attempted in the seven- teenth century, the restoration of science. Vid. Leibnitz, Epist. ad Tlwms, i. THEOR Y OF MA TTER. 43 secondary qualities having disappeared, and all primary ones* having been resolved into motion and tendency to motion, the sciences which relate to phenomena appear to be resolved into the general doctrine of motion. But if this be true the universe can it is said present to us nothing but one great dynamical problem. Motion, and force the cause of motion, belong essen- tially to the domain of mechanics: and if chemical affinity be a cause of local motion, that is, if in virtue of its action f a particle of matter finds itself at a given time in a position dif- ferent from that which it would else have occupied, chemical affinity is not really distinct from mechanical force (which looked at from the dynamical point of view includes everything which is a cause of motion); whereas if it be not a cause of motion the enquiry at once presents itself of what is it? In illus- tration of this view we may refer to any chemical experiment. If an acid is dropped into a glass containing any vegetable blue, the colour is changed to red. But to say this is to say that the liquid when the acid is introduced into it begins to act on the luminifer- ous vibrations which exist near it in a different manner from that in which it had previously acted. The whole change, whether we call it a chemical phenomenon or not, consists in the introduction of new forms of motion in virtue of the action of mechanical force. 7. From considerations of this kind it appears to follow that a complete explanation of all phenomena would introduce no principles beyond those with which the science of mechanics is conversant. And in truth if the conclusion drawn had been that all phenomena might, if our knowledge of nature were sufficiently extensive, be reduced to cinematical considerations (using the word cinematics in the large sense in which it is equivalent to the doctrine of motion), I do not see how on our fundamental hypothesis we could refuse to assent to it. But the conclusion drawn by the maintainers of the all-sufficiency of a mechanical philosophy is something different from this and as I conceive the error they appear to have committed is to be sought for in this discrepancy. But before entering into the discussion of this point, I will make a few remarks on certain points in the history of what may be called the theory of matter. * That is, all that are commonly enumerated as primary qualities. 1* As, for instance, in the phenomenon of crystallization. 44 SOME REMARKS ON THE 8. If we suppose the maxim that secondary qualities are to Tbe explained by means of the primary to have^ been accepted (either in that or in some equivalent form), or if not formally accepted, at least unconsciously assumed, at a time when the idea of mechanical force was as yet very imperfectly appre- hendedthe natural result of this state of things is the forma- tion of an atomic theory. For in order to individuate the constitution of any given body, we could only have had recourse to the configuration or motion of its parts. Gold, to return to our previous example, was said to be yellow in virtue of such and such a configuration of its parts; since except configuration there appeared to be no disposable circumstance*, if I may so speak, whereby gold was in its intimate constitution to be distinguished from silver or from anything else. ^But this configuration must be independent of the body's visible and external form, since changes of the latter do not affect the body's sensible qualities. Hence it must be a configuration of small parts, and we are thus at once led to the primitive form of the atomic theory. In this the atoms possess the primary qualities of larger bodies they are of various forms and act if the expression may be used by their forms, not by being centres of attractive forces. Such was the atomistic system of the school of Democritus f a system which we know found no little favour among the scientific reformers of the seventeenth centuryf. As an instance of the influence it exerted, I need only mention the great work of Cudworth, in which it is pre- sented apart from the atheistical doctrines with which it had often been connected. Cudworth goes so far as to affirm that Democritus and his followers had corrupted and degraded the atomistic system which was originally altogether free from any irreligious tendency, and which he sought to restore to its first estate. But as the imperfections of the atomic system became maxri- * Specific differences of motion seem for more than one reason not to have been used in giving an account of the differences of bodies. t See for a more favourable and, I think, a juster view of the philosophy of Democritus than that which we commonly meet with in the writings of modern historians of philosophy, teller's Philosophic X>er Oriechcn, I. 10. J The physical theories of Des Cartes, though not properly atomistic, since ho proceeded on the hypothesis of a plenum, yet in many respects are akin to those of which we are speaking. THEORY OF MATTER. 45 fest, and on the other hand mechanical conceptions came to Ibe more developed, a new form of this system arose. The atoms, retaining their forms and those which are commonly called their primary qualities., were now supposed to act as centres of attractive force, in other words, each atom was to the rest a cause of motion. But as the ordinary "primary qualities" of bodies may as we have seen be analysed into conceptions which involve nothing beside motion and force, this new form of the doctrine may clearly be considered merely as a state of transition to that which is now known by the title of Boscovich's theory*. To Boscovich appears to belong the credit of having perceived that if the atoms were conceived of simply as unex- tended centres of force the primary qualities of bodies might sufficiently be accounted for without supposing them to result from the primary qualities of their constituent atoms a mode of explanation of which, though there has been something like a return to it in some recent speculations, it may be observed that it explains nothing. Boscovich's theory seems to have been so completely in accordance with the direction in which mathematical physics have of late been moving, that it was adopted as it were unconsciously almost all modern investiga- tions on subjects connected with molecular action are in effect based on his views, though his name is, comparatively speaking, but seldom mentioned. And this theory, (whether or not the hypothesis of the existence of discrete centres of action be or be not essential to it, a question connected with that which in former times caused so much perplexity, namely, the nature of continuity, and which it is not necessary to my present purpose to consider), is in truth the highest developement which the mathematical theory of matter has as yet received it is that on which the pretensions of mathematical physicists to vindicate for their own methods the right, so to speak, if not the power, to explain all phenomena mainly depend. Adopting for the sake of definite conception the received form of this theory, that * It is, I believe, known that Boscovich's fundamental idea was deduced hy a not unnatural filiation from the monadism, of Leibnitz. Yet the scope and limits which he proposed to himself differ essentially from those of the German phi- losopher, inasmuch as they are essentially physical Moreover, the latter would have objected on the principle of sufficient reason to the want of any thing to indi- viduate the atoms of Boscovich; and, at least in the latter years of his life, to the "Feme Wirkung," on which the whole theory depends. 46 SOME REMARKS ON THE namely in which, the centres of force are discrete and at in- sensible distances from each other, I now shall attempt to show what ulterior deyelopements it admits of, and how by means of these the error noticed at the close of the last Section, namely, the confounding the admission that all phenomena are to be explained cinematically with the assertion that they can all be explained mechanically may be met, and, as it seems to me, sufficiently refuted. 9. I begin by observing that though we speak and shall continue to do so of the action of matter on matter, yet that no part of the views I am about to state depends on the hypothesis we adopt touching the nature of causation. They would remain unchanged whether we accept a theory of pre-established har- mony, or one of physical influence, or whether we abstain from all theories on the subject This being understood, we may, I think, lay down the axiom that whatever property we ascribe to matter, we may also ascribe to it, the property of producing in other portions of matter the former property. Of this axiom the present state of Boscovich's theory affords a familiar illustra- tion. Every portion of matter is locally moveable, therefore we may ascribe to any portion of matter the power of producing motion in any other, hereby giving rise to the whole doctrine of attractive and repulsive forces. At this point we have hitherto stopped, but for no satisfactory reason. We may proceed farther, and we are therefore bound, in constructing the most general possible hypothesis, to do so : we may ascribe to each portion of matter the power of engendering in any other that which we call force, in other words the power of producing the power of actuating the potential mobility of matter. It is not a priori at all more easy to conceive that A should have the power of setting J5 in motion, or of changing the velocity it already has, than that G should have the power of enabling A to act on j?, or of changing the mode of action which A already possesses. And let it be observed, that the new power thus ascribed to G is as distinct from force, as force is from velocity. The two are related as cause and effect, but formally are wholly independent. Now unless this hypothetically possible mode of action can be shown to have no existence in rerum naturd, it is clear that the inference from the conclusion that no phenomenon can be THEORY OF MATTER. 47 imagined not resoluble en derm&e analyse, into local motion to the assertion that mechanical force is the only agency to be recognised in the material universe is altogether illusory. For matter may act on matter in a manner wholly distinct from force, and yet this kind of action shall, ultimately and indi- rectly, manifest itself in modifications of local motion. Further- more, if for an instant we call this kind of action (force) 2 , we shall at once be led to recognise a hypothetically possible mode of action of matter on matter which in accordance with analogy we shall call (force) 8 , which consists in the power of modifying (force) 2 * And so on, sine limite. 10. If we compare the language in which the relation between mechanical force and chemical affinity is commonly spoken of, we shall I think perceive its analogy with that which I have used in describing the mode of action which we have called (force) 2 . Its chemical affinity is spoken of as something which suspends or modifies the action of force, as something distinct from it, but which yet interferes with its effects. Or again, if in physiological writings we observe the manner in which vital action* is described we recognise, or seem at least to do so, the possibility of referring its effects to that mode of action which we have called (force) 3 . I do not however wish to lay much stress on these similarities, because I think the kind of reasoning we have pursued shows more satisfactorily than they can do, that if chemical affinity and vital action are not resoluble into force, they must be referred to some of the modes of action we have pointed out. It would be useless to remark on the many points of specu- lation which here present themselves. The expansion of bodies by heat may however be particularly mentioned, because not- withstanding what has been learnt with relation to the theory of heat, nothing like a mechanical explanation of this phenomenon has as yet been discovered. It seems to depend not on the introduction of new mechanical forces, but on a modification of those which already exist ; such modification, in cases of ordi- nary conduction, being propagated from one part of the body to that which is next it. It is easy to conceive that by an altera- * I am, of course, not to be understood as suggesting a materialistic explana- tion of phenomena of thought or volition. 48 SOME REMARKS, &c. tion in the function which expresses the mutual action of the molecules, the body may pass into a new state of equilibrium in which the average distance between adjacent molecules may be increased or diminished. If such an explanation could be esta- blished, we should have a case of the action of (force) 2 , 11. In conclusion, it may be well to remark that mathe- matical analysis is conceivably as applicable to these new modes of action of matter on matter as to ordinary questions in dynamics. It is, however, easily seen that as in these we deal chiefly with differential equations of the second order, and in merely cinematical questions with equations of the first only, so contrariwise when we introduce higher powers of force (so to call them) we shall correspondingly have to do with equations of higher orders. I venture to predict with a degree of confi- dence, which doubtless I shall not communicate to many, that if we ever succeed in establishing a mathematical theory of che- mistry, it will be as much conversant with equations of the third or of a higher order, as physical astronomy is with equations of the second. REMAKES ON THE FUNDAMENTAL PRINCIPLE OF THE THEORY OF PROBABILITIES*. I WISH to make an addition to the remarks on the foundation of the theory of probabilities which were offered some years since to the notice of the Societyf. My intention in doing so is to consider, in what way the proposition, which I conceive to be the fundamental principle of the theory, may be the most clearly and conveniently expressed. This principle may for the moment be thus stated : " On a long run of similar trials, every possible event tends ultimately to recur in a definite ratio of frequency." Our conviction of the truth of this proposition is, I think, intui- tive, the word being used, as in all similar cases, with reference to the intuitions of a mind, which has fully and clearly appre- hended the subject before it, and to which therefore to have arrived at the truth and to perceive that it has done so are inseparable elements of the same act of thought. If we endeavour to translate the proposition just stated into ordinary philosophical language, we may in the first place remark that the phrase " similar trials," expresses the notion of a group or genus of phenomena to which the different results are subordinated as distinct species. If the trial is the throwing of a die, this may be regarded as the generic character; the occurrence of ace, deuce, &c. constituting different species. Thus much is clear ; but it is less obvious how the idea expressed by a " long run of trials" in " definite series of experiments," and the like, is to be ex- pressed, so as to make the analogy between the fundamental principle of the theory of probabilities and those of other sciences more obvious than it has hitherto been. The idea in question * Transactions of the Cambridge Philosophical Society, Vol. IX. p. 605. [Read Nov. 13, 1854.] *h Transactions of the Cambridge Philosophical Society, Vol. VIII. p. i. [p. t of this volume.] 4 50 REMARKS ON THE FUNDAMENTAL PRINCIPLE is not readily expressed in any way, "because in its own nature it is negative and indefinite. The phrases I have just quoted imply merely the absence of the limitations inseparable from individual cases, or from any finite number of such cases, whether contem- plated as actually existent or as about to be developed within definite limits of space and time. When individual cases are considered, we have no conviction that the ratios of frequency of occurrence depend on the circum- stances common to all the trials. On the contrary, we recognise in the determining circumstances of their occurrence an extra- neous element, an element, that is, extraneous to the idea of the genus and its species. Contingency and limitation come in (so to speak) together ; and both alike disappear when we consider the genus in its entirety, or (which is the same thing), in what may be called an ideal and practically impossible realization of all which it potentially contains. If this be granted, it seems to follow that the fundamental principle of the theory of probabilities may be regarded as included in the following statement ; "The conception of a genus implies that of numerical relations among the species subordinated to it. 1 ' 2. But in what relation, it may be asked, do these con- ceptions stand to outward realities ? How can they be made the foundation of a real science, that is, of a science relating to things as they really exist ? We are by such questions led back to what was long the great controversy of philosophy ; I mean the contest between the realists and the nominalists. The former in asserting the reality of universals did not maintain that what we think of when we use a general term is an actually existing thing. Like every one else they admitted, that in one sense nothing can exist but the individual, nevertheless they held that universals are not mere figments of the mind, but that they have a reality of their own which is the foundation of the truth of general, propositions. To assert therefore that the theory of pro- babilities has for its foundation a statement touching genera and their species, and is at the same time a real science, is to take a realistic view of its nature. And this I believe is what, on consideration, we cannot avoid doing. If it be said that the grouping phenomena together is merely a mental act wholly disconnected from outward reality and alto- OF THE THEORY OF PROBABILITIES. 51 gether arbitrary, it may be replied that no mental act can be so. Why and how facts and ideas correspond is no doubt one of the great questions of philosophy ; "but the answer to it is surely to be developed from the consideration, that man in relation to the universe is not spectator ab extra, but in some sort a part of that which he contemplates, and that the rebus avolsa ratio, which is in truth the fundamental postulate of nominalism, is therefore inconcessible. The thoughts we think are, it is true, ours, but so far as they are not mere error and confusion, so far as they have anything of truth and soundness, they are something and much more. The vertices essendi (to recur to the language of the schoolmen) is the fountain from whence the veritas cognoscendi is derived. The meaning which these phrases were intended to convey is expressed in more modern language by Leibnitz in the passage which I have cited in the note*. In every science the fact and the idea correspond because the former is the reali- zation of the latter, but as this realization is of necessity partial and incomplete or rather because in the same fact are simulta- neously realized a variety of separate ideas, separate, that is, as we conceive them this correspondence is but imperfect and approximate. It is only when in thought we remove the action of disturbing causes to an indefinite distance, that we can con- ceive the absolute verification of any h priori law. Only on the horizon of our mental prospect earth and sky, the fact and the idea, are seen to meet, though in reality the atmosphere is everywhere present. Everywhere it surrounds and interpene- trates the 777 /jL6\awa on which we- stand ; making it put forth, and sustain all the numberless forms of organization and of life. The indefinitely prolonged series of trials, which enters into the ordinary statement of the fundamental principle of the theory of probabilities, is analogous to the infinite and infinitely smooth horizontal plane, which would enable us to verify the first law of motion. 3. The simple negative notion of the absence of disturbing forces is perpetually confounded with that of a tendency inherent * C'est Dieti qui est la derniere raison des clioses, et la connaissance de Dieu ii'est pas moms le principe des sciences, cjue son essence et sa volc-nte* sont les principes des e^tres. [Erdmami, p. 106.] A little further on he adds : C'est sanc- tifier la philosophic, que de faire couler ses ruisseaux de la fontaine des attribute da Dieu. 42 52 REMARK^ &c. in a series of successively developed results to restore the balance of frequency of occurrence, when this has been by accidental cir- cumstances temporarily deranged. It is commonly thought that this notion, which, as we know, is the foundation of many unsuccessful attempts to circumvent fortune, is sufficiently refuted by saying, that what is past can exert no influence on what is yet to come. But in reality the past influences the future in a thousand different ways ; and it is only in idea that we can secure the possibility of an indefinite series of trials, of which those which we regard as the permanent circumstances are not progressively, however slowly, undergoing alteration. The dice box for example wears smooth, and the edges of the die are rounded; and though, in this example, we cannot say what result is facilitated by the change, yet this is not always the case. Such progressive alterations may tend so to alter the ratio of frequency of occurrence, as to restore the balance which the result of past trials has disturbed. There is thus nothing absurd in the notion of a restorative and balancing tendency, though the grounds on which it is commonly assumed indicate much con- fusion of thought* It would for instance be perfectly reasonable to inquire, whether in the succession of seasons hot years are not oftener followed by cold and cold by hot, than vice versd. Such questions indicate a branch of the theory of methods of observa- tion to which hitherto but little attention has been paid, REMARKS ON AN ALLEGED PROOF OF THE METHOD OF LEAST SQUARES, CONTAINED IN A LATE NUMBER OF THE EDINBURGH REVIEW. IN A LETTER ADDRESSED TO PROFESSOR J, D. FORBES**, MY DEAR SIR, THE review of Quetelet's Let-fores h 8, A, R. le Due rtynant de Saoce Colourg et 6fotha, which appeared in the July Nu'mber of the Edinburgh Review, contains a new demonstration of the method of least squares which ought not, I think, to pass unnoticed. If it is correct, it is so much simpler than those which have hitherto been received, that it ought to supersede them ; and if not, the sooner its incorrectness is pointed out the tetter. Some years since, in a paper published in the Cambridge Transactions for 1844, I made an analysis of all the demon- strations, or professed demonstrations of the method of least squares, with which I was then acquainted, and I therefore read this new one with more attention than you perhaps have given to it. The reviewer gives some account of the history of the sub- ject, and remarks that the demonstration of the least squares was first attempted by Grauss, but that his proof is no proof at all, because it assumes that in the case of a single element the arithmetical mean of the observed values is in all cases the most probable value, " a thing to be demonstrated, not as- sumed." Gauss afterwards gave another demonstration, which is perfectly rigorous ; but of this the reviewer takes no notice, though it i.i mentioned in at least one of the works on the theory of probabilities which he has recommended to the attention of students. However, in the proof which the re- viewer refers to, which is contained in the tract entitled Theoria * Philosophical Magazine, November, 1850., 54 REMARKS ON AN ALLEGED PROOF OF Mottis Elliptiei, Gauss undoubtedly does assume that the arith- metical mean is the most probable value in the case of direct observations of a single element. From this assumption, he shows that the probability, that the magnitude of an error lies between x and x 4- dx, must be r VTT h being an indeterminate constant. It follows from this, that the results of the method of least squares are always the most probable values that can be assigned to the unknown elements. Without referring to the Theoria Mottis, you can see the de- tails of Gauss's reasoning in a paper by Bessel, of which a translation appeared in Taylor's Scientific Memoirs. The re- viewer is right in saying that Gauss was not entitled to assume that the arithmetical mean is the most probable value. But when he speaks of this as a thing to be proved, and not as- sumed, we are led to suppose that he believes that subsequent writers have actually proved it. In truth this appears, not only from his statements, but also from, the illustrations of which he has made use. Thus he states that if shots are fired at a wafer which is afterwards removed, and we are asked to determine from the position of the shot-marks the most pro- bable position of the wafer, " the theory of probabilities affords a ready and precise rule, applicable not only to this but to far more intricate cases;" and he goes on to say that it may be shown that the most probable position of the wafer is the centre of gravity of the marks. Now this result is only then true when the law of probability of error, which is implied in Gausses assumption, really obtains; so that, according to the reviewer, the demonstration of the principle of least squares must amount to showing that this law obtains universally ; or, which is the same thing, that the arithmetical mean is always the most probable value in the case of direct observations of a single element. If this can be proved, it is doubtless a very curious conclusion.; but it is at any rate certain that Laplace has not proved it, of whom however the reviewer asserts that he has given a rigorous demonstration of the principle of least squares. From one end of Laplace's great work to the other, there is nothing to justify the assertion that the THE METHOD OF LEAST SQUARES, 55 centre of gravity of the shot-marks is the most probable position that can be assigned for the wafer; that is, that the concurrent existence of the deviations or errors which must have taken place if the wafer really occupied this position is more probable than that of those which are similarly implied in any other hypothesis as to its place. If you find anybody sceptical as to this, pray ask them to point out the passage, either in the introductory essay, or in the work itself, or in the supplements. What, then, did Laplace demonstrate ? something so unlike this, that one is disposed to wonder how he can have been thus misunderstood. The method of least squares is simply a method for the combination of linear equations, of which the unknown quantities are the elements to be determined; the constant term of each being a direct result of observation, and therefore affected by an unknown error, while the coefficients are supposed absolutely known, If there are more equations than requisite, that is, more than elements to be determined, what is the best way of combining them? In the first place, they must clearly be combined by some system of constant multipliers, else the resulting equa- tions, not being linear, would generally be insoluble. This condition, however, though absolutely necessary in practice, is in no way derived from the theory of probabilities. It is a merely practical limitation. The question thus narrowed is simply to determine the system of factors to be employed for obtaining the value of any particular element. The factors must of course be such that, in the final equation, the coeffi- cient of this element may be unity, and those of the others severally equal to zero. These conditions being fulfilled, we get a value for the ele- ment in question which is affected by an unknown error, namely the sum of the errors of observation multiplied respectively by the corresponding factors. The mean arithmetical value of this sum may in theory at least be determined, if we know the law of probability of error for each observation ; and Laplace calls that system of factors the most advantageous which makes this mean value a minimum. If, however, the law of probability of error is unknown, the mean value of the error cannot be determined. Nevertheless, if the number of 56 REMARKS ON AN ALLEGED PROOF OF observations is very large, this mean value approximates to a certain limit, the form of which is independent of the law of probability. The essence of Laplace's demonstration consists in its enabling us to determine this limit. When this is done, it may easily be shown that the most advantageous system of factors, those, namely, which make this limiting mean value of the error a minimum, will give the same value to the element to be determined as the system of final equations obtained by employing the method of least squares, provided equal positive and negative errors are equally probable. And the same is of course true with respect to the remaining elements. Thus this system of final equations gives to each element a value affected by a smaller average error than any other linear system, if the number of observations is sufficiently large. \It nowise follows that these values are the most pro- bable; that is, that the errors which must have been com- mitted if these are the true values, form a combination ^ priori more probable than the errors which in like manner have been committed if any other set of values are the true ones. The most advantageous set of factors for determining any element depends only on the coefficients of the equations to be dis- cussed, and not on their constant terms, which are the direct result of observation. Thus these factors are determinable, & priori, before the observations are made. But it is only after the observations have been made that the most probable values of the elements can be found, and then only if we know the law of probability of error. Laplace has pointed out the differ- ence between the two investigations. This difference, however, the reviewer does not seem to have apprehended. He plainly supposes that Laplace proves the results of the method of least squares to be the most pro- bable results, which can only be the case, as Gauss had *in effect shown, if a special law of error obtains. He therefore undertakes to prove, that for all kinds of observations this is actually the only possible law. But for the supposed authority of Laplace, he would pro- bably have perceived that nothing can be more unlikely than that the errors committed in all classes of observations should follow the same law ; and that at any rate this proposition, if true, could only be proved inductively, and not by an & priori THE METHOD OF LEAST SQUARES. 57 demonstration. For It is beyond question distinctly conceiv- able, that different laws may exist in different classes of ob- servation; and that which is distinctly conceivable is h priori possible. So that we cannot prove it to be impossible, though we may be able to show empirically that it is not true. You will probably agree with me in thinking that a wrong notion of Laplace's reasoning lies at the root of the reviewer's new demonstration. But we now come to the demonstration itself. The assumption that the law of error is in all cases the same, is, we are told, justified by our ignorance of the causes on which errors of observation depend. The law " must necessarily be general, and apply alike to all cases, since the causes of error are supposed alike unknown in all." Two remarks are suggested by this statement: in the first place, that our ignorance of the causes of error is not so great but that we have exceedingly good reason to believe that they operate differently in different classes of observations ; and in the second, that mere ignorance is no ground for any infer- ence whatever. Ex nihilo nihiL It cannot be that because we are ignorant of the matter we know something about it. Or are we to believe that the assumption is legitimate, inas- much as it in a manner corresponds to and represents our ignorance? But then what reason have we for believing that it can lead us to conclusions which correspond to and repre- sent outward realities ? And yet the reviewer at the conclu- sion of his proof asserts, that, on the long run, and exceptis excipiendis, the results of observation " will be found to group themselves according to one invariable law." Thus the assumption, though "it is nothing more than the expression of our state of complete ignorance of the causes of error and their mode of action," leads us by a few steps of reasoning to the knowledge of a positive fact, and makes us acquainted with a general law, which is as independent of our knowledge or our ignorance as the law of gravitation. Let us, however, suppose it to be true that the law of error is always the same, and that equal positive and negative errors are equally probable. To determine the special form of the law, the reviewer employs a particular case he supposes a stone to be dropt with the intention that it shall fall on a given mark. Deviation from this mark is error ; and the pro- 58 REMARKS ON AN ALLEGED PROOF OF bability of an error r may be expressed by the function or /(; 2 4-2/ 2 ), the origin of co-ordinates being placed at the mark. It is of course supposed that equal errors in all directions are equally probable. We have now only to determine the form of/ This the reviewer accomplishes in virtue of a new as- sumption, namely, that the observed deviation is equivalent to two deviations parallel respectively to the co-ordinate axes, " and is therefore a compound event of which they are the sim- ple constituents, therefore its probability will be the product of their separate probabilities. Thus the form of our unknown function comes to be determined from this condition, viz. that the product of such functions of two independent elements is equal to the same function of their sum." Or in other words, we have to solve the functional equation But it is not true that the probability of a compound event is the product of those of its constituents, unless the simple events into which we resolve it are independent of each other ; and there is no shadow of reason for supposing that the oc- currence of a deviation in one direction is independent of that of a deviation in another, whether the two directions are at right angles or not. Some notion of an analogy with the composition of forces probably prevented the reviewer from perceiving that, unless it can be shown that a deviation y occurs with the same comparative frequency when x has one value as when it has another, we are not entitled to say that the probability of the concurrence of two deviations x and y is the product of the probabilities of each. Without this sub- sidiary proof, the rest of the demonstration comes to nothing. The conclusion to which it leads is in itself a rednctio ad a"b~* surdum. Of the above written functional equation the solution is /(# 2 ) =e wa)9 , m being a constant, so that the probability of an error of the precise magnitude x is a finite quantity; and I need not point out to you that it follows from hence, that the probability of an error whose magnitude lies between any assigned limits is equal to infinity, a result of which the interpretation must be left to the reviewer. He may have thought that the exponential factor is the essential part of the expression THE METHOD OF LEAST SQUARES. 59 V-7T and that the others might, for the sake of simplicity, be dropt out. But whatever his views may have been, his conclusion is unintelligible. The demonstration may, however, be amended so as to avoid this difficulty, and we will suppose that the reviewer meant something different from what he has expressed. Let f (a? 2 ) dx be the probability of a deviation parallel to the axis of abscissae, of which the magnitude lies between x and x + dx. Then/ly 2 ) dy is similarly the probability of a deviation parallel to the axis of ordinates, and lying between y and y -f dy. Thus the probability that the stone drops on the elementary area dxdy, of which the corner next the origin has for its co-ordi- nates x andy, seems to be f(%?}f(y*} dxdy\ and as all devia- tions of equal magnitude are equally probable, this probabi- lity must remain unchanged as long as the sum of the squares of x and y remains the same ; so that we have for determining the unknown function the equation of which the solution is /(<*)= A*-*; and as the deviation must of necessity have some magnitude included between positive and negative infinity, we must have Hence m must be negative; if we call it A 2 , it is easy to show that A is equal to -pr ; so that finally VTT , VTT which is what may be called Gauss's function. But to this demonstration, though it leads to an intelligible conclusion, the original objection still applies : the probability that the stone drops on the elementary area dxdy is not, gene- 60 REMARKS ON AN ALLEGED PROOF OF rally speaking, equal to/(cc 2 )/(y 2 ) dxdy, so that the equation for determining the form of the function, namely, is not legitimately established. To illustrate this, let ^ (xy) dxdy be the probability that the stone falls on the elementary area in question; then the con- dition that the probability of a deviation of given magnitude is constant will be expressed by r (xy) = y" be the new values of the co-ordinates, we find, after some simple reductions, a cos a" a sin a" 77 = . ** > cos a cos a ' cos a cos a Squaring these and adding, f ' 2 JL. 2/" 2 - .__ as; ____. cos 2 a cos 4 a' cos a cos a' cos a" * cos a cos a' or cos a cos a cos a Now this being symmetrical between a, a', a", will hold equally true of the three points of intersection, and it is the equation to a circle passing through the origin which is the focus, whose diameter coincides with the axis of cc, and whose radius is _ a 2 cos a cos a' cos a!' * The chief advantage of this method besides its simplicity is, that it gives us very readily the radius of the circle, and the position of the diameter which passes through the focus. It is easily seen that the distances from the focus of the three points of intersection of the tangents are respectively r __^ ~~" " ~~~ cosacosa'' ~~" cos a cos a" ' ~~~ cos a' cos a' 7 * The area of the triangle formed by the intersection of the tan- gents, can be expressed by an elegant symmetrical function of ON SOME PROPERTIES OF THE PARABOLA. 67 tana, tan a r , tan a", that is, of m, m', m". Since the lines joining the origin with the vertices of the triangle make angles a, a', a" with the diameter of the circle or the axis of x, the angles they make with each other are a! a, a" a, a" a, and the area of the triangle will be i rr' sin (a' -a) + J/r" sin (a" - a') - i rr" sin (a" - a). Substituting for r, /, and r" their values, this becomes a 2 f sin (a' a) sin (a" a') sin (a" a) 2 [cos 2 a" cos a cos a' cos 2 a cos a' cos a" cos 2 a' cos a cos Expanding the sines and making obvious reductions, we get a 2 (tan a! tan a , tan a" tan a! tan a tan a" 2 [ cos 2 a" ^ cos 2 a "*" cos 2 a 7 or grouping differently, and putting sec 2 a for - g- , and so on, cos a 2 {tan a (sec 2 a' - sec 2 a") + tan a' (sec 2 a" sec 2 a) Hence -f- fly> 8 + [ft +ft Taking the chromatic variations of the two terms in the usual way, and equating each to zero, we find The first of these equations becomes unnecessary in the par- ticular case considered by Mr Airy, viz. that in which y l = ; the second is identical with that given by him at p. 245, when attention is paid to the signs. Taken together they determine OF TELESCOPES AND MICROSCOPES. 73 the relative positions of three given lenses, which shall form a combination achromatic for rajs of any degree of obliquity. In the particular case in which the focal distances of all the lenses are equal, and the intervals a^ & 2 , &c. are also equal, the general equations degenerate into a system of simultaneous equa- tions in finite differences. They are then Eliminating & n9 we get #- (P The general solution of this will be A being a root of the recurring quadratic equation a? 2 (pa + 2) x + 1 = 0, c and Cj are to be found by the conditions The general solution of the system of quasi-equations em- ployed in the enquiry must involve some functional operation which degenerates into the radical contained in A. It would be perhaps worth considering how far we might be able to present this operation in a distinct form, defined and distinguished by a particular symbol; but the subject is not one which can be discussed at present. At any rate, we see that the research of the general expression for y n is one of considerable difficulty. The greater part of the investigation given by Mr Airy in the conclusion of his paper, with respect to the achromatism of microscopes, becomes unnecessary by employing the general expression given above fory 4 . His object is to determine the distance of an object-glass of given focal length from a diaphragm whose distance from the field-glass of a given eye-piece of three lenses is given. Let a be the distance of the diaphragm from the field-glass ; therefore we have z t = a y l9 and putting this value for # x , we get an expression for y 4 of the form y 4 = y^R. The chro- matic variation of this is to be zero, and consequently that of its logarithm j 74 ACHROMATISM OF EYE-PIECES, <&c. Now Mr Airy lias shown that, (adopting the notation of this paper) Ay, Sa _ */,! - mtm a + ft] + a fa + a,a, [pj> + a A l> A + PA] + i, Let f (x, y, &) c be the equation of the free surface ; then F being a function of (#, y, #), all that is requisite for the force at any point of the new free surface to be normal is, that shall be its equation. Let F= J(f'x)*+(f'yY +(/'*)* J then ^; Ju FLUID PARTICLES. 77 V r^=a constant, which we may take for unity; therefore ^V. Eesolving this force along the axes, X=7'; whence X=f'x, and so F=/>, Z=fz. & is the increment of pressure = Sp ; multiplying the three equations (1) by X, Y, Z, and adding, we get or putting d for S, dp = Xdx + Ydy + Zdz .................. (2), the equation of equilibrium of an homogeneous and incom- pressible fluid, whose density is unity. An objector to Clairaut's reasoning might urge, that this result, though certainly sufficient, was not shown to be neces- sary : he might argue, that a way has been shown of building up a fluid mass ; but that it has not been proved that every fluid mass is capable of resolution into the smaller masses, by means of which alone Clairaut investigates the conditions of equilibrium. Unless it be made a direct postulate, that every fluid mass in equilibrio will continue in equilibrio, when the part of it contained "between the free surface and any level sur- face is removed, it is difficult to see how this objection can be met, except by showing that the property assigned by Huygens to a free surface, viz. that the force is normal to it, belongs to every surface of equal pressure, and that consequently Clairaut's reasoning is in reality independent of any construction or reso- lution of a fluid mass into successive strata. When we assert, with Clairaut, that a fluid mass in equilibrium is not disturbed by the addition of a stratum producing equal pressure, we imply that the reaction produced at any point of the surface of A, by the pressures exerted over the rest of the surface, i. e. the effect of the transmitted pressures, is normal to it. For we know that the forces at the surface are so ; and unless the inference first stated is correct, there could be no equilibrium. It hence appears, that Clairaut's axiom is equivalent to this Equable pressure produces a reaction normal to the surface on which it is applied. But if the force at a surface of equal pressure were not normal to it, there could be no equilibrium, because 78 EQUILIBRIUM OF MUTUALLY ATTRACTIVE it Is only by the transmitted pressures that it can be esta- blished. Clairaut, as his views are represented by Mr Ivory, says nothing of the transmission of pressure ; but it is impossible to investigate fluid equilibrium without tacit or expressed re- ference to some distinctive character of fluidity; and in the principle he makes use of, the idea of the transmission of pres- sure is essential. It appears, then, that the force at a surface of equal pressure is normal to it ; and this conclusion is little else than a different way of putting the principles employed by Clairaut. We are now enabled to dispense with any process of constructing a fluid mass. On referring to the mathematical reasoning employed above, we shall easily see that, substituting two infmitesimally near surfaces of equal pressure for the consecutive free surfaces of Clairaut, the result we arrive at is simply the symbolical ex- pression of the principle just laid down, viz. that the force at a surface of equal pressure is normal to it. A very little at- tention will show, that (2) is true in every case of fluid equi- librium, and that it is completely equivalent to the principle which it represents. In translating, so to speak, his funda- mental idea from the infinitesimal to the fluxionary conception, that namely of successive generation, Clairaut has tacitly intro- duced a new condition, namely, that a surface of equal pressure will necessarily be a free surface of equilibrium, the superin- cumbent part being removed. Mr Ivory remarks " The investigation of Clairaut is clear and definite. It evidently assumes that there is no cause tend- ing to disturb the equilibrium of A, except the action of the forces at the surface of A upon the matter of SA. On this account his method fails when there is a mutual attraction be- tween the mass A and the stratum SA. If the mass A attract the matter of the stratum SA, and cause it to press, it follows necessarily that the matter of SJ. will react, and by its attrac- tion will urge the particles of A to move from their places. In this case, therefore, the equilibrium of A is disturbed by a force which Clairaut has not attended to; and unless the effect of this new force is counteracted, the body of fluid A -f- $A will not be in equilibrium. The principle of the method suggests a remedy for this omission, for it is easy to prove that the equi- FLUID PARTICLES, 79 liTbrium of A will not "be disturbed by the attraction of the stratum SA, if the resultant of that attraction on every particle in the surface of A be directed perpendicularly to it." This reasoning satisfactorily shows, that if a fluid mass of attractive matter be increased by a stratum producing equal pressure over the free surface, the equilibrium will be destroyed unless a certain condition is fulfilled, of which the symbolical expression is P, Q, JR being the attractions, parallel to the axes of co-ordi- nates, of an element of that part of a fluid mass which is external to a given level surface. But the necessity of this condition cannot be proved, unless it is shown to be impossible in any way to increase the mass A, without destroying the equilibrium, supposing it not fulfilled. All that has been shown is, that the mass cannot be increased by a stratum producing equable pressure over the free surface. Now, generally 'speaking, the mass so increased will not fulfil the condition of having the forces at the new free surface normal to it, those acting at the original free surface being of course so. We cannot, therefore, affirm that we have fallen on a case in which the ordinary con- dition is fulfilled, without producing equilibrium. If, however, we dispense with the limitation, that the stratum added shall produce equable pressure, we lose the simplicity of Clairaut's method, nor can we make any use of his principle, except by setting aside the construction he employs, which confines him. to the particular case in which a surface of equal pressure is potentially a free surface. This has already been done, and the result is the general equation of equilibrium. It remains to show, that it is in all cases sufficient. It is admitted to be sufficient in the case of a fluid acted on by forces tending to fixed centres. We shall endeavour to reduce the general case to this. Conceive a body acted on by a force directed to a fixed point. It may be so placed, as to remain at rest under the action of the force, that is, the resultant of the force upon it is equal to zero. In this position of the body, the centre of force is some point within it. Let the body, remaining in the same position, diminish sine limite, being always similar to itself, the resultant of the .force upon it is always equal to zero ; and ultimately, when the body 80 EQUILIBRIUM OF MUTUALLY ATTRACTIVE, &c. becomes a physical point, it coincides in position with the centre of force, and is in the same state with respect to the action of other forces upon it, as if this force did not exist, This being granted, conceive a homogeneous mass of fluid composed of mutually attractive particles, the free surface of which fulfils the required equation Let the attractive power of each particle "be conceived trans- ferred to a fixed centre of force coinciding with it. Then the action of all the other particles on one particle is precisely re- placed by that of the fixed centres ; and it has been shown, that the resultant of the action of the centre coinciding with a par- ticle on that particle, equals zero. Hence, the supposition we have made does not change, in any way, the forces acting on any particle of the mass. Were the system in its present and former state respectively to move, the motions would be widely different ; but in the arbitrary position we have placed it in, the action on it is precisely the same in the two cases. Now, the single equation given above assures its equilibrium, when we regard it as a system acted on by forces directed to fixed centres; and as the hypothesis by which we are enabled to look upon it in this way nowise affects the forces acting on it, it follows, that the system considered as acted on by mutual attraction must be in equilibrium. Consequently, a mass of homogeneous fluid, the particles of which are mutually attrac- tive, will always be in equilibrium when the free surface fulfils the single condition implied in the general equation obtained above. The same reasoning applies to the case of any fluid, elastic or incompressible. If this demonstration be thought satisfactory, the question raised by Mr Ivory, as to the sufficiency of the general equation, must be looked upon as settled. The suggestions here made with respect to the new condition tacitly introduced in Clairaut's reasoning, will, it is thought, enable us to trace the source of the difference of the view taken by Mr Ivory, and that gene- rally entertained. In one form or other, it seems to recur in every way in which that distinguished mathematician has treated the subject. MATHEMATICAL NOTE*. IN Vol. I. p. 205 1, there were found for the co-ordinates of the point of intersection of two tangents to a parabola, the ex- pressions y = m (a + a'), x = maa', a, a' being the tangents of the angles which the tangents to the curve make with the axis of y. From these expressions it fol- lows, that if y 1? y 2 , &c. a? l3 # 2? &c. be the co-ordinates of the angles of any re-entering polygon of 2n sides circumscribing a parabola, and Also, the continued product of the abscissae of the points]of intersection of any number of tangents, is equal to the continued product of the abscissae of the points of contact, provided no three points of intersection lie in the same straight line. Let x r , x", x' 1 ', &c. be the abscissas of the points of contact, then it is easily seen, from the equation to the parabola, that ce'=W 2 , e" = ma" 2 , a?"' = ma!"*, &c. the continued product of which is x'x"x m . . . x (n) = wVV'V" 2 . . . a (tt)2 . And if x 19 # 2 , x s , &c. be the co-ordinates of the points of inter- section of the tangents, we have cc x = ma' a", x 2 = ma"a'", x 3 = ma f/ a A/ ', &c. the continued product of which is ^ , m n /y /2 /y" 2 "' 2 rv ( "> 2 vOj^Gg.*^ . . . JU n fib . , 01 C(, ...Ot 3 which is equal to the preceding expression. It is necessary to limit the intersections in such a way that no three shall lie in the same line, because otherwise some one of the a's in the second series would appear more than twice. * Cambridge Mathematical Journal, No. VII. Vol. II. p. 48, November, 1839. 1* Page 63 of this Volume. 6 VARIATION OF NODE AND INCLINATION THE following method of finding the variations of the in- clination and longitude of the node, is more convenient than that given in Pratt' s Mechanical Philosophy ', p. 336. Adopting the notation usual in the lunar theory, we have * = &sin(0-ry) (3), d*% ILZ dR also = I- - -4 = =5 ( 2 ) . d? r* ds ^ J In the disturbed orbit, (1) and its first derived equation will be true, as if the elements were invariable ; which * gives the equation fn * dJc 7 //\ \ d , d0 7 dk dR and p -7- K -j- == -7- , r & ^ ^7 ' 9 d6 , . dy _ 1 " """"' dk ___ 1 na dR, ,g, ^~&/Wr^ dy which agree with the known results, Jc being = tan J, or sin J, g^aw proximb) and 7 being what Mr Pratt denotes by ii. (The squares, &c. of I are neglected throughout.) 62 INVESTIGATION OF THE ABERRATION IN RIGHT ASCENSION AND DECLINATION*. THE following investigation of the formulae for Aberration in Right Ascension and Declination will Ibe found to be more simple than that given in Maddy's Astronomy, p. 214. Let E = o>. 1st. For the Aberration in Declination : produce 8N to a point Q, such that 8Q=* 90, and join TQ, TK If A be the coefficient of aberration, and AS the aberration in declination, AS = - A sin ST. cos T8Q = - A cos TQ, as T8Q is a quadrantal triangle. * Camlridge Mathematical Journal, No. IX. Vol. n, p. iao, May, 1840, INVESTIGATION OF THE ABERRATION, &c. 85 But cos QT=* cos TN. cos QN+ sin TN. sin QN. sin and cos 7!^= cos TV . cos v -ZV+ sin rr . sin nn JV. cos = sin cos a cos sin a cos a>. Also, sin TN. sin TJW = sin TV . sin T) + cosScos sin o>}. 2nd. For the Aberration in Eight Ascension : produce to a point R, such that JVS = 90, and join .5$, RT. Then, if Aa be the aberration in right ascension, a = -- ? sn.cos = ----R COS COS O But cos ^3^= cos TV . cos JB qp + sin Tw . sin R T cos 5 qp = sin sin a + cos a sin cos G>. Consequently, A & a -. -- j s i n gin a + cos a sin cos a>}, COS l J ON THE LINES OF CURVATURE ON AN ELLIPSOID*. THE following investigation of the Lines of Curvature on an Ellipsoid has the advantages of symmetry and of giving a distinct geometrical conception. The artifice on which it de- pends may, it is thought, be found useful on other occasions. The symmetrical equation to the lines of curvature is (5 2 - c 2 ) xdydss + (c 2 - a 2 ) ydzdx+(a* -V*} zdxdy = (1), (see Mathematical Journal, Vol. I. p. 142), where xyz are con- nected by the equation to the surface, 2 n ,. # y Z / , x Put ^ = ^ f* = ^ j = to .................. (A). mr 77 T / T dv dw abo 7 7 Inen xdydz ^avu.b - 7 =c~7=.= == udvdw, Vv Nw 4 Nuvw Hence, after the substitution and multiplying by becomes (Z> 2 - o 2 ) wrfyrfw + (c 2 - a 2 ) with the relation Differentiate (3) ; then, since js _ C 2 + C 2 _ ^ + ^ _ &2 ^ Oj * Cambridge Mathematical Journal, No. IX. Vol. n. p. 133, May, 1840, a s - 7/) w duetto = 0..,(3), ON THE LINES OF CURVATURE ON AN ELLIPSOID. 87 we get (5 2 - c 2 ) ud(dvdw) + (c 2 ~a 2 ) vd (dwdu) + (a*-t>*)u>d(dudv)=Q ...... (5). Now this is satisfied by the assumptions -^, ~, =j-.... ....... (B), / ff h /, g, A, being constants. But from (4) we deduce du + dv + dw Q ........................ (6), and (B) gives du =fdu dv div, dv = ff dudv dw, dw = h du dv dw. Hence f+g+h^O ........................ (7), which establishes a relation among the otherwise arbitrary con- stants/, g, Ji. Now (B) implies the existence of two linear equations in u 9 v, w. Hence, a particular solution of (1) is two linear equa- tions connecting the three variables. But the given equation (4) is linear ; hence the solution in question is the one congruent to the problem. To find the other relation in u, v, w, eliminate the differen- tials from (3) by means of (B), and there is (8). Equations (4) and (5), with the relation (7), contain the com- plete solution of the problem. It is obvious that the apparent want of homogeneity of (B) is wholly immaterial. Keeping in mind the values of u, v, w, given by (A), we see that the geometrical interpretation of (8) is, every line of cur- vature on an ellipsoid lies on a conical surface of the second order, of which the vertex is the centre of the ellipsoid. To determine the constants, let the line of curvature pass through a point, for which the values of u> v, w> are w 15 v# w v we have 88 ON THE LINES OF CURVATURE Hence, after a slight reduction, a quadratic in * , of winch the roots are real and of unlike signs. This is obvious, for u v v^ are essentially positive, and a, Z, c, "being in order of magnitude, the signs of 5 3 c 2 and c 2 a 2 arc opposite. Similarly, | is determined by a quadratic, whoso roots are always real and of opposite signs. Thus two lines of cur- vature pass through every point on the surface of the ellipsoid. Let us now consider the envelope of the surfaces represented V (8). Differentiating (7) aud (8) foi-yj g, 7t, we get ) - v d ff + (a-5 2 ) Jd = ...... (10), / dh = Q ..................... (11). Z being an indeterminate factor, we may put Z=(S a -c 2 )^, Z=(c'- 2 )J, Z=(a s -5 s )J; J *s whence, taking the values of f,g, h, to substitute them in (8), we deduce VP^T? Vw + Vc a - a 2 ^v + Va^^T 8 Vw = ...... (12). As the signs of the radicals are independent, this represents four planes; but c 2 a? is negative. Hence the possible part of these planes is their traces on the plane of x&, for which v = 0. Thus we get the two straight lines V6* - c a Vw + VS^^F Vw = ............... (13), and for the points where they meet the ellipsoid, u + w=*l ........................... (H), whence a^J 2 =5- - 1 a c 0.2V" AN ELLIPSOID. 89 These values belong to the umlbilici of the ellipsoid ; a result easily anticipated. When they are introduced in (9), it "becomes 2 2 ^ = 0, and similarly ^ = 0. / ^ Hence (8) reduces to ^ = (16), and represents the principal section of the ellipsoid, which passes through the greatest and least axes. In this case then, as our analysis would lead us to anticipate, the lines of curvature coincide ; a result which, although well known, seems not very accurately demonstrated by Leroy. After having shown (p. 309 of the second edition) that the two directions of curvature coincide at the umbilical points, he proceeds to integrate, and fjfj passes from -^ = to y = h, and thence, determining the con- stant, to y = ; which last represents the line of curvature sought. But -j- has been shown to have the value 0, only for CbtX/ the umbilical points, and we are therefore not at liberty to pass by integration from these to any other points at which this may not hold. Were the process legitimate, it would lead to the strange conclusion, that the lines of curvature through an umbilicus are necessarily plane curves. As there appears to be still some difficulty with regard to the theory of these singular points, we may enquire whether, in order to determine the lines of curvature through any point whatever, more is requisite than to substitute its co-ordinates in the general equation of the lines of curvature, and thus to get two values for the arbitrary constant; whether the result can ever be indeterminate, except when the lines, as at the extremity of an axis of revolution, are so in reality. In this view we see at once, that the process given by Leroy after Poisson for de- termining the directions of curvature at an umbilicus, is simply the ordinary method for ascertaining the position of the branches of any curve at a multiple point ; and that the result arrived at, is not that more than two lines of curvature pass through an umbilicus, but that every point which, with reference to the surface, is umbilical, is, with reference to the lines of curvature, a multiple, or more generally a singular point. These sug- 90 ON THE LINES OF CURVATURE gestions may, perhaps, show how we must determine the lines of curvature which pass through an umbilicus, a problem dis- tinct from that solved by Leroy of finding the directions of curvature. Many curious properties may be deduced from the equations we have arrived at. Thus, if we take on two concentric and confocal ellipsoids, a series of pairs of corresponding points, (such as are spoken of in the enunciation of Ivory's theorem,) and if the locus of the points on one of the ellipsoids is a line of curvature, then that of those on the other is so too. Again, the traces on the tangent planes at the extremities of the three axes, made by one of the cones represented by (8), are an ellipse and two hyperbolas respectively. The areas of this ellipse, and of the ellipses conjugate to the two hyperbolas, are so related that their continual product is constant for the same ellipsoid, and for all ellipsoids of the same volume. The method of demon- strating these two theorems is so obvious, that it scorns unneces- sary to enter more fully on either. It still remains to be shown how we pass from (8) to the projections of the lines of curvature on the co-ordinate planes. The symmetry -of the problem is destroyed by the transition ; but as it is in this shape that the results are commonly ex- hibited, we shall dwell rather more upon it than would other- wise have been necessary. Putting C=a?-~ 6 2 , B = c 2 a 2 , A = V c 2 , and eliminating u, v, w, successively between (4) and (8), there result A\ fB 9 (A B\ fC A\ A - I -? w = (-- (f Put J g r U f) w -f B 0\ (A B\ _B _I_,_^ M __ A . B G , A B ~ = ' ~ = l ' ~~ .(17). Then ON AN ELLIPSOID. Thus we get the relations 91 (19). mg Teh = Hence, > = 1 = y ( -j j- ) > ' mf mg h\l k ' * and consequently similarly, A B- _0_A ff ~~m I . k .(20). By means of (18) and (20) the equations (17) Tbecome OUe Tcu Iv = mv Jew = Ak- Bl Amk Bm - Ok Blm .(21); ~ 01 -Am) which, restoring their values to u 9 v 9 w, may be written G Jc If Jc cf Put y -5 = m, ; /. j==m^ 9 and then we get or u = mc?(tf with similar equations for the projections on the other co-ordi- nate planes. This result is identical with the known one in Leroy, p. 304, or Hymers, p. 201 . It is hoped that the novelty of treating symmetrically a non- integraHe equation in three variables, will be admitted as an excuse for the length to which this paper has extended itself. MATHEMATICAL NOTES 1. THE area of a polygon of a given number of sides, circum- scribing a given oval figure, will be the least possible when each side is bisected in the point of contact. This elegant proposition, given in the Senate-House Problems for 1836, may be easily demonstrated as follows ; Let A 9 BO) CD, be consecutive sides of the polygon. Produce AB, DO, to meet in JE; then BG must, by the con- dition of the minimum, be in such a position that EBO is a maximum. Kefer the oval to EA, ED, for axes, then the equation to the tangent BG is y'dx x'dy = ydx xdy, y and x being the co-ordinates of the point of contact P. Put x' = j and so x Also area of JEJBO= ^ O y sin E. , T (ydx xdyY . . ,,, . . . . Hence, -r~ -5 ^~ is a maximum (the minus sign is im- material) . Differentiate, considering x as independent ; then ydx-xdy 2 dxdy d The last factor only gives a solution ; * Cwnbridge Mathematical Journal, No. IX, Tol. n. p. 142, May* 1840. MATHEMATICAL NOTES. 93 that is, PM being parallel to EG, EM^^EB, and .% or BO is bisected in the point of contact P. The same is true of any other side, and therefore every side is "bisected in the point of contact. 2. Let p, p', be two forces into which a given system on a rigid body may be resolved, a, 0, their least distance, and in- clination of their directions ; pp a sin 6 is invariable. (Senate- House, 1833.) Let the line a meet the directions of p and p in P and P' respectively. At P apply two forces equal and parallel to p', and opposite each other. Thus the system of forces is replaced by the couple pa, and by the force at P, which is the resultant of p and p '. Kesolve this, the resultant, along the axis of the couple and in its plane. Then the former component can arise only from the resolved part of p, asj/ is wholly in the plane of the couple. Also, as the shortest distance is perpendicular to both lines, it follows that the arm of the couple is perpendicular at P to the plane which contains the two forces p and p \ Hence 6, their mutual inclination, is that of p on the plane of the couple, and therefore p sin is the part of the general re- sultant resolved along the axis of the couple. Then, if the general resultant makes an angle with the axis, we have in the usual notation pp a sin = G-R cos < = R . G cos $. Now 6r cos <)E>, as is known, or as may be easily shown, = Gr v the minimum maximorum moment of the system ; therefore pp'a sin = GJR, which is constant. ON THE TAUTOCHRONE IN A RESISTING MEDIUM*. OtJB olbject is to reduce the problem of the tautochronous curve, when the resistance is equal to 7cv* or to Jiv -f 7^ 2 , to the cases in which it is a cycloid, viz. when the resistance is equal to zero or to Tiv. In vacuo and the necessary and sufficient condition of tautochronism is / = Ax. Hence generally d* r J o is independent of z v provided z* = A . Fz. Now, when It = 7cv 8 ? there is vdv 7 o dx = kv = a -7- , ds J ds ' ^ J ds a dx S"' therefore therefore and Put dz = e"* s cfej and let z and s be equal to zero together, therefore 3 i (1 - e' u ] , fs = J?i, * Cambridge Mathematical Journal, No, X, Vol. n. p. 153, November, 1840, ON THE TA UTOCHRONE IN A RESISTING- MEDIUM. 95 , ,_ and V 2<7 t J l Therefore for tautochronism tf = A. Us. But J&=rl- therefore -r # = a# == -7- (1 &#), -a. 6fc5 ^j x e""^ s therefore ^j -y = a =^ , & e s or J = a(e**~l) .............. . ..... (1), the equation of the tautochronous curve. We shall have, if t = when s = s 1? = 0! COS ogr t. must become, when s is expressed in ^, a result easily verified, for . 1 fos (1 The coefficient of dz in (a) is of course that of d?& in (/3), which shews that if the equation of motion were or s= 96 0^ THE TAUTOCHMONE IN A RESISTING MEDIUM. the equation in z would be This is precisely tlie form of the equation of motion on a cycloid when E = Jiv. And as in that case f x deduced from it is independent of s v so in this it will be independent of # 15 and consequently of s r Hence the curve whose equation is (1), is tautochronous, not only when B = Jcv* 7 but also when E = hv 4- Kv*. For Laplace's abstruse solution, see the first book of the Mtoamgue C&leste^ or Mr Whewell's Dynamics, ON THE INTEGRATION OF CERTAIN DIFFERENTIAL EQUATIONS*. No. I. IT is shown in the theory of the earth's figure, that if the pressure and density at any point be connected "by the equation dp = Jcp dp, where Jc is a constant, then the ellipticity of the surface may be deduced from the solution of the equation This equation is not easily integrated. La Place, in the eleventh book of the Mecanique C&leste (v. 51), gives a solution of it, but without demonstration ; and the lacuna thus left is not supplied in the works on the subject generally made use of in Cambridge. Mr Gaskin has however effected the integration of whenj? is integral, in finite terms (vide Hymers' Diff. E% p. 53), and the proposed equation is a case of this one. But perhaps a more direct analysis is preferable, as it enables us to extend our method to two or three classes of equations of all orders. One of these will be considered in the present paper another, the solution of which admits of a remarkable symbolical form, will be given in the next number of the Journal. We shall begin with the particular equation which occurs in. the theory of the earth's figure, both because from its physical * Cambridge MofthGinaiical Journal, No, X, Vol. IT, p, 169, November, 1840, 7 98 INTEGRATION OF DIFFERENTIAL EQUATIONS, application it has an interest for some who care but little for pure analysis, and because it will exemplify the general method. Let y ... {nfa-lJ-GK + gV^O ............... (3), n (n - 1) - 6 = n (n - 1) - 3 (3 - 1) = (n - 3) (n 4- 2) , A (w-3) (w + 2)a n + ^ 1 or 5 as n is odd or even, and we see at once that ^b n x n =5 5 cos gx 4- & t sin ^ or changing the constants, S&^ n =:(7 sin (go? -fa) ...................... (7). Also by (5), for by (6), 4 2 n the complete solution, which may be written thus, a ) ...... (jo). INTEGRATION OF DIFFERED We now proceed to the more general As before, we shall get {(-!) -^(^-1)1 Now n (n 1) ~p (p 1) = (w ^>) (n which is the fundamental principle of our analysis, (n-jpj^+^p- 1)^ + 2*0^ = ......... (14). Assume (n + (n-~p + 2) l n ............ (15), and (^~p + 2)(w+^-3)5 w +^V 2 = ......... (16), Again, assume )c^ ............... (17), and so on successively* Thus we shall get a series of equations, of which ^-lJ^ + ^^O ......... (18), is the general type, where ^ is even. If p is even, let p~p>, .*. p /* 1 = 1. If it is odd, let j) s=r /A + 1, .*. jp + ^ = 1 : and in both cases (18) becomes and therefore S4aj M =(7sin(^+a) ..................... (19). Let n- be any two consecutive equations ; then - A* + 1)4 =(*-.? + /*),. ............ (20), 2 ^ 1, + 1 ....... (21), 72 j? DIFFERENTIAL EQUATIONS. = - -j (n + 2 -p + - = # I s ( - * + M) *.*". Bj the application of this formula, y or 2a n # w niay be de- duced "by a series of regular operations from G sin (j# + ), If p is even, 2 (p - /A) + 1 gives the scries 1, 5 5 9, &c. If it is odd, the series is 3, 7, 11, &c. Particular cases may "be solved by (22) with considerable facility* By inspection we have y = CMsin (gx + a) + cos (gx + a) [ i ( l x J for the solution of The solution of -y4 + =s J sin (g + a) H cos (jo? + a) > d a ?-r*y& IS I '" . 5(7 INTEGRATION OF DIFFERENTIAL EQUATIONS 101 The second line is equivalent to 2 2 ) - - sin (qx + )-* cos (qx + a) J , and thus / / "IK y == C -Isin (jaj + a) H -- cos (gx + a) ^ s * n (. Here we have n {(- 1) (n - 2) -^(^-1)} " Let .% w (w-^ - 1 + 3) (n+^ - 2 - 3) l n + fb^ = ; and generally w (n - j? - 1 + v) (n +j? - 2 - v) 4 + fl^ = ...... (24), where v is divisible by 3* Ifj? is so too, let p = z/, /. w p l + i/ = ^ l and ^-fp 2 v n 2. If ^ 1 is divisible by 3, let j? 1 = z/ ? .% p 1 + z; = 2 and jp 2 v = .1 ; and in both cases (24) becomes and therefore 5Z n cB* fulfils the equation ^ = ........................ (25). 102 INTEGRATION OF DIFFERENTIAL EQUATIONS. Again, if n (n -^ - 1 + v - 3) (n +p - 2 - v + 3) l n + fl^ = 0, n (n-p - 1 + v) (n+ p - 2 - v) & + ;= 0, is dx* z = Ojf* + Cj* sin f ~ qx + a) , \ ' and by (26), , *> T.- i. or y = * + , which giTes for the complete solution of the proposed equation. INTEGRATION OF DIFFERENTIAL EQUATIONS 103 It is olbvious that analogous equations exist in all orders, and that when^> is of certain forms, may be integrated in finite terms. f It will be sufficient, after what has been said for the cases of m = 2 and = 3, to state the results of the general investigation ; they may be very readily deduced by the same method as that we have already used. The process succeeds when either of the factors p or p 1 is divisible by m, and the general formula of which (22) and (26) are cases, is ...... (27). Particular cases may however be easily solved without re- ference to this formula ; thus, if we had we should proceed as follows : n (- 1) {(-2) (n-B) -4 .3} a n - cfa^ = 0, n (n - 1) (re - 6) (n + !)- cfa^ = 0, ' 3 sin (gx 4- a), 3 3 and a = ~b n 5 A = T}* i ( /. y = *%b n & n 4 Sw (w 1) (n 2) Zy*> n 4 , f 3 3 hin (gx + a) H cos I S 35 104 INTEGRATION OF DIFFERENTIAL EQUATIONS: The principle of our analysis, it has already been remarked, is contained in the equation n (n 1) p (p 1) = (n p) (n + p 1) ; and this consideration suggests an extension of it. For it is obvious that the coefficients of a n in - differ only in this, that where the first has the factors (n m + s -h 2) (n m + 5 + 1), the second has ^? (^ 1). Thus the same transformation applies; and if either p, or p - 1 is divisible by m, the solution of may be made to depend on that of and thus effected in finite terms. The formula of reduction in this case is a little more com- plicated than those already given, and we will not dwell longer upon it, our object being rather to point out the integrability of certain classes of equations than actually to integrate them. The equation n (n 1) p (p - 1) == (n -p} (n +p - 1) is a particular case of n(n-p) -p (p - /*) = (n -p) (n + p - /*), and the latter will give us various formulae of reduction accord- ing to the value of p,. Thus INTEGRATION OF DIFFERENTIAL EQUATIONS. 105 may be reduced to ^_ 1 dy 2 cfe 2 S5S + ^- ? for the coefficient of a w in tlie former is w (w - 2) p (p - 2) = (w ^) (w +

^=Q ...... (B). The factor n 2 may be safely neglected. But n I IB essential, because the solution of the auxiliary equation d z & dz __ da? " dx gives (n - 1) (nb n + gb^ = 7 - and would be incomplete if we omitted the first factor. From (a) we get 106 INTEGRATION OF DIFFERENTIAL EQUATIONS. and, as except when n l, there is unless ^ = 0, , ~ /. a, = *,+ -i 4 ........................ (<*') Now the solution of the auxiliary equation is * = Ci + etf*\ and from (a') we deduce # = H -- z\ ^ qx therefore y = (l + ~) (o, + c a O is apparently the solution of the proposed equation. But it will be found not to satisfy it, unless c t = 0, and then is only a particular solution. The reason is, that in laying clown (a') as generally true, we imply that *&* + qb^ = is true for %! whereas the equation which contains the i s d* solution of shows that 5 t is not necessarily connected with 5 ; and that if we assume such connection, we get only a particular solution. Hence our formula of reduction implies the connection of Z> a and J ; while their independence is implied in the general solution of the auxiliary equation, to which this formula is applied ; and these contradictory suppositions lead to an erroneous result. To put c^O, is to connect J and J 1? or, which is the same thing, to neglect the factor n 1 ; and the value of y thus got is therefore a solution, but not the complete solution of the pro- posed equation. To complete it, we must, bearing in mind the independence of 5 , recur to (a), which is always true, " ^o--^ INTEGRATION OF DIFFERENTIAL EQUATIONS. 107 and from (a'), which is true for n = 1, we get -6 -I -1 "I 0' Now &_j is obviously = 0, /. , = --,; and these two quantities are independent of a 1? ^ 2 , &c., 2^/ is a particular solution, and is the complete solution of the proposed equation. The method of proceeding suggested by this example is to obtain a solution, neglecting all factors analogous to (n 1), and then to complete it by reference to the assumptions of trans- formation, such as (a), which hare been made use of. The equations which we hare solved are not a very numerous nor perhaps an important class. But one of them, at least, is susceptible of a physical application of great interest ; and so few equations of the higher orders are integrable in finite terms, that the discussion of those which are, has always some degree of value. ON THE INTEGRATION OF CERTAIN DIFFERENTIAL EQUATIONS*. No. II. IN the last number of the Journal f, a method was giveri for the investigation of a class of differential equations,, Iby means of successive reductions. The present communication contains solutions of some ana- logous equations effected "by a similar process. The results will however he exhibited in a very different form. We begin by taking a particular case of the equations in question, where p is an integer. Let n + ka^ m = 0. Assume a n as [n m + 1 (p - 1) m] {n m + 1 - (p - 2) m} f(Jc) lacing some function of 7c, to be determined hereafter . Then W = (n m + 1 pm] {n m + 1 (p 1) m} ..,(n-m + l-m)f(k)b n _ m ...... (4). * Cambridge MatJiematical Journal, No. XI, Vol. II, p, 193, February, 1841, t Page 96 of this volume, INTEGRATION OF DIFFERENTIAL EQUATIONS. 109 If we substitute these values In (2), every factor of (4) and every factor, except the last, of (3), will disappear, and the re- sulting equation will be n(n-l)...(n-m + 2) (n - m + 1) t> n + kb MH = ...... (5), This is what (2) would be, were p = 0. Hence y = S l n & n d m y fulfils the equation -=- + ley = 0, to wliich (1) would, in that case, be reduced. Let y = X be the ordinary form of the solu- tion of the last-written equation. Then we must obviously have where (f> (&) may be any function of 7c. A very little attention to the mode of integrating linear equations with constant co- efficients will show, that in X, x always occurs in conjunction JL with k m . If we put X=^.A n x\ we must consequently have where N is a function of n, except for values of n < m, when it is an arbitrary constant ; therefore Recurring to (3), inverting the factors and multiplying and dividing by m p , we shall easily deduce the following equation, i- (p - 1) N.f(Tc) + [Jc] The form of these p decreasing factors naturally suggests the idea of making /(&)="*; and if we then put (k) =&~~, we get m / \ m 110 INTEGRATION OF DIFFERENTIAL EQUATIONS. n M , and .-. The factor m* may obviously be neglected, and we shall X therefore have, on replacing S5^ n by $ (ft) -2T, i e. by -^ , the & m following equation, for the solution of (1), y = X being that of 4- % == 0. If m = 2, X= sin (A/(^) a? -f a}, and ^- = |. Hence is the solution of This result is given in Hymers 1 Diff. Equations, and is, I believe, due to Mr Graskin. If the proposed equation were we should immediately conclude, from analogy, that its solution must be n +ka n _ m = 0. Assume a *~~*' ^ ^ - INTEGRATION OF DIFFERENTIAL EQUATIONS. Ill Therefore as before. and a n z=m~ p m J \ m m-l Here we make /(&) = P, and (&) = Ji m , no complementary function being added. Hence, precisely as before, we find that = 3r* X is the solution required. Let us now consider the more general equation, If y = 2 . a n x n , there will be i ... (n s + 1) (n -~spm) (n s I).,* This equation is analogous to (2); Ibut m 1 is replaced "by $. Assume, therefore, " ^= (^ s pm){n s - (j? 1) m}.,. (n s -f m) Tc^l^, and n.. . (n s -f l)(n 5) (n 5 1)... (n m + 1) # n + ^J^m = 0. Hence, as before, where JT denotes the same function of x that it did in the former case. Consequently l n = N^>(Jc)Jc m 9 and 112 INTEGRATION OF DIFFERENTIAL EQUATIONS. s_ We must, it is evident, make < (&) = k~*\ and then rp jr Hence 2/ = jrp s is the solution of the equation d s y for as before the factor m may Tbe neglected. A particular case of this result is that in which s = 0. If with this value of 5 we have m = 2, the equation to be integrated takes the form and the solution is Equation (l) is the most general one in which the coefficient of a n differs in one factor only from what it is in the case of But our method is applicable in other cases. Let us resume the equation discussed in the last number of the Journal, By the usual method of mating y = Say n , we get n (n - 1) ... (n - m + 3) {( - m + 2) (n -m + l)-p.(p- 1)} a n or n(n l)...(n wi + 3) {( m + 2 _ + ^ w = ...... (9). It may be remembered that we found it necessary that p or p 1 should be divisible by TW. Suppose then that p is so divisible, and that the quotient is j. OF DIFFERENTIAL EQUATIONS. 113 TUu ...... (10), (p m)} ...... , (there are q factors in both numerator and denominator) ; equa- tion (9) becomes n (n - 1 ) . . . (n - m + 3) (n - m + 2) (n - m + 1 ) ~b n + ~kb n -m = J and, as in the two preceding cases, we shall have and b n =zN. (10) may be written thus, as jp = fn - m + 2\ m \ g J w " Now m J \ m But if we make (&) == &"% then n -m+ Let us also put /(&) = k*~ m j therefore (^r^) - Nk ' = N die 114 INTEGRATION OF DIFFERENTIAL Hence y = jp (/ T ^ s 4p) .................. (H) is the solution required. It admits also of another form, which it may be worth while to remark. ' J " \ _ ( : therefore -j- 7 dfc \ aw * /_ OT + 2 _ fn m 4- 1 V m Here we must make <]>(k} = k m and /(Jfc) = /fc a m , and then and B = -^( ...... ) therefore = ^ The value of y, deduced from the development of (12), can of course differ only in a factor of some function of Je from that which is given by (11), and it will easily appear, on comparing the values of a n in the two cases, that this factor is H***. Let us now consider the case in which p is not divisible by m, while p - 1 is so. And let p - 1 = gm. The two factors of (9) on which our reduction operates, viz. (n m -f 2 p] (n m + l +p}, may be written thus, {n ~ m + 2 + (p - 1)} {n - m + 1 - (p - 1)}, or (n-m + Z + qm) (w m + 1 jw). INTEGRATION OF DIFFERENTIAL EQUATIONS. 115 The change which this will introduce in the process, is not difficult to perceive. We must assume n m + 1 n m + 1 m n m -f- 2 n m -j- 2 f- 1 f- q m m * as before, the transformed equation will be Now dk* k " = therefore If then we make /(&) = & m and (&) = & TO ? we get ~)-&c. ^^m \ m J ' " . &c. Hence, finally, As an illustration, let us take the case of the equation which occurs in the theory of the figure of the earth, Here m = 2, ^> = 3, p 1 = 2: hence j = 1, and as jp is not a multiple of m, the formula (13) is to "be used. 82 116 INTEGRATION OF DIFFERENTIAL EQUATIONS. It is in this case, as JT= C sin y(k) x + a}, eT 1 r 74 d (7 sin {V(&) ! + }"! a* - P J' therefore y = JC [a* 4 cos {V(&) a + a} - sin (V (A) a + a}], CtrC or integrating the first term by parts, But 7_1 2/fc^ 4-, sin IV (*) a; + a} = - cos (V(fc) + a} aA; ^ + ^si x therefore, if fcO = O x , / = (7 a [sin (V (&) J 4- a} Q J + cos {V(fc) + } - 2 sm (V(*) a + a}], is tte required solution. Equation (13) corresponds to (11) : but there is another form of the solution in the case of p ~ 1 = #w, which we shall just mention, and which is the counterpart of (12). It is > , It would be a needless repetition to go through the steps which lead to this result. All the operations indicated in these symbolical solutions are practicable. This will appear by considering the nature of the function X, which, in its most general form, consists of the sum of terms, of which the type is INTEGRATION OF DIFFERENTIAL EQUATIONS. Ill If we make k = K, this will become which may be integrated any number of times for #, and con- sequently, if it is multiplied by any rational and integral func- tion of K, it may still be integrated by parts as often as we dJc please. Now will become md nr ~*~'' 1 die \ and as m and s are #* integral, the method of parts applies, provided s is not greater than m 1, which it is in none of our formulas. Fourier's expression, by means of definite integrals for the fc th differential coefficient of any function, would enable us to extend our solutions to the cases in which p is fractional. But merely analytical transformations of the results at which we have arrived are not of much interest, and the methods of effect- ing them are direct and obvious. Equation (1) admits of another symbolical solution besides the one already given. It is easily seen, that if a n = (n pm + 1) . . . (n m + 1) l n , which is what (3) is, when/(&) = 1, (-3- applies to all that follows it.] Hence we shall clearly have d 1 d 1 d 1 -rr ^~" e&c x 1 " 1 ' 1 dx x m ~~ l dx x m ~ l for the solution of (1). Similarly, the solution of (6) is * "" * aJa; x m " 1 6?C x* Many applications and modifications of the method we have employed will readily present themselves, but the subject is not of sufficient importance to deserve a fuller discussion. It is not difficult to multiply artifices, by means of which particular equations may be solved, but the results will, generally speak- ing, be of little value. ANALYTICAL DEMONSTRATIONS OF DE MATTHEW STEWART'S THEOREMS IN 1746 Dr Matthew Stewart, the father of Dugald Stewart, published his " General Theorems." He was at that time a candidate for the chair of mathematics at Edinburgh, then vacant by the death of Maclauriii ; and his success is attributed to the celebrity which these remarkable propositions immediately ac- quired. They were enunciated by Dr Stewart without demon- strations, and remained undemonstrated till 1805. Mr Q-lenie, in the Edinburgh Transactions for that year, has given a geo- metrical method "by which the General Theorems and other similar results may be established. But as yet they have not I "believe, been proved, except by Geometry; and in an article in the 17th volume of the Edinburgh Review, ascribed to Playfair, they are strongly recommended to the attention of analysts. It is hoped, therefore, that the follow- ing attempt will have some degree of interest. We shall begin by establishing a general proposition, from which all the theorems in question, and many others, may be deduced. LEMMA. If f is a rational and integral function of smd> = < + ] , A T n J Cambridge Mathematical Journal, No. XII. Vol. n. p. 171, May ; 1841. STEWART'S THEOREMS. and therefore to - A- {/( + Jhr) -/# - A- 0. Now A" 1 = ^a m sin mntfr + 25 m cos mn, (m integral) . Hence /(<) + .,. -f/ (^ -f ^ - 2?r J = 2a w sin ^m< ~f &c. \ vi> / Let the index of the highest power of sin<^> or cos $ in f "be p; then it is easily seen that when/^ is developed, &s it may always be, in a series of sines and cosines of the multiple arcs, p will be the largest arc that can enter into tlie development. But if n is greater than p, mn will be greater than pj>, except when m is zero. Hence the development Sa, n sin mnp, + ^ 2?r = 6 ... a constant, . Q, E. D. The constant 5 will of course be the sum of the constant parts of the developments of /<, &c. ; and as these are all equal and are n in number, it will be n times the constant term in /<. 1 f 2jr Now by Fourier's theorem this is equal to I /< . $j&, as in- S^r J deed is obvious. Hence /^ + .../(< + ^ 2r) - which is our fundamental formula. The first of Stewart's propositions is the following ; From any point in the circumference of a circle draw per- pendiculars p, j? x , &c. to the sides of a regular ^-sided polygon circumscribed about it ; then, if r is the radius, = 5wr* ...... (1) ...... n>3 120 STEWARTS THEOREMS. DEM, Let the assumed point subtend at the centre an angle from the adjacent point of contact. Then H ) K &c. &c. and by tlie general formula, n f 2ir S (1 - cos <) 9 = (1 - cos ) 3 d$ ...... n being > 3. -ttTT J Now "(l - cos <) 9 cZ< = 2 4 ("si ^ and * S/-* \a ?l (l~-COS<^>) 8 -~, and therefore 2!%? 9 = 5nr 3 . Q. E. D. This is a particular case of the second proposition in which the assumed point is not confined to the circumference of the circle, but may have any position whatever. Let I be its dis- tance from the centre ; then 3^V...^>3 ............... (2). DEM. In this case p = r Z cos <, &c. = &c. .-. S/ = nr 3 - 37- 2 Z 2 cos + 3rl* 5 cos 2 ~l*$ cos 8 . But /cos $<<= 0, Ico?d = 7r 9 j '" cos 3 d = 0; J o / o J and Q. E. D. In the third proposition, a regular n-sided polygon is in- scribed in the circle, and lines c, o v &c. are drawn from its corners to a point assumed in the circumference j then Sc 4 = 6nr* (3). STEWARTS THEOREMS. 121 DEM. The assumed point and adjacent corner subtending an angle < at the centre, we have c 2 = 2r 2 (1 - cos <) ; .-, 2c 4 = 4/2(1- cos <) 2 , /. c 4 = 6ft/. Q.E, D. The fourth proposition includes the third. The assumed point may now have any position we please. Let I Tbe its dis- tance from the centre. Here we hare and Sc* = n (r 4 + V + 2r 2 Z 2 ) - 4rZ (r 2 + Z 2 ) 2 cos + 4r 2 Z 2 S cos 2 . By the values above given for S cos and X cos 2 <, this becomes 2c 4 = nr 4 + 4^r 2 Z 2 + nV ........... . ....... (4), which is the proposition in question. In the fifth proposition we return to the circumscribed poly- gon, and our object is to determine the sum of the fourth powers of the perpendiculars. As before, and therefore 8S/ = 35w.r 4 ........................... (5). Q. E. D. In the general case, when I is the distance of the assumed point from the centre, 2p 4 = nr * - 4/Z cos < + 6rT 2 cos 2 < - 4rZ 3 2 cos 3 + I* % cos 4 0, w 3 . 1 3n 122 STEWARTS THEOREMS. or Spn+m + nl* .................. (6). This is tlie sixth proposition. The seventh, is, for the m til power of the perpendiculars, what the first and fifth are for the third and fourth powers respec- tively. It is this : S^n*"- 1 -*"- 8 :"V ...... (7) ...... (>). f DEM. Let 6 = 2m . 2m 2 ... 2 ' m . m 1 . . . 1 ' Q. E. D. If the assumed point is at I distance from the centre, This, it is easily seen, will reduce into the following form, 1.2 2 '1.2.3.44.2 (8), which is the eighth proposition (n > m). Lastly, let us consider the 2m th powers of the chords in the case of the inscribed polygon: we have already in the third proposition found the value of their sum when m = 2. As before, c 2 = 2r 2 (1 cos <) ; .% Sc 2 * 1 = 2V* 2 (1 - cos 0)'". That is, as we have already seen, m. m 1 ... 1 (9). v ; STEWART'S THEOREMS. 123 We have thus gone through Dr Stewart's properties of the circle, and have arrived at his results by a simple and uniform method. It is evident that there is no limit to the number of geome- trical theorems which may be deduced from the general formula : almost every curve will afford interpretations, if the word may be so used, of our analytical conclusions. Thus in the ellipse : If any n radii vectores be drawn from the centre at equal angles to one another, the sum of the squares of their reciprocals is equal to n times the square of the reciprocal of that radius vector which is equally inclined to the major and minor axes. For we have ~ 2 = p(l~e 2 cos 2 <); ^1 n e 2 ^ , I ,., 1 2\ S -2 = 2 - p 2 cos 2 < = n p (1 - \e*}, 11 9 93" and i = cos 2 T ; 4 therefore &c, Q. E. D. It is to be regretted that we have hardly any idea by what considerations Dr Stewart was led to the curious theorems which bear his name. It is said, indeed, that he was engaged on geo- metrical porisms when he discovered them, and we are told that he would have published them under the title of porisms, but for his unwillingness to interfere with a subject which the re- searches of his friend, Dr Simson, seemed to have appropriated. Whether they are in reality porismatic, is a question on which it would not be worth while to enter. The fundamental formula of our analysis is perhaps not new ; the geometrical applications which we have made of it appear to be original. NOTE ON A DEFINITE INTEGEAL. /I log sin 9 dd, obviously the same as that of * ' 7*2 / log cos 0dff, was first assigned by Euler, and may be obtained /O in the following manner. By Cotes's theorem, * 2 -~l = (3 9 ~l)(* 2 ~ 2*003^ + ^ v m J \ m J ...... (1). Let z = 1, then - ...... . m 2 m 2 Take the logarithms of both sides, and divide by m, then log m -f 2 (m 1) log ^ Let m become infinite, and = -=- : the first side becomes equal dx *- to lo S ^ ' for \~m~J =: wlie:a m = o ' an( ^ tlie second is trans " formed into the definite integral / log sin x ~ . dx \ therefore ^ ^o ^ I logsina;~cfe = logj. J o -4 Let = a?^; ^ * Cambridge Mathematical Journal, No. XII. Vol. n. p. 282, May, 1841, NOTE ON A DEFINITE INTEGRAL. 125 .(1). COB. 1. Integrating by parts, we get [log smOdB = 6 log sin - The integrated part vanishes at both limits, COR. 2. Let sin^sse"**; therefore the limits of x are and oo ? COR. 3. In this last integral, if we expand the denominator f 1 and as I e" 1 ** a? c?cc = - z 9 we find TT, 111 1.3 1 1 1.3.5 1 1 - REMAKES ON THE DISTINCTION BETWEEN ALGEBRAICAL AND FUNCTIONAL EQUATIONS*. THE distinction which it is usual to make "between alge- braical and functional equations will not, I think, bear a strict examination. It is generally said that an algebraical equation determines the value of an unknown quantity, while a func- tional equation determines the form of an unknown function. But, in reality, the unknown quantity in the former case is a function of the coefficients of the equation, and our object in solving it is simply to ascertain the form of this function. Thus it appears, that in both cases the forms of functions are what we seek. Let us therefore consider the subject in a more general manner, and endeavour to find a more decided point of dis- tinction. The science of symbols is conversant with opera- tions, and not with quantities; and an equation, of whatever species, may be defined to be a congeries of operations, known and unknown, equated to the symbol zero. Every operation implies the existence of a base, or something on which the operation is performed in the language of Mr Murphy, a subject. But the base of an operation is often the result of a preceding one. Thus, in log a? 2 , the base of the operation log is x 2 , itself the result of the operation expressed by the index on the base x. This in its turn may be considered as the result of an operation performed on the symbol unity. But in every kind of equation there is a point at which the farther analysis of symbols into operations on certain bases becomes irrelevant ; and thus we are led in every case to recog- nize the existence of ultimate bases. * Cambridge Mathematical Journal, No. XIV. Yol, ill. p. 92, February, 1842. NATURE OF FUNCTIONAL EQUATIONS. 127 To solve an equation of any kind, is to determine the un- known operations by means of the known. If one symbol is said to be a function of another, it is, in reality, the result of an operation performed upon it. Thus the idea of functional dependence pervades the whole science of symbols, and on this idea the following remarks are based. In order to classify equations, we can make use of two considerations : 1st. The nature of the operations which are combined together; 2nd. The order in which they succeed one another in the congeries of operations which is made equal to zero. Let us illustrate these remarks by some examples. If we have an equation of the form. a?+ax + 5 = ........................ (1), the bases are a and "b ,* the operations are, first, the unknown one denoted by a?, and then certain known ones denoted by the index, the coefficient, &c. All these are what are called alge- braical operations. If again we have an equation of the form i ........................... w. the base is x; the operations are, first, the unknown one denoted by y, which is a function of #?, then the operation -*-, and lastly, certain algebraical operations. From the pre- (L3C sence of the operation ~j- , this is called a differential equa- (%X tion. Equations (1) and (2) are discriminated by the nature of the operations combined, on our first principle of classifi- cation. But in one important point these equations agree. In both, the unknown operation is performed immediately on the bases ; the known are subsequent to the unknown : but in what are called functional equations this is not so. Thus, in the equation <)>(mto)+ x-Q ........................ (3), the base is cc, the unknown operation is <, which is performed, not on a?, but on the result of a previous operation. In the 128 NATURE OF FUNCTIONAL EQUATIONS. preceding example the previous operation is known ; but this is not essential. Thus in $$x 4-^=0 ........................... (4) the previous operation denoted by the right-hand < is unknown. The operation -j- may enter into equations where the un~ doc known operation is not performed on the "base. Thus we may have an equation of the form (5). Equations (3), (4), (5) are functional equations; (3), (4) are ordinary functional equations ; (5) is a differential functional equation ; (3) is said to be of the first order, (4) of the second. The introduction of the functional notation appears to be sometimes taken as the essence of functional equations ; but if we wrote (1) and (2) thus, 5) + & = .................. (1)', a*0)-a = .................. (2)', they would still be perfectly distinct from (3) or (4) or (5). The name functional equation is not happy; it refers "to the notation, and not to the essence of the thing. A question now arises : To what class shall we refer equa- tions in finite differences ? These are generally of the form F(x,y*,y**.-) = * .................... (6), where y x is an unknown function, say $ (a?) of x \ so that (6) may be written thus, Here the unknown operation is <, which in the case of '(x is performed, not upon the base a?, but on x+1. Thus it appears, that equations in finite differences are only a case of ordinary functional equations of the first order : and this is the reason why, in researches on functional equations, we per- petually meet with cases in which they may be reduced to equations in finite differences. NATURE OF FUNCTIONAL EQUATIONS. 129 The preceding remarks contain, I think, the outline of a natural arrangement of the science of symbols. It is not difficult to overrate the importance of a mere classification; but I hope to be able to show, that the considerations now suggested are not without some degree of utility. As the distinction between functional and common equations depends on the order of operations, it follows that, when part of the solution of an equation does not vary with the nature of the operation subjected to the resolving process, this part is applicable as much to functional equations as to any other. The special application of this principle to the discussion of a class of differential functional equations will be the object of a subsequent paper. In the preceding remarks, operations of derivation, such as D, A, &c. are supposed to be replaced by functional operations in every case in which this can be effected. MATHEMATICAL NOTES*. 1. IN the Examination Papers for 1834, the following problem is given: "If the chord of a conic section, whose eccentricity is e, subtend at its focus a constant angle 2a, prove that it always touches a conic section having the same focus whose eccentricity is e cos a." A solution of this problem by a peculiar analysis will be found in a preceding article ; but the following method may be found not uninteresting. Let TI 5 r 2 be radii vectores to the ends of the chord, (/> a, < -j- a, the corresponding angles vectores, p the perpendicular from the focus on the chord ; /. p x chord = r^ sin 2 a, 1 ^ ( r * ~*~ r Q 2 ~~ ^ r i r * cos ^ a ) * " p "~ r t r 3 sin 2a // 112 \ = cosec2a A /(-?+* cos2a) , V / \ a) 1 _ ^T ~~ ; > ~ "~ <5cc - For cos 2a put 1 2 sin 2 a ; then, by a few obvious steps, - j(i ^_ 20 cos a cos 6 + e e cos 2 a). _p Z cos a ^ T x Put a = ; then the chord becomes the tangent, and But the general form coincides with this, if we put I cos a = X and e cos a = e ; for then - ==.:- * Cambridge Mathematical Journal, No. IX, Vol. in. p. 94, February, 1842. MATHEMATICAL NOTES. 131 Hence p is generally the perpendicular on a tangent of an ellipse of eccentricity e cos a. Hence the chord touches such an ellipse. The latus rectum is diminished in the same ratio as the ec- centricity. 2. The relation between the long inequalities of two mu- tually disturbing planets, may be easily found without having recourse to the development of the disturbing function. Let m, m f , be the masses of the planets, a, a', the major axes of their orbits, n, ri, their mean motions, 7^, h', twice the areas described in I" ; then we have flF , /Jr n ~ ~J 9 n -4 ? a f a* fi being the mass of the Sun, in comparison with which the masses of the planets are neglected, so that it is the same for both. Taking the logarithmic differentials of these equations, and replacing the differentials by differences, we find n 2 a J ri 2 a' But by the principle of the conservation of areas, mh + m'K = const so that m&h + m'&K = 0. Now the orbits being supposed circular, we have , Ah 1 Aa AA' 1 Aa' hence T^^T' ^^i^ 1 Therefore we have A^ ^ A^ ; __ Aa a' __ A/a A' __ m' a'^ f n * ri "" Aa A * a ~~ AA' * A "" m "^i ' and and r are the inequalities due to the disturbances, so n n that their ratio is thus given. 92 ON THE SOLUTION OF FUNCTIONAL DIFFERENTIAL EQUATIONS*. IT is well known that the solution of a considerable class of differential equations may be effected by means of differentiation. Olairaut's equation is a particular case of this class. We will begin by considering it. ... (1) where f denotes any given function. Differentiating (1), we get (o?+/'_p) g = ........................ (2), hence # = 0, or a?-f/p==0 ........................ (3). The first of these equations gives the complete integral. Being twice integrated it becomes y = ax + & ; and on substi- tution in (1), we get &=/a, therefore y=ao? +/&... (4) is the complete integral of (1). It has always been supposed, in this and similar cases, that f must necessarily be a given function. But this condition is not essential: a differential equation, e.g. such as (1), will, when solved, give y as a function of x. Now the function^ which enters into (1) may, instead of being given, as is usually the case, be in some way -dependent on the function which y is of x. Thus the form of / is unknown, until that of the latter function has been determined. It is evident that accord- ing to the classification proposed in the last number of the Journal, (1) is in all such cases a functional equation. For the unknown operation y is performed onj?, which is itself the result of the unknown operation ^ performed on x (we suppose Cambridge Mathematical Journal, Ho, XV. Vol. in. p. 131, May, ON THE SOL UTION OF FUNCTIONAL JQ UA TIONS. 133 To differential functional equations, ordinary methods of solution do not, generally speaking, apply, "because they re- quire a knowledge of the forms of the functions on which they operate. But in the case before us, the differentiation and subsequent substitution, by which (4) was derived from (1), are independent of any knowledge of the nature of/. Consequently (4) is always true. Let us suppose, for instance, that / = inty, m being a constant; then = Q .................. (5). We are of course obliged to introduce a functional notation: (4) in this case becomes tyx = ax m^ra ........................ (6). In order to determine tya, put x = a ; then - T 2 /rr\ and yx = ax - - a ....... . ....... (7), which is a solution of (5) . In the ordinary cases of Clairaut's equation, the factor x-\-fp = leads to the singular solution; and so it does when /is an unknown function. Thus, in the example just considered, as /' = witjr', we shall have m^r^rx = x ................ ,r. ........ (8). Of this a solution is Hence we get, by integration, On substitution it is found that (7=0, therefore is a new solution of (5), and perfectly distinct from (7). 134 ON THE SOLUTION OF If w= 1, (5) and (7) "become respectively ? = .................. (5') ? (7'); in this case, (8) admits of a variety of simple solutions. Thus we shall have &c. = &c. as singular solutions of (5'). The preceding remarks are sufficient to indicate the exist- ence of a class of functional equations, to which a considerable portion of the theory of singular solutions may be applied. They appear therefore to possess some interest with reference to this theory, independently of the method they suggest for the solution of such equations. In fact the theory can hardly "be considered complete, unless some notice is taken of the equations of which we have been speaking. They have been excluded from it, because the func- tion /, which they involve, is not, as in the ordinary case, a known function. But this, it has been already remarked, is not an essential distinction. On the other hand, the method by which the singular solution is in the common theory deduced from the complete integral, does not apply to the cases now considered. It appears unnecessary to point out the reason of this difference. With regard to the class of differential equations, which, like Clairaut's, separate into factors on differentiation, we may refer to Lagrange's Legohs sur le Galcul des Fonctions, 1. 16 me . He there shows that if a differential equation of the first order can be put into the form M^fN, where M and JVare the values of a and b deduced from F(xyab] = 0, then, when differentiated, it will resolve itself into two factors, one of which leads to the singular solution, and the other to FUNCTIONAL DIFFERENTIAL EQUATIONS. 135 the complete integral. (The latter is, as may readily Toe seen, The demonstration of this proposition is probably familiar to the majority of my readers, and I shall therefore not dwell upon it. Similar considerations apply to equations of higher orders. Generalizing the remarks already made, we see that in the equation the function / need not be a given one ; it may be, in any way we please, dependent on the function which, in virtue of this equation, y is of x. In all such cases the equation in question is functional. Nevertheless, Lagrange's reasoning applies as much in these as in other cases. Let us take one or two ex- amples of what has been said. The following problem may be proposed. Any point P of a certain curve is referred to the axis of x in M, and to that of y in N. MP is produced to Q ; PQ is taken equal to a, and NQ touches the curve. Find its equa- tion. Let x, tyx be the co-ordinates of the point where NQ touches the curve. and as P is a point in the curve, ON=ir{NP), or .(10). This is the equation of the problem. Differentiating it, we get <^ f x = 0, The former equation gives the complete integral, but, for a reason I shall hereafter notice, leads to no tangible solution of the problem ; the latter corresponds to the singular solution. 136 02V TEE SOLUTION OF In cider to solve it, assume a -, ; & & 3 X then ^ -77- = -5- and = "V Y ifr'o? % 2 and therefore % 2 o? = ^ +2 . Then = ^ i = 4 , where C is arbitrary ; U*+l %+2 C therefore %# = Cir. is a function of #, which does not change when z + 1 is substituted for #. "We confine ourselves to the only simple case, that in which it is an absolute constant ; then * a & n n ^. ^0? = - = - ... (10 a) T Cx x ^ ' and ^|ra? = &log- ... c being an arbitrary constant. o On substitution, we find 6 = ae ; therefore ........................... (11) c/ is a solution of the problem. This is the equation of a logarithmic curve, which has there- fore the required property. The method employed to resolve the equation in %o?, namely, is applicable to every equation of the form .................. (12). Every such equation may be at once reduced to the follow- ing equation in finite differences, .-.tO^O ...... ............ (13)- This reduction is in reality a particular case of an important transformation due to Mr Babbage, which often enables us to solve functional equations of the higher orders. FUNCTIONAL DIFFERENTIAL EQUATIONS. 137 In (12) we may write for %#, <}>f(jf l x. Hence ^xffi^x &c. = &c., and (12) Tbecomes F($x.$fx ... x by u.^ and replace # by #, we shall ob- tain (13). It must be admitted, that it is difficult to prove that the generality of (12) is not restricted by these transformations. They are however often useful, and serve to illustrate what was remarked in the last number, with respect to the affinity of functional equations, and equations in finite differences. If, instead of (10), we had taken the more general equation (15), where A is an arbitrary constant, precisely the same method would have applied. In this case the factor fy"x = would have led to the result ^x = ax -f j8, and by substitution /3 = a -f /3 + A, therefore a+A=Q, or/3=oo. Now in the case we have been considering, the former con- dition is not fulfilled ; hence we must have /3 = oo , and the geometrical interpretation of the complete integral is a right line at an infinite distance from the axis of abscissae. We not unfrequently meet with similar cases, in which the complete integral becomes nugatory" or impossible in the pro- cess of introducing the necessary relation between its constants * Under particular conditions, however, this difficulty does not occur, and then we obtain, what in the ordinary methods of discussing functional differential equations, appears to be a conjugate solution, unconnected with any other; (15) would be an instance of this, were a + A = 0, 138 ON THE SOLUTION OF I shall next consider a celebrated problem, first proposed "by Euler, in the Petersburg!* memoirs. In a certain class of curves, the square of any normal ex- ceeds the square of the ordinate drawn from its foot by a certain quantity a. Let y 2 = tyx be the equation of the curve. The subnormal is therefore fafrx, and the equation of the problem conse- quently is ^(x + ffix)=Tfrx + %fy r xy-a ............ (16). Differentiating this, we get ^ (x 4- | ^r'a?) = ^'x * y or The first of these two equations leads to the singular solu- tions. In order to solve it, let then rfx - %xx + x = 0. Hence by the transformation already noticed, whence U K = Pz + where Pz and P^ are functions of z 9 which remain unchanged when increases by unity ; therefore u^ = Pz + (z + 1 ) P^. Hence we have W* = Pfl f ,1 - * i r p JL PV \ ^ or 16 re( l uirec *- solution. therefore ydy = P^ (Pss + P^ + zP^z) dz ; and integrating by parts, we get f - P f (2 A for a general solution of the proposed problem. (The parame ter a is involved in P. FUNCTIONAL DIFFERENTIAL EQUATIONS. 139 Let us suppose Pz and P^z constant ; x = "b + az ; therefore y* = 2aa? + (7. On substitution we find a 2 = a. Thus, in order to a real result, we must suppose a negative, e. ff. let a = & 2 ; then (7 ....................... (18), the equation to a parabola, which accordingly is a solution of the problem, and the only simple one it admits of. When a = 0, it becomes two straight lines parallel to the axis. The other factor 1 + fyty'x = gives, on integration, ..................... (19), the equation to a circle ; but 011 substitution, we find which leads to no result, unless a = 0. A solution of this problem, by Poisson, is given at p. 591 of the last volume of Lacroix's great work. It is apparently equivalent in point of generality to (17) ; and the author points out its incompleteness in the case of a = 0. The pre- ceding views show distinctly the nature and origin of the new solution which then presents itself, Mr Babbage also has considered this problem at the end of his second essay on the Calculus of Functions (vide Phil. Trans. 1816, p. 253). But I believe it will be found that his solution is erroneous. Notwithstanding the length this paper has already reached, I must endeavour to point out, as briefly as possible, my reasons for thinking so. Mr Babbage confines himself to the case of a = 0. He begins by demonstrating the existence of a relation, equivalent, excepting a difference of notation, to ^ {x -f fytyx} = ^r'a?, but in doing this, loses sight of the other factor 1 4- J^r"a? = 0. 140 ON THE SOLUTION OF This relation shows that ^rx is constant, for a series of points in the curve, and therefore, Mr Babbage reasons, we may con- sider it as a constant in (16), which thus becomes an equation in finite differences. He integrates it on this supposition, and adds an arbitrary function of tfx, which has been treated as an absolute constant. The result is therefore -f f . f^ or *-**+/(, .................. (, which is an ordinary differential equation. This process appears to have been suggested by an incorrect analogy with the way in which arbitrary functions are intro- duced into partial differential equations. ' A little consideration would have convinced Mr Babbage, that by integrating (16) as an equation in finite differences, he only passed discontinuously from one ordinate of the curve to another, and therefore could not obtain a continuous relation between x and y. The legitimate result of his process is merely 3 2 x where n is any positive or negative integer. This is quite different from In exemplifying equation (20), Mr Babbage first supposes dy\ __ dy dx/ dx J and thus obtains the equation of a straight line parallel to Ox, as a solution of the problem, which undoubtedly it is. In his next example f(y -f^l = a s . By making the con- stant of integration imaginary, he gets 2/ 2 = a 2 or 5 , the equation to a circle. But although this is also a real solution, it has no connection with the relation ty' {x 4- 1- ty'x] = ty'x, from which it appears to be derived. It is, as we have seen, a particular case of the complete integral. Consequently if the method FUNCTIONAL DIFFERENTIAL EQUATIONS. 141 pursued had been correct, it could not have given this solu- tion. The preceding pages appear to contain the germ of a general theory of differential functional equations ; a subject of great extent, and ultimately, perhaps, of considerable importance. But it cannot be denied, that hitherto the Calculus of Functions has not led to many results of much interest. Its value arises chiefly from the wide views it gives of the science of the com- bination of symbols. MATHEMATICAL NOTE*. Problem from the, Papers of 1842. If F(x, #,#) = $ ( w > v > w }> where F is homogeneous of the n il1 degree in a?, y, , and x "~ " w]lpre wheie It is obvious that "Fxj o r>, (fcc = 2 I J ax ; w x J Q & and that if n is odd, fo, which is of course a rational and in- tegral function of circular functions (sines and cosines) of x, must be developable in a series of sines exclusively, and if n is even in a series of cosines exclusively. Thus, we may assume fx Now, as J is not infinite when x = 0, the lowest power of CO x which can enter into fx must be not < n, call it m, and develope in powers of x every sine or cosine which appears on the second side of the last written equation. We must have %Aa m ~*^ 0, %Aa m ~*~ 0, &c. ... J w- 1 equations if m is even, and ^ (m 1) if it is odd. Let us now consider the definite integral r J o Integrating it repeatedly for r } we get r J o .-otar sin rx dx = tan"" 1 - , x a EVALUATION OF CERTAIN DEFINITE INTEGRALS. M5 = _,. tan' 1 - + a ]ogV(a 2 + O + 0, a and generally .. (2), where JP (ra) does not become infinite for a = 0. Replace r by every quantity represented in (1) by the general symbol a. Multiply each result by the corresponding coefficient A, and add. Then, in virtue of the conditions, %Aa m ^ = Q, %Aa m ~*=Q, &c. we shall have Put a = 0, then tan" 1 - = - , according to whether a is oc & > or < than zero. Thus we have From hence, the truth of our theorem is obvious. Of the ambiguous signs outside the symbol of summation, the upper is to be taken when n is of the forms 4p ? or 4p + 1. When a is positive, we must take the upper of the am- biguous signs under the 2. It will be remarked, that in obtaining (3) we have elimi- nated all the constants at once, instead of getting rid of them one by one by particular conditions at each successive inte- gration, and that the generality of this method enables us to recognize a class of definite integrals, which are all deduced f from the" known value of I e^ x cos r& dx. J o 10 146 E VAL UA TION OF CER TAIN DEFINITE INTEGRALS. Equation (3) admits of several remarkable applications. Thus let us suppose n = 3 and fa = sin ax sin "bx sin ex : then fx ~ J {sin (a + 6 + c) a? sin ( a + 5 + o) a? sin (a 5 + c) oj sin (a + 5 c) x}* Consequently, we have "by (3), sin aa? sin 5# sin ca? 7 7r f2 _ / \ 2 ~/ 7\2~ / \ 2 i * f J - 3 - -, where, as in trigonometrical formulee, 25 = a + 1 4- c : the upper sign is to be taken when the quantity to which it is affixed is > 0. T ., . Likewise f 00 sin ao; sin bx sin c# 7 TT /.--_._. x I - dx= (1 + 1+1 + 1), J Q X $ where the signs follow the same rule as in the former case; the different unities involved being the zero powers of s, s a, &c. Let us now suppose that fx = &m m x cos zx ; the correspond- . , , f 00 sin m x cosmic 7 . ,, , 7 . ing integral, viz. / - 5 - db occurs in the theory of j a? probabilities. Its value is given at p. 170 of Laplace's TMorie des ProlabiUtSsj where it is obtained by a method founded on a transition from real to imaginary quantities. The nature of what are called imaginary quantities is certainly better tinder- stood than it was some time since ; but it seems to have been the opinion of Poisson, as well as of Laplace himself, that results thus obtained require confirmation. In this view I confess I do not acquiesce; but if only in deference to their authority, it may be desirable to show how readily imaginary quantities may be avoided in estimating the value of the integral in question. sin w a?=: t \ cos mx -^ cos (m-2] o? + &c.i if m is odd, and = Jain mx ~~ sin (m - 2) x + &c.i if it is even. TT ^^ ^ /9\ f Hence, by (3), J sn m a; cos a? EVALUATION OF CERTAIN DEFINITE INTEGRALS. 147 Let us suppose & > m ; then the lower of each pair of am- biguous signs must be taken, and the expression within the brackets may be written thus 1 As, from the nature of the case, m and n are either both odd or both even, if m is even, n I is odd, and therefore (m z) n ~~ l (z m) n ~ l ; and thus, in every case, (j) equals (1 - IT 1 ) + mr 1 . . . [Dz = (z + 2) say,} and this is A m D~ m (z + m) 71 " 1 = A TO (2? m) 1 *" 1 = 0, since m is > w 1. Consequently cos zx dx == 0, when z is > m, a remark not made by Laplace ; when m = n = I, its truth is known. As (j) = 0, add it, multiplied by r _ !^ -m-i to &s value of the integral already found, therefore where the series stops whenever the next term would intro- duce a negative quantity raised to the power % 1. This is easily seen to be true, for every such term will have a different sign in (#), and in the definite integral, and thus on addition, all such terms will disappear. Equation (4) is Laplace's form ; the discontinuity of the function is now expressed, not by ambiguous signs but by the stopping short of the series at different points. By a similar method, we find that which might have been deduced from (4) by integrating both sides without introducing any complementary quantities. This remark is general; having once established the general form of (4) for any given value of n, we may deduce from it that which corresponds to any other value of w, simply by differ- entiation or by integration, without bringing in any constants. 102 148 EVALUATION OF CERTAIN DEFINITE INTEGRALS. I conceive that this remark is general, and if so we may dif- ferentiate on both sides with fractional indices. Let the index be ~j?, then as &* cos zx = x p cos the first side of (4) will become f sin w & -, / 7r\ , J a* cos (zx -p -J &?; and the second will be Thus, we get J If we take n = m this is equivalent to Laplace's general formula (p], at p. 168 of the Thforie. The method of this paper leads to some elegant results when /CO e^dx, but it is enough to point out this application, which involves no difficulty what- ever. MATHEMATICAL NOTE*. Stability of Eccentricities and Inclinations. The equation for proving the stability of the eccentricities and inclinations of the planetary orbits may, as has been shown by Laplace, be deduced from the principle of the conservation of areas, joined to the fact of the invariability of the major axes. Let m be the mass of a planet, Ji be twice the area de- scribed by the radius vector about the sun in an unit of time projected on the ecliptic, and i the inclination of its orbit to the ecliptic; then, by the principle of the conservation of areas, S (mh] = const. Every term under the sign of Summation is positive, be- cause all the planets move round the sun in the same direction. But Ji = { a (1 - e 2 ) } 4 cos i = a* (1 - e 2 ) 4 (1 + tan 2 i)~*. Now e and i are small at the present time; hence, if we neglect their, fourth power and the products of their squares, we have Hence S {mc$ (1 \& -| tan 2 i) } = const. But since the major axes have no secular inequalities 5 (ww$) = const. ; hence the preceding equation is equivalent to S (ma? $ + wwfc tan 2 i} = const. Now the left-hand side of the equation being small at the present time, the second side is also small, and therefore the first side is always small, and therefore X (ma* e 3 ) and 5 (mc$ tan 2 i} are both always small. * C &e. are arbitrary, provided they include all values of the variables which satisfy the given inequality. Again, if in the given multiple integral the limits were determined "by the single relation f* ^ Ji, joined to the con- ditions that a?, y, &c. were to have no values less than certain assigned limits; e.g. if we were to consider only positive values of the variables, the formula (I) would still apply with a slight modification. The inferior limit of integration with respect to u would "be arbitrary, provided it included all the values which could be given to f. by admissible values of the variables, while the inferior limits of integration for cc, y y &c. would be determined by the particular conditions of the case. Let us take, as an example of the method, the integral fdxfdy...x a ~' 1 i/ ) -*2 c ~\..f( cos a (mx+ny+. . . w) 77V &1 J o J o Jo (A 1 <0). /DO ,-00 Let .F= I dxl dy ... e" CWMri * Blr< '- ) cos a (0*0 4 rcy + ... - u) ; ^0 ^ / r then J T =cosawl c?^ ^ J Jo - /. I dx\ dy ,. t Jo Jo sn aw o wtich we may put equal to Q- cos aw + .2?" sin OM. First to find the yalue of Gf- = f *&? f * dy ... e^^^^-^ cos a (ma? + wy + ...). J ^ Develope the cosine; the result is composed of terms, each containing sines or cosines of all the variables. Also "by the formulae a I Jo /' >o cos a mx dx f k sin a mx dx = we see that every factor whether sine or cosine introduces on integration a factor of the form r-o - - . Moreover a sine factor introduces a in the numerator, a cosine factor a or b or, &c Let P represent the continued product of a, 5, c ? &CL and D that of m(a 2 +a 2 M& 2 +a 2 ), &c. Then ^ = ^2(71, where /*, is a positive integer less than the whole number of variables in E, and equal to the number of factors in the rtfi denominator of -, , and O is some coefficient. ab ... A little consideration shows, that if we develope (- + T+ ...... I \a o / 154 EVALUATION OF DEFINITE MULTIPLE INTEGRALS. in a series of powers and products of 1 1 al .................. ' and neglect all terms involving powers above the first of these a, quantities, the result will Tbe = 2 (7=- . Consequently _ -7 -- Tt~ T - -J.-. ab ... o> o a o c j 5 - T denoting the combinations two and two of the quan- tities -, T ; an ^ so f ^" ie res "k ^ or * n the development $5 c? of (-4 H- 2? -f ) m where m is not greater than the number of quantities A, , &c. there is always a term involving no power above the first of any of these quantities, and its coefficient is 1 . 2 . . , m. This term is obviously the sum of the com- binations m and m together of -4, B, &c. 3 and that its coefficient is equal to 1.2.3 ...m, appears from the polynomial theorem, viz. , Thus g--P- 4 - Ihus <*- where p i? jp^, &c. are the alternate coefficients of the equation v * -j^t?*- 1 + &c. ^^ v + p t = 0, whose roots are a, 5, c, &c. (I suppose to be the number of variables in E.) In precisely the same way we should find JT- Next to find the values of TOO -GO I Cr cos g udv. and H cos a uda. J o Jo T^t 7T~ f * cosa^g JLJCL J.JL I -7 3 -^r j A \ct> "4" g ) f o *4" g ) "cosauda EVALUATION OF DEFINITE MULTIPLE INTEGRALS. 155 the upper sign is to be taken when u is > 0, By differentiating this for a, we have the value of /"" _ _ J (cf +aty(V+ 0, and TT 1 when u is < 0. But ^ - op M + &c. ct = 0. Hence ,F= 0, when w is < 0, and therefore in (E) we may make \ = 0, so that the limits of integration for u are and 1i Again cf -f p^ 1 -f &c. = (a -f a) (a + 5) . . . and 2a(5 2 -~a 2 )(c 2 -a 2 ) ... = (2a. Therefore __ - and J 7 - ~ Z STT - - r .... .......... (10), mn ... (o a) (c a) ,.. v y and therefore finally #m... (b a) (c a) ... If a = b = c = &c. = J[, the first side of this equation (11). * ' 156 EVALUATION OF DEFINITE MULTIPLE INTEGRALS. As a verification of our analysis we may remark, that in this case for we have, what is probably a known result, and which at any rate may Tbe easily proved, 2 77 - f/ a) , ..... ~^ff^>(^) when a ==& = &*.== 4, (& a) (0 a) ... r() v where -F ( * } (4) denotes the p ih derived function of F(A] : and this formula applied to the case where F(a] =e~ au gives the above written result. By differentiating. (11) for a* 5, c 9 &c. X, //,, v, &c. times respectively, X, /*, z>, &c, being integral or fractional, and dividing by m x , n*, p v , &c. we should obtain the value of which would include every case of (3). But the investigation would be complex, and I shall therefore only indicate it In a future number of the Journal I may perhaps apply the method to some other cases, and particularly with regard to such multiple integrals as fdxfdy ... $ (ay ..,)/[* (xy -Ol^tx^ -Oli &o- the limits being given by the series of inequalities, X - > ^ < k, &c. > &c, < &c. In theory, such an integral is reducible to a multiple integral of as many variables as there are limiting inequalities. But it is not easy to find cases in which this reduction can be actually effected. MATHEMATICAL NOTES*. 1. IF a plane passes through any point of a surface, and makes any function of the intercepts it cuts off from the axes, a maximum or a minimum when it touches the stirface, this maximum or minimum value is constant for all points of the surface ; and, conversely, if for every point of a surface, a given function of the intercepts of the tangent plane is constant, this function is, with reference to any single point of the surface, a maximum or minimum for the tangent plane. This appears at once from the following considerations: a?, y^ z "being a point in the surface, o? , y , # , the three inter- cepts, $ (o? , y , # ) the given function, if we seek to determine the surface so that shall "be a maximum or minimum, we lave the equations + +-=1 (i), fC Iff. & n f Ui(p & / 1\ jj, being a factor. From these equations we get and therefore the differential equation of the surface is * (5); Cambridge Mathematical Journal, No. XIX, YoL IV. p. 47, Kovembfer, 1843* 1 58 MA THEM A TICAL NO TES. for by the ordinary equation of the tangent plane we have 1 m I I mm dF t dF m dF x ' y Q ' s " ~dx * ~dy * dz ' F=0 being the equation to the surface. Again, if we seek to determine the surface, so that shall be constant, L e. to find the envelope of all the planes represented by (1), we haye (1), (2), (3), (4), as before, and in addition Thus, as before, ^ =/ 1? y = / 2? ^Q = f 3 y an( 3- the equation of the surface may be got by integrating (5), and determining the constant so that the result may coincide with (6). And the identity of the equations connecting w^y^z^ and x, y^ 0, in the two cases proves our proposition and its converse. Take as an example the ellipsoid * y Z ^ TT & O C -1+72 + -o = l. rierectf= , ?/. = . = a o c Q x * u z Therefore the tangent plane to any point of the ellipsoid 1 <* V y g <> passing through that point. 2. To find the value of _ A _ , _ A K-^K -,)...(-,) ^ K- 2 )K- 1 when a x = a 2 = &c. = a. Let 0.^ = 0+^, a 2 = and so of the rest ; .% A fa \- f - r - - - j -- i y l(.-*0 ...... (**-*y(*i-*J ...... (^- + &C. MATHEMATICAL NOTES. 159 + &c. fay Taylor's theorem. Now, when ^ 1 =^ 2 =&c.=0, the coefficient off ( a will vanish if p > n. And whatever the values of z, the coefficient off (p} a vanishes if p < n : for we know that iL =0 (~ \ * Tc being < n, and = 1 when Tc = n ; which was to be found. NOTE ON A DEFINITE MULTIPLE INTEGRAL*. IN the XVIIF 1 number of this Journal Mr Boole pointed out the incorrectness of a theorem given by M. Catalan. The following pages contain a brief demonstration of the result to which he was led. Both he and M. Catalan made use of what is generally known as Liouville's theorem, and thus perhaps rendered their analysis less simple than it would otherwise haye "been* Let us transform the integral by the assumption and by (n 1) other linear relations connecting x^ ...... x n and u ...... w n? and such that Then, as is well known, dx l ...... dx n is to be replaced by du ...... dw n , and thus ... + a n x n } ^fdujkujdut ...fdu n ... (1). Let the integrations on the first side of this equation include all values of the variables which do not transgress the limits were J5 is supposed to be greater than A. Then, as !><3? = Sw the corresponding limits on the second side of the equation are * Cambridge Mathematical Journal, No. XX. Yol. rv. p. 64, February, 1844. NOTE ON A DEFINITE MULTIPLE INTEGRAL. 161 Transform the integral in x by assuming sc* ^~ r cos c/.| , Xn -" r sin t/^ cos t/f. . * ^?, = "" : ^* sin c/- * , sin c/*, i and that in w, by similar assumptions, u^ = r cos 0' 1? u 2 = r sin 0^ cos 0' 2 . . . w n = r sin f x ... sin 0^. I may be allowed to mention that this transformation, which appears to have been given for the first time by Mr Boole, in the last number of the Journal, had occurred to me before I had seen his paper. His analysis leads at once to the conclusion that dxi ... dx n is to be replaced by This may be also proved by successive substitutions in the man- ner pointed out in the case of three variables by Mr A. Smith, in the first volume of the Journal. Thus (1) becomes, since Sa? 2 =Sw 2 = r 2 , r^dr /sin' 1 " 2 t dff^.. fd6^f{r (a, cos l + ... a n sin l . . . sin 0^) cos i The limits for and & are the same. That this equation may subsist for all values of A and J5, it is necessary and sufficient that /sin* 1 "" 2 0j d& 1 . . . fdffn^ffr (a t cos 9 i 4- . . . a n sin 9 l . . . sin 7l _ 1 ) } With respect to the limits of and 0', it is not difficult to perceive that if l ... 6 n _ l are taken between the limits and TT, &i ... av~i w iH receive all the values of which they are capable, namely, all that included between r and + r ; and that the same set of values cannot occur more than once. But in order that x n may vary from to r, it is necessary to extend the superior limit of n _ 1 from TT to 2?r. Thus the limits of M _ 1 are and STT, while those of the other variables 1 ...0 n - 2 are and TT. And similarly for '. On the second side of (2) we have the factor ^ (A). 11 162 NOTE ON A DEFINITE MULTIPLE INTEGRAL. Let n be odd, then we shall have (-8)...4.2 (-4)...3.1 n-3 Similarly r&FffW pBurv, dff, - - , &c. = &c. J o ^ Lastly 2 rr vl 2 a ;- 2 - Again, if w be even, we get -8 And this, multiplied ."by |Aj4 , or unity, gives, as before, W-jgl; and thus (2) becomes, making r - 1, ["sin"" 8 O^dSi \'wsi e *6jl6 t ... | 'dB^+ffa cos ^ + ... Ac.) ^ n ^ n J o ?',) (3). This is the general result given at the end of Mr Boole's paper, and -which includes the others. NOTES ON MAGNETISM*. No. I. A GEOMETRICAL construction, Tby means of which the action of a small magnet on a distant particle of free magnetism may be readily determined, is mentioned, in a memoir by Weber, on the Bifilar Magnetometer (S&ientific Memoirs -, n. p. 270). It is due to Gauss, but I do not know where he has demon- strated it. A proof of it may be acceptable to some readers of the Journal. I begin by enunciating the construction in question, which will be easily understood without a figure. Let AB be a small bar magnet, c its centre, P the particle of magnetism oil which it acts. Join cP, draw PD perpendicular to it, meeting cP produced in Z>. Let cQ = JcD. Join PQ. Then PQ or QP (according to the sign of the magnetism of P and the direction of the poles of AB) is the direction of the action of AB on P; and p ~ is its magnitude, M being the measure of the mag- netism of AB, m that of the magnetism of P. The dimensions of the magnet being small, and its length in the direction of its axis being much greater than its breadth or thickness, we may proceed as follows. Conceive the magnet to be composed of a series of intense particles ranged along its axis. Let s be the distance of any one of them from c, pds the measure of its magnetism. Also let cP= r, and call the angle cP makes with cD, 0. * Cambridge Mathematical Journal, No, XX. Vol. IV. p. 90, February, 1844. 112 164 NOTES ON MAGNETISM. Then the distance of the particle in question from P is (r 2 + $ 2 2rs cos 0)*, and consequently its action on P is magnetic attraction being supposed to follow the ordinary law. The component of this action along cD } is muds . 2r$ COS <7) 2 This is approximately equal to YftitL 9 ~ (1 + 3 - cos 6) (r cos s) ds, or to ~ [r cos + (3 cos 2 1) 5} ds ; and therefore the total action of AB on P, parallel to cD, is m cos f 7 3 cos l f //jsas, the integrals being taken along the whole length of the magnet. Consequently since the aggregate magnetism of a magnet, each element being taken with its proper sign, is zero. Again, fpsck being the moment of the magnetism of AJB, is the measure of its magnetic power, or what we have called M. Consequently the action parallel to cD is (? COB'*- I) ...................... (1). The action of the element at 5, perpendicular to cD, is . r Sm ' or, approximately, (1 + 3 cos0)Bin0&. The total action perpendicular to cD is, therefore, BMm . ^ i sm0 cos5 ........................ (2), NOTES ON MAGNETISM. 165 The equation to the resultant of these two forces is, since the line passes through P, y r sin _ x r cos 6 3 sin 6 cos 6 ~~ 3 cos 2 0-1 * For y = 0, or at the point Q, X Q = r cos - Jr sec (3 cos 2 - 1), or X Q = -J-y sec 0, Now cD = r sec 0, therefore which proves the first part of the construction. Next, to find the magnitude of the whole action, square and add (1) and (2), then (1-6 cos 2 + 9 cos*0 +9 sin 2 or i + scoB is the magnitude sought. Now PQ = r (sin 2 + (cos e - -I sec 0)*}* = r(l-$+4sec0)* = ^r (sec 2 6 + 3)*, and c@ = Therefore = (1 + 3 co c(^ Consequently, if jB be the magnitude of the resultant sought, PQ which was to l)e proved. The preceding formulas enable us to determine all the cir- cumstances of the mutual action of two magnets, which are such. as to fulfil the conditions of our hypothesis. For instance, in the memoir Intensity vis Magneticce Terres- iris, Q-auss has shown that if a magnet and a needle Tbe placed at right angles to one another, then the moment of rotation of the needle due to the action of the magnet is approximately twice as great when the line of the axis of the magnet passes 166 NOTES ON MAGNETISM. through the centre of the needle, as it Is when the line of the axis of the needle passes through the centre of the magnet. In neither case has the transverse force (2) any tendency to produce rotation. Its effect is destroyed Iby the fixed centre of the needle. Let y "be the distance of any element of the needle from the centre, mdy its magnetism. Then, in the first case, we shall have 6 = ^ approximately, (E being the distance between the Jti centre of the magnet and that of the needle). Consequently the force (1) may be expressed by the formula 2 - 3 } since cos 2 = l~0 2 nearly; " &J J ' which, neglecting y 2 , becomes the moment of this round the centre of the needle is - ^3 ? jLt and the total moment is, therefore, , . .-, where M In the second case, = for every element of the needle, and the moment sought is, therefore, MM' . ,,. - - Mm ...... jsg ......... , since (1) then becomes -- 5- , and the sign is immaterial. The moment in this case is, there- fore, one half of what it was before, which was to be proved. This result is included in the following general investiga- tion, in which we shall ascertain the moment of rotation due to the action of one magnet or needle upon another, whatever be their relative positions. * The data by which we shall suppose the position of the magnets to be determined, are the distance between their Centres, the angles which the axis of each respectively makes with the line joining their centres, and the angle between the axes themselves. The last element may be readily replaced by the dihedral angle between two planes which intersect in the line joining the centres of the magnets, and one of which passes NOTES ON MAGNETISM. 167 through the axis of one of tlie magnets while the other passes through that of the other. Let the axis of the magnet, whose action on the other is to be calculated, Tbe taken as axis of x, the centre of the magnet being the origin of co-ordinates. Let x, y, z be the co-ordinates of any point in the other magnet. Then, if r 2 = Again, let #, 5 ? c be the co-ordinates of the centre of the second magnet, a, /3, 7 the cos angles its axis makes with the co-ordi- nate axes, p the distance of any element dp from the centre, X> F, Z the forces on dp parallel to the axes of co-ordinates, m the intensity of dp. Then , Z = g xzdp. Let 6^, H, K be the moments of these forces about lines drawn through the centre of the second magnet parallel to the axes of co-ordinates, Kow cc Hence, neglecting the square, &c, of /> 3 ~ BMm = ~ [3aca - (3a 2 - JS 2 ) 7} pdp, where J5 2 ==a 2 168 NOTES ON MAGNETISM. Integrating for p, we find, for the total moments, ,. WOT , . , (H) = gj- a (ca. -07) + -gp- 7, . ~ . , MM' M' being for the second magnet what M is for the first. Let L be the resultant of these three moments, then Consequently, since a a + $* + be the angle between the axes themselves. Then a S cos and aa + 1/3 + 07 jB cos ^' ? also a = cos <. ConseoLuently {9 cos 2 !? sin 2 ^ 6 cos 6 (cos ff cos0 - cos<9) + sin 2 or i = --gp- {14-3 cos 2 5 - (3 cos cos 5' - cos <) 2 } Let % be the dihedral aingle already mentioned, then cos = cos cos & + sin sin 0' cos %. Consequently the last equation becomes MM' L = ~gr-(l + 3 cos 2 ~- (2 cos cos 0' - sin sin 0' cos the required expression. ON A MULTIPLE DEFINITE INTEGRAL*. IN the eighteenth number of the Journal, I pointed out the mode in which Fourier's theorem may be employed in the evaluation of certain definite multiple integrals. The theorem generally known as Liouville's, and another of the same de- gree of generality, were readily deduced from the considera- tions then suggested. I proceed to another application of the same method. THEOE. v-i '/ftUf T ''/ T j J& &U ab ... ] h if + (ma the integral being supposed to involve v variables x, y, &c., and the limits being given by the inequalities mx + ny + * - > Ti and ^ Ji. (Negative as well as positive values of the variables are admissible.) DEM. Recurring to the general theorem stated at the com* mencement of the paper already mentioned, we see that the integral whose value is sought is equal to f J Develope the cosine in a series of products of sines and cosines of simple arcs. Evefy term involving a sine disappears on * Cambridge Mathematical Journal, No, XXI. Yol. IV. p 116, May, 1844. 170 ON A MULTIPLE DEFINITE INTEGRAL. integration, as the limits extend from oo to + GO . Consequently the last written expression "becomes 1 f*' /. 7 f 7 f COSOMWOJ 7 f* cos any 7 I fu du I cos cm &a I 5- ace I = ^ ay. . . NOW J 3 3- C?W = e -*a ^ Ct a- ^Q^ and thus the integral becomes ub . . . J & / j^^ which was to Tbe proved, It may "be well to verify this result in a particular case. Let v = 2 ; then we have to prove that f r, f(mx + y*y) _ ma + nb f h ' fudu ]<**l*y () 2 } {u 2 + (ma - n j.i Consequently u J and thus the integral sought is seen to be equal to ma + nb n ' fudu 7T which was to be proved. Similar considerations apply in the case of more variables, and doubtless by induction our general result might be established. But the method we have followed, besides being more analytical, is also very much simpler. 172 ON A MULTIPLE DEFINITE INTEGRAL. Another result of this same kind of analysis I shall indicate without a demonstration, which there will "be no difficulty in supplying. ffofdy ... e" a ^~ l ^"'f(max -f nby + ...) the limits being given by + ... A and In conclusion, it may be well to remark, that the analysis of which we have made use is not unfrequently applicable to questions which though not difficult in principle are neverthe- less somewhat perplexing in practice. ON A QUESTION IN THE THEORY OF PROBABILITIES*. THE following question affords a good illustration of the methods employed in the more difficult parts of the theory of probabilities. In a paper presented to the Philosophical Society, I applied the kind of analysis we are about to make use of, to the celebrated Rule of Least Squares. There is, in fact, a close analogy between the two investigations. Laplace's solution of the present question is obtained by a process similar to that which he had employed when treating of the best method of combining discordant observations. What is the probability that the sum of the times which each of n persons has respectively yet to live will amount to a given time T? Let p x f dx s be the probability that the p ih person will live precisely a time x p longer, < p denoting some function of o^, which is necessarily such that as it is certain that he will die at some time or other. Let a? 1? a? 2 , ... x n be so related that x 1 + x si + ...x n = T. The probability of this particular combination is or & n x n cos a (T-Sa?) ... (a) J o / corresponds to a = 0, and is, therefore, unity ; and that when n is large (a) diminishes rapidly as a increases. Consequently /OO (a) da. depends, when n is very large, on the elements for which a is very small. This consideration enables us to employ an approximate value of (a). Let T=*t + m,m being a disposable quantity; then /< /.oo (a) =cos at dx^ ... I dx^x^ ... n x n co$a(m-'%x) J J o ,00 QO + sin at dx v . . . I dx n fax^ . . . <{> n x n sin a (m - 2k:) , ^ ^0 which may be thus written, (a) = cos at Or + sin atS. In order to obtain approximate values of G- and H, expand cos a (m - 2j) and sin a (m - Sec) ; they become respectively, a being very small, - |a* (m - S#) 2 and a (m - THEORY OF PROBABILITIES. 175 / -00 $xxd& K\ o J o /-co then, since I $xdx = 1, we stall Iiave J H~xdx and I x*xdx. Jo ^o We have stopped the approximation to the value of G at the second power of a. Had we gone farther, and retained only the principal term in the coefficient of each power of a, a similar result, viz. one which may be assumed as coincident with the exponential function, would, there is little doubt, have been obtained; while the coefficient of each power of a in H would be negligible in comparison of the corresponding power in G-. Some remarks on this, or at least on a cognate question, will be found in the paper already mentioned. As a verification of the approximation we have employed, which is in effect the same as that of Laplace, let us suppose that the functions 1} < 2 ... are all of the same form <, and that . x n ^ which do not transgress the limits Now and thus we have P= e T dTf Q dx t . . . f Q dx n ^ , the limits being given by Hence, it is easily seen that In order to compare this with the approximate expression (jp), I remark that T~n-l renders P a maximum; assume, therefore, THEORY OF PROBABILITIES. 177 Now, by Stirling's theorem, or by that which Binet proposes to substitute for it (yide Journal de T Mcole Polytechnigue^ xvi. p. 226), we have, when n is very large, T (n) Consequently (n) "" V{27r ( - 1)} ,/? being a certain function of t and n. Therefore Now the coefficient of f in ft is 1 -1. 1.2' and that of t Is 6 11 611 n^l^ (w~l) 2 (n-l)'J 1.2.3,4' Hence the coefficient of f in e~V& ^ s 1 that of f is 1 I 1 1 12 178 THEORY OF PROBABILITIES. and, lastly, that of t* is 1 __ g ___ 2 ___ 1_ 4 ( n _ i) + 4 (n - 1) 1 . 2 . 3 - I) 2 4 (n - 1) 11 1 1.2.3.4 (%-l) 2 . "^ JL r 1.2 Now if, in forming the approximate expression, we reject all ( t } p 1 terms o the form J~ j , where q is different from zero, IV ( n ) ) n i. e. if we look on > / v as a quantity all whose powers are to V(w) u "be retained, except when divided by any power of n, the / t V" 1 value of e~M M -- r- may be taken as equal to V n l) J u ^ 1 f which, as similar results would have Ibeen obtained had we pursued the investigation farther, might be shown to be equal to and thus P = P is the probability that T is equal to n 1 + tf : writing tf 2 tf -f 1 for t in ^- and reducing;, we find that, within the n * 1 limits of the approximation, it may also be assumed as the f f probability that t is equal to n+t: also - r=- ... 2-JP'j % JL yj and thus which, as in our case, F = 2 and jT= 1 is precisely equivalent to the result deduced from the general formula (p)* The legitimacy of some parts of the pteceding approximation may be questioned; as quantities which are neglected may, THEORY OF PROBABILITIES. 179 tinder certain conditions, be larger than those which are retained : and, as the result coincides with that of the general method, the doubt thus suggested appears to extend to the latter. The sub- ject of approximation by means of definite integrals is certainly not free from obscurity. The method of this paper extends m.m. to the case in which. we seek to determine the degree of improbability that the average length of the reigns of a series of kings shall exceed by a given quantity the average deduced from authentic history. The ap- plication of considerations of this nature to historical criticism appears to have been first made in Sir Isaac Newton's Chrono- logy. They are doubtless entitled to much attention ; but any attempt to evaluate their legitimate influence, would, for more than one reason, be unsatisfactory. 122 ON THE BALANCE OF THE CHEONO- METER*. IT is well known that a common watch goes more slowly when its temperature is raised, and versd vice. The reason of this is that the elasticity of the balance-spring decreases with the increment of temperature and increases with its decrement. Neglecting the mass of the spring and the connection of the, balance with the other parts of the watch, we may take as the equation for determining the oscillations of the balance, ^ + ^ = a? + 1 u ' where e depends on the form and elasticity of the spring, and I is the moment of inertia of the balance. The time of oscilla- 6 tion depends, of course, on the ratio y, e being, as we have said, a function of t the temperature. In order, therefore, to the equable rate of the watch, it would be necessary that / should be such a function of t, that -*. may be constant. In the balance of a common watch I is sensibly constant. Hence the inequality of which we have spoken. In the chronometer the balance is so constructed that its figure alters when the temperature varies. The figure repre- * Cambridge Mathematical Journal, No. XXL Yol. iv. p, 133, May, 1844. BALANCE OF THE CHRONOMETER, 181 sents a common form of the chronometer balance. The arc AB, which carries a weight at 0, is formed of two concentric laminse of different kinds of metal, the outer lamina being the most ex- pansible. These two larninse are securely united in their whole length, so that an increase of temperature necessarily distorts the arc AB into some form like AB'. Similarly for ab. Con- trary effects are produced by a decrease of temperature. Thus, the moment of inertia I decreases as t increases ; as e also does. And thus we are enabled, by suitable adjustments, to make y, at least approximately, constant. It would, I believe, be impossible, without some hypothesis, to determine the form which AB assumes under the influence of a change of temperature. The following suppositions are probably sufficiently near the truth to be applicable when the variations of t are not excessive. Let us suppose the laminas to be cylindrical and concentric, and bounded by four plane surfaces, two of which are perpen- dicular to the axis of the cylinder, while the other two, which form the boundaries at A and B, pass through the axis. These conditions being fulfilled, whatever the value of t may be, it is clear that the variation of form can depend on two elements only, namely, the radius of the cylinder, and the angle which AB subtends at its centre. To determine these, we assume that the middle filament of each lamina expands as it would do if free. In the normal state, let 2e, 2e' be the thicknesses of the outer and inner laminae respectively, r the radius of the boundary of the two laminse, /, p the coefficients of expansibility of the outer and inner laminae (jju > p!}, 6 the angle subtended at the centre. The radii of the middle filaments are, therefore, T -f 6, r e ; let their lengths be I and T, then For an increase of temperature t, let r and become r x and X : then we shall have = (r, + 6) e lf e and e' being so small that their variations may be neglected. 182 BALANCE OF THE CHRONOMETER. ^ T 4- e 7 1 + pt Mence - , = 7? 7 r r x 6 I I and therefore which, as p and // are very small, is approximately Consequently, for a first approximation, Ar-A^* ..... . .................. (1), where Ar = r 4 - r, A/* = ^ //, and r = e + 4 . Again T^ = I - ? + (pi - /r) t and therefore But pi and the last term is negligible. Therefore (2). We distinctly perceive from (1) and (2) why the effects of distortion are so considerable in comparison with those of simple expansion ; it is because the expressions of Ar and A# have the s'mall quantity r in the denominator. AB becomes a larger arc of a smaller circle. To apply these results: we suppose that when tf the centre of A B coincides with the central point ; and AB, "being securely fastened at A, continues perpendicular at that point to the line OA, consequently its centre remains in that line. Let G be its new position, then 00 r = Ar, If m be the mass of AB, its moment of inertia about was mv* ; BALANCE OF THE CHRONOMETER. 183 T B about 0' it is m^ Sm^/x t nearly. Let Cf be the centre of gravity of AB : then, in the triangle O r Gr, we have OG*~00'*+(0'GY+ 200'. 0' & cos i^, since 00'* or (Ar) 2 may be neglected, then, approximately, (0&y=(0'&)*~2Ar.O r a cos 0, T y-uxv sin 40, ^ sini^ , and, as 0' G = r, * * = 2r ^~ nearly, "" we have (OG)*- (0 ; ff) 2 = - 2r An Now the moment of inertia, round is equal to that round 0' increased by m {(OG} Z - (0' G}*} ; hence, finally, ............... (3), 1^ being the moment of inertia of the arc AS. (In accordance with the rest of the approximation the ex- pansion of A is neglected.) Again, we will suppose the weight at G to be a material particle, and that the angle AOO is equal to <. Then, J 2 being the moment of inertia of this weight, whose mass we will de- note by m 2 , we shall have r s , t (1- cos<) .............. (4). Consequently, as the Inertia of the bar OA does not undergo any sensible alteration, and as every thing which has been proved of OAB is true of Oab 7 we have, finally, AI=~4A/*~ t jm^l - S -2p) +m 2 (1 -cos )l ...... (5). It appears that the variation of e is exactly proportional to t : so that e becomes e (I- vt}> v being some constant. Consequently ^ i - j 0.1 ^ e (1 ^) e we must have, in order that -~ -r~^ = -j , ...... (6). In calculating the value of J we may take into account the moment of inertia of a OA] moreover, instead of the approxi- 184 BALANCE OF THE CHRONOMETER. mate expression m^ for the moment of inertia of AB> we may employ a mote accurate one involving the quantities e and e'; the approximate expression is sufficiently accurate for the deter- mination of A J. The adjustment for compensation is effected- Tby shifting the weight m 2 along AB ; that is, by altering the value of ^> until (6) is fulfilled. On the hypothesis we have made, the value of I for t = is not affected Iby the change of . In determining the approximate expressions (5) and (6), we have neglected all terms in which Aju, occurs not divided by T ; all terms involving A/* multiplied Iby e or e ; all terms into which any power of /i or p! enters. In consequence of the last restriction t can only rise to the first power in the result* If this were absolutely correct it would follow that, if the compensation were effected for a particular value of t, it would subsist accurately for all values of t. For instance, if we give t equal values, positive and negative, the decrease of I in the one case ought to be equal to its increase in the other. But when t is considerable, it is found that there is a sensible deviation from this result; and, assuming that the expression for Ae does not in any perceptible manner involve powers of t, it follows that that of AJ must do so. Any term involving f (and, a fortiori, any higher powers of that quantity), must be very small, since t always occurs multiplied by p or p!\ but it may, nevertheless, sensibly affect the chronometer's daily rate. On the usual construction, the balance oscillates 216,000 times in twenty-four hours. Consequently a very slight change in the moment of inertia of the balance will become perceptible in that period. In order to obviate the consequent error, it has been proposed by Mr Dent, a distinguished chronometer-maker of the present BALANCE OF THE CHRONOMETER. 185 day, to alter the form of the "balance. The figure represents one of those which he proposes to substitute for that in common use. It would be easy to determine the corresponding expression for A/, to the degree of approximation of our previous results. As, however, the comparison of the merits of the two forms must depend on the terms involving f, it may be well to reserve it for another opportunity. If there appears reason to believe that our hypotheses represent the facts with sufficient accuracy to encourage us to proceed farther, I hope to resume the subject in the next number of the Journal, NOTES ON' MAGNETISM*. No. II. IN order to a distinct understanding of the results obtained in the last number of the Journal, it will be desirable to con- sider the established conventions with respect to the signs of the symbols which we had occasion to employ. North magnetism is assumed to be positive ; hence, of course, south magnetism must be considered as negative. The measure M of the magnetic power of a bar magnet is, as we have seen, equal to f/j,sds, ^ being the magnetism of the element ds, which is situated at a distance from the origin equal to s. The limits of the integral are such as to include the whole length of the magnet. The position of the origin is arbitrary : we may conveniently place it at the centre of tne magnet, but the value of //*sds is the same whether this be done or any other point be taken. For let the origin be shifted through a distance a, so that s = s* a, then (s f a) ds' =//j,s r ds' a and as all the integrals extend throughout the length of the magnet fpds' = 0, and therefore ffjLsdsf^s'ds' or M=M', which was to be proved. But the value of Jjjisds changes its sign if the direction in which s is measured changes. Let I be the length of the magnet ; then, s being measured in one direction, say from left to right, we have ri M= fj,sds. J o * Cambridge Mathematical Journal, No. XXI. Vol. iv. p. 139, May, 1844. NOTES ON MAGNETISM. 187 Now suppose that s r = I ~~ s, then the limits are interchanged and ds r = ds; consequently ' = - M s' being measured in the direction opposite to that of s, or from right to left. The magnetism of a magnet may thus be always represented "by a positive quantity. Any two points in the axis of a magnet may be taken as its poles. But although the position of the poles is matter of con- vention, yet relatively to one another, one is the north and the other the south pole. The physical character by which they are distinguished is this : if a particle of north magnetism be placed in the pro- longation of the axis from south to north, it is repelled from the magnet. Contrariwise, if it be placed in the prolongation of the axis towards the south. Further, we must integrate fjisds from south to north, i. e. s must be taken as positive when /nds lies to the north of the origin, in order that M may be positive. This may be shown by supposing a particle of north magnetism m placed in the prolongation of the axis towards the north, and at a distance r from the centre of the magnet. If we assume that from south to north is positive, ,, , . ,, , . %Mm , . . . the action ot the magnet on m is - ; and as- this action is repulsive its expression will be positive, and therefore M is so. If we had assumed from north to south to be positive, the action of the magnet would have been represented by -- -5 , and as this is positive, M will necessarily be negative. So that, in order to make the measure of the magnet's power positive, we must take the direction S . . . JV as positive. Consequently the angle must be measured from it. We suppose it measured in the usual manner, viz. in the unscrew direction. The general expression for the moment of rotation due to the action of one magnet on another is much simplified when the two magnets are supposed to lie in one plane. 188 NOTES ON MAGNETISM, The dihedral angle % is then zero, and consequently the equation L = *Sj{L {i + a eos 2 - (2 cos 6 cos ff - sin 6 sin ff cos % ) 2 }* JL\J "becomes + 4 sin 0' cos 6 sin (9 cos 0' - sin 2 sin 2 0'}* . The quantity between the brackets is equal to 4 sin 2 0' cos 2 + 4 sin & cos 9. sin Q cos 0' + sin 2 cos 2 0'. Consequently L = ^L' (sin cos & + 2 sin 0' cos 0). jt This result may "be readily established by an independent pro- cess, which the reader will find no difficulty in supplying. The last result may be put in the following form : Professor Lloyd, in the 19 th volume of the Memoirs of the Royal Irish Academy, has investigated this case of the mutual action of two magnets. His result is (mutatis mutandis) - 8 sn ~ This differs from the last written result, merely because, in the Professor's analysis, and 0' are measured in opposite directions. If we replace in Prof. Lloyd's result by 2?r 0, it becomes as before. The general formula affords a simple solution of the follow- ing problem. The position of a magnet, and that of the centre of a needle being given, to place the needle in the position in which the moment of rotation due to the action of the magnet is a maximum. By the formula established in the last number of the Journal, we have L = -3- {1 + 3 cos 2 - (3 cos cos 0' - cos <)*}*. NOTES ON MAG-NETISM. 189 We suppose the magnet and needle to be in the same plane. In the figure let be the centre, SON the line of the axis of the magnet, the centre of the needle. Project on ON in D, take OE = 2 OD, draw JEflP perpendicular to SN meet- ing 02V', which is at right angles to 00 in N', ON' is the line in which the axis of the needle must be placed, its north pole being turned towards N r . In order to prove this, we haye only to remark that the angle 6 or CON is constant, the position of being given ; consequently the condition to be fulfilled, in order that the moment of rotation L may be a maximum, is 3 cos 6 cos & cos = 0. Now as

= 0, or the required condition is fulfilled. There are three particular cases worth noticing : (1) = 0. In this case G lies in the axis ON, D coincides with it, and ON r is perpendicular to ON, and therefore parallel to JEN'. Consequently the point N f is removed to an infinite distance, and ON 1 is therefore perpendicular to ON. The , _ . 2MM f , . ., corresponding value of is - , and is the maximum maximorum. (2) 6 = 0'. In this case the magnet and needle are parallel 190 NOTES ON MAGNETISM. to one another. The quadrilateral CDUN' is a parallelogram, EN' is equal to D (7, and consequently tan GOD : tmN'OE :: HO : OD :: 2 : 1. But tfOZ) = and since CON' is a right angle. Consequently tan = 2 cot 0, or tan = 4/2- The corresponding value of 3 cos 2 6 is therefore unity ; and consequently we have, in this case, (3) = - . Here D (and therefore E] coincides with 0, 2 while ON' lies in the axis OS'. Consequently N f is at <9, and the needle is therefore again perpendicular to the magnet. In this case ,. MM 1 The first and third cases were noticed in the last number of the Journal. In the second, the value of L is a mean proportional between what is in the other two cases, in the last of which it is a minimum maximorum. If we were required, for a given position of (7, to find the position in which the needle would be in equilibrium, or the moment L equal to zero, we might have recourse to Grauss's construction already mentioned; for if the needle be placed along the line in which the magnet tends to attract or repel (7, as the dimensions of the needle are small, every element would approximately be attracted or repelled along this line, and therefore the total action would be destroyed by the resistance of C. Thus there are always two directions for every position of <7 f ; one of maximum moment and the other of equilibrium : these two directions are at right angles to one another. MATHEMATICAL NOTE*. THIS definite integral is evaluated in a curious manner by M. Bertrand in Liouvilles Journal. The demonstration I am about to give of his result, is somewhat different in form from that which he made use, of. The method employed in an ingenious paper which appeared in the last volume of the Journal (in. p. 168), will apply to the integral we are about to consider. Let fu Then *A. r "I* X t U Consequently x x +u u ux] and therefore d j, 1 f 1 xdx u I" 1 dx u f 1 dx __ 1 f 1 xdx u I" 1 dx "~ ITw 2 J 1 + ^ 1-fw 2 J l + ^ 2 "~l Integrate for w, from to 1, /(I) ~/(0) = log 2+ log 2 -/(I). Cambridge Mathematical Journal, No, XXI. Vol. rv. p. 143, May, 1844* 192 MATHEMATICAL NOTE. But/(0)=0, since log 1 = 0. Therefore /(I) -log a. Therefore ^ - Iog2 . - The singularity of this method, and its applicability in other cases giye it interest : but, as the writer of the paper already noticed pointed out to me, the integral may be got by assum- ing x = tan y ; it then becomes [i J o and, by his fundamental equation, IT Plog (l + ta Jo =: log l +tan - - , , fir \ t 1 tan y 1 + tan T y = 1 + - KL = - \4 ^/ 1-ftany 1 and therefore i whence the truth of M* Bertrand's result is obvious, MEMOIR OF THE LATE D. F. GREGORY, M.A. FELLOW OF TRINITY COLLEGE, CAMBRIDGE*. THE subject of the following memoir died in his thirty-first year. He had, nevertheless, accomplished enough not only to justify high expectations of his future progress in the science to which he had principally deyoted himself. But also to entitle his name to a place in some permanent record. Duncan Farquharson Gregory was born at Edinburgh in April 1813. He was the youngest son of Dr James Gregory, the distinguished professor of Medicine, and was thus of the same family as the two celebrated mathematicians James and David Gregory. The former of these, his direct ancestor, is familiarly remembered as the inventor of the telescope which bears his name ; he lived in an age of great mathematicians, and was not unworthy to be their contemporary. Of the early years of Mr Gregory's life but little need be said. The peculiar bent of his mind towards mathematical speculations does not appear to have been perceived during his childhood; but, in the usual course of education, he shewed much facility in the acquisition of knowledge, a remarkably active and inquiring mind, and a very retentive memory. It may, perhaps, be mentioned here, that his father, whom he lost before he was seven years old, used to predict distinction for him ; and was so struck with his accurate information and clear memory, that he had pleasure in conversing with him, as with an equal, on subjects of history and geography. In his case, as in many others, ingenuity in little mechanical contrivances seems to have preceded, and indicated the developement of a taste for abstract science. * Camh*idge Mathematical Journal, No. I. Vol. iv. p. 145, November, 1844, 13 194 MEMOIR OF MR GREGORY. Two years of his life were passed at the Edinburgh Academy; when he left it, being considered too young for the University, he went abroad and spent a winter at a private academy in Geneva. Here his talent for mathematics attracted attention ; in geometry, as well as in classical learning, he had already made distinguished progress at Edinburgh. The following winter he attended classes at the University of Edinburgh, and soon became a favourite pupil of Professor Wallace's, under whose tuition he made great advances in the higher parts of mathematics. The Professor formed the highest hopes of Mr Gregory's future eminence : those who long after- wards saw them together in Cambridge, speak with much in- terest of the delighted pride he shewed in his pupil's success and increasing reputation. In 1833, Mr Gregory's name was entered at Trinity College in the University of Cambridge, and shortly afterwards he went to reside there. He brought with him a very unusual amount of knowledge on almost all scientific subjects : with Chemistry he was particularly well acquainted, so much so that he had been at Cambridge but a few months when it was proposed to him by one of the most distinguished men in the University to act as assistant to the professor of Chemistry ; which for some time he did. Indeed, it is impossible to doubt that, had not other pursuits engaged his attention, he might have achieved a great reputation as a chemist. He was one of the founders of the Chemical Society in Cambridge, and occasionally gave lectures in their rooms. He had also a very considerable knowledge of botany, and indeed of many subjects which he seemed never to have studied systematically : he possessed in a remarkable degree the power of giving a regular form, and, so to speak, a unity to knowledge acquired in fragments. All these tastes and habits of thought Mr Gregory cultivated, to a certain extent, during the first years of his residence in Cambridge, of course in subordination to that which was the end principally in view in his becoming a member of the University, namely, the study of mathematics and natural philosophy. He became a Bachelor of Arts in 1837, having taken high mathematical honours : more, however, might, we may believe, MEMOIR OF MR GREGORY. 195 have been effected In this respect, had his activity of mind per- mitted him to devote himself more exclusively to the prescribed course of study, From henceforth he felt himself more at liberty to follow original speculations, and, not many months after taking his degree, turned his attention to the general theory of the com- bination of symbols. It may be well to say a few words of the history of this part of mathematics- One of the first results of the differential notation of Leibnitz, was the recognition of the analogy of differentials and powers. For instance, it was readily perceived that or, supposing the y to be understood? that just as in ordinary algebra we have, a being any quantity, This, and one or two other remarks of the same kind, were sufficient to establish an analogy between -j- the symbol of differentiation and the ordinary symbols of algebra. And it was not long afterwards remarked that a corresponding analogy existed between the latter class of symbols and that which is peculiar to the calculus of finite differences. It was inferred from hence that theorems proved to be true of combinations of ordinary symbols of quantity, might be applied by analogy to the differential calculus and to that of finite differences. The meaning and interpretation of such theorems would of course be wholly changed by this kind of transfer from one part of mathematics to another, but their form would remain unchanged. By these considerations many theorems were suggested, of which it was thought almost impossible to obtain direct demonstrations. In this point of view the subject was developed by Lagrange, who left undemonstrated the results to which he was led, in- timating, however, that demonstrations were required. Gradually, 132 196 MEMOIR OF MR GREGORY. however, mathematicians came to perceive that the analogy with which they were dealing, involved an essential identity; and thus results, with respect to which, if the expression may be "used, it had only been felt that they must be true, were now actually seen to be so. For, if the algebraical theorems by which these results were suggested, were true, because the symbols they involve represented quantities, and such opera- tions as may be performed on quantities, then indeed the analogy would be altogether precarious. But if, as is really the case, these theorems are true, in virtue of certain funda- mental laws of combination, which hold both for algebraical symbols, and for those peculiar to the higher branches of mathematics, then each algebraical theorem and its analogue constitute, in fact, only one and the same theorem, except quoad their distinctive interpretations, and therefore a demonstration of either is in reality a demonstration of both*. The abstract character of these considerations is doubtless the reason why so long a time elapsed before their truth was distinctly perceived. They would almost seem to require, in order that they maybe readily apprehended, a peculiar faculty a kind of mental disinvoltura which is by no means common. Mr Gregory, however, possessed it in a very remarkable degree. He at once perceived the truth and the importance of the principles of which we have been speaking, and proceeded to apply them with singular facility and fearlessness. It had occurred to two or three distinguished writers that the analogy, as it was called, of powers, differentials, &c., might be made available in the solution of differential equations, and of equations in finite differences. This idea, however, probably from some degree of doubt as to the legitimacy of the methods which it suggested, had not been, fully or clearly developed : it seems to have been chiefly employed as affording a convenient way of expressing solutions already obtained by more familiar considerations. To this branch of the subject Mr Gregory directed his _ * The values of certain definite integrals are to be looked upon as merely arithmetical results ; in such cases we are not at liberty to replace the constants involved in the definite integrals by symbols of operation. In other cases we are at liberty to do so, and this remarkable application of the principles stated in the text, has already led Mr Boole of Lincoln, with whom it seems to have originated, to several curious conclusions. MEMOIR OF MR GREGORY. 197 attention, and from the general views of the laws of combina- tion, of symbols already noticed, deduced in a regular and systematic form, methods of solution of a large and important class of differential equations (linear equations with constant coefficients, whether ordinary or partial) of systems of such equations existing simultaneously, of the corresponding classes of equations in finite and mixed differences ; and lastly, of many functional equations. The steady and unwavering apprehension of the fundamental principle which pervades all these applica- tions of it, gives them a value quite independent of that which arises from the facility of the methods of solution which they suggest. The investigations of which I have endeavoured to illustrate the character and tendency, appeared from time to time in the Cambridge, Mathematical Journal. In this periodical publication Mr Gregory took much interest* He had been active in establishing it, and continued to be its editor, except for a short interval, from the time of its first ap- pearance in the autumn of 1837, until a few months before his death. For this occupation he was for many reasons well qualified; his acquaintance with mathematical literature was very extensive, while his interest in all subjects connected with it was not only very strong, but also singularly free from the least tinge of jealous or personal feeling. That which another had done or was about to do, seemed to give him as much pleasure as if he himself had been the author of it, and this even when it related to some subject which his own researches might seem to have appropriated. - This trait, as the recollections of those who knew him best will bear me witness, was intimately connected with his whole character, which was in truth an illustration of the remark of a French writer, that to be free from envy is the surest indi- cation of a fine nature. To the Cambridge Mathematical Journal, Mr Gregory con- tributed many papers beside those which relate to the researches already noticed. In some of these he developed certain parti- cular applications of the principles he had laid down in an Essay on the Foundations of Algebra, presented to the Eoyal Society of Edinburgh in 1838, and printed in the fourteenth volume of their Transactions* I may particularly mention a paper on the 198 MEMOIR OF MR GREGORY. curious question of the logarithms of negative quantities, a question, which, it is well known, has often Tbeen discussed among mathematicians, and which even now does not appear to be entirely settled. In 1840, Mr Gregory was elected Fellow of Trinity College; in the following year he "became Master of Arts, and was ap- pointed to the office of moderator, that is, of principal mathe- matical examiner. His discharge of the duties of this office (which is looked upon as one of the most honourable of those which are accessible to the younger members of the University) was distinguished by great good sense and discretion. In the close of the year 1841, Mr Gregory produced his "Collection of Examples of the Processes of the Differential and Integral Calculus;" a work which required, and which manifests much research, and an extensive acquaintance with mathematical writings. He had at first only wished to super- intend the publication of a second edition of the work with a similar title, which appeared more than twenty-five years since, and of which Messrs. Herschel, Peacock, and Babbage were the authors. Difficulties, however, arose, which prevented the fulfilment of this wish, and it is not perhaps to be regretted that Mr Gregory was thus led to undertake a more original design. It is well known that the earlier work exercised a great and beneficial influence on the studies of the University, nor was it in any way unworthy of the reputation of its authors. The original matter contributed by Sir John Herschel is especially valuable. Nevertheless, the progress which mathematical science has since made, rendered it desirable that another work of the same kind should be produced, in which the more recent im- provements of the calculus might be embodied. Since the beginning of the century, the general aspect of mathematics has greatly changed. A different class of problems from that which chiefly engaged the attention of the great writers of the last age has arisen, and the new requirements of natural philosophy have greatly influenced the progress of pure analysis. The mathematical theories of heat, light, electricity, and magnet- ism, may be fairly regarded as the achievement of the last fifty years. And in this class of researches an idea is prominent, which comparatively occurs but seldom in purely dynamical enquiries. This is the idea of discontinuity. Thus, for instance, MEMOIR OF MR GREGORY. 199 in the theory of heat, the conditions relating to the surface of the "body whose variations of temperature we are considering, form an essential and peculiar element of the problem; their peculiarity arises from the discontinuity of the transition from the temperature of the body to that of the space in which it is placed. Similarly, in the undulatory theory of light, there is much difficulty in determining the conditions which belong to the bounding surfaces of any portion of ether; and although this difficulty has, in the ordinary applications of the theory, been avoided by the introduction of proximate principles, it cannot be said to have been got rid of. The power, therefore, of symbolizing discontinuity, if such an expression may be permitted, is essential to the progress of the more recent applications of mathematics to natural philosophy, and it is well known that this power is intimately connected with the theory of definite integrals. Hence the principal im- portance of this theory, which was altogether passed over in the earlier collection of examples. Mr Gregory devoted to it a chapter of his work, and noticed particularly some of the more remarkable applications of definite integrals to the expression of the solutions of partial differential equations. It is not improbable that in another edition he would have developed this subject at somewhat greater length. He had long been an admirer of Fourier's great work on heat, to which this part of mathematics owes so much; and once, while turning over its pages, remarked to the writer, "All these things seem to me to be a kind of mathematical paradise. 1 * In 1841, the mathematical Professorship at Toronto was offered to Mr Gregory: this, however, circumstances induced him to decline. Some years previously he had been a candidate for the Mathematical Chair at Edinburgh. His year of office as moderator ended in October 1842. In the University Examination for Mathematical Honours in the following January, he, however, in accordance with the usual routine, took a share, with the title of examiner, a position little less important, and very nearly as laborious, as that of moderator. Besides these engagements in the University, he had been for two or three years actively employed in lecturing and examining in the College of which he was a Fellow. In the fulfilment of these duties, he shewed an earnest and constant 200 MEMOIR OF MR GREGORY. desire for the improvement of his pupils, and his own love of science tended to diffuse a taste for it among the "better order of students. He had for some time meditated a work on Finite Differences, and had commenced a treatise on Solid Geometry, which, unhappily, he did not live to complete. In the midst of these various occupations, he felt the earliest approaches of the malady which terminated his life. The first attack of illness occurred towards the close of 1842. It was succeeded by others, and in the spring of 1843, he left Cambridge never to return again. He had just before taken part in a college examination, and notwithstanding severe suffer- ing, had gone through the irksome labour of examining with patient energy and undiminished interest. Many months followed of almost constant pain. Whenever an interval of tolerable ease occurred, he continued to interest himself in the pursuits to which he had been so long devoted ; he went on with the work on Geometry, and, but a little while before his death, commenced a paper on the analogy of differ- ential equations and those in finite differences. This analogy it is known that he had developed to a great length; un- fortunately, only a portion of his views on the subject can now be ascertained. At length, on the 23rd February 1844, after sufferings, on which, notwithstanding the admirable patience with which they were borne, it would be painful to dwell, his illness terminated in death. He had been for a short time aware that the end was at hand, and, with an unclouded mind, he prepared himself calmly and humbly for the great change ; receiving and giving comfort and support from the thankful hope that the close of his suffering life here, was to be the beginning of an endless existence of rest and happiness in another world. He retained to the last, when he knew that his own connection with earthly things was soon to cease, the unselfish interest which he had ever felt in the pursuits and happiness of those he loved. A few words may be allowed about a character where rare and sterling qualities were combined. His upright, sincere, and honourable nature secured to him general respect. By his intimate friends, he was admired for the extent and variety of his information, always communicated readily, but without a thought of display, for his refinement and delicacy of taste MEMOIR OF MR GREGORY. 201 and feeling, for Ms conversational powers and playful wit; and lie was beloved by them for his generous, amiable dis- position, his active and disinterested kindness, and steady affection. And in this manner his high-toned character acquired a moral influence over his contemporaries and juniors, in a degree remarkable in one so early removed. To this brief history, little more is to be added ; for though it is impossible not to indulge in speculations as to all that Mr Gregory might have done in the cause of science and for his own reputation, had his life been prolonged, yet such speculations are necessarily too vague to find a place here ; and even were it not so, it would perhaps be unwise to enter on a subject so full of sources of unavailing regret. ON THE SOLUTION OF EQUATIONS IN FINITE DIFFERENCES*. THE partial differential equations which occur in various "branches of mathematical physics are, for the most part, of such forms that solutions of them may be obtained without much diffi- culty. As is well known, the great difficulty in almost all such cases consists in the necessity of determining which of all possi- ble solutions satisfies the particular conditions of the problem on which we are engaged. It seems that before the time of Fourier's researches on heat, the course which mathematicians had uni- formly followed was, first to obtain the general solution of the equation of the problem, and then to determine by particular considerations the arbitrary functions which it involved. This course undoubtedly would be the most direct and analytical, were there any general method for determining the form of the functions in question : as, however, there is none, the analytical generality of the first part of the process is in many cases sterile and useless, Fourier's methods, which depend essentially on the linearity of the partial differential equations which occur in the theory of heat, consist in assuming some simple solution of the equation of the problem, in deducing from hence a more general solution of it, and in determining successively and by means of particular considerations the arbitrary quantities thus introduced in such a manner as to satisfy all the conditions of the question. The general solution with arbitrary functions does not make its appearance in his process ; and the reason why it is so much more manageable than the other appears to be, that it is far easier to determine arbitrary constants in accordance with certain * Cambridge Mathematical Journal, No. XXII. Vol. iv, p. 182, November, 1844- ON THE SOLUTION OF EQUATIONS, ETC. 203 conditions than arbitrary functions. There will, generally speak- ing, Tbe an infinite number of arbitrary constants, and it is therefore necessary to treat them in classes. The ingenious synthesis by which this is effected by Fourier, in the different problems discussed by him in the Theorie de la Chaleur, forms one of the most interesting parts of that admirable work. The same kind of reasoning is made use of by Poisson, in his re- searches on similar subjects : and there can be little doubt that the methods of Fourier, developed and extended as they have been by subsequent writers, will long continue to be an essential element in the application of mathematics to physical researches. Similar methods may be made available in the solution of equations in partial finite differences. Such equations do not, it is true, present themselves very often, as the continuity of the causes to which natural phenomena are due, leads rather to differential equations than to those in finite differences. In fact, I am not aware of any subject, except the theory of probabilities, in which we meet with problems whose solution depends on that of an equation in partial finite differences. In this theory, however, such problems are not uncommon. One of the most interesting of them, both in its own nature and historically, may serve as an illustration of the application of the methods of Fourier to finite differences. This problem, which has engaged the attention of several writers on the subject of probabilities, and of which a solution was among the earliest efforts of Amp&re, is that of the duration of play. Professor De Morgan has spoken of this solution and of that of Laplace, as being of the highest order of difficulty: that which I am about to enter on has, I think, a decided advantage in this respect The problem itself may be thus stated: Two persons, M and JV", have between them a number a of counters : they play at a game at which M J s chance is p, and JV's q. The losing player gives one counter to the other, and they are to play on until one or other have lost all his counters. What is the probability that the party will terminate in M 's favour after any assigned number of games, N being supposed to have originally x of the a counters ? Let y^ be the probability that M will win the party at the (z + l) th game. If he win the next game (of which the pro- 204 ON THE SOLUTION OF EQUATIONS bability is_p), this becomes y^:^} if he lose it (of which the probability is j), it becomes y x+l . K ^ I3 and therefore This is the equation of the problem. It is clear that #o* = 0, 2^ = ........................ (2), as the party ceases as soon as M or N has a counters. Again, yao = unless x = l, and y^p ............ (3); for if N hare more than one counter he cannot lose them all at the nest game ; and if he have only one, his chance of his being left without any is p. Let us assume y XK = tfv x ........................... (4), a being arbitrary. Then =pv^ + qy^ ...................... (5). Of this a solution is (6), where and o> are arbitrary, and /M such that 2 *J(pg) cos fjb = a (7) : this form of solution is therefore real if of is less than In order that (4) may satisfy the conditions (2), we must have a i> = C&inco Q, v a C(%-\ sin (pa + G>) = (8). It is impossible to satisfy these two conditions without making (7=0, which would give a nugatory result, unless sin /m = or p _. j r being an integer. Let us therefore assume this value for //, ; and then, by (7), T7T a = 2 VC? = 0, and then, substituting the values of v x and a in (4), we get sin xfcos Y (10). tt \ fc / OF FINITE DIFFERENCES. 205 This value, in which G and r are arbitrary, satisfies (1) and (2), and in consequence of the linearity of these equations they will be satisfied by a sum of similar values, and we shall thus have a more general solution, viz, (ii). If in this we put z = 0, we have (12). Now, by (3), this is to be equal top for#=l, and to for the a 2 values of a?, 2.3 ...... a 1. There are thus a 1 conditions for (12) to fulfil, and therefore we have, extending the summation 2 from r 1 to r = a - 1, the following system of equations : #1 sin + ...... + Q-i sin ir 0=asin + ...... + CL iS in2 TT c& & = ..................... r ^ - - (13). A a From these a 1 equations we have to determine the a 1 quantities 0^... <7 a _ t . In order to do this, multiply the first T 7* equation by sin - TT, the second by sin 2 - TT, and so on, (r being an integer less than a), and add. Then, as may be easily shown, the coefficient of every one of the quantities C, except C r9 will in the resulting sum be equal to zero, while that of Of will be. Consequently (13) is equivalent to the svstem of equations included in the general formula m (U), and consequently (11) becomes SG ~L g+l /ffA^ T TX f T y*=- (4M)~(f ) Sr 1 sin * sin j T (, cos which is the required probability. 206 ON THE SOLUTION OF EQUATIONS We may deduce from this formula, by indirect considera- tions, one or two analytical theorems. For it is obviously im- possible that the party should terminate in Jfef's favour in less than x games, as x is the number of counters he must win from N. Consequently n yx f v \ S/^sin -TT sin TT! cos -TT) =0 ......... (16), 1 a a \ a J for all integer values of z less than x 1. Again, M may win the party at the x ih game, if he win x games in succession, the probability of which is p*. Hence, putting z = x 1, we have r TV f r \ x ~ 1 S^sin-irsin 9r[cofl-ir) , 1 a a \ a / ' or S/"" 1 sin - TT sin ir( cos - TT J = ^ ...... (17). These formula may undoubtedly be established by other methods, but I have thought it worth while to point out this way of deducing them, from the analogy it bears to that in which many remarkable theorems are obtained by Poisson, in his Theorie de la Ckaleur, namely by considering the nature of the quantities which his formulas represent. This mode of establishing analy- tical theorems by considerations founded on the interpretation of our results, is one of the most curious features of the more recent methods of treating physical questions* To (16) and (17) another theorem may be added, by the following consideration. Jf, if he win, must win the party either in x games or in x -f and even number of games. For If he lose k games he must win back k games and x more or there must have been x + %Je games in the party. Hence his chance is zero whenever + l=a?+2& + l, and therefore v* y - rx f T N 33 "^ 2^ sin- w sin ir{ cos - TT =0 .......... (18), 1 a a \ a J ^ " Jc being any positive integer whatever. When a is infinite, the sums contained in the last three equations become definite integrals. Let r j. JL-L 9r 7t i a I - TT = fa tnen - = d sin x$ (cos } s d(j> = ....... ....... (19), Jo (z being integral and less than x 1), fir I sin< sincc<(cos<)*"V<=-; ........... (20), J o 2 I sin< s>mx(j> (cos ) x +^d$ = Q . ........ (21). J o If, instead of seeking the probability that M will win the party at the (z + l) th game, we wished to find that of his winning it after z or more games shall have been played, we should only have to sum (15) for z from z to infinity. Calling this new probability u^, we should thus get . r . TX sin - TT sin TT If in (22) we put z equal to zero, we have then the proba- bility of M" s winning the party at the first, second, &c. games, i.e. of his winning it at all. Writing simply u x for % 0? we shall thus get . r . TX sm - TT sm f t a/ o> IsFow of this probability we can obtain, as is well known, a much simpler expression. For it is easily seen that we shall have u x =pu x _^qu x+ (24) for every value of as, provided that, instead of considering u x as the probability that M will win the party, we make it denote the probability that he either has won or will win it. As it is impossible that he can have already won it while x differs from zero, this alteration does not affect the value represented by u x except for the case of x =0. In this case the value of u x , as expressed by (23), will be zero, as the party is at an end, M having already won it. But according to the proposed 208 ON THE SOLUTION OF EQUATIONS modification, the new value of U Q will be unity, and therefore we have for the initial and final values of u x) tt = l, u a =0 ...................... (25). The necessity of this modification arises from this, that otherwise the relation expressed by (24) would not be in all cases true. For when x = 1, we should have u l = qu^ whereas the true value is of course From (24) we have (introducing the relation j? +# = 1), .~ + (fT ....................... (26)? a and ft being arbitrary constants : and thence, by (25), we get ()_()' ^ and consequently u x = ;~ a (27). This expression is therefore, except for x = 0, equivalent to (23), into which however the relation already mentioned, viz. that p + q = 1 has not as yet been introduced. When p and % are equal, (27) becomes while (23) similarly becomes . r . rx sin - TT sin TT u - *** a a **~a\ , r 1 COS- 7T a or % = - S*" 1 cot - -~ sin TT .......... ,. (29). a a a 2 a ^ ; Comparing (28) and (29), we have the following theorem; writing x for a x* ^ = 2r 1 cot^|sin7r ............... (30), the tipper sign to be taken when r is odd. OF FINITE DIFFERENCES. 209 This theorem, like the preceding ones (16), (17), &c., re- quires x not to transgress the limits x = 1, x = a 1. In the case supposed (viz. when p and ^ are each equal to ^), (22) becomes X *4 ,._ t . 7 7T . TiD f T U XK = - 2,* COt -~ S1H 7T COS - 7T ** a * a 2 a (31). But as the party cannot be won in less than x games, U XQ = u x* while is less than x, and therefore cot - sin Tr/cos-Trj ......... (32), of which (30) is a particular case, If, instead of seeking the probability that at the (z 4- 1)* game N would lose the party, by losing the last of his x counters, we had sought that of his having at the termination of this game any assigned number of counters Jc, the following method might have been made use o Let y^ be the probability in question. It is clear that it will satisfy, as before, equations (1) and (2). But instead of (3), we shall in this case have o = 0, unless x = Jc 1, andy fc Lo = -?)} (3). Equation (11) therefore, which depends merely on (1) and (2), will still obtain; but instead of the system of equations (13), we shall have the following: sin a = &c. = &c. == Cj sin = &c. = GI sin &c. = &c. sin - TT 7T+...4-OL : 7T+...+ sn sin TT -f - 4- w-- sin . (13'). 14 210 ON THE SOLUTION OF EQUATIONS From whence, by the same system of factors as before, we deduce the general formula sin TT cos -TT (14') ; a a for the factors corresponding to the two equations whose first (k 1) r inembers are different from zero, are sin TT and Cb Bin- ^-TT, and the sum of these is 2 sin TTCOS-TT. a a a Consequently the expression of the probability sought will be (accenting the y for distinctness), x-Tc S^ sin - Train -IT cos -9T) ...(150- a a \ a J It is an obvious consequence of the discontinuity of the limiting conditions of the problem, that this expression does not reduce itself to (15) when k is taken equal to zero. For the same reason it is not applicable when k is equal to unity : and on the other hand, it is not to be greater than a 2. It is unnecessary to trace the different corollaries deducible from thfe last written equation, as it has been introduced merely to illustrate the facility with which our method discusses any proposed modification of the question of the duration of play. One point, which is perhaps worth notice, is the symmetri- cal manner in which x and jp, k and j, enter into (15') : the result, however, which is the interpretation of this symmetry may probably be obtained by general considerations. A more general question would arise from supposing it possible for M to win or lose at each game any number of counters not greater than a. The method we have been illustrating would apply to this question, but the solution of it involves that of an algebraical equation of a degree superior to the second. Another part of the subject, namely, the numerical calculation of the expressions already obtained, would not be consistent with the design of this paper. When z is sufficiently large, all the summations with respect to r may be reduced to their first and OF FINITE DIFFERENCES. 211 last terms, unless a is extremely large, in wMcli case other methods of approximating (those, namely, of Laplace), may "be made use of. Enough has probably been said to show the facility which the method I have proposed is capable of giving to questions of acknowledged difficulty. I am not aware that it has been before pointed out ; but as I am not at present able to refer to any work on the subject, I cannot speak confidently on this point. [x } a are integral throughout.] 142 GENERAL THEOREMS ON MULTIPLE INTEGRALS*. IN Liouville's theorem for the reduction of a certain class of definite multiple integrals, the integrations comprise all positive values of the variables which do not transgress a limiting inequality, which either is of, or may easily be re- duced to, a linear form. Take for illustration the case of two variables, and let mx + ny<.h be the limiting inequality in question, m, n and h being positive. Then, geometrically, mx + ny = h is the equation of a straight line which forms the base of a triangle of which the intercepts of the positive half axes of co-ordinates are the sides, and our integration extends over the whole surface of this triangle. A similar interpreta- tion may of course be given in the case of three variables. But to return to that of two. Let mx +- ny h" cut the axis of a? in the point M and that of y in the point N: conceive another straight line m'x -h n'y = h r ; m', n ', h' being also all positive ; and let it cut the axes in M', N' respectively. Let us suppose for distinctness that - , is greater than , . Then, if the value of y/ be intermediate between those of the two ft, fractions 7 and -7, it will be easily seen that the two lines m n must intersect in some point A, lying in the positive quad- rant of co-ordinates, and that we shall have a quadrilateral OMAN'y (0 being the origin of co-ordinates,) formed by the axes and by the two bounding lines. If now we integrate any function of x and y for all positive values of the variables * Cambridge and Dublin Mathematical Journal f Vol. I. p. r, 1846. GENERAL THEOREMS, &c. 213 not transgressing the two inequalities mx + ny(mx+...,, A y being all positive ; and < and <, any func- tions whose values may be represented within the limits of inte- gration by Fourier's theorem. Let the value of the integral in question be J; then, by con- siderations analogous to those of which I made use In a paper which appeared at the commencement of the last volume of the Journal*, we shall have i /** [ h < r 00 f /= / udu I (frjUjdUjl da I da f .Gr, where 6r =/ dx ... I ^a J o Jo and the lower limits of integration with respect to u and u may be any negative quantities. I remark in the first place, that /< r r r I da. I da t Gr = J I da I da { H, where J ^ ^ -oo " oo * Page 150 of this volume. 214 GENERAL THEOREMS r 09 r* Jo *H and therefore j- / iwdfo 6u.du I da I da t H. ITT J J J^ */_ Let 5"= JST cos (aw + a,w,) + sin (aw + a,tO . Then it wiH easily "be seen that K = N L = W f where, if we take the case of three variables, , / am -f am. an + a,n t am + apn ap + a i p l JV = abc 1 7 \ a o a c an + a,n l ap + a,p\ ' I o a b ap + a,p, am H- a n t ap + ajT C Cld b c D = {a 2 -h (am + a^) 2 } (S 2 + (Precisely the same law of formation of these quantities would obtain if we were to take any number of variables. I have taken the case of three merely for distinctness of repre- sentation.) Putting for cos (au + afr) and sin (au+a^) their expo- nential values, we find that ED = a ... Jl - V(- I and as ON MULTIPLE INTEGRALS. 215 {a + V(- Now, assume that _ (& 4 am 4- a v m,) ... _ ^L __ , (&+ am-j-a,mj (c4-j? + a,^) ...... (2); where F& 9 F ac , &c. are independent of a and a,. This assump- tion is justifiable because it introduces ' disposable quan- tities F } viz. as many as there are combinations two and two of the r quantities a, 5 ... c, and it will be easily seen that there are the same number of conditions to be satisfied. Consequently as 6 (M+V)V(-i) = C os (ccw+ aV) 4- V(- 1) sin (aw 4-aV), we shall have - S= Fa { a b ~~ (& m + a / m ;) ( an + a ^/)} COS ( au + a A) + {a (an 4- a,ra y ) 4- 5 (am 4- #,m y )} sin (an + a^) divided by {a 2 4- (am + aW) 2 } [V 4- (a 4 a'^') 2 } Let us next assume u = mx 4- ny, u t mx + n t y, x and y being here two new variables ; also a' = am 4- a,m /5 and /3' = a^ 4- a^ : then the coefficient of F^ in the expression of ^wiH become (ab - a f /3 f }^(ax + ffy} + (a^ + 5aQ sin ( g f a? + ffy) Moreover dudu^dad^ will be replaced by dxdyda!d$ ; and therefore, as we have I rh rn r +a> r +QO j = / ^J^l faujHuA da da,!!, we shall have J= j-i *2tF& (mx 4- wy) ^ (m 216 GENERAL THEOREMS where the sign of summation extends to all the quantities F, and where *-' cos aa*+a+>a. sn From the known integrals coBqg.^g r~aw.cas.fa _ -* 22 ~ ' the upper signs to Ibe taken when x is positive, it follows that If x and y are both positive, the bracket becomes 1 + 1 + 1 + 1 or 4 ; if a? only be negative, it becomes 1 1 1 + 1 or 0; if y only be negative, it becomes 1 1 + I 1 or ; and similarly if both x and y are negative. Thus generally *- or 0. There are, indeed, exceptional cases ; as if y be zero, a? being positive, when M = 2-7r 2 e~ aa; , and similarly if x be zero, y being positive ; and again, if x and y are both zero, when M = 9r 2 : but of these, as we are about to multiply M by the element dxdy, it is unnecessary to take account. Therefore, in inte- grating for x and y y we include only positive values of the variables ; and as u and u are not to be greater than h and Ji t respectively, x and y must be such as not to transgress the inequalities mx + ny^k, mx + n ( y < Ti r Thus we find that n t y] e - a *~*y, the limits being given by the two above-written inequalities. It appears, therefore, that the integral (1), when there are two limiting inequalities, is reducible to the sum of a series of double Integrals. This result is analogous to that which is obtained in the case of the function < (mx+ ,..jps) $-*<*, in the paper already re- ferred to. It remains to determine the form of the quantity F^. This is done at once by multiplying equation (2) by (a + am + a y m,) (5 + cm + a^,) , ON MULTIPLE INTEGRALS. 217 *;nd replacing a, a, by values which, make both these factors vanish. It hence appears that _ {c (mn l m t n) + a (njp, %jp) + b the denominator being the continued product of r 2 factors, each of the same form as the one written down. Of course the other quantities F are obtained in the same manner. Let us now take the more general case in which there are s limiting inequalities, s being less than r, and in which the function to be integrated is the inequalities in question being We shall arrive at a perfectly analogous result in this more general case. In the first place the integral sought may be thus written, 1 /*! [ ha f 03 f 00 faUidUi ... $& s du s da^... da 8 Gr, where 7T J J J J 00 / CX3 = I dx . . . I dz e"" aa! -- a cosa 1 (m 1 fl?+. . .pjw$ . . .cosa s (?^^+. . .p^-u J o J o Now a little consideration will convince us that where -00 .00 ^ *. + |.+ W rfoL... da a & = - d^... da,H, J Jo ^ J~ j -< i. co fe... I dfe^-^-'-^cos "0 for if we take the expression cos [a? 2am 4 ... + Sap Saw], make a x negative, add the resulting expression to the original one : then in the two terms thus got make a 2 negative, and as before add the results, we shall, continuing this process, get in all 2* terms, which will be found to be equal to 2* times the continued product of the cosines involved in 6% 218 GENERAL THEOREMS Effecting the integrations indicated in H, we see that NCOS Sat + N* sin where D = {a 2 + (Sam) 2 } . . . {^ + (Sap) 2 }, and N and JV' follow the same law of formation as in the particular case already considered, except that for - - , &c. a ^ we substitute ~ , &c. With this remark we perceive that H = [a + //(- 1) 2am} . . . {c + V(- 1) Sa The assumption now to be made is that (a + Sam) . . . (c 4- Sap) A ^ '' where A is the product of every set of s factors taken out of the whole number of r factors a 4- Sam, ...c and F is independent of a x . . . a 8 . rra. -n^.1. T. T .T l...r S-fl... _ T I here will thus be - - - disposable quantities Fj which will be found to be the number required to make identically true. Consequently we shall have v cos Sa-^ 4- v sin Saw where S is the product of s factors of the form a 2 + (Sam) 2 ; and v and i/' are formed just as in the case of 5 = 2 : that is to say, we shall have where G t is the sum of the products of every combination ~A that can be made of the a quantities ^^ , &c., taken t and a t together. ON MULTIPLE INTEGRALS. 219 In order to simplify the expression v cos Saw -f- v sin S&w _ ' let us denote the 5 quantities Saw, &c.. which are involved in it, "by the single symbols /^ ... /3 S , and assume U^ = S'/^, W 2 = 2?ft 2 CC, . . . U s = Sm^, the sign of summation S extending only to that set of s out of the r quantities x ... z, which corresponds to the factors in- volved in the denominator S. Of course a?, y, &c. are here, as before, new variables. (In the case of s = 3, for instance, these assumptions will be of the form It follows from this that ^ ... du s da 1L ...d& 8 will be re- placed by db ... dy . d/3^ ... {Zft, and that the factors in S will take the simpler form a 2 + /S/, 5 2 4-/3 2 2 3 &c. ; while Saw will I&Q- The integrations with respect to /3 extend, like those for a, from cc to + oc. Let r+ J J - / cos Then, from the obvious analogy between the forms v and v, and those of the developments of cos 2rf3x and sin S/&c respectively, it follows that if x, y, &c. are all positive, W^ife-**-**- (1 + 1 + ) there being twice as many units within the brackets as there are terms in the development of sin (f + . . . f a ) , or of cos (/ x 4- . . . f s ) , that is to say, twice 2 8 "" 1 or 2 s . Moreover, if any one, as a?, of the quantities x, y, &c., is negative, M ' = ; and this, whether it alone is negative or any others, are so too. For if x^ x', let its coefficient ^ be as- sumed equal to j8/, when the expression of M ' becomes of the same form as if x were positive, except that v and v are changed 220 GENERAL THEOREMS "by having /3/ wherever ^ occurred previously. Now none of the quantities /3 can occur raised to any power, and therefore every term involving ft t will change sign when /3 is replaced by /3/. Hence we shall have M^TT'e-^'-ty' (1 1 ...)> there being as many negative units as positive within the brackets, since in the development of sin (f^ + . . . f 8 ] or cos (j^ + ... f 8 ] there are 2*^ terms independent of the sine of f^ and 2 s "" 2 terms which involve that quantity, and which therefore change sign when j does so. Hence the quantity within the bracket, and conse- quently M t , is equal to zero if x be negative ; and so, of course, for the other variables y . . . z. M vill, in particular cases analogous to those already noticed, assume exceptional or limiting values, but of these we need not take account. And thus we arrive at the following remarkable theorem : The definite integral ofx variables x ... z / O dx ... / O dz^ (m x x -f ... p 4 z) ... $ 8 (m B x+ . whose limits are given l>y s inequalities m^+.-.p^^hj, ...m s :s:-f ...p s z5h can generally le expressed as a linear function of 1.2...S integrals of s variables each. The form of each of these integrals may "be deduced from the original integral ty omitting from it any set of r s of the variables, and similarly the form of the limiting inequalities may ~be got by omitting the same set of vari- ables from the original inequalities (r > s). In certain cases, however, when the constants a, m, &c. have particular values, the theorem fails because the assumption (2') becomes illegitimate. This failure is indicated by certain of the quantities F becoming infinite. To determine the form of F> we have merely to multiply (2') by A 3 and then to equate to zero all the s factors of which A is composed. All the quantities F, except the particular one under consideration, will then dis- appear, and we have s equations determining the s quantities a, ON MULTIPLE INTEGRALS. 221 Hence it will appear that F is equal to a fraction whose nume- rator is unity, and denominator equal to the value assumed by the product of the remaining r s factors, when the values already assigned for the quantities a, &c. have been substituted for them ; a result which it is obvious can be immediately ex- pressed in the notation of determinants. F will therefore become infinite if our equating the s factors by which it is divided in (2') to zero will make one or more of the remaining r s factors vanish. Let it make t of these factors vanish; then equating these t factors also to zero, we get in all s -f t equations, which are equivalent to s independent ones. Therefore any set of s out of these s + t equations will satisfy the remaining t eqiiations. Hence 1.2 ...t of the quantities j^will become infinite, and therefore the second side of (2') will consist of finite terms and of a finite quantity expressed in the form of the sum of that number of infinite terms. This indetermlnation of course indicates a change in the form of the function, the general character of which the reader will have little difficulty in perceiving. But the consideration of these particular cases, some of which are interesting, must be de- ferred to another occasion. I am inclined to believe that the process developed in this paper will admit both of simplification and extension. For the exponential function we may substitute with certain modifi- cations any function of ax + .,.cz, in accordance with a result given by Mr Boole in his very interesting Memoir on a new Method in Analysis, which is published in the Transactions of the Royal Society. (This result would include the one which I obtained in the last volume of the Journal, from which however it might be deduced.) Thus, if in the theorem established in this paper we replace #, J, ... c by lea, Jcb ... fo, 7c being a wholly arbitrary quantity, we may, comparing the coefficients of its powers, deduce new theorems from the given one. Developing the first side of the equation, the coefficient of k n will be ^ 222 GENERAL THEOREMS, dc. and in the second it will be the sum of a series of terms of the form rrr as it is manifest that Fwill become ^ . Hence, if K ty (ax + ... cz) be such a fanction that its development may be substituted for it in the integrations, we shall have k (m l x+...p 1 z) ... m x n a, cos az cos ##. J 00 J -00 Or fm^ = t^M whence ty m z = %^ m , % being independent of m, and therefore, by Fourier's theorem, 1 f * $ m o? = - I %^ m cos xz dz, TTJ o The required solution. * Cambridge and Dublin MaAhematiml Journal, Yol. TO. p. 103, ON THE AREA OF THE CYCLOID*. To the Editor of the Cambridge and Dublin Mathematical Journal. SIR, The determination of tlie area of the Cycloid, so easily effected by modem analysis, was regarded Tby the geo- metricians of the seventeenth century as a problem of no small difficulty. Mersenne was the first who attempted a solution: lie was however unsuccessful. It was proposed by him in despair to Eoberval in 162S, who also failed in his attempt at that time. About seven years afterwards, however, Eoberval overcame the difficulty, and communicated his good fortune to Mersenne. In a letter to Descartes, Mersenne made mention of Ro- bervaVs discovery of the area of the Cycloid as a great feat in geometry, simply stating the result obtained by Eoberval, without giving any clue to the method. Descartes, solving the problem himself with little difficulty, communicated his method in reply to Mersenne, with some supercilious remarks about the supposed difficulty of the problem. Fermat and other mathematicians of that day exercised their ingenuity in the same question. A solution of the problem by pure geometry, which was some time ago communicated to me by Mr E. L. Ellis of Trinity College, possesses so great a superiority over any of the geometrical methods of these early mathematicians which I have seen, that I think it may be acceptable to those readers of your Journal who take an interest in the history of mathematics. " The motion of the generating circle may be resolved into two uniform motions, a motion of translation parallel to the * Cambridge and Dublin Mathematical Journal, Vol. ix. p. 263, 1854. ON THE AREA OF THE CYCLOID. 225 directrix and of rotation round Its own centre. The area gene- rated "by the describing point may be considered as generated by these two motions : that of translation nowise affects the motion of rotation, and the area due to the latter is the same as if the former did not exist, that is, it is equal to the area of the generating circle. Contrariwise the motion of rotation does affect the area due to that of translation, inasmuch as in virtue of it the distance of the describing point from the di- rectrix is varied : the mean distance, viewed as depending on the motion of rotation, is e<|ual to the radius of the generating circle, and the corresponding area is therefore a rectangle, the base of which is the space slided over and altitude that radius ; and, as this space is the circumference of the generating circle, the area in question is equal to twice the area of that circle : on the whole, therefore, the area of the cycloid is equal to three times that of the generating circle. " The reason is just the same as that by which what are called Guldinus's properties are established. We here resolve the motion of a describing point into motions parallel and perpendicular to the abscissa ; the latter generates no area, the former generates a rectangular area having for its base the abscissa and for its altitude the mean value of the ordinates; that is, the ordinate of the centre of gravity of the arc, which is a known result. The only difference to be attended to in the two cases relates to the mode in which the average is to be taken." Mr Ellis has remarked, that the same method may be ex- tended to the determination of the areas of the hypocycloid and epicycloid. I am, Sir, Tour obedient Servant, WILLIAM WALTON. Cambridge^ July 3,1, 1854. 15 SUE LES INTEGKALES AUX DIFFERENCES FINIES*. Ox pent evaluer Vint^grale (1) /^/rfy.../efe^(o?,y, ...,), dans laquelle les variables x 3 y, . . . , doivent prendre toutes les valeurs positives qui satisfont & 1'inegalite (2) ^ (a?, y, ., en remplacjant dans la formule (1) la fonction < par une fonction discontinue, qui devient egale & z&o pour toutes les valeurs des variables non comprises dans la formule (2). On pent alors etendre les integrations depuis zero jusqu'a, 1'infini, ce qui sim- plifie beaucoup les calculs. Je crois que c'est & M. Lejeune-Dirichlet qu'est due 1'idfe de cette mani&re d' ^valuer les integrales multiples ; c'est ainsi qu'il a obtenu, il y a quelques ann^es, une generalisation trfes- remarquable d'un th^orfeme dtl ^, Euler. La th^orie des integrales definies nous fourtiit plusieurs moyens d'exprimer les fonctions discontinues ; je me suis servi, pour cet objet, du tn^orfeme de Fourier. Au moyen de ce tho- r^me, j'ai determine, dans un petit M^moire insure dans le Journal de Math&matigues de Cambridge f, les valeurs de deux integrales multiples. La premifere de ces integrales revient la generalisation qu'a donnee M. Liouville du resultat de M. Dirichlet ; mais je crois que la seconde est nouvelle. * Extraat du Journal de Matkmatiques pure$ et appliquges, Tome IX. 1844. t Page 150 of this volume. SUR LES INTEGRALES, ETC. 227 La facility avec laquelle j'avais obtenu ces resultats me fit penser qu'on pourrait peut-tre appliquer une m^thode sembla- ble aux differences finies ; les resultats auxquels je suis parvenu par eette consideration font le sujet de ce qui va suivre. En suivant 1'analogie qni existe entre les differences finies et les differences infiniment petites, on voit qu'& 1'integration multiple, il faut substituer des sommations par rapport a toutes les variables qui entrent dans la fonction donne"e. Soit $ (a?, y) cette fonction* Je designe par 2 6 Sj? (x, y] la quantity suivante (b a et d c etant des nombres entiers et positifs), ^> (a, c] + (a + 1, c) + ... < (&, c) + <(a, c +1) + .............. 0(5, c + 1) 5, d}. H est visible que cette notation pourrait s'e*tendre a un nombre quelconque de variables. Le th^or^me de Fourier se remplacera par la formula suivante, dans laquelle 5 x et x a sont des nombres entiers et positifs, 1 C v (3) fx = I da ^ a fu cos a (a; ze). 7rJ On peut done poser x = a, =&-j- 1, -.. = 5; mais si 1'on donne a x (qui doit toujouxs 6tre nn nombre entier) une valeur quelconque non comprise dans ces limites, on aura rda S fu cos a (x u] = 0. .. La demonstration de ce th^orfeme est si facile, qu'il n'est pas ncessaire de s'y arr^ter ; je ferai seulement observer en passant qu'elle suppose que la fonction fit, ne devienrie mfiriie pour aucune des valeurs D&ignons par {x}* la fonction ^ . ^ : toutes les fois 1 (X) que p est un nombre entier et positif, nous aurons 1 ... x+jp 1- 152 228 $UM LES INTEGEALES Cela pos ? entrons en matifere. Je vais chercher la valeur de la somme multiple (4) S 2 ... 2 {r {y} 2 - 1 ... {ar/ dans laquelle 1'^tendue de la semination est donnee par I'infi- galitd (5) # + #+... -f z^h. D'apr&s l'ide fondamentale de notre analyse, je remplace dans la formule (4) la fonction f(x +y + ... + #) par 1 f 7 " I da %lfu cos a (x -f y -f . .. 4- # w). Done nous aurons, en changeant Tordre des sommations, (6) -2J> f^aS S ...2j^r i fy} 2 -\..{0r i cosa(^+y+.^ " J (i est un nombre negatif quelconq[ue). Les sommations par rapport S, x, y, etc., peuvent ^ present s'^tendre jusqu'it Tinfini. Nous allons determiner les valeurs de S* {x}^ cos ax, et de 2^ {o;f -1 sin ao?. Soit # = e*^ 1 ^; nous aurons, par un thforfeme connu, 2 2 a-* cos aa? = -i + L. , puisqu'on a + e tc.=-JL. et i + + etc. = a ^ a Pareillement on a ~ et de 15, AUX DIFFERENCES FINIES. 229 En dcrivant dans cette quation "* an lieu de z, elle deviendra TW^ S " 1 + etc - ==r ^(^?- Ajoutant cette Equation a la derni&re, nous aurons, a cause de (8) (On doit remarquer que {0}* = 0, puisque T (0) a une Yaleur infinie.) Ensuite, , cause de (a z) (a z~ l ) ~l2a cos a + c& 2 , nous aurons Mais, puisque = cos a 4- V 1 sin a, est feale & - ~T * Done, r (l-Sac 2 nous aurons finalement os > - r () (1 - 2a cos a + a 8 ) ,. 8 * 230 SUR LES INTE&RALES On trourera de la mme mani&re que (10) A present faisons a = 1. Alors nous aurons 1 cosa _ ro ^ rt _ r 2 J A ^ a et les Equations (9) et (10) deyiendront / \ \y + . . . + cz ^ A, sans an moins lui donner une forme beaucoup plus compliqu^e. A present, designons, suivant la notation usit^e, par [a?] p la fonction r . ^ - ^-r (nous aurons, quand p sera un nombre entier, [afp =i x . x 1 ... a; p + 1), et tachons d' ^valuer la somme suivante, dans laquelle cc peut prendxe toutes les valeurs _p 1, jp, j?+l, etc., tandis que ^ peut prendre toutes les valeurs # 1, ^ ? ^ + 1? e tc. 3 et ainsi de suite pour les autres variables. L'etendue des som- mations est determine par 1'indgalit^ dans laquelle Ti est ^gale a p + g + . . . r + un nombre entier. Nous allons premiferement trourer les valeurs de S^j. [oO*" 1 cos aa?, et de Si^Li M^" 1 s i n <^^ Puisque nous avons il s'ensuit que AUX DIFFERENCES FINIES. 233 Bempla9ons z par z~ l , nous aurons, en ajoutant les deux rfeultats et en posant a = 1, [p I]*"" 1 cos a (p - 1) + [p] p ~ l cos op + eta 2 c'est-i-dire nous aurons et pareillement II est facile de voir, en suivant S, peu pres la m6me route qu'auparaTant, que ces deux Equations reviennent & celles-ci : cos Tr-a- (18) S p -. x [ aJ - * f P r \ } sin -Hj- (TT a) av (19) S;., [scr 1 sin as = T Q,) - \j - - I . (2 sin-) Cela pose, on peut facilement s'assurer que la somme dont nous cherclions la valeur est gale a Par consequent elle est gale, en Tertu des formules (18) et (19), 4 et de la nous aurons finalement (S M ^...S 1Hl [ (20) _r(p)r( g )...r 234 $UIt LES INTEGRALES Les deux resultats (17) et (20) suffisent pour montrer 1'esprit de notre analyse, rnais je vais encore 1'appliquer a un autre exemple. t Eyaluons 1'expression (21) 2 S ... ^ Q 1 ^ ne u 1 2a cos a + a 2 XL renferme pas a, est gal a 7 - 77 - ^ T, - 2W1 n - N (a- 8)... (a c) (l~a)(l aJ)...(l ac) ou a puisque 1 a $a + cf Sab etc. __ (1-a 2 ) (1-oS) ... (l-oo)*" * Nous voyons done que, pour toutes les valeurs positives de u et pour u = 0, fH' (24) _^ I ~ (a-i) ... (a -c) "" (S^a) "... (6-c) Si w est n^gatif, faisons u = u, nous aurons N = cos OLU Sa . cos a (u 1) + etc. a&...ccos a (u' v), 238 SUR LES INTEGRALES . 1 x J - * et le terme de - __ q U1 n e renfermera pas a. sera 1 2a cos a -f o gal a "(g-i)... (g-c) (1-a 2 ) (l-gj)...(l-ac) * En supposant que %' ne soit pas moindre que z^, il est visible * such that its difference from a given arc shall be rectifiable. Of this problem he gave a solution in the twentieth volume of the same journal. The principle of the solution consists in the transformation of a certain differential expression by means of an algebraical and rational assumption which introduces a new variable. The transformed expression is of the same form as the original one, but is affected with a negative sign. By integrating both we are enabled to compare two integrals, neither of which can be assigned in a finite form. It is difficult, however, to perceive how Fagnani was led to make the assumption in question; a remark which applies more or less to his subsequent researches on similar subjects. The theorem which has made his name familiar to all mathematicians, appeared in the twenty-sixth volume of the 1 Giornale.' In its application to the comparison of hyperbolic arcs we find some indications of a more general method* We have here a symmetrical relation between two variables, x and * Those o M. Gauss, which, would doubtless have Been exceedingly valuable, have not, I believe, been published. They are mentioned in a letter from M. Crelle to Abel. Tide the introduction to the collected works of the latter, p. vii. 240 ON THE REGENT PROGRESS OF ANALYSIS. z, such that the differential expression f(x) dx may be written in the form z dx. It follows at once that /() dz = xdz, and consequently that //(a?) das+jfte) dz = {{xdz + zdx} -= The remarkable manner in which the idea of symmetry here presents itself, suggested to Mr Fox Talbot his ' Researches in the Integral Calculus. 3 In applying his methods to the division of the arc of the lemniscate, Fagnani obtained some very curious results, and has accordingly taken for the vignette of his collected works a figure of this curve with the singular motto, * Deo veritatis gloria.' 3. In MacLaurin's Fluxions, and in the writings of D'Alembert, instances are to be found where the solution of a problem is made to depend on the rectification of elliptic arcs, or, as we should now express it, is reduced to elliptic integrals. But of these instances Legendre has remarked that they are isolated results, and form no connected theory. MacLaurin is charged, in a letter appended to the works of Fagnani, with taking from the latter without acknowledgement, a portion of his discoveries with respect to the lemniscate and the elastic curve. 4. In 1761, Euler, in the c Novi Comrnentarii Petropolitani ' for 1758 and 1759, published his memorable discovery of the algebraical integral of the equation mdx _ ndy ~~ m and n being any rational numbers. He says he had been led to this result by no regular method, t c sed id potius tentando, vel divinando elicui,' and recommends the discovery of a direct method to the attention of analysts. In effect his investigations resemble those of Fagnani : he begins by assuming a symmetrical algebraical relation between the variables, and hence finds a differential equation which it satis- fies. In this differential equation the variables are separated, so that each term may be considered as the differential of some ON THE REGENT PROGRESS OF ANALYSIS. 241 function. With one form of assumed relation we are led to tlie differentials of circular, and with another to those of elliptic integrals, and so on. It is in this manner that Dr Gudermann, in the elaborate researches which he has published in Crelle's Journal, has commenced the discussion of the theory of elliptic functions. 5. In the fourth volume of the Turin Memoirs, Lagrange accomplished the solution of the problem suggested by Euler. He integrated the general differential equation already mentioned by a most ingenious method, which, with certain modifications, has remained ever since an essential element of the theory of elliptic functions. He proceeded to consider the more general equation dx ___ dy where X and Y are any similar functions of x and y respectively, and came to the conclusion, that if they are rational and integral functions, the equation cannot, except in particular cases, be integrated, if they contain higher powers than the fourth. He also integrated this equation in a case in which X and Y involve circular functions of the variables. It had been already pointed out in the summary of Euler' s researches, given in the Nov. Com. Pet. Tom. vi., that if JTand Y are polynomials of the sixth degree, the last-written equation does not in general admit of an algebraical integral, since, if so, it would follow that the solu- tion of the equation ^ = - ?/ 9 , which (as the square of 1 -f- 3Cr 1 -f" y 1 + x 3 is a polynomial of the sixth degree) is a particular case of that which we are considering, could be reduced to an alge- braical form. Now this solution involves both circular functions and logarithms, and therefore the required reduction is impossible. This acute remark* showed that Euler's result did not admit of generalisation in the manner in which it was natural to attempt to generalise it. It was reserved for Abel to discover the direction in which generalisation is possible. 6. The discovery of Euler, of which we have been speak- ing, is in effect the foundation of the theory of elliptic fane- * M. Blchelot, in one of his memoirs on Ahelian or hyper-elliptic integrals, quotes it, in a slightly modified form, from Euler's Opmcula. 16 242 ON THE RECENT PROGRESS OF ANALYSIS. tions, as the generalisation of it by Abel, or more properly speaking, the theory of which Euler's result is an isolated frag- ment, is the foundation of our knowledge of the higher trans- cendents. We may therefore conveniently divide the subject of this report into two portions, viz. the general theory of the comparison of algebraical integrals, and the investigations which are founded on it. Mathematicians have been led, by comparing different transcendents, to introduce new functions into analysis, and the theory of these functions has become an important subject of research. The second portion may again be divided into two, viz. the theory of elliptic functions, and that of the higher trans- cendents. This classification, though not perhaps unexceptional, will, I think, be found convenient. 7. About sixteen years after the publication of Lagrange's earlier researches on the comparison of algebraical integrals, he gave, in the New Turin Memoirs for 1784 and 1785, a method of approximating to the value of any integral of the form j__ ? w here P is a rational function of x and R the square root of a polynomial of the fourth degree. I shall consider this important contribution to the theory of elliptic functions in con- nexion with the writings of Legendre. At present, in order to give a connected view of the first division of my subject, it will be necessary to go on at once to the works of Abel, and to those of subsequent writers. In the history of any branch of science the chronological order must be subordinate to that which is founded on the natural connexion of different parts of the subject. I shall merely mention in passing, that in 1775, Landen published in the Philosophical Transactions a very remarkable theorem with respect to the arcs of a hyperbola. He showed that any arc of a hyperbola is equal to the difference of two elliptic arcs together with an algebraical quantity. In 1780 he published his researches on this subject in the first volume of his Mathematical Memoirs, p. 23. This theorem, as Legendre has remarked, might have led him to more important results. It contains the germ of the general theory of transformation, the ON THE RECENT PROGRESS OF ANALYSIS. 243 eccentricities of the two ellipses being connected by the modular equation of transformations of the second order*. It is on this account that in a report on M. Jacobi's Fundamenta Nova, contained in the tenth volume of the Memoirs of the Institute, Poisson speaks of Landen's theorem as the first step made in the comparison of dissimilar elliptic integrals. Several writers have accordingly given Landen's name to the transformation commonly known as Lagrange's. 8. We have seen that even Lagrange failed in obtaining a result more general than that which had been made known by Euler, and yet, as we now know, Euler* s theorem is but a particular case of a far more general proposition. But in order to further progress, it was necessary to introduce a wholly new idea. The resources of the integral calculus were apparently exhausted; Abel, however, was enabled to pass on into new .fields of research, by bringing it into intimate connexion with another branch of analysis, namely, the theory of equations. The manner in which this was done shews that he was not unworthy to follow in the path of Euler and of Lagrange. I shall attempt to state in a few words the fundamental idea of Abel's method. Let us suppose that the variable x is a root of the algebraical equation fx=Q, and that the coefficients of this equation are rational functions of certain quantities , 5, . . . c, which we shall henceforth consider independent variables. Let us suppose also that in virtue of this equation we can express certain irrational functions f of x as rational functions of x } a, ft, ... c. For in- stance, if the equation were a? + ax + - (a 2 1) = 0, it follows A that Vl 3? = a -ha;. So that any irrational function of the form F (x Vl a?} can be expressed rationally (F being rational) in x and a. * Tide infra, pp. 263 and 289. t It must be remembered that an algebraical function is either explicit or implicit : explicit, when it can be expressed by a combination of ordinary algebrai- cal symbols ; implicit, when we can only define it by saying that it is a root of an algebraical equation whose co-efficients are integral functions of x. Thus y is an implicit function of x if y^+yy + i =o. The remarks in the text apply to all alge- braical functions, explicit or implicit. 162 244' CUV THE RECENT PROGRESS OF ANALYSIS. From the given equation we deduce Tby differentiation the following. where cc, /3, ... 7 are rational in x, a, 5, ..., c. Let y he one of the functions which- can he expressed ration- ally in a?, &c. 5 it follows that ydx = Ada + Bdl + ...+ Cdc> where A, B^ ... G are also rational in x, &c. The equation fx=Q will have a number of roots, which we shall call x l9 x z , ... o? M . It follows that where the indices affixed to y, A, &c. correspond to those affixed to x, so that y^ for instance, is the same function of x tnf-jf" ot m of c Now AI+ - +d-fj, is rational and symmetrical with respect to x^ ... Xp, therefore it can be expressed rationally in the co- efficients of /(aj)=0, and therefore in a, Z>...c. We will call this sum J? a5 and thus with a similar notation for b, &c. we get The second side of this equation is from the nature of the case a complete differential, and it is rational in a, 5, c, &c,; it can therefore be integrated by known methods; and if we denote c i)? we S^ Jf being a logarithmic and algebraic function of &, &, &c., which we may suppose to include the constant of integration. ^ (a?) is in general a transcendental function, while a, 5, &c. are necessarily algebraical functions of a\ 5 ... 5 cs^, and the result at which we have arrived is therefore an exceedingly general formula for the comparison of transcendental functions. The simplicity and generality of these considerations entitle them to especial attention : it cannot be doubted that the ap- plication thus made of the properties of algebraical equations to ON THE REGENT PROGRESS OF ANALYSIS. 245 the comparison of transcendents will always Tbe a remarkable point in the history of pure analysis. A very simple example may perhaps illustrate what has Ibeen said. Let us recur to the equation B + (^-l)=0, .......... ,....(1), and suppose that Differentiating the first of these equations, we find that (%x -f a) dx -f (x -f a) da = 0. Comparing this with the general expression of dx^ we perceive that and as , da qdx = - ^ 2 so that Let a? t and a? 2 "be the two roots of our equation, we have thus to find the value of snce ^ + & 2 = a. Hence y x dS^ + ^ 2 (x) dx + $ (y) dy + ... + () dz = d [xy ... z} 9 and thus give us U (x) dx + U (y) dy + ... + [< (z) dz = asy ...* + 0. Precisely the same remark, though expressed in a different notation, is the foundation of M. Hill's memoir, published in 1834, on what he calls ' functiones iteratse.' It will be found in Crelle's Journal, XI. p. 193. A much more general theorem might be established by similar considerations: they are of course applicable whether the function cf> be algebraical or trans- cendent. In the course of his researches, Mr Talbot recognised the Important principle, that the existence of n 1 symmetrical algebraical relations among n variables may be expressed by treating them as the roots of an equation, one of whose coeffi- cients at least is variable, the others being either constant or functions of the variable one. Unfortunately he did not pass from hence to the more general view, that the existence of n p symmetrical relations may be expressed in a similar manner if we consider p of the coefficients of the equation as arbitrary quantities. Had he done so, it is possible, though not likely, that he would have rediscovered Abel's theorem ; but as it is, lie has never introduced, except once, and then as it were by accident, more than one arbitrary quantity. Thus only one of * It must "be remembered also that Mr Talbot admits Mmself to have been anti- cipated to a considerable extent by the publication of Abel's theorem. ON THE RECENT PROGRESS OF ANALYSIS. 251 his variables is independent, and consequently, in more than one instance, his results are unnecessarily restricted cases of more general theorems. The character of his analysis will be perceived from "what has been said. If \Xdx be the transcendent to be considered, X being an algebraical function of x, he makes the following assumption X-/(o), v being a new variable, and /a rational function. From this assumption he deduces an algebraical equation in x, the co- efficients of which are rational functions of v. This equation then is one of those of which we have spoken, by means of which the function to be integrated can be expressed in a rational form. Taking the sum with respect to the roots of this equation, we get It must be remarked that many forms might be assigned to the function/ which would give rise to a difficulty, of the means of surmounting which Mr Talbot has given no idea. If x and v are mixed up in f(xv), it is manifest that we cannot integrate f(xv) dx, since v is a function of x, which if we eliminate we merely return to our function X. We must therefore express ^f(xv) dx in the form Vdv, V being a function and, as Abel has shown, an integrable function of v. Abel has given for- mulse by means of which this reduction may be effected in all possible cases. But there is nothing analogous to this in the writings of Mr Talbot, and consequently he could not, setting aside the defect already noticed, obtain results as general as many previously known. In Mr Talbot' s investigations,/^) dx is such that %f(xv) dx may be put in the form V {\xdx} + F 2 S {' 2 xdx} + &c., fax, fax, &c. (of which (f>\x, (j>\x, &c. are the derived functions) being rational functions of x. Then ^(j>x = a rational function of v by a well-known theorem. Let the form of this function be ascertained, and let us denote it by %u. Then differentiating, and hence ) dx = [F,x> + F 2 %> + . . .] dv, "252 ON THE REGENT PROGRESS OP ANALYSIS. and tlic second side of this equation is of course rational and intcgrablo. But the form of the f unction /(ow) is unnecessarily restricted in order that this kind of reduction may be possible. Nevertheless, Mr Talbot's papers, from their fulness of illus- tration and the clear manner in which particular cases of the general theory are worked out by independent methods, will be found very useful in facilitating our conceptions of the branch of analysis which forms as it were the link between the theory of equations and the integral calculus. In Mr Talbot's second memoir (Phil. Trans. 1837, part 2. p. 1) he has applied his method to certain geometrical theorems. Three of them relate to the ellipse, and are proved by the three following assumptions : (I eV] 4 1 ex - = 1+ V or == -=- - re , or == -=- , or = - 1 x } vv 1 These assumptions are all cases of the following : I of where a, a 1 , c, c 1 are arbitrary quantities. The results of this assumption are completely worked out by Legendre (ThSorie des Fonctions Elliptiques, ill. p. 192) in showing how the known formulas of elliptic functions may be "derived from Abel's theorem. Mr Talbot's first theorem is a case of the fundamental formula for the comparison of elliptic arcs. This remark has reference to an inquiry which Mr Talbot suggests as to the relation in which his theorems stand to the results obtained by Legendre and others. In conclusion, it may be well to observe that Mr Talbot has remarked that, apparently, a solution discovered by Fagnani of a certain differential equation cannot be deduced from. Abel's theorem; but as this solution may be easily derived from the ordinary formula for the addition of elliptic integrals of the first kind it is manifestly included in the theorem in question. II. 11. I now come to the history of researches into the pro- perties of particular classes of algebraical transcendents. The earliest, and still perhaps the most important class of these ON THE RECENT PROGRESS OF ANALYSIS. researches relates to the transcendents which are commonly called elliptic functions or elliptic integrals. For a reason which will be mentioned hereafter the latter name seems pre ferable, and it is sanctioned by the authority of M. Jacolbi, though the former was used by Legendre. Elliptic integrals then may be denned as those whose differentials are irrational in consequence of involving a radical of the form \f{a + /3x + 7^ 2 + S# 3 4- e 4 }. But it may perhaps be more correct to say that all such integrals may be reduced to three standard integrals, to which the name of elliptic integrals has been given. In the Turin Memoirs for 1784 and 1785, p. 218, Lagrange considered, as has been already mentioned, the theory of these transcendents. He showed that the integration of every func- tion irrational in consequence of containing a square root may P be made to depend on that of a function of the form -= , P being rational, and R the radical in question ; and that If under the sign of the square root x does not rise above the fourth degree, it may ultimately be made to depend on that of Ndx where -ZV^is rational in x\ He thus laid the foundation of that part of the theory of elliptic transcendents in which a proposed integral is reduced to certain canonical or standard forms, or to the simplest combination of such forms of which the case admits. In Legendre' s earliest writings on elliptic functions there is nothing relating to this part of the subject. Having thus, in the simple manner which distinguishes his analysis, reduced the general case to that which admits of the application of his method, Lagrange proceeded to prove that if we Intro- duce a new variable whose ratio to x is the subduplicate of the ratio of 1 j? B a? a to 1 #V, the last written integral is made to depend on another of similar form, but In which p and q are replaced by new quantities p 1 and g 1 . If p is greater than j, p 1 will be greater than p, and d, which as we know re- presents an elliptic arc, and shows how other functions, for instance the value of the hyperbolic arc, may be expressed by means of it, and of its differential coefficient with respect to the eccentricity c. The memoir does not contain much that is now of interest. After writing it he became aware of the existence of Landen's researches; and in a second memoir appended to the first gave a demonstration of Landen's principal theorem. This demonstration is founded on Legendre's own methods, and he deduces from it the remarkable conclusion, that if of a series of ellipses, whose eccentricities are connected by a certain law, we could rectify any two, we could deduce from hence the rectification of all the rest. The law connecting the eccentricities of the ellipses is that which would be obtained "by making use of Lagrange's method of transformation, with which accordingly this result is closely allied. Legendre's next work was an essay on transcendents*, pre- sented to the Academy in 1792 and published separately the year after. It contains the same general view as that which is * A translation of It appeared in Leybourne's Mathematical Repository, Vols, II. and nr. The original I have not seen it has long been scarce. ON THE RECENT PROG-RESS OF ANALYSIS. 255 developed in the first volume of the Exercises de Calcul In- tegral, which appeared in 1811. 12. The theory of elliptic functions, as it is presented to us by Legendre, may conveniently be considered under the following heads: a. The reduction of the general integral, Pdx + ^4- OB*' in which JP is rational to three standard forms, since known as elliptic integrals of the first, second and third kinds 45 ". This classification, though the reduction of the general in- tegral had, as we have seen, been, already considered by La- grange, is I believe entirely due to Legendre. If we consider how much it has facilitated all subsequent researches, we can hardly over-rate the importance of the step thus made. It may almost be said that Legendre, in thus showing us the primary forms with which the theory of elliptic integrals is conversant, created a new province of analysis : he certainly gave unity and a definite form to the whole subject. For the three species of functions thus recognised Legendre suggested the names of nome, epinome and paranome, the name of the first being derived from the idea that it involves, so to speak, the law on which the comparison of elliptic integrals depends. But these names do not seem felicitous, nor have they, I believe, been adopted. To this part of the subject an important theorem relating to the reduction of elliptic integrals of the third kind, whose parameters are imaginary, seems naturally to belong. * These three forms are r -<^V J o s '" Legendre always replaces x by sin 0, so that the integrals become [ $ d$ /> _ Jo Vi~c* S inV Jo A/I - The radical *ji-& sin 3 < is often denoted by A. The constant c is called the modulus; the second constant n (in the third kind) is called the parameter. The modulus may always be supposed less than unity, and if c=sin e, then c is the angle of the modulus, 256 ON THE RECENT PROGRESS OF ANALYSIS. ft. The comparison of elliptic integrals of the same form differing only in the value of the variable, or as it is often called, the amplitude of each. This part of the subject divides 1 tself into three heads, corresponding to the three classes of integrals. The fundamental results are to be found in the memoirs of Euler, of which we have already spoken. By Le- gendre however they were more fully developed. It is interesting to observe that Legendre suggested that the discovery of Euler (namely that the differential equation dx dif admits an algebraical integral, f(x) being the polynomial a -f fix + y# 2 + $x* + ex 4 ) might be generalised, if we consider the differential equation dx dy dz + '" + = * He remarks that this is perhaps the only way in which, it can be generalised. 7. Theorems relating to the comparison of different kinds of elliptic functions. One of the most remarkable of these is the relation between the complete integrals (those, namely, in which the variable x is unity) of the first and second kind, the moduli of which are complementary; that is, the sum of the squares of whose moduli is equal to unity. Legendre's demonstra- tion of it is rather indirect, but many others have been since given. Another theorem may be mentioned, that the complete integral of the third kind can always be expressed by means of the complete integrals of the first and second. A third and most important result shows that in elliptic integrals of the third kind we may distinguish two separate species, and that to one or other of these any such integral may be reduced. A memorable discovery of M. Jacob! has greatly increased the importance of this subdivision, of which we shall hereafter speak more fully. This part of the subject is, I imagine, entirely due to Legendre. 8. The evaluation of elliptic integrals by means of ex- pansions. . The method of successive transformations. The idea of ON THE RECENT PROGRESS OF ANALYSIS. 25? tliis method originated, as we liaye seen, with Lagrange. It is developed at great length by Legendre, with a special reference to the modifications required in applying it to the different species of integrals. As Lagrange had shown, the series of transformed integrals extending indefinitely both ways conducts us, in whichever direction we follow it, towards a transcendent of a lower kind than an elliptic integral, or in other words, towards a logarithmic or circular integral. There are thus two modes of approximation, one of which depends on a series of integrals with increasing moduli, and the other on a series whose moduli decrease. Thus for the three species of integrals there will be in all six approximative processes to be considered, In the case of the elliptic integral of the third kind, we have to determine the law of formation of the successive parameters w, ft 1 , &c. f. Eeductions of transcendents not contained in the general formula (e.g. I ) to elliptic integrals. V J V 1 gf) y. Lastly, applications to various mechanical and geometri- cal problems. This analysis, however slight, will give an idea of the contents of that part of the Exercices de Calcul Integral which relates to elliptic functions. In the third volume there are tables for facilitating the calculation of integrals of the first and second kind: they are accompanied with an explanation of the manner in which they were constructed. The ninth table is one with double entry, the two arguments being the angle of the modulus and the amplitude. 13. In 1825 Legendre presented to the Acad&nie des Sciences the first volume of his TrattS des Fonctwns Ellip- tiques. A great part of this work is precisely the same as the Exercices de Calcul Integral. By far the most important addition to the theory of elliptic functions consists in the dis- covery of a new system of successive transformations quite dis- tinct from that of Lagrange. In the earlier work Legendre had shown that a certain transcendent might be expressed in two ways by means of elliptic integrals of the first kind. Comparing the two results, he obtained a very simple relation between the two elliptic 17 258 ON THE RECENT PROGRESS OF ANALYSIS. integrals. Their moduli are complementary ; while tlie ratio of the A's in the two integrals can be expressed rationally in terms of the sine of the amplitude of one. This circumstance seems to have suggested to Legendre the possibility of generalising the result. He accordingly assumed a relation between the amplitudes of two integrals, of which the equation subsisting in the theorem of which we have been speaking is a particular case ; and showed from hence that a simple relation perfectly similar to that which he had obtained in the particular instance existed between the two integrals, viz. that they bore to each other a ratio independent of their amplitudes. Their moduli are connected by an algebraical equation, but are not comple- mentary. This circumstance therefore now appeared to be un- essential, though in the Exercices the investigation is intro- duced for the sake of exhibiting a case in which an integral may be transformed into another with a complementary mo- dulus. Legendre thus obtained a new kind of transformation, which might be repeated any numlber of times or combined in an in- finite variety of ways with that of Lagrange. To illustrate this he constructed a kind of table a c damier analytique.' In the central cell is placed the original modulus c. All the moduli contained in the same horizontal row are derivable from one another by Lagrange' s scale of moduli ; those in each vertical row Tby the newly discovered scale. He seems to have been very much struck by the infinite variety of transformations of which elliptic integrals admit. The integral of the first kind is especially remarkable, because of the simplicity of the relation which connects it with any of its transformations, viz. that their ratio is independent of the amplitudes. Legendre's second work was, as we have remarked, pre- sented to the Academy in 1825, but it was not published till 1827. In the summer of 1827 M. Jacobi announced in Schu- macher's Astronomischen Nachrichten, No. 123, that he was in possession of a general method of transformation for elliptic integrals of the first kind. He was not acquainted with Le- gendre's discovery of a new scale, and as an illustration of the general theorem gave two cases of it, the first being equiva- lent to Legendre's method of transformation. Thus much was announced in a letter to M. Schumacher, dated June 13th ; ON THE REGENT PROGRESS OF ANALYSIS. 259 but in one of a later date (August 2nd) lie gave a formal enunciation of his theorem, but without demonstration. The two communications appear consecutively (AsL NacJi. vi. p. 33). In No. 127 of the Nachricliten, VI. p. 133, II. Jacobi gave a demonstration of his theorem. If we can so determine y in the terms of x as to satisfy the differential equation dij __ I dx V(T - /) (1 - AY) ~ 3? V(l _ a?) ^ - Xjtf) (J/ being constant) , it is manifest that we shall have (F denoting the elliptic integral of the first kind) F (fax) = 3IF(\y), pro- vided that y and x vanish together. The question therefore is, how may the differential equation be satisfied, for it is clear that by means of a solution of it we transform the elliptic in- tegral F(kx) into another, viz. into F(\y). M. Jacobi shows that if y be equal to -p., Hand F being integral functions of x, the differential equation will be satis Sed, provided U and F fulfil two general conditions, the second of which is found to be declucible from the first. He then makes an assumption which is equivalent to assigning particular form.- to U and F, and thence shows, by a most ingenious method that these forms of 27 and F are such as to fulfil the first of tin required conditions, which, as has been said, implies the other. He thus verifies, ^ posteriori^ the assumed value of the func- tion y. In proving that the forms assigned for Z7and Fhave the required property, it is necessary to pass from an expression of the value of 1 y in terms of x to one of 1 Aj in terms of the same quantity. This is done by means of a remarkable property of the functions U and F, namely, that if in both x be replaced by j-~ , -^ or y will (the constants being properly KX> F 1 adjusted) become j- 7 or . Therefore, in any form in which the relation connecting y and x can be put, we may replace x by j-~ , provided we at the same time replace y by . K3G *\*y This has been called the principle of double substitution, and 172 260 ON THE RECENT PROGRESS OF ANALYSIS. by means of it we pass from the expression of 1 y to that of 1 y an( l thence obtain that of 1 - \y. It is to be ob- \y served that this principle is used merely to prove a certain pro- perty of the functions U and V. Of course, as the change of x into j implies that of y into x r in the finite relation between K3Q y these qtiantities, the same thing will be true in the differential equation by which they are connected, a remark which may very easily be verified. But, on the other hand, it by no means follows that because it is true in the differential equation there- fore any assumed finite relation between y and x having this property is the integral required. The property in question therefore does not enable us to verify any assumed value of y. This remark has reference to a communication from Le- gendre which appears in No. 130 of Schumacher's Nachrickten, VI. p. 201. In it he gives an account of M. Jacobi's researches, and an outline of the demonstration of which we have been speaking. I find it impossible to avoid the conclusion that this great mathematician mistook the character of the demonstration in question, and th&t to him it appeared to be in effect a mere verification of the assumed value of y by means of the principle of double substitution. He remarks that the direct substitution of the value of y in the differential equation is impracticable, but that M. Jacobi had avoided this substitution by means of * une propri6t6 particuli&re de cette Equation qui doit tre com- mune aux integrates qui la represented.' This property is the principle of doiible substitution; and after showing that it is true of the differential equation, the writer proceeds thus : ' Ce principe une fois pos, rien n'est plus facile que de verifier Tequation trouv^e y = ^, car par la double substitution on obtient la ra&me valeur de y a un coefficient prfes qui doit tre gal & Turiit^ ;' and, after a remark to our present purpose im- material, concludes, c Ainsi se trouve dSmontrfe g&idralement 1'dquation y = -TT ainsi que, etc.' As we have seen, such a verification would be wholly in- conclusive, nor is the essential point of M. Jacobi's reasoning, ON THE REGENT PROGRESS OF ANALYSIS. 261 namely, that the assumed forms of U and F satisfy the general condition, laid down at the outset of Ms demonstration, here adverted to. In 1828, Legendre published the first supplement to the Traite des Fonctions Elliptiqites, &c. It contains an account of the researches of M. Jacobi, and of a memoir by Abel inserted in the third volume of Crelle's Journal. The account here given of M. Jacobi's demonstration is fuller and more ex- plicit than that already noticed. It leaves, I think, no doubt of the error into which Legendre had fallen. Xo notice what- ever is taken of the first part of M. Jacobi's reasoning: and after remarking that the differential equation is satisfied when the double substitution is made, he goes on, fi Tout se reduit done a faire cette double substitution dans 1'int^grale y = - -y et \ examiner si elle est satisfalte.' After showing that it is so, he adds, 'Par ce proce'de tres simple il est constate qtie liqua- tion y = - -~- satisfait ... a Fequation differentielle dont 1'in- tegrale est F (Jcfy = pF (i-^r), etc.' (Trait, des Fonct. Ell. ill. p. 10.) Legendre remarks, that although M. Jacobi's demonstration rests on c un principe incontestable et tres inge'irieux,' it is still desirable to have another verification of so important a theorem. He accordingly gives an original demonstration of it, which is however more nearly allied to M. Jacobi's than to him it seemed to be. This demonstration had already been hinted at in his communication to the NacJirichten. The principal difference is, that while M. Jacobi proved generally that if the first of the two required conditions were satisfied, the second would also be so, and then showed that the forms assigned to U and F satis- fied the first condition ; Legendre shows the assigned forms are such as to satisfy both conditions, on the connection between which it is therefore unnecessary for him to dwell. In the third supplement to the Traite des Fonctions Elliptiques, Le- o-endre has given another demonstration of M. Jacobi's theorem, remarking that it is both more rigorous and more like II. Jacobi's than that which he had first given. I have thought it necessary to make these remarks, because it has been said that it was in the supplements to Legendre's work that the 262 ON THE RECENT PROGRESS OF ANALYSIS. demonstration of this theorem received * le dernier degr de rigueur*.' 14. In 1829 M. Jacobi's great work on elliptic functions, the Fundamenta Nova Theorize Functionum Ellipticarum> was published at Kcenigsberg. It contains his researches not merely on the theory of transformation. Tout also with respect to other parts of the subject. But the great problem of transformation is the fundamental idea of the whole work ; the other parts are subordinate to it, or at least derived from it. The subject is treated with great fulness of illustration and in a manner not unlike that of Euler. SI. Jacobi begins by considering the possibility of trans- forming the general transcendent whose differential coefficient is unity divided by the square root of a polynomial of the fourth degree. Subsequently, having shown that this transcendent may be transformed by introducing a new variable y equal to the quotient of two integral functions of x, and also that the general transcendent may be reduced to one of the form he proceeds to consider the latter in detail. The first step of this reasoning, viz. the possibility of the transformation, depends on a comparison of the number of the disposable quantities in the assumed value of y with that of the conditions required, in order that the quantity under the radical in the transformed expression may be equal to the square of an integral function of x multiplied by four unequal linear factors. It is shown that the number of disposable quantities exceeds by three that of the required conditions. But, as Poisson has remarked in the report already mentioned (Mem. de VInstitut. x. p. 87), and as M. Jacobi himself intimates, this does not amount to an absolute ^ priori proof of the possibility of the transformation ; non constat but that some of these conditions may be incompatible. Granting however the possibility of putting the quantity under the radical in the required form, it is shown, as in Schumacher's Journal, that this condition is not only necessary * Verhnlst, Traits Eltmentaire des Fonctions Mli$tigue$. ON THE RECENT PROGRESS OF ANALYSIS. 263 Lut also sufficient, or, in other words, that it involves the second condition already mentioned. The transcendent _:^_-^ - -- may Tbe transformed JV(i-/) (i-xy) by assuming y = - Z7 being composed wholly of odd powers of a?, and F of even powers of it. If the degree of U be greater than that of F ? the transformation is said to be of an odd order, and of an -even order in the contrary case. This being premised, 11. Jacobi discusses the particular cases of the transformations of the third and of the fifth order. The first is the same as that of Legeiidre. It is shown that if we put where u and v are constants connected by the following equa- tion U* - V* + 2w {1 - wV} = 0, we shall get dy _ v 4- 2* 3 _ cfa _ i in which & = u 4 and \ = ?A The equation connecting u and v is called the modular equation. The ' principle of double substitution' may be illustrated by writing ^ for x in the expression for y 3 which then becomes, U "JS ^ according to the principle in question, -^ . / If we seek to show that the assigned value of y actually satisfies the differential equation just stated, we begin "by find- log the value of 1 y. Eeducing this value by means of the JUT equation between u and v y we can put it in the form (1 #) -y- > E being an integral function of x and F, as heretofore the denominator of the expression for y. The value of 1 + y is hence got by changing the sign of x and y, while that of 1 v*y is ob- tained by simultaneously replacing x and y respectively by and and reducing. Similarly for 1 -I- v 4 y* Hence it will 7 appear that 264 ON THE RECENT PROGRESS OF ANALYSIS. (I-/) (l-y) = (!-*') (1_ M V)^ (a), where $, like E, is integral. By differentiating and reducing, we then show that , v + %u 3 S 7 ^___^ and combining these two results obtain the required verification. The essence of M. Jacobi's demonstration consists in showing that if the value of y in terms of x is such that an equation of the form (a) subsists, then necessarily dy , S &>='"?* (/3) ' where /-tis a constant; the existence of the two equations (a) and (/?) being equivalent to the two conditions of which we have already spoken (p. 259). In the particular case we are now con- sidering, V 15. After considering the transformation of the fifth order (in which the modular equation is M. Jacobi prepares the way for a more general investigation by introducing a new notation. This step is one of the highest importance. We have been in the habit of calling $ the am- plitude of the integral r r J n vi } let this integral be called u. The new notation is contained in the equation <=anm; or if we call sin <, x, so that u = then x = sin am u. A new notation is in itself merely a matter of convenience : what gives it importance is its symbolizing a new mode of con- sidering any subject. We had hitherto been accustomed to look on the value of the elliptic integral as a function of its amplitude, to mate the amplitude (if the expression may so be used) the in- dependent variable. But in reality a contrary course is on many accounts to be preferred, "We have in the more advanced part ON THE RECENT PROGRESS OF ANALYSIS. 265 of the theory more frequently occasion to consider the value of the amplitude as determined by the corresponding value of the integral than vice versd ; and it therefore becomes expedient to frame a notation by which the amplitude may be expressed as a function of the integral. In a paper in the ninth volume of Crelle's Journal by M. Jacobi, which, like many of his writings, contains in a short compass a philosophical view of a wide subject, he has made use of the analogy between circular and elliptic functions to illustrate the importance of the new notation for the latter. When the modulus of an elliptic integral of the first kind is equal to zero, the integral becomes dx which, as we know, is equal to the arc whose sine is x 9 or to sin" 1 ^. Now this is a function which we have much less often occasion to express than its inverse sin x, and we accordingly always look on the latter as a direct, and on the former as an inverse function. Yet in the case of elliptic functions, the func- tional dependence for which we had an explicit and recognised notation, viz. that of the integral on the amplitude, corresponds to that which in circular functions has always and almost neces- sarily been treated merely as an inverse function. The origin of this discrepancy is obvious; our knowledge of the nature of circular functions is not derived from the algebraical integrals connected with them, and therefore these integrals are not brought so much into view as in the theory of elliptic functions the corresponding integrals necessarily are ; but it is certain that while the discrepancy continued to exist the subject could never be fully or satisfactorily developed. The maxim " verba vestigia mentis " is as true of mathematical symbols as of the elements of ordinary language. We shall see hereafter that Abel took the same step in his first essay on elliptic functions. At present I shall only re- mark, that one of the earliest consequences of the new notation was the recognition of a most important principle, viz. that the c inverse function' sin am u, that is, the function corresponding to sin u in circular functions, is doubly periodic, or that it retains the same value when u increases by any multiple either of a certain real or of a certain imaginary quantity. Now 263 ON THE REGENT PROGRESS OF ANALYSIS. M. Jacobi has shown that no function* can be triply periodic, and therefore these inverse functions possess the most general kind possible of periodicity, a property which gives them great analytical importance. Following M. Jacobi, we shall henceforth give the name of elliptic functions to those which are analogous to circular func- tions. It is on this account better to call Legendre's functions elliptic integrals than, as he has done, elliptic functions (vide ante : p. 253). By the new notation we are led to consider a great variety of formula analogous to those of ordinary trigonometry. The sine or cosine of the amplitude of the sum of two quantities may be expressed in terms of the sines and cosines of the amplitudes of each, &c.f ; and we have only to make the modulus equal to zero to pass from what has sometimes, though not with much propriety, been called elliptic trigonometry to the common properties of circular functions. M, Jacobi gives a table of formulas relating to the new elliptic functions, and proceeds to apply their properties to the problem of transformation. It was in this manner that he had treated the problem in the NackricJiten. As in his earlier essay, he assumes y equal to a rational function of #, whose coefficients * i. e. no function of one variable, h The fundamental formulse are , x sin am it cos am vA. ain v + sin am v cos am wA am u sm am (u + 0) = cos am (u + if) = i - K* sin' 5 am u sm" am v cos am u cos am v - sin am u sin am vA am wA am v T - k* sin 2 am u sin' 2 am v ' A am A am v - & 2 sin am u sin am v cos am u cos ara v i - k* sin 2 am u sin^ am v 3 being the modulus, and A am u= *Ji - & 2 sin 2 am u. If T r s d

(u) corre- sponding to the function denoted in the Fundamenta Nova by sin am w, while f(u) and F(u) correspond respectively to cos am u and A am u. This notation has the defect of appro- priating three symbols which we cannot well spare. On the other hand it is certainly more concise than M. Jacobi's. He then verifies the fundamental formulae for the addition of the new functions, and goes on to show that they are doubly periodic*. He next considers the expressions of <$>na, &c. in ^>a, &c. 5 and proceeds to prove the important proposition that the equation of the problem of the division of elliptic integrals of the first kind is always algebraically soluble. In order to illustrate this, which is one of the most remark- able theorems in the whole subject, it may be observed, that as .any circular function of a multiple arc can be algebraically ex- pressed in terms of circular functions of the simple arc, so may *j>na,Jna, Fna be algebraically expressed by means of (j>a y /a, Fa. Conversely, as the determination (to take a particular fiinction) of sin a in terms of sin na requires the solution of an algebraical equation, so does that of $a in terms of $wa. The equation which presents itself in the former case is, as we know, of the nth or of the (2w)th degree as n is odd or even* But the equation for determining a rises to the (W 2 )th degree in the former case, and in the latter to the (2^ 2 )th* We may however confine ourselves to the case in which n is a prime number; * The formulae in question differ from those already given, only because Abel's r dx form of the elliptic integral is I . ...... - , wMch "becomes the same as Legendre's on making e?=: i. The double periodicity of the functions is ex- pressed by the formula with similar formulae for /and F. The quantities m and n are integral, and i ^ 5jr=r2 =2 I jo 182 276 ON THE RECENT PROGRESS OF ANALYSIS. since if it be composite the argument of the circular or elliptic function may first be divided by one of the factors of n, and the result thus got by another, and so on. Thus setting aside the particular case of ^=2 3 we shall have to consider, in order to determine sin a or ju, being the number of the roots x, y, . . . z. For the sum within the bracket being a rational and symmetrical function of the roots, is necessarily expressible in the coefficients of the equation, and the same is therefore of course true of %#, or of any of the other quantities to which it is equal. ON THE REGENT PROGRESS OF ANALYSIS. 277 If, therefore, by means of the relations which we know to exist among the roots of the equation to "be solved we can esta- blish the existence of a system of such functions, %, %'? %"? & G -? each of which retains the same value of whichever root we suppose it to be a function ; and if by combining these functions we can ultimately express x in terms of them, the equation is solved, since each of these functions may be considered a known quantity. Such is the general idea of Abel's method of solution. The principle on which it depends, namely, the expressibility of any unchangeable function %, is one which is frequently met with in investigations similar to that of which we are speaking. M. Gauss's solution of the binomial equation is founded upon it. I have already remarked that an important simplification of Abel's process was given by M. JacobL The result which M. Jacobi has stated without demonstration may be proved by means of a theorem established by Abel in the fourth volume of Crelle's Journal, p, 19-4. M. Jacobi shows the existence of a system of % 3 functions p, %', &c. 5 by combining which we can immediately express the values of the roots. In the last of his Notices on elliptic functions we find, as has been said, the explicit determination of all the roots. The formula given for this purpose is, like the former, undemonstrated, and I do not know whether any demon- stration of it has as yet been published; but from a note of M. Liouville, in a recent volume of the Comptes Hendus, we find that both he and M. Hermite have succeeded in proving it. But in whatever manner the solution is effected it will always involve certain transcendental quantities, which are intro- duced in the expressions of the relation subsisting between the different roots. The solution can. therefore be looked on as complete, only if we consider these to be known quantities. They are the roots of a particular case of the equation to be solved. They relate to the division of what are called the com- plete integrals. We may therefore say that the general case is reduced to this particular one. But the latter is not, except under certain circumstances, soluble, though the solution of the equation on which it depends can be reduced to the solution of certain other equations of lower degrees. But for an infinity of particular values of the modulus, the 278 ON THE REGENT PROGRESS OF ANALYSIS. case in question is soluble by a method closely analogous to that used by M. Grauss for the solution of binomial equations. Thus for all such values the problem of the division of elliptic inte- grals is completely solved. The most remarkable of these cases corresponds to the geometrical problem of the division of the perimeter of the lernniscate. Abel discovered that this division can always be effected by means of radicals, and further, that it can be con- structed by the rule and compass in the same cases (that is for the same values of the divisor) as the division of the circum- ference of a circle* Of this discovery we find Abel writing to M. Holmboe, "Ah qu'il est magnifique ! tu verras*." In order to form an idea of the nature of the difficulty which disappears in the case of which we are speaking, let us suppose that we have to solve the algebraical equation which is repre- sented by the transcendental one (f> (30) = 0, in the same manner as the equation 4o? 3 3a? = is represented by sin (30) = 0. The roots of 4x 3 3x = 0, are, setting aside zero, . 2-7T . 47T sm T , sm T . Those of the former algebraical equation, which, as we know, is of the ninth degree, are, beside zero, 2m* 4m ~8~' ^~3~ 2 (o> 4- m) 4 (co - - , 2 (CD + 2m) , 4 (o> + 2W) < - - 9 $ - _ - 9 where i = V 1. * It is riglit to mention tnat M. Libri has disputed Abel's title to the theory of the division of the lemniscate. I shall, however, not enter on the merits of the controversy which arose on this point between him and M. Liouville. The reader will find it in the seventeenth volume of the Comptes Rendus. It appears that M. Gauss had himself recognised the applicability of his method to the equation arising out of the problem of the division of the perimeter of the lemniscate (vide Uecherches Aritkmetigues,, vii. p. 429. I quote from the translation published at Paris, in 1809)* ON THE RECENT PROGRESS OF ANALYSIS. 279 To satisfy ourselves that these are the roots required, we observe that (ma -f- ntxi} for all integral values of m and n. Hence the general form of the roots of our equation is {(- l) w+w -f mo> + nm}. E. g. The non- j. -U i a. j . t 5o> + 2m. T J . ,, , , 4 f ct> -f -&n) tabulated root 9 is equal to our sixth root

4- 2m, and the sum of 3 and 2 is an odd number. On considering our table, we observe that it consists of 3 -f 1 horizontal rows, each containing 3 1 terms, and that the argu- ments of the terms in each row are connected by a simple relation ; that of the second being double that of the first. If we were to replace 3 by any odd number j?, we should get an equation of the p 2 degree, whose roots, setting aside zero, might similarly be arranged in p + 1 rows, each of p 1 terms, the arguments of the terms in each row being as 1, 2 3 3, &c, Moreover, sin - is rationally expressible in sin , and o & 11 2 P7T 27T , T . generally sm ^ - is so in sin - , n and p being any in- Jt'Fl "-p J. Ai'fl ~r -L tegers we please. So too are all the terms in each horizontal row of our table, whether for the particular case we have written down, or for that of any odd number, rationally expressible in the first term. Hence it may be shown that when the divisor %n -f 1 is a prime number, an equation whose roots were the terms in any horizontal row could be solved algebraically, by a method essentially the same as that of Grauss, just as we can solve the equation the type of whose roots is sin ^ . But to construct this equation, i. e. to determine its coefficients, requires the so- lution of an equation of the same degree as the number of horizontal rows, V. e. of the degree 2?& + 2, And this equation is in general insoluble. The difficulty we here encounter may be expressed in general language, by saying that although we 280 ON THE REGENT PROGRESS OF ANALYSIS. can pass from one root to another along each horizontal row, yet we cannot pass from row to row. Our table, however, has the remarkable property, that sup- posing, as we may always do, 2% 4-1 to he a prime number, all the roots are rationally expressible in terms of any two not lying in the same row. This depends on a property of the function <, which it is very easy to demonstrate, and it is inti- mately connected with the relations which exist among the terms of the same row. If, then, which is the case for an infinite variety of values of the modulus, we can express any root rationally in terms of another of a different row, say in - - - , all the roots "become &n -f* J- rational in terms of $ ^ . Moreover, it appears that not AtTb ~T~ JL only are the roots all expressible in one, but they are so in such a manner that the functional dependencies among them fulfil a certain simple condition, which, as Abel shows in a separate memoir (Crelle, IV. p. 131 ; or Abel's Works, I. p. 114), renders every equation, all whose roots are rationally expressible in terms of one, algebraically soluble. To take the simplest case, the arc of the lemniscate may be f dx represented by the integral . If 6 be the function in- r Jvl or verse to this integral, we have the simple relation between roots of different rows, < - = i - , co being in this case &YI -j- JL ATI Hr 1. equal to -ST. To apply what has been said to the solution of the general equation for determining - , &c., by supposing n to increase sine ON THE RECENT PROGRESS OF ANALYSIS. 281 limite, and are therefore analogous to the expression of sin which we have already mentioned. The second contains the development of what had already teen pointed out with respect to the lemniscate, so far as relates to the division of its perimeter by any prime number of the form 4m + 1. In an interesting note which M. Liouville communicated to the Institute in 1844, and which is published in the eighth volume of his Journal, p. 507, he has proved generally that the division of the perimeter of this curve can always be effected whether the divisor be a composite or prime number, real or complex (that is, of the form p + V qr, p and gt being integers). In order to do this, it was only requisite to follow m . m . , the reasoning by which Abel has shown that the equa- tion which presents itself in the problem of the division of the circumference of the circle is always resoluble. Thus, as M. Liouville has remarked, his analysis is implicitly contained in Abel's. This memoir also contains Abel's theorem for the transfor- mation of elliptic integrals of the first kind. It is equivalent to that of M. Jacobi ; nor is the demonstration, though presented in quite a different form, altogether unlike M. Jacobi's. Abel begins by considering the sum of a certain series of functions whose arguments are in arithmetical progression. He shows that the sum of this series is a rational function of its first term. If we call this sum (multiplied by a certain constant) y, and the first term a?, then y is such a function of x as to satisfy the differential equation already mentioned, viz. or rather an equation of equivalent form. In fact y is m . m . the same function of x that it is in M. Jacobi's theorem. Thus the sum of the series of elliptic functions is itself, when multiplied by a constant, a new elliptic function, having a new modulus, and whose argument bears a constant ratio to that of the first term of the series. It appears also that for the sum of the elliptic functions we may, duly altering the constant factor, sub- stitute their continued product. Thus, beside the algebraical expression of y, there are two transcendental expressions of, it, both of which are given by M. Jacobi in the Fundamenta 282 ON THE RECENT PROGRESS OF ANALYSIS. Nova. At the close of the memoir Albel compares his result with the one in Schumacher's Journal, No. 123, and mentions that he had not met with the latter until his own paper was terminated. 19. In the 138th number of this journal, Abel resumed the problem of transformation, and treated it in a-more general and direct manner than had yet been done. This memoir appeared in June 1828. M. Jacobi, in a letter to Legendre, has spoken in the highest terms of Abel's demonstration of the formulae of transformation: he says, "Elle est au-dessus de mes eloges, comme elle est au-dessus de mes travaux." An addition to this memoir, establishing the real transformations by an independent method, appeared in Number 148 of the same journal. These two papers are printed consecutively in the first volume of Abel's Works, pp. 253, 275. In the first of these two remarkable essays, Abel makes use of the periodicity of the function 0, or, as he here denotes it, X#, to determine h priori what rational function of x, y must be in order that the differential equation dx ___ = a may be satisfied. [I have altered his notation for the sake of uniformity.] Let -fyx be the function sought, then considering y^tyx as an equation determining x in terms of y> he shows that certain relations necessarily exist among its roots. Let \Q be one of them and \0 f another, it will readily be seen that we may put dff = d6, since each is equal to _ dy " Hence ff = 6 4- a, a being the constant of integration, or, which is the same thing, being independent of y. Hence \0 being one root, every other root is necessarily of the form \ (0 + a). Again, we see from hence that which is to Tbe true for all values of 0, and which therefore ON THE RECENT PROGRESS OF ANALYSIS. 283 implies the existence of a series of equations, of which the tjpe is where Jc is an integer. Hence \(6 + Jca) is a root, whatever in- tegral value we may give to Jc. But the equation y = tyx has but a finite number of roots, and therefore the values of the general expression X (6 + Jca) must recur again and again, This consideration throws light on the nature of the quantity a ; it must in all cases be an aliquot part of a period (simple or com- pound) of the function X#. All the values of X (9 + Jca) got by giving different values to Jc are roots ; but the converse is not necessarily true ; all the roots are not necessarily included in this expression. But it is not difficult to perceive that all the roots are included in a more general expression, viz. M^+^i^i+^aS ... k n a n ), and conversely, that all the values of this expression are roots. The number n is indeterminate : we may have formulas of the form y = ^rx, in which n is unity, others in which it is two, &c.; but in all cases a is an aliquot part of some period of X0, and Jc is integral. It is easy when the roots of y = ^rx are known, to express y in terms of 8. For let fx tyx = J -=r , /and F being integral functions. Then is (yp q being the coefficient of the highest power ofx in yFxfx) an identically true equation; whence, to determine y in 0, we have only to assign a particular value to x, or to com- pare the coefficients of similar powers of it*. This then determines the form which the ftmction y must necessarily be of: the question which Abel goes on to discuss is this : Under what circumstances will a function of the form thus determined h priori be such a function as we require? The character of the reasoning by which this question is treated is similar to that of the method by which Abel had, in his second memoir on elliptic functions, verified the form which, without assigning any reason, he had there assumed for the function y. The second essay is singularly elegant. If <& denote the * I have not noticed an ambiguity of sign at the outset of tliia reasoning^ as given by Abel, as for the purposes of illustration it is immaterial. 284 ON THE RECENT PROGRESS OF ANALYSIS. function Inverse to the integral . o = , and 6 e the Jvl u* 1-AV corresponding function for the modulus c, then, on introducing the Inverse notation, the differential equation dy __ dx becomes of course d& = ad0, with x = <]> C and y = <&#'. Hence for a given increment a of 0, that of & is aa. Let us take the simplest case, and suppose y to be a rational function of x; then, as a? or $> C 6 remains unchanged when 6 in- creases "by a period of the function c , y does so too ; that is

c , or in other words, a times a period of

k and <}> e have each a real period, here denoted by 2 c respectively, and each an imaginary period ^^i and ^r e i re- spectively, 'BTfc and tsr c being both real. Let 6 receive first the increment 26> C3 and secondly the increment in c i, then, by what has been said, m, n, jp, j being certain Integers. But can these two equations subsist simultaneously? Not generally, since if we eliminate a and equate possible and impossible parts, we get two relations among a^'sr^'sifc, which are continuous functions of the two quantities Jc and o. Hence both are determinate ; and if we wish c to remain indeterminate, we must either make in and q equal to zero, in which case a is impossible, or, making n and j? equal to zero, assign a real value to it. When a is real we have G>Jc 'OTfc a = m = q , <* c * ^c and hence the remarkable conclusion, that <*>k < C : :: q : m, CTfc "CTc X m and y being integers. * 13T here is in M, Jacobi'a notation iK r > so that 00 = (0-f- sww }- ntvi) t wi and n being any integers* ON THE RECENT PROGRESS OF ANALYSIS. 285 The commensurability of the transcendental functions , ^Jfc ^C Is therefore a necessary condition, in order that an integral with modulus c can be transformed into one with modulus 7c, the regulator a being real and c indeterminate. And it may be shown that this condition is not only necessary but sufficient. Similar considerations apply to the case in which a is impossible. Simple as this view is, it leads to many consequences of great interest. The function q, of which we have already spoken * v integer). Hence in multiplying an integral, the multiplier must be an integer, if y is rational in x, except for particular values of c. In the paper of which we are speaking Abel has applied precisely similar considerations to the case in which x and y are connected by any algebraical equation. Passing over one or two shorter papers, one of which has been already referred to at p. 276, we come to a Precis of the theory of elliptic functions, published in the fourth volume of Crelle's Journal, p. 236. The work of which it was designed to be an extract was never written, and the Precis itself is left unfinished. A general summary was prefixed to it, from which we learn that the work was to be divided into two parts. In the first elliptic integrals are considered irrespectively of the limits of integration, and their moduli may have any values, real or imaginary. Abel proposes the general problem of determining all the cases in which a linear relation may exist among elliptic integrals and logarithmic and algebraical functions in virtue of algebraical relations existing among the variables*. His first step is to apply his general method for the com- parison of transcendents to elliptic integrals, which may be * In the assumed relation, the amplitude, or ratter the sine of the amplitude of each elliptic integal, is to be one of the variables, and not a function of one or more of them. 286 ON THE RECENT PROGRESS OF ANALYSIS. done by what is called Abel's theorem, in at least two different ways: the one, that of which he now makes use; the other, that which we have seen is applied to the case of four functions by Legendre in his third Supplement. He next determines the most general form of which the integral of an algebraical differential expression of any number of variables is capable, provided it can be expressed linearly by elliptic integrals and logarithmic and algebraical functions. The result at which he arrives admits of many important ap- plications. It is, that the integral in question may be expressed in a form in which the sine of the amplitude of each elliptic integral and the corresponding A, and also the algebraical and each logarithmic function are all rational functions of the varia- bles and of the differential coefficients of the integral with re- spect -to each. He proceeds by an interesting train of reasoning to establish the remarkable conclusion, that the general problem which we are considering may ultimately be reduced to that of the trans- formation of elliptic integrals of the first kind. The problem of this transformation is then discussed, and by a method essentially the same as that of which he had made use in his paper in Schumacher's Journal. The appearance however of the two investigations is dissimilar, because no reference is made to elliptic functions (as distinguished from elliptic integrals) in the first part of the Pr&cis. The relations therefore which exist among the roots of y tyx are established by considera- tions independent of the periodicity of elliptic functions ; though it is not difficult to perceive that they were suggested by the results previously obtained by means of that fundamental pro- perty. It is shown, that if the equation y = tyx, where tyx is a rational function, satisfy the differential equation (A), then this equation, considered as determining x in terms of y, is always algebraically soluble. As the multiplication of elliptic integrals may be considered a case of transformation (that, namely, in which the modulus of the transformed integral remains un- changed), this theorem may be looked on as an extension of that which we have spoken of (p. 275) in giving an account of Abel's first memoir on elliptic functions. The two theorems are proved by the same kind of reasoning. The second part of the memoir was to have related to cases ON THE RECENT PROGRESS OF ANALYSIS. 287 In which tlie moduli are real and less than unity; of this tow- ever only the summary exists. Abel proposed to introduce three new functions, the first corresponding to that which he had previously designated by 6*. He now denotes it by X#. The second and third functions are apparently what the second and third kind of elliptic integrals respectively become, when, in- stead of x, we introduce the new variable #; x and 6 being of course connected by the equation x = \d. The double period- icity of the function X and its other fundamental properties having been established, it was his intention to proceed to more profound researches. Some of his principal results are briefly stated. I may mention one, that all the roots of the modular equation may be expressed rationally in terms of two of themf * One of the last paragraphs of the summary relates to func- tions very nearly identical with those which M. Jacobi discusses at the close of the Fundamenta Nova, and which he has desig- nated by the symbols H and . The second volume of AbePs collected works consists of papers not published during his life. Two or three of these relate to elliptic functions. The longest contains a new and very general investigation for the reduction of the general trans- P cendeiit, whose differential is of the form -=, P being, as usual, V R rational and B a polynomial of the fourth degree ; together with transformations with respect to the parameter of integrals of the third kind* 20. Having now given some account of the revolution which the discoveries of Abel and Jacobi produced in the theory of elliptic functions, I shall mention some of the prin- cipal contributions which have been made towards the further development of the subject since the publication of the Funda- * In the Precis Abel Las adopted tlie canonical form of the integral of the first kind made use of by Legendre and M. Jacobi ; so that the quantity under the radical is (i-iB 3 ) (i- , and deducing from hence its other properties. It has been re- marked that the continued products of Abel and M. Jacobi are derived from considerations which, although cognate, are yet distinct; those of the latter being singly infinite, while Abel's fundamental developments consist of the product of an infinite number of factors, each of which in its turn consists of an infinite number of simple factors. Thus we can have two very dissimilar definitions of the function by means of con- tinued products. M. Cauchy, who has investigated the theory of what he has termed reciprocal factorials, that is, of continued products of the form {(l+o?) (1+te) } 1(1 + to" 1 ) (1 + af*) ......} which is immediately connected with M. Jacobi's developments^ has accordingly set out from the singly infinite system of pro- ducts, and has deduced from hence the fundamental properties of elliptic functions (Gomptes Sendus, xvn. p. 825). Mr Cayley, on the other hand, has made use of Abel's doubly infinite products, and has shown that the functions defined by means of them satisfy the fundamental formulae mentioned in the note at page 266, which, as these equations furnish a sufficient definition of the elliptic functions, is equivalent to showing that the continued products are in reality elliptic functions. He has ON THE RECENT PROGRESS OF ANALYSIS. 293 therefore effected for Abel's developments that which M. Cauchy had done for M. Jacobi's. Mr Cayley's paper appeared in the fourth volume of the Cambridge Mathematical Journal, but he has since published a translation of it with modifications in the tenth volume of Liouville's. On the same subject we may mention a paper by M. Eisenstein (Crelle's Journal, XXVII. 285). 24. M. Liouville has in several memoirs investigated the conditions under which the integral of an algebraical function can be expressed in an algebraical, or, more generally, in a finite form. This investigation is of the same character as that which occurs in the beginning of Abel's last published memoir on elliptic functions (vide supra, p. 286). But while Abel's re- searches are more general than M. Liouville's, the latter has arrived at a result more fundamental, if such an expression may be used, than any of which Abel has left a demonstration. He has shown that if y be an algebraical function of x, such that lydx may be expressed as an explicit finite function of x, we must have I ydx = t+A log u + B log v 4- 4- log w, A y -B, ... (/being constant, and t, u, v, ...w algebraical functions of x. The theorem established by Abel in the memoir refer- red to includes as a particular case the following proposition, that if r ... 4- Clogw, then t, u, v, .. w may all be reduced to rational functions of x and y. Combining these two results, it appears that if lydx be ex- pressible as an explicit finite function of x^ its expression must be of the form 1 4- A. log u 4- B log v 4- ... 4- log w f where , u, v, ..* w are rational functions of x and y, or rather that its expression must be reducible to this form*. * An equivalent theorem is stated by Abel in Ms letter to Legendre for im- plicit as well as explicit functions (Orelle's JbuntoZ, vi.)% 294 ON THE REGENT PROGRESS OF ANALYSIS. After establishing these results in the memoir (that on ellip- tic transcendents of the first and second kinds), which will Tbe , found in the twenty-third cahier of the Journal de VEcole Polytechnigue, p. 37, M. Liouville supposes y to be of the P form ~7r5 , where P and B are integral polynomials, and hence f P deduces the general form in which the integral l-y^ dx may necessarily be put,. provided it admit of expression as an ex- plicit finite function of x. r p He shows from hence that if I -j^ <%& cannot be expressed by an algebraical function of x, it cannot be expressed by an explicit finite function of it, and finally demonstrates that an elliptic integral, either of the first or second kind, is not ex- pressible as an explicit finite function of its variable. In a previous memoir inserted in the preceding cahier y M. Liouville proved the simpler proposition, that elliptic integrals of the first and second kinds are not expressible as explicit algebraical functions of their variable (Journal de VEcole Poly* technique, t XIV, p. 137). His attention appears to have been directed to this class of researches by a passage of Laplace's * Theory of Probabilities/ in which the illustrioBS author, after indicating the fundamental, and, so to speak, ineffaceable dis- tinctions between different classes of functions, states that he r dx had succeeded in showing that the integral . . is 6 6 J*/l+ax 2 + l3x* not expressible as a finite function, explicit or implicit, of x. Laplace however did not publish his demonstration. In his own Journal (v. 34 and 441), M. Liouville has since shown that elliptic integrals of the first and second kinds, con- sidered as functions of the modulus, cannot be expressed in finite terms. 25. In the eighteenth volume of the Gomptes Rendu (Liouville' s Journal, ix. 353), we find in a communication from M. Hermite, of which we shall shortly have occasion to speak more fully, a remarkable demonstration of Jacobi's theorem. It is stated for the case of the first real transformation, but might of course be rendered general. This demonstration depends ON THE RECENT PROGRESS OF ANALYSIS. 295 essentially on the principle already mentioned (p. 276), that any rational function of a root of an algebraical equation which, has the same value for every root of the equation is rationally ex- pressible in the coefficients. The equation to which this princi- ple is applied is that to which we have so often referred, viz. y = j considered as an equation to determine x in terms of y, and by means of it, M. Hermite shows at once that a certain rational function of x is also a rational function of #, the form of which is subsequently determine! M. Hermite goes on to prove other theorems relating to ellip- tic functions. As elliptic functions are doubly periodic, we may determine certain of their properties by considering to what conditions doubly periodic functions must be subject This view is mentioned by M. Liouville in a verbal communication to the Institute (Comptes Rendus, t. xix.). He states that he had found that a doubly periodic function which is not an absolute constant and has but one value for each value of its variable must be, for certain values of it, infinite ; that from hence the known properties of elliptic functions are easily deduced ; and that by means of this principle he had succeeded in proving the expressions of the roots of the equation for the division of an elliptic integral of the first kind, which M. Jacobi had given without demonstration in Crelle's Journal*. I am not aware that any development of M. Liouville's view has as yet ap- peared. In the recent numbers of Crelle's Journal there are many papers by M* Eisenstein on different points in the theory of elliptic functions. Among these I may mention one which con- tains a very ingenious proof of the fundamental formula for the addition of two functions, derived from the differential equa- tion of the second order, which each function must satisfy. Other contributions to the theory of elliptic functions might be mentioned ; some of these, not here noticed, are referred to in the index which will be found at the end of this report. But in general it may be remarked that the form which the subject has * M. Liouville lias mentioned that M. Hermite had demonstrated the formulas in question in a different manner. 296 ON THE RECENT PROGRESS OF ANALYSIS. assumed, in consequence of the discoveries of Abel and M. Jacobi, is that which it will probably always retain, however our knowledge of particular parts of it may increase. What has since been effected relates for the most part to matters of detail, of which, however important they may be, it is difficult or im- possible to give an intelligible account. 26. It does not fall within the design of this report to con- sider the various applications which have been made of the theory -of elliptic functions ; but I shall briefly mention some of the geometrical interpretations, if the expression may so be used, which mathematicians have given to the analytical results of the theory. The lemniscate has ? as is well known, the property that its arcs may be represented by an elliptic integral of the first kind, the modulus of which is = , The problem of the division of its A/2 * perimeter is accordingly a geometrical interpretation of that of the division of the complete integral, and was considered by mathematicians at a time when the theory of elliptic functions was almost wholly undeveloped. Besides Fagnani, whose re- searches with respect to the lemniscate have been already noticed, we may mention those of Euler, who however did not succeed in obtaining a solution of the problem. Legendre, who seems to have attached considerable importance to geometrical illustra- tions of his analytical results, assigned the equation of a curve of the sixth order, whose arcs measured from a fixed point represent the sum of any elliptic integral of the first kin$ and an algebraical expression. He showed also that an arc of the curve might be assigned equal to the elliptic integral, but in order to this both extremities of the arc must be considered variable, so that in effect the integral is represented by the difference of two ar<5S measured from a fixed point (Traite des Fonctions Ellvptigues, i. p. 36). M, Serret, in a note presented to the Institute in 1843 (Liouville's Journal, vm, 145), has proved a beautiful theorem, viz. that the sum and difference of the two unequal arcs, inter- cepted by lines drawn from the centre of Cassini's ellipse to cut the curve, are each equal to an elliptic integral of the first kind, and that the moduli of the two integrals are comple- ON THE RECENT PROGRESS OF ANALYSIS. 297 mentary. In the lemniscate, which is a case of Cassinf s ellipse, one of these arcs disappears, and the moduli of the two integrals are equal, each being the sine of half a right angle. So that M Serret' s theorem is an extension of the known property of the lemniscate. M. Serret has since considered the subject of the representa- tion of elliptic and hyper-elliptic arcs in a very general manner. His memoir, which was presented to the Institute and ordered to be published in the Savans Strangers, appears in Liouville's Journal, X. 257. He had remarked that the rectangular co- ordinates of the lemniscate are rationally expressible in terms of the argument of the elliptic integral which represents the arc, for if we assume / K + # 3 8 ass= V2a j-p we shall have ~j====, Vl +* and if between the first two of these equations we eliminate %, we arrive at the known equation of the lemniscate *. So that if we state the indeterminate equation ds? + dy* = Z.df, (x, y and Z being real and rational functions of #), the lemniscate will afford us one solution of it ; and every other solution will correspond to some curve whose arc is expressible by an elliptic or hyper-elliptic integral. Of this indeterminate equation M. Serret discusses a particular case. He succeeds in solving it by a most ingenious method, which is applicable to the general equation, and shows from hence that there axe an infinity of curves, the arcs of which represent elliptic integrals of the first kind. M. Serret's researches however have not led him to a geometrical representation by means of an algebraical curve of any integral of the first kind, though his results are generalized in a note appended to his memoir by M. Liouville. In order that the curve may be algebraical, it is necessary and sufficient, * On reducing the integral I - f to the standard form of elliptic integrals* /A* we find that it is an elliptic integral of the first kind, of which the modulus is the sine of 45, 298 ON THE RECENT PROGRESS OF ANALYSIS. as M. Liouville has remarked, that the square of the modulus of the integral should be rational, and less than unity. In a subsequent memoir (Liouville's Journal., x. 351) he has very much simplified the analytical part of his researches, and in the same Journal (x. 421) has proved some remarkable properties of one class of what may be called elliptic curves. In the fourth number of the Cambridge and Dublin Mathe- matical Journal (p. 187), M. Serret has developed this part of the -subject, and has also given a general sketch of his pre- vious papers. M. Liouville (Oomptes Rendus, xxi. 1255, or his Journal, x. 456) has given a very elegant investigation of an analytical theorem established by M. Serret. In the fourteenth volume of Crelle's Journal (p, 217), M. G-udermann has considered the rectification of the curve called the spherical ellipse, which is one of a class of curves formed by the intersection of a cone of the second order with a sphere* He has shown that its arcs represent an elliptic integral of the third kind. In the ninth volume of Liouville's Journal (p. 155), Mr W. Roberts proves that a cone of the second order, whose vertex lies on the surface of a sphere, and one of whose external axes passes through the centre, intersects the sphere in a curve whose arcs will, according to circumstances, represent any elliptic inte- gral of the third kind and of the circular species ; or any elliptic integral of the same kind and of the logarithmic species, pro- vided the angle of the modulus is less than half a right angle ; or (subject to the same condition) any elliptic integral of the first kind ; or lastly, by a suitable modification, any elliptic integral of the second kind. The cases here excepted may be avoided by introducing known transformations. The cases in which the arcs represent elliptic integrals of the first kind, Mr Roberts has previously mentioned in the eighth volume of Liouville's Journal (p. 26B). He has since given in the same Journal (x. 297), a general investigation of the subject, in which it is supposed that the vertex of the cone may have any position we please. M. Verhnlst has represented the three kinds of elliptic integrals by means of sectorial areas of certain curves, and the function T by the volume of a certain solid. It is manifest, however, that it is incomparably easier to do this than to represent these transcend- ents by means of the arcs of curves. ON THE RECENT PROGRESS OF ANALYSIS. 299 Besides one or two other papers I may mention a tract "by the Abbe Tortolini, on the geometrical representation of elliptic integrals of the second and third kinds. This tract, however, I have not seen. Lagrange long since proved (vide Theorie des Fonctions Ana- lytiques, p. 85), that by means of a spherical triangle a geome- trical representation of the addition of elliptic integrals of the first kind may easily be obtained, and that hence by a series of such triangles we are enabled to represent the multiplication as well as the addition of these integrals. M. Jacobi has given (Crelle's Journal, III. p. 376, or vide Liouville's Journal, x. p. 435) a geometrical construction for the addition and multiplication of elliptic integrals of the first kind. It is founded on the properties of an irregular polygon inscribed in a circle, and the sides of which touch one or more other circles. It is to be remarked that Legendre, in giving an ac~ count of this construction in one of the supplements to his last work, has only considered its application to multiplication and not to addition, and has been followed in this respect by M. Ver- hulst, whose treatise on elliptic functions has been already men- tioned. In consequence of this, M. Chasles was led to believe that until the publication of his own researches, no construction for addition excepting that of Lagrange was known. But he has recently (Comptes Eendus, January 1846) pointed out the error into which he had fallen. 27. In the Transactions of the Royal Irish Academy (ix. p. 151), Dr Brinkley gave a geometrical demonstration of Fagnanfs theorem with respect to elliptic arcs, and in the sixteenth volume of the same Transactions (p. 76), we find Lan- den's theorem proved geometrically by Professor MacCullagli. M. Chasles has considered the subject of the comparison of elliptic arcs by geometrical methods, and with great success. His fundamental proposition may be said to be, that if from any two points of an ellipse we draw two pairs of tangents to any confocal ellipse, the difference of the two arcs of the latter respec- tively intercepted by each pair of tangents is rectifiable. Or, what in effect is the same thing, if we fasten a string at two points in the circumference of an ellipse, and suppose a ring to move along the string, keeping it stretched, and winding it 300 ON THE RECENT PROGRESS OF ANALYSIS. on and off the arc which lies between its two extremities, the ring will trace out a portion of an ellipse confocal to the former. If for the first ellipse we substitute an hyperbola confocal with the second, the sum of the arcs will be constant. From hence a series of theorems is deduced, remarkable not only for their elegance, but also for the facility with which they are obtained. They furnish constructions for the addition and multiplication of elliptic integrals. The whole of this investigation, of which an account is given in the Oomptes Eendus (Vol. XVII. p. 838, and VoL XIX. p. 1239), shows, like others of M. Chasles, how much is lost in treating geometrical questions by an exclusive adherence to what may be called the method of co-ordination. Invaluable as this method is, it yet often introduces considera- tions foreign to the problem to which it is applied*. III. 28. The first outline of a detailed theory of the higher transcendents was given by Legendre in the third supplement to his Traiti des Fonctions Elliptiques. He proposes to classify the transcendents comprised in the general formula f /(s)&_ J (x a) Va?, the first class being that in which the index of this degree is three or four; the second that in wjiich it is five or six, and so on. The first class therefore consists of elliptic integrals; all the others may be designated as ultra-elliptic. This epithet, however, which was proposed by Legendre, has not been so generally used as hyper- elliptic, which was, I believe, first used by M. Jacobi. M. Jacobi, however, has proposed to call the higher transcendents Abelian integrals. The principle of Legendre's classification is to be found in the mininum number of integrals to which the sum of any * M. Chasles has also considered the subject of spherical conies, as well as that of the lines of curvature and shortest lines on an ellipsoid. The latter has re. cently engaged the attention of several distinguished mathematicians MM. Jacobi, Joacjiimsthal, Liouville, MacCulIagh and Roberts may be particularly mentioned. ON THE RECENT PROGRESS OF ANALYSIS. 301 number of them can Tbe reduced. As we know, this number is unity in the case of elliptic integrals, and- by Abel's theorem we find that it is two in the first class of the higher transcendents, three in the next, and so on. Following the analogy of elliptic integrals, Legendre pro- posed to recognise three canonical forms in each class of hyper- elliptic integrals, and thus to divide it into three orders. The sum of any number of functions of the first kind will, when the required conditions are satisfied, be equal to a constant ; that of any number of the second and third kinds respectively will, under similar conditions, be equal to an algebraical or logarith- mic function. Much the greater part of the remainder of the supplement consists of a discussion of the particular transcendents It contains a multitude of numerical calculations, and if the writer's age be considered (he was then almost eighty), is a very remarkable production. By means of the numerical calculations he recognised, as it were empirically, the values to be assigned in different cases to the above-mentioned constant : what these values ought to be, he did not attempt' to determine h priori. At the close of the supplement we find a remarkable reduc- tion of an integral, apparently of a higher order to elliptic in- tegrals. The method, employed has been generalised by M. Jacobi, in a notice of Legendre's Supplements, inserted in the eighth volume of Crelle's Journal (p. 413). 29. In the ninth volume of Crelle's Journal (p. 394), we find a most Important paper by M. Jacobi ( Considerationes Ge- nerales, &c.} which may be said to have determined the direction in which the researches of analysts in the theory of algebraical integrals were to proceed. The writer proposes two questions, both suggested by the cases of trigonometrical and elliptic functions. First, as in these cases we consider certain functions to which circular and elliptic integrals are respectively inverse, and which are such that func- tions of the sum of two arguments are algebraically expressible in terms of functions of the simple arguments, what are the cor- 302 ON THE REGENT PROGRESS OF ANALYSIS. responding functions to which the hyper-elliptic or Abelian integrals are inverse, and how by means of them can Abel's theorem be stated? Secondly, as in the same cases we obtain algebraical inte- grals of differential equations, whose variables are separated, but which nevertheless can only be directly integrated by means of transcendents*, what are the differential equations of which Abel's theorem gives us algebraical integrals ? These two ques- tions are, it is obvious, intimately connected. M. Jacobi first takes the particular case in which the poly- nomial under the radical is of the fifth or sixth degree. If we call this polynomial X, it follows from Abel's theorem, that if ' dx xdx we shall have the equations faa + fab = fax + fay + fax 1 + fay\ where a and 5 are given as algebraical functions of the indepen- dent quantities x, y, a? 1 , #\ Let x + a + ^>J = u + u l faa + fab=v+ v\ a = X (u + u 1 , v + v l ) 1 , v+v l ). =o, of wMcli the algebraical integral is aj *Ji-y*+y*Ji-&=C. Each term of this differential equation is a differential of a transcendent function ON THE REGENT PROGRESS OF ANALYSIS. 303 Hence the functions \ (u + u 1 , v + v l ) and \ (u -f w 1 , v + -y 1 ) are expressible as algebraical functions of These then are functions to which the integrals are in a cer- tain sense inverse, and which have the same fundamental pro- perty as circular and elliptic functions. In the general case of Abel's theorem, we introduce (when the degree of the polynomial is 2m or 2m 1), m 1 functions analogous to X, each being a function of m 1 variables. These functions will, it may easily be shown, have the fundamental property just pointed out for the case in which m is equal to three, Again, the differential equations of which Abel's theorem gives us algebraical integrals, are, if the degree of the polynomial JL be five or six, the following : dx dy dz ___ + vl ' xdx vdv zdz and generally, if the degree of the polynomial be 2w or 2m 1, there are m 1 such equations, the numerators of the last con- taining the (m 2)th power of the variables. M. Jacobi concludes by suggesting as a problem the direct integration of these differential equations, so as to obtain a proof of Abel's theorem corresponding to that which Lagrange gave of Euler's (vide ante, p. 241). 30. Another important paper by M. Jacobi is that which is entitled De Functionibi^s duarum Variabilmm guadrwpliciter periodicis, etc. (Crelle, XIII. p. 55). It is here shown that a periodic function of one variable cannot have two distinct real periods. In the case of a circular function, though we have for all values of x sin x = sin = sin m and n being any integers, yet 2w?r and 2wrr do not constitute two distinct periods, since each is merely a multiple of 2w, which 304 ON THE RECENT PROGRESS OF ANAL YSIS. is the fundamental period of the function. But if we had for all the values of x we should also have where m and n may be any integers, positive or negative* Hence ma -f n/3 may, provided a and /S are incommensurable, which is implied in their being distinct periods, be made less any assignable quantity, so that we may put where e is indefinitely small, and this manifestly is an inad- missible result. Accordingly we see that one at least of the periods of elliptic functions is necessarily imaginary. Again, similar reasoning shows that in a triply periodic func- tion, that is in one in which we have f(x) =f{x+m (cc-f/3 V 1) -f ?n'(a'+/3V 1) +m"(a"+/3'V 1)} for every value of x, m, m r , m' being any integers, and in which the three periods a + /3 V 1, &c, are distinct, we can make /()=/( +e) by assigning suitable values to m, m, m"; e being as before less than any assignable quantity. Hence as this result is inadmissi- ble, it follows that there is no such thing as a triply periodic function. Whenever therefore a function appears to have three periods they are in reality not distinct, and so a fortiori when it appears to have more than three. But now we come to a diffi- culty. For M. Jacobi proceeds to show that if we consider a function of one variable inverse to the Abelian integral (a + fix) dx X being of the sixth degree in cc, this function has four dis- tinct and irreducible periods. His conclusion is that we cannot consider the amplitude of this integral as an analytical function of the integral itself. Tn the present state of our knowledge, this conclusion, though seemingly forced on us by the impossibility of recognising the existence of a quadruply periodic function of one variable, is not, I think, at all satisfactory. The functional ON THE REGENT PROGRESS OF ANALYSIS. 305 dependence, the existence of which we are obliged to deny, may be expressed by a differential equation of the second order ; and therefore it would seem that the commonly received opinion that every differential equation of two variables has a primitive, or expresses a functional relation between its variables, must be abandoned, unless some other mode of escaping from the difficulty is discovered. It is probable that some simple consideration, rather of a metaphysical than an analytical character, may here- after enable us to form a consistent and satisfactory view of the question, and this I believe I may say is the opinion of M. Jacobi himself. The same difficulty meets us in all the Abelian integrals : as in" the case of those of Legendre's first class, namely where JSTis of the fifth or sixth degree, so also generally, the in- verse function has more than its due number of periodicities. Abel, in a short paper in the second volume of his works, p. 51, has in effect proved the multiple periodicity of the func- tions which are inverse to the integrals to which his theorem relates. The difficulty to which this gives rise did not strike him, or was perhaps reserved for another occasion. M. Jacobi next proves that his inverse functions of two variables are quadruply periodic, but that quadruple periodicity for functions of two variables is nowise inadmissible. A difficulty however seems to present itself, which is sug- gested by M. Eisenstein in Crelle's Journal, viz. that if for each /dx -y= (vide supra, p. 302), has an infinity of magnitudes real and imaginary, and the same is the case for y, it is by no means easy to attach a defi- nite sense to the equation u = <>x + x make the latter quantity appear indeterminate is undoubtedly a difficulty ; but it is, so to speak, a difficulty collateral to M. Jacobf s theory^and therefore need not prevent our accepting it. 306 ON THE RECENT PROGRESS OF ANALYSIS. elliptic functions, viz. the reduction and transformation of the integrals themselves, and the theory of the inverse functions. But before considering these I shall give some account of what has been done in fulfilment of the suggestion made by M. Jacobi at the close of the Gonsiderationes Generates. Mathematicians have succeeded in effecting the integration of the system of differential equations to the consideration of which we are led by Abel's theorem, and which is commonly designated by German mathematicians as the " Jacobische system;" its existence and its integrability having been first pointed out by M. Jacobi. In Crelle's Journal (xxm. 354), M. Richelot, after modifying the form in which Lag-range's celebrated integration of the dif- ferential equation of elliptic integrals is generally presented, ex- tended a similar method to the system of two differential equations which occurs when we consider the Abelian transcen- dents of the first class. He thus obtains one algebraical integral of the system. In the case of Lagrange's equation one integral is all we want; but in that which M. Richelot here discusses we require two. Now if in the former case we replace each of the variables by its reciprocal, we obtain a new differential equa- tion of the same form as the original one, and integrable there- fore in the same manner; and if in its integral we again replace each new variable by its reciprocal, that is by the original variable, we thus, as it is not difficult to see, get the integral of the original equation in a different form. That the two forms are in effect coincident may be verified a posteriori. But the same substitutions being made in M. Richelot's equations, which are of course those we have already mentioned at p. 302, the first of them becomes similar in form to the second, arid vice versa the second to the first. Thus the system remains similar to itself; and if in the algebraical integral we obtain of it we again replace the new variables by their reciprocals, we fall on a new algebraical integral of the original system ; this integral being, which is remarkable, independent of that previously got. Thus the system of two equations is completely integrated. Extending his method to the general system of any number of equations, M. Richelot obtains for each two integrals, but of course these are not all that we want. At the conclusion of his memoir M. Richelot derives from Abel's theorem the algebraical integrals of the " Jacobische system." ON THE RECENT PROGRESS OF ANALYSIS. 307 Though in this memoir M. Bichelot only obtained by direct integration two of the m I algebraical integrals of the " Jaco- "bische system," yet lie put the problem of its complete inte- gration into a convenient and symmetrical form. As there are m variables and m 1 relations among them, we may suppose each to be a function of an independent variable t. Lagrange, as we know, in integrating the equation dx ... d V /> Vr ' introduced such an independent variable by the assumption dx which of course implied that == VZ This assumption is unsymmetrical, and it is therefore difficult to see how to gene- dx ralise'it. Buif if we assume -?- == - , we shall of course have - at x y ^ = and therefore t is symmetrically related to x and dt y x y. Let Fu = be an equation whose roots are x and ^, then, as we know, when u=x, F"u=>x-~y : and, whea % = ^ ? j"u~ y c, so that using an abbreviated notation VF Nothing is easier than to generalise this result For instance, the " Jacobische system' 3 of two equations is xdx , ydy , zdz L. i Si. JL. . ... _ Now if Fu = have x, y, z for its roots, the two preceding equations may, in virtue of a very well-known theorem, be re- placed by the three following, ^ = dt~~ F'x* dt~ F'y* dt 202 308 ON TEE RECENT PROGRESS OF ANALYSIS. which, introduce an independent variable ^ symmetrically re- lated to a;, y and z ; and so in all cases. M. Hichelot* then takes a symmetrical function of x, y, ... #, viz. their sum, and by means of the last written equations arrives at an integrable differential equation of the second order, the principal variable being the said sum and the independent variable being t. From the first integral of this equation it is easy to eliminate the differentials, and we have thus an alge- braical relation in x, y, ... &, from which, in the manner already mentioned, M. Eichelot deduces another. We now see that if we could find any other symmetrical function which would lead to an integrable equation we should get a finite relation among the variables. In the next volume of Grelle's Journal M. Jacobi took the following function as his principal variable, p, being a root of JT= or fx = if we suppose X=fx. Call- ing this function v, we get a simple differential equation in v and t, and a corresponding integral of the system. Now fx~Q has 2m or 2m 1 roots, and we only want m 1 integrals. The integrals therefore which we get by making /z the first, second, &c. root of fx = are not all independent. * As M. Richelot's method of demonstrating Euler's theorem is more sym- metrical and far more easily remembered than Lagrange's, it ought, I think, to be introduced into all elementary works on elliptic functions. The equation to be integrated being ^ a + fte + ^ + ^ +&?_ dt -x Then Let p=ac+2/ Then after a few obvious reductions Hence the algebraical integral sought. "^It may easily be expressed in other forms. ON THE RECENT PROGRESS OF ANALYSIS. 309 In the twenty-fifth volume of Crelle's Journal M. Eichelot resumed the subject of his former paper, and discussed it in a very interesting memoir. This fundamental or principal result may be said to be a generalisation of M. Jacobi's. It is that in the function p, may have any value whatever. The resulting differential equation, though rather more complicated than when, with M. Jacobi, we suppose p, a root of fa = 0, is still very readily in- tegrable. We have thus an indefinite number of algebraical in- tegrals, since the quantity p is arbitrary, but of course not more than m 1 of them are independent. In the same volume of Crelle's Journal, p. 178, there is a curious paper by Dr Hsedenkamp, in which the algebraical in- tegrals of Jacobi's system are for the case of a polynomial of the fifth degree under the radical deduced from geometrical con- siderations. It is shown that in a system of curvilinear co- ordinates (those of which MM. Lame* and Liouville have made so much use), the equations of the system are the differential equations given by the Calculus of Variations for the shortest line between two points. Consequently the finite equations of a straight line are the integrals sought. This very ingenious consideration is afterwards generalised. 32. In the twelfth volume of Crelle's Journal, p. 181, M. Eichelot has considered the Abelian integral of the first class. The principal result at which he arrives is, that the only rational transformation by means of which such an integral may be changed into one of similar form is linear in both the variables which it involves. By means of this substitution, he trans- forms, under certain conditions, the integral in question into a form analogous to the standard forms of elliptic integrals. The subject of the division of hyper-elliptic integrals of each class into three genera is also considered, and the same prin- ciple of classification as Legendre's is made use of. The paper concludes by pointing out an error which Legendre committed in the application of his principle. Legendre had thought that the formula of summation given by Abel's theorem for integrals of the form f^= coul(1 not iavolve a logarithmic function. J Vx, disappear, but while x is merely known as the polynomial of the Xth degree, we cannot decide whether the integral is to be referred to the second or third kind. I may mention here a very elegant result due to M. Jacobi. It appears in the thirtieth volume of Crelle's Journal, p. 121, and is a generalisation of the fundamental formula for the addition of elliptic arcs. With a slight modification it may be thus stated. If \ (uv} (vide ante, p. 302) are algebraical functions of transcendental functions which involve but one variable. M. Hermite's subsequent researches have embraced a much more general theory than that of the Abelian integrals, namely, ON THE RECENT PROGRESS OF ANALYSIS. 313 that of the integrals of any algebraical function whatever. Thus his views bear the same relation to Abel's general theory, deve- loped in the Savans Strangers, that those of M. Jacobi in the Consider ationes Generates do to Abel's theorem. All that has yet been published with respect to them is con- tained in the Comptes Rendus, XVITI. p. 1133, in the form of an extract of a letter from M. Hermite to M. Liouville. This ex- tract is reprinted in Liouville's Journal, IX. p. 353. It was com- municated to the Institute in June 1844. Following the course of M. Jacobi's inquiries, M. Hermite proposed to determine what are the differential equations of which AbePs investigations give the complete algebraical inte- grals. When this is done it suggests the nature of the inverse functions which are to be introduced. The number of these functions will of course vary in different cases, just as in M. Ja- cobi's less general theory. Let us suppose this number to be denoted by % then each function will involve z being the quantity correspond- ing to , viz. its vertical descent below its initial position. Now suppose the circle to move horizontally in its own plane with a velocity equal at every instant to the velocity, along the arc, of one of the points, the direction of the motion of the circle being towards the right or the left accordingly as the point is to the right or the left of the vertical diameter. If the former point be chosen, then the velocity of the circle will be constant, if the * Walton's Problems in Elementally Mechanics., p. 745. ON THE TAUTOCHRONISM OF THE CYCLOID. 327 latter point, it will Tbe variable. In either case, the path of the point- selected obviously becomes a cycloid, and it is easily seen that the velocity of the point towards the lowest point of the circle is destroyed by the motion of the circle itself, while the velocity at right angles to this direction is doubled: consequently the whole velocity of the point will have, for its square, 2pz or 2^', and we have thus a perfect representation of cycloidal mo- tion under the action of gravity. But it is obvious from the fundamental hypothesis that the two points will reach the lowest point of the circle at the same time: that is to say, the descent to the lowest point of the cycloid is tautochronous. The analogy to the descent to the lowest point of a circle along its chords is in the essential point complete ? but here the motion is not along the chord but along the arc, and is compli- cated with the motion of the circle itself. In both cases it is easily seen that a medium, the resistance of which varies as the velocity j does not affect the tautochronism. ON NAPIER'S RULES* To the Editor of the " Quarterly Journal of Mathematics.' Some time ago, my friend Mr B. L. Ellis sent me the lowing remarks upon ISTapier's rules. NAPIER'S rules for tlie solution of right-angled spherical tri- angles are generally presented merely as a memoria technica / and when so presented do not exhibit the principle upon which they depend. To investigate and exhibit that principle is the purpose of this paper. LEMMA. If the three sides of a tetrahedron are right-angled triangles, no right angle being at the apex, then the base is also a right- angled triangle. Let OAB be a right-angled triangle in -4, and similarly OA C. Let the third side 0GB be right-angled in G. Then will the base AGB be also right-angled in G. * Since Napier's method of investigating his rules was independently discovered by Mr Ellis, the Editor is of opinion that this Essay, written by the Dean of Ely, will not be without interest to the readers of this volume. ON NAPIERS RULES. 329 .-. AC* + Off = AS*, or the angle at is a right angle. Q.E.D. Con. It follows from this that the dihedral angle A is a right angle. Considering the figure OABC, we observe that it is in some sort symmetrical with regard to the line OB. OB being at right angles to the plane OAC, and OA at right angles to the plane CAS; and as the dihedral angle 00 is a right angle, so also is the dihedral angle SA. Now the three lines OA, OB, 00 evidently represent any right-angled spherical triangle; and similarly the three lines JBO, SA, BO represent another, between which and the former a certain relation exists. One angle, namely, the dihedral angle SO, is common; the side ABO is the complement of the dihe- dral angle OA ; the hypothenuse OBO is the complement of the side BOO] the side OB A is the complement of the hypo- thenuse A OS i and, lastly, the angle SC is the complement of the side A OG. Hence this conclusion. If a v 2 , a s , & 4 , a Q represent the parts of a right-angled spherical triangle taken in order, and begin- ning at the Lypothenuse, then 7T IT 7T , and a 2 are the parts also taken in order and beginning at the hypothenuse of another right-angled spherical triangle. If 330 ON NAPIER'S MULES. therefore to characterize the former triangle, we introduce a new 7T set of quantities jp, such that a 1 + j p 1 = a 2 +j? 2 ==a 5+j ? 5 = ~, the original triangle being characterized by p v jp a , p s , p# p 5 the secondary triangle is similarly characterized by jp 3 , p# p# p v _p 2 ; and as the secondary triangle gives rise to a third, and so on, we thus see that every right-angled spherical triangle is one of a system of five such triangles. It is obvious, that the transformation just employed de- pends upon the circumstance that the complement of the com- plement of an angle is the angle itself, and would succeed equally if the parts of the secondary triangle, which are the com- plements of those of the first, had been any function (/) of them, provided that/ 2 = 1. Eeturning to the figure we observe, that if, instead of taking a point B in OB we had taken one, as -4, in OA, we should by a similar construction have got another of the four spherical tri- angles, which with the original one make up the system of which we have been speaking. Further, if in BG we assume any point as a first centre related to B as B to 0, and make a similar assumption of a fifth centre in A (7, the system will be .complete* But as a figure so drawn would be complicated, it is better to adopt a different plan. Since BA is at right angles to OA, and lies in the plane OAB, it is easy to represent the corelate spherical triangles, to which the system of lines of which we have been speaking gives rise. Let BAG (fig. 2) be the original triangle right-angled in A. ON NAPIER'S RULES. 331 Produce BO to B, and A to J.', making ^' = BB' = ~- ; and on Then AB'G' is the secondary triangle; and that it is a right-angled triangle may be proved, independently of what has been already said, Tby joining BA\ For since AA 1 is a quadrant, and A a right angle, BA is as well as BB 1 a quadrant, and therefore B is the pole of A'B r , and therefore B' a right angle. The hypothenuse of the secondary triangle A! is the complement of A 0, a side of the first : the angle G is common to the two triangles: GB' is the complement of the hypothenuse BG : B'A = angle B'BA and is therefore the complement of the angle ABC: lastly, the angle BAG has its complement measured by the arc AB. Again producing B'A to (9, B' being a quadrant, and similarly GA to G r , GO being also a quadrant, we obtain a third triangle 00' A and finally completing the figure (the obvious details of demonstration are omitted) we get a reentering pentagon, all the angles of which are right angles, each being the right angle of one of the system of five right-angled triangles. From this point of view it is plain that Kapier's rules may just as naturally be considered as the statement of similar pro- perties of five associated triangles, as in the usual mode, namely, as the statement of dissimilar properties in. one triangle; and thus we obtain the rationale of their uniformity. Every relation between a middle and opposites, is the relation Between two sides and the hypothenuse: every relation between a midcUe and adjacents is the relation between two angles an. Therefore the ratio of inequality between the given rotation round IT, measured by the sun's diurnal motion in his orbit, and rotation, round P, which measures the retardation, is then greatest when the resultant axis Q lies as near as possible to II, or, in other words, when P Q is a minimum. Now, 8 being the pole of the arc ZTi> it will easily be seen that the angle QZTL is a right angle and the angle QZP is an obtuse angle, while near the equinox the angle ZQP must always be acute. Therefore PQ is least when Q lies in PZ, and then II does so too. In other words, the retardation is least when II culminates at sunrise, that is, at the equinoxes. The common expression for retardation may easily be de- duced from the expression furnished by what has been said, ^.^ msinUQ * JSTow first published. A SOLUTION OF PROBLEM IX. OF THE FIRST BOOK OF NEWTON'S PRINCIPIA*. THE fundamental principle of the proof I gave Dr Goodwin f Is, that as the time varies as the increment of area, or as the square of the radius into the increment of angle, the force varies as the increment of angle, so that tlje acceleration in an instant dt is equal to ~ dO. The following application of this principle ft is perhaps more simple. Let "be the centre of force, P a position of the moving body. Take PR at right angles to PO and equal to j dt. Join RP l > P f being the position of the body at the next instant. Similarly draw P'R' at right angles to P'O and equal to PR. Join jR'P", P" being a third position of the body. Draw RQ parallel to R'P'. Therefore the angle at R being equal to the increment of the angle vector, QP is * Dictated to tlie Editor of tHs volume and now first published, t Goodwin's Course of Mathematics. 338 SOLUTION OF PROB. IX. B. I. OF THE PRINCIPIA. equal to the acceleration into dt produced by tlie central force during the instant dt. Likewise PP' is the space described with the initial velocity in the time dt. Compounding this with the acceleration we have P'Q equal in magnitude and coincident in direction with PP". Therefore in the triangles RQP f , RP'P", we have, neglecting quantities of the third order, RQ equal and parallel to JB'P', QP' equal and in same direction with P'P", and therefore the third side HP equal and parallel to in order that its aperture avoirdupois. It is therefore consid< een the calix and the ad- which is usually made in England, kin g there is the essential stood that the Eomans may have^ly from the extremity of that which they were in the h? z mem ^ calix, it passes at of pipes. It has already been ~ voman system of distribution is bore are now made by exte" -^ution through pipes inserted into of an inch bore are made r tiie rogation was made. To prevent Fifteen English feet of - nt of tlie pi? 6 was made of brass, and avoirdupois. I ha^ actecl > that f r ^ feet the diameter of the because they i1 me same as that of the calix to which it was improved r- aS one f Frontinus's complaints that the aquarii latin,? 1 sometimes insert pipes without using a calix, and thereby e an opportunity, to use a homely phrase, of playing tricks and so on. Maclaurin was, I believe, the first to correct Koenig: he has not always had credit for this priority. Thus Dr Carpenter in his Physiology, informs us that Lord Brougham, not satisfied with Koenig's determination, took into account cer- tain small quantities previously neglected, and showed that the coincidence between theory and observation was absolute. ON THE FORM OF BEES' CELLS. 355' Lord Brougham's own remark is that the deviation between Koenig and Maraldi had always been ascribed either to an error of measurement or to an inaccuracy in the construction of the cells. It is certainly not so ascribed by Maclaurin, who remarks it could only have arisen from Koonig's not having carried his computation far enough. Throughout Lord Brougham's elabo- rate discourse on the matter, one circumstance is omitted, namely, Maraldi' s statement that the angles, according to his measure- ments, were 110 and 70. Either the habit of an advocate's mind, for Lord Brougham may be regarded as counsel for the bees, or his not having read Maraldi' s paper, must have been the cause of this omission. Nor has he mentioned the advantage which Maraldi conceived to result from the angles being what they are: namely, that only two plane angles occur throughout the structure. He quotes at second-hand the opinion of Boscovich that the angles could not be measured with the supposed degree of precision, I have not seen what Boscovich says, but have no doubt that Lord Brougham has misunderstood him : for he makes Boscovich's opinion to be, that Maraldi merely deduced his result by assuming that the dihedral angles are all equal to 120, that is, he makes Boscovich accuse Maraldi of dishonesty, whereas, in reality, he probably only repeated what Maraldi in effect says : that he deduced the precise angles from assuming that the angles in the trapezium were precisely equal to those of the rhomboid, his measurements giving the same value to both, namely, as I have already said, 110 and 70 6 . The matter has been so long confused that it is worth while to quote Maraldi's own words. * Chaque base d' Alveole est fonn6e par trois rhombes presqi^e toujours ^gaux et semblables, qui suivaftt les mesures quo nous avons prises, ont les deux angles obtus chacun de 110 degrs et par consequent les deux aigus chacun de 70 degr&s*.' * * r * * * * Ces six m&mes c6ts des trois rhombes sont autant de bases sur lesquelles les Abeilles ^levent des plans qui forment les six c6t<5s de chaque Alveole. Chacun de ces c6t<% est un trapeze qui a un angle aigu de 70 degrs, 1'autre obtus de 110 degr^s, et les deux angles du trapfeze qui sont du cdt, and of the place of the liquid, are all matters of frequent occurrence in Spanish. Perhaps we ought to assume a diminu- tive form onilvilluSj the final Z being lost with the shifting back of the accent. Calibre. The word is principally used, speaking of the bore of a cannon, or the diameter of a column ; but it has another meaning which must be the original one. The word is used by masons, carpenters and workers in metal, for the tool or model which goes round anything and enables them to see if it be of the right size; its form, &c. vary in different cases, but it is always something that embraces or clips what it is applied to. It therefore seems clear that calibre is simply clipper, the latter word being borrowed from England or Holland, for I do not know that it exists in modern German. If a Frenchman pro- nounced clipper half a dozen times it would run into something not to be distinguished from calibre. Compare the English word caliper, and canif from knife; two things of the same size are said to be of the same calibre, because they would fit the same ; and hence the other sense of the word, which there is no occasion to derive from the Arabic. Caviar. "Kavidpi is certainly not a Greek word. I imagine the Greeks intended the four vowels to represent the full sound of the Italian or Prove^al u, and that the word was originally curie or curata, meaning simply cured roe. When it came to the West from Greece, the perplexing number of vowels caused the hardening of the u into v. Compare, for an analogous change of a travelled word, the French -word fashion derived SOME THOUGHTS ON THE FORMATION OF A CHINESE DICTIONARY, AND ON THE BEST MODE OF PRINTING CHINESE. IN A LETTER TO THE REV. J. POWER, M.A., FELLOW OF CLARE HALL, AND UNIVERSITY LIBRARIAN*. Cantantes licet usque minus via Iscdit eanrns. MY DEAK SIR, THE study of Chinese is hindered Tby many difficulties, the nature of which is not generally understood. The expense of printing Chinese with our kind of types is one of them. Much has been done in this matter by Breitkopf, of Leipsic, who has been employed by the American mission. Some of the results were exhibited at the Crystal Palace, but I have not been able to learn anything of the details of the analysis to which he subjects the Chinese characters. More recently Professor Brockhaus has proposed, that in order to get a complete and inexpensive Chinese dictionary, we should have recourse to Lithography. Even so the undertaking would be a great one, and it is very desirable that such a dictionary should Tbe arranged in the most convenient and useful manner. Brockhaus proposes to follow the ordinary Chinese arrange- ment ; according to which the characters are distributed under 214 radicals, remarking in favour of doing so, that it is conve- nient to be in accordance with the Chinese practice f. No doubt this Is true; but the objections to this mode of proceeding are considerable. In the first place, it is essentially unscientific ; the radicals are chosen on no definite principle. They seem to be, as Kant somewhat too boldly asserted of Aristotle's categories, aufgerafft, gathered up at random from all sorts of sources. In the second place, the relation of the characters to the radicals under which they stand, is arbitrary and uncertain. In some cases the relation is one of mere resemblance, in others * Previously printed for private circulation. t The system of 214 radicals is not of high antiquity, and has not been always followed since its first introduction. THOUGHTS ON THE FORMATION, rfr. 401 the radical is a component part of the character, while occa- sionally it is very difficult to see what the connexion between them is. Again, there is no principle of arrangement under each ra- dical, except according to the number of strokes of which the character is composed; and in a complete Chinese dictionary there would be on an average 200 characters under each radical, and in some cases more than 1000. The result of this is, that it is necessary to have a supplementary index of characters, of which the radical is difficult to recognize, and in this there is no principle of arrangement, except the number of strokes. Gallery, of whose improvements I am about to speak, after mentioning the way in which Chinese dictionaries are con- structed, remarks, that it is not wonderful that so few persons attain to a knowledge of the language, and they, only after years of painful labour. In fact, to be able to use a dictionary, is a great part of the whole business of learning to read Chinese. The principle on which Gallery proceeds, he derived from his instructor, G-ontjalves, whom he speaks of as the ablest of Chinese scholars, and who has published several works, in which it is followed, though he has nowhere fully explained it. Most of these works are in Portuguese, which is perhaps the reason why they seem not much known. Gallery's own work is in a mixture of Latin and French. It was printed at Macao, and it is said that most of the copies were accidentally destroyed. It consists of two parts; the first introductory, the second, a dictionary of perhaps 13,000 characters, arranged according to his own method, in which the principle of Gon and one of the words for /, are both pronounced eu, and the character for the latter is the same as that for the former, with the addition of that for mouth. A similar instance has given rise to one of the innumerable foolish things which are said and repeated about the Chinese. Ho happens to mean corn, and concord or comfort, and the character for it, in the latter sense, is the same as in the former, with the same addition as in the preceding instance. The mouth is here used diacritically, yet people have been found to s'ay, that the grossness of the Chinese is shown by their having a character for happiness which indicates that they have no higher idea of it than mere eating. It so happens that the same sound ho also means a child's crying, and in this sense also is represented by corn and a mouth differently placed in relation to one another. What authority can there be for an interpretation in the former case which is obviously inapplicable in the latter ? This, however, is a less offensive error than those into which the early missionaries fell, in seeking for the doctrines of Christianity in China. I can only allude to their interpretation of the word yang. If a Chinese were to say that the English are a particularly selfish people, because the same symbol denotes unity and per- * Oases might be pointed out in which both elements are representatively pho- netic, so as to form a sort of reduplication. In some cases it may be said thafc both elements are at once phonetic and logical, Thus assuming that the two ele- ments of pi, to compare, are both pi spoons, the idea of comparison results from their similarity. In other cases the same element is phonetic and logical. Thus tsien, a small coin, consists of t&icn, small, and kin, metal or coin, Compare our word ' groat,' 406 THOUGHTS ON THE FORMATION OF sonality*, lie would only imitate tlie example of errors long gravely maintained by European scholars. No more effectual mode of getting rid of these errors presents itself than making a complete analysis of all compound characters, in order after- wards to recognize and classify the different principles which hare guided their formation. Eemusat's remark is perfectly just, that the Chinese characters are formed in a variety of ways, and that nothing Tbut confusion can result from any attempt to analyse all on the same principle. Chinese etymology consists, as Humfboldt has observed, of two parts, that of the characters, and that of the spoken language. Both parts involve great difficulties, and as yet neither has been treated scientifically. The former part is particularly attractive: there is no more amusing book than a Chinese dictionary. Perhaps my saying so may remind you of the painter's reflection, c chfe dolce cosa & la perspettiva.' One instance may be enough to show you the sort of interest I mean. The character for heart, which expresses generally all mental operations, combined with that which represents an enclosed and divided field, means to think or consider : we have here a graphic representation of the Latin * contemplor,' formed, as there seems little reason to doubt, from 'templum,' in the sense in which the augurs used the wordf- A very interesting part of the study of the Chinese character would be the comparison of it with Egyptian hieroglyphics, not in order to revive the old notion of an historical connexion between them between the flowery region and the lands of the lotus and the papyrus but in order to see how similar problems have in the two cases been dealt with. It has been said that 'Thoth was wiser than Fo' (Fo-Hi), which may be true, but still the comparison is worth making. It is curious, that while in Egypt the feet or legs seem to be the symbol of activity, so as to give a verbal signification to the symbol with which they are associated, or, in some instances, rather grotesquely joined, the hand should often serve the same purpose in China. Other points of analogy might doubtless bo * Guesses at Truth. "t Grimm's derivation of templvm, from the same root as tepeo, malting It refer to the sacrificial fire, seems open to more than ono objection : in tlio firwt place, the augurial sense of the "word appears to be the primary one, and this connects It with r^fjoffof ; and in the second, it would then be particularly strange that the house of "Vesta should not have been a templum. A CHINESE DICTIONARY. 407 Indicated, though the less complicated forms of the Egyptian characters can hardly admit of phenomena so various as those which are presented by the Chinese* In the latter, for instance, "i believe we might trace that curious principle of language, which, for want of a "better name, may Tbe called i the principle of intelligibility,' of which we cannot have a better instance than the conversion of 'mandragore' into 'main de gloire;' I mean, that in Chinese groups of strokes, originally forming only a part of a complex picture, have probably, in some cases from accidental suggestions, shaped themselves into the likeness of other simpler characters. With regard to the etymology of the spoken language, such an arrangement as Gallery's is of the greatest value. When we find different sounds associated together under the same phonetic element, we may, special cases being set aside, conclude that they are, with reference to the Chinese organs of speech, cognate sounds, and thus establish the laws by which our investigations are to be guided. The change of Ch into T, which we are familiar with as the peculiarity of the Folrien dialect, is one of the most obvious phenomena thus made manifest*. To return from this digression, to the formation of an index to a Chinese dictionary. I should propose to form a list of all simple characters, and of all in which there could be any serious doubt or uncertainty as to their analysis. The latter would not be a very large addition to the number of the list. Gallery's own estimate is that his 1039 phonetic elements result from about 300 primary ones. A small number, and those capable of further analysis, of his classifying characters do not belong to his phonetic list; but if we say, that all the elements he employs cannot much exceed 300, we shall not perhaps be far from the truth. Taking account of omissions, accidental and otherwise, we may perhaps say, that 500 elements would appear in our list. I admit this seems a small number; but Gallery speaks of having gone through almost all the Chinese characters, and having omitted * Compare with this the change of tr into cr, in craindre from, foemere, vewtre (Hymn on JEulalia) from vinctre, &c. The same thing is seen in the English cor- ruption of ask into oat. Is not triticum formally equivalent to KpiBy ? Compare also the Greek and Latin names for Carthage j Oxniantown and Ostmantown, (Worsaae, Danes and NortJimcn in England.) 408 THOUGHTS ON THE FORMATION 01? only such primary characters (lie calls them by perhaps a better name, 'indivisible or fundamental characters') as were rarely found, or useless. To allow 200 for such omissions seems suf- ficient. However that may bs, I should propose the formation of such a primary list, and its being printed in a tabular form, as a frontispiece to the index. Gallery has done this with his 1039 phonetic characters, and they are all visible (and of a suf- ficient size for clearness) at one opening of the book. These being arranged in the manner already mentioned, the remainder of the index is to be placed under these, as keys or headings. Under each I would place, in order, all with which it combines : first, all the simple characters ; then, all the binary characters, and so on. No doubt there would be a good deal of repetition in this ; every compound character would be entered twice at least; those consisting of three elements, three times; and so on. But the advantage of being able to find any character you want with comparatively little trouble, as soon as you have recognized one of the elements it is composed of, seems to out- weigh this disadvantage, and perhaps about 80 quarto pages would be enough for the index to a dictionary of thirty or forty thousand characters, It is to be observed, and this I think a very important part of the plan, that I do not propose to use compound characters at all. The original 500, or whatever the number may be, would be the whole number of characters used, and therefore of types required. You are reading, we will suppose, a Chinese book, and come to a character you do not know. Seeing that It consists of woman, mouth, and heart, you look for it under any one of these three characters, and in a little while find the other two grouped side by side with a number which enables you to refer to the body of the dictionary. There is but one character in the language made up of these three elements, and therefore in order to recognize it without ambiguity, it is not necessary that you should actually sec it before you in the index, or that you should even be told there what you already know, that, to speak heralclically, woman oc- cupies the dexter chief, and mouth the sinister, " Some cases undoubtedly there are probably only few in which the same elements, differently arranged, form different characters; but A CHINESE DICTIONARY. 409 nothing can be easier than to devise diacritical signs ; by which the same group of characters should be made to refer, without possibility of mistake, to the different compound characters. Take a simple instance already noticed. Under corn you find mouth, accompanied by an arrow pointing to the right, and again a mouth with the arrow pointing upwards. The reference to the first would be, to ho in the sense of comfort^ and to the second to ho in the sense of crying ; for in the first case mouth stands to the right of corn, and in the second above it. This principle once admitted, namely, that a character may be as clearly recognized by means of its elements alone as if a fac simile of it were given, may of course be applied much more widely than merely to forming the index of a dictionary. It seems to furnish the solution of the chief difficulty by which the study of Chinese has hitherto been impeded. For we thus get a mezzo termine between the unintelligibility of Chinese written with Roman characters, and the impracticable expense of a complete fount of Chinese type. Even if we had the 3000 elements**, which the ingenuity of Breitkopf has devised, where should we, here in Cambridge at least, find a compositor suf- ficiently learned to put them together ? The reason why so many more elements are required to imitate compound characters, than are necessary, if we content ourselves with simply representing them, is of course the varia- tion in size and shape, requisite in order to give uniformity in these respects to the compound character. This uniformity of size and contour is a matter of Chinese taste with which in books intended for European use we need not trouble ourselves. It is, by the way, an inconvenient taste even for the Chinese, because in order to make complex characters distinct, the simpler ones must be unnecessarily large. If it be said that the Chinese would never become accustomed to characters of the proposed kind, we may answer that even if this be so, the necessity of printing in Europe books intended for Chinese use is not very obvious. Whatever may have been the case formerly, there can hence- forth be probably no difficulty to hinder the printing whatever is intended to be read in China, at presses (which, by the way, * Am I right in thinking there are 3000 type elements ? or are there only 3000 punches ? 410 THOUGHTS ON THE FORMATION OF is an incorrect phrase in speaking of Chinese printing) esta- blished in the towns to which Europeans have now free access. And it must be remembered that we could scarcely hope to pro- duce in Europe what the Chinese would account a handsome book. The softness of impressions from wood can hardly be imitated with metallic type, and Chinese paper cannot, I believe, be used in our printing presses. A collateral advantage, resulting from what is now proposed, would be the facility of learning to read Chinese. The difficulty Of analysing the characters would be removed, and when once a student was able to read a book printed in the new method, the transition to the usual characters would not cause more dif- ficulty than Greek contractions, or than the ligatures in Sanscrit, Another advantage would be that as the characters would follow one another in regular order, accompanied only by brackets to form them into groups, and by a few simple diacritical signs, any ordinary compositor would be able to set them up. Pro- bably it would not be found very difficult to distinguish the phonetical elements by printing them with red ink, which to beginners would be a great assistance. By similar means we might distinguish the same character, according as in any sen- tence it presented itself as a noun or as a verb. Brockhaus has proposed a different way of printing Chinese for European use, namely, in Homan letters with a numerical reference under each word to its place in the dictionary he wishes to see made. There are two or three objections to this plan. In the first place, difficult as it is to remember Chinese characters, it would be found much more difficult to remember the meaning of a number, even with the help of the pronunciation, because num- bers give very little for the mind to fasten on, and can never be exclusively associated with a single class of ideas. Imagine the difficulty of remembering the plot of a story the persons in which were denoted only by numbers. Again, in all questions relating to what I think must, by and bye, form an interesting part of comparative philology, namely, the theory of the Chinese characters, such a plan would be useless, even if we could suppose that all Chinese scholars agreed to use the same dictionary. Lastly, this method would form no introduction to the study A CHINESE DICTIONARY. 411 of works printed in the Chinese character, a class to which the great mass of Chinese literature must always belong. A man might give years to the study of this Stratford atte Bo we Chi- nese without being able to read the commonest characters. A more radical reform has sometimes been proposed, namely, simply to print Chinese In Eoman letters. Why are not the absurd Chinese characters laid aside ? has been asked in much the same tone as the question one occasionally hears, of why legal terms and forms should be used in conveyances ? Get rid of these, it is said, and any deed might be written on a single sheet of paper. The answer in the two cases is much alike. If it had been possible during the last six hundred years to enforce brevity in legal instruments, not only the practice but the theory of conveyancing would be very unlike what they now are. The complicated relations which have grown np amongst us, the various subtle modifications of which the idea of property has been found susceptible, could never have been developed on such a system. Not only the outward form, but that which the form represents, would have been different. Our thoughts, and the mode in which we express and record them, act and react on one another. The influence of writing on the history of language, which has been made the subject of an interesting essay by William llumboldt, has been greater in China than anywhere else. The hand and eye have, so to speak, brought into subjection the voice and ear ; the reason of which is to be sought partly in the original nature of the language, and partly in the general dif- fusion of education. The language of China, especially the written language, is in many respects what it is, in virtue of the character, which we cannot now give up without introducing ambiguity and con- fusion*. To a certain extent the IWman letters may be used, and this has already been done, as for instance by Morison and Gon9alvcs. The dialogues of the latter are particularly valuable, from giving both the Mandarin and the Canton pronunciation. * The best plan by which indeed many of the difficulties would be removed, would be to write all compound characters like fractions, I mean with the pronun- ciation of the whole character above, and those of its elements below a horizontal line. Lin, a wood, for instance, would be denoted by . 7 mou, mou 412 THOUGHTS ON THE FORMATION OF So far as the question, as to giving up the Chinese character, can be decided by authority, it seems sufficiently settled. I may refer particularly to what is said on this subject in Mr Kidd's work on China. One kind of influence exerted by the character is sufficiently peculiar to deserve mention. We have many words whose meaning has been changed in consequence of a mistake caused by accidental resemblances of sound. Such, for instance, has been, at least in popular use, the case with demean. Johnson even thought the secondary meaning had the authority of Shakespeare. Again, there are words whose meaning has been influenced by i juxta-position,' by their occurring, so to speak, in contact with others. Such are implicit, and buxom*. But both kinds of influence may concur in Chinese. Not only the sound of the word, but also the way in which the sound is expressed, by bringing the character into constant association with another, may influence the meaning. Take as an instance the character already noticed, composed of hearty woman, and mouth. The two latter characters alone form a binary character, pronounced ju, and meaning even as, sicut* This binary character is the phonetic element of the ternary one, of which it forms the upper part. The latter is pronounced shu, and means goodness or kindness. But it is related that Confucius taught that this word is the summing up of all mo- rality ; that it means the state of mind in which a man interests himself in the happiness of others, even as in his own. This development of the meaning of the word was, it is pro- bable, merely the result of an acccidental coincidence of sound, and of the selection of the one character to be the phonetic element of the other. But, error or not, this opinion as to the meaning of the word has perpetuated itself; and what in this case ia referred to the authority of Confucius, has probably happened tacitly in many others. One of the difficulties in making a Chinese dictionary arises from the number of compound words, that is, words each of which means something separately, but when grouped together express a single idea. Kemusat went so far as to say, that the compound word WEB polysyllabic, and that each character merely represented a syl- lable. This question is scarcely worth the attention which has * So too in German EJie, A CHINESE DICTIONARY. 413 "been given to it ; "but the important point is, that the unwary scholar frequently endeavours to give separate translations to each element of compound words. These are given in the "best dictionaries ; but there may "be some difficulty in making a list of them complete and easy of reference. In printing it would be well, I think, to connect the elementary characters Tby a hyphen. The matter is so peculiar that you will not object to my giving you an instance of the errors it is apt to produce. In Eemusat's version of one of the 'Four Books,' as they are called, of Confucius, it is said that Confucius lived in accordance with the seasons and with the earth and water. The meaning of this is certainly not clear ; but i water-earth ' simply means ' climate. 5 The habits of Confucius were not in accordance with the earth, whatever that may mean, nor with the water, but simply, which is quite intelligible, with the climate. The instances which in the Notes to Humboldt's letter to him Kemusat quotes from other languages ('horseman' is his English instance), are not quite parallel, for though no gram- matical form indicates the relation between their parts, yet ideally one of them is a substantive, and the other a modifying adjective: whereas in Chinese the compound word is a new formation, of which the meaning is suggested only by those of its parts. 'Elementa guodammodo man exit in composite*,' we cannot define the matter more precisely. We see here, as in the formation of the Chinese characters, and in the structure of the language, the tendency to merely external union. There is contact and combination, but no interpenetrating compound growth, and the whole resembles not a picture but a mosaic. The same remark might be made as to Chinese style, which is all compact of set phrases and antitheses. I cannot enter on all the matters of detail connected with the Index, of which I have endeavoured to give you an outline. My ideas of them are of course very imperfect. Gallery's merits, with respect to Chinese lexicography, are doubtless great, both in his exposition of the ultimate dissection of the characters, and in showing, more clearly and fully than had been done before, the presence of a phonetic element in the great majority of characters. His assumption, that a set of phonetic elements were deliberately * S. Thomas Aquinas de Princfyriis. 414 THOUGHTS ON THE FORMATION, Ac. and simultaneously invented, is unphilosophical, and seems to have led him into his principal error, that in all compound cha- racters one element is phonetic. This error is decidedly opposed to competent Chinese autho- rities, and would place us, if we adopted it, in the dilemma of either rejecting obviously correct analyses, or of setting aside the laws by which the affinities of sounds are governed. I must here conclude these remarks. They are the result of your kindness, which has led to my seeing the works on sub- jects of Chinese literature recently added to the Library, and has thus recalled my thoughts to matters which my increasing illness had made me lay aside. You know the circumstances in which I write, or, to speak more accurately, dictate. Vive et vale. Yours very truly, E. L. ELLIS. March 17, 1854. VALUE OF ROMAN MONEY*. GrRONOVius's estimate of the value of Roman money is vitiated by two principal errors : his doctrine that 100 denarii went to the pound weight of silver, a doctrine connected with his theory that the proper and direct meaning of sestertium is two pounds and a half of silver, but which is contradicted both by testimony and by the denarii, which like the bricks in Richard II. are alive to this day to witness to the contrary ; and his confounding the pound Troy with the Roman pound. The errors tend to balance, one making the denarius too little in value, and the other making our currency of too small value; but his result is of course mere haphazard, to say nothing of his neglecting the question of alloy. The basis of the calculations in the Dictionary of Antiquities is much more satisfactory, but the calculations themselves are wrong. The articles Sestertius and Denarius do not take into account that our shilling circulates as a counter above its in- trinsic value. The value of the denarius is determined by com- paring its weight of fine silver with that of the shilling. Now as our coinage since 1816 is at the rate of 66s. to the pound, the result is the same as if the price of silver had been taken to be 66J. per ounce standard, which certainly is not its real price. The rate of coinage was purposely fixed above the variations of the bullion market to prevent melting. Sixty- two pence is the price commonly assumed in calculating the par of exchange, and is rather a large average price. Taking the data given in the article Denarius, and this price of silver, the denarius of the end of the Republic is worth (not 8*6245df. as it is there made) but 8"099c?., or in round numbers, not S^d. but Bd. The error will be nearly the same in the value of the later denarius. * Journal of Classical and Sacred Philology, Vol. r. p. 92. 416 VALUE OF ROMAN MONEY. The value of the sestertium resulting from the value of the denarius which I have quoted is 8. 19s. 8y some error of calculation it is reduced to 8. 17,9, Id.; the real value is 8. 85. S^d. 9 so that the two mistakes, like Gronovius's, tell against one another. It is curious that the later value of the denarius gives the sestertium 7. 7s. 7j<#., a sura in 7 as the other in 8. In the article Aureus, the writer says that the sovereign con- tains 113*12 grains of fine gold. It really contains (neglecting the third place of decimals) neither more nor less than 113 grains. The result is that he gives the aureus as l. 3s. Id., and a little more than a half-penny, instead of as nearly as possible l. Is. 2d. The following is an outline of my calculation : Required the price of 60 grains of silver, ||ths fine, at 62 d. per ounce, standard. (1 ounce = 480 gr.) x = 60 If (Standard being ft ths fine.) -r> , . 31 x 29 Keducmg, ^ = "3^37' 31 x 29 = 30 2 - 1 = 899, 3 x 37 = 111, x = 8'099c?. = value of early denarius, 250 denarii = 1 sestertium, 240 pence =1; 809'9 101*23 .'. value of sestertium = - = - = 8*435, = 8. 8s. Sd. 4 or 8. 8s. 8^d. nearly. The later denarius is 52'5 gr. or 8'75 of the earlier, and the sestertium is in the same proportion. THE COURSE OF MATHEMATICAL STUDIES* THE seventh query f, so far as It relates to the limits beyond which, it is not expedient that the undergraduate course of ma- thematics should extend, seems naturally to form a part of a more general question, namely, how the whole time given to the study of mathematics may "be most advantageously employed , In order to discuss this more general question, it is necessary to consider on what grounds the study of mathematics is made to form part of our system of education, I, The grounds are two-fold: mathematics are studied as ancillary to natural philosophy and as a means of training and developing the mind. In the latter point of view they are chiefly valuable, because they deal with necessary and not con- tingent truth $. Of every necessarily true proposition which the * Cambridge ffniwnity Commission, 1852. Evidence on Mathematical Studiea and Examinations, p. 222, f The seventh query is : Would you be disposed to recommend the limitation of some of the subjects included in the present range of the examinations, for Instance to omit such propositions and applications of the Calculus of "Variations, of the theories of Definite and Elliptic Integrals, of the Planetary and Lunar theories, of the theories of Heat, Electricity, and Magnetism, of the undulatory theory of Light, a& require for their treatment a very refined and laborious analysis? Might such higher mode of treating these subjects be advantageously reserved for examination for special prizes at periods subsequent to the Degree of B.A. ? Would not the concentration of the attention of Students upon a smaller number of sub- jects, and those restricted within narrower limits, tend to increase the accuracy and raise the character of their knowledge, and to bring their instruction more completely within the grasp of the public and recognized teaching of the Univer- sity? $ This applies to mixed as well as to pure mathematics j the necessity of the conclusion being, however, in the latter absolute, and in the former hypothetical, - rd ! iiiroOfaws foayKcuoy, 27 418 THE COURSE OF MATHEMATICAL STUDIES, mind distinctly apprehends as such, tlie contradictory is seen to Tbe inconceivable ; this inconceivableness of the contradictory "being ex parte mentis the criterion of necessary truth. Never- theless, although when we think of any simple proposition in arithmetic or geometry, we perceive not merely that it is true, but that it must of necessity be so, this is nowise the case with respect to all demonstrated or demonstrable results. The in- tuition, so to speak, of the ablest mathematician is confined within a narrower circle than that of the truths which he can prove. He may satisfy himself of the cogency of each step of the demonstration, and yet the essence of the conclusion the fundamental principle of its truth remains unseen. The on is manifest, but the Sion obscure ; and consequently a proposition contradictory to that to which he ha& been led does not appear to him an absurdity, but simply an untruth. It might, for what he sees, have been true, though he knows that actually it is not, and thus while he is aware that his conclusion is true neces- sarily, yet still it seems as if it were so only contingently and as a matter of fact, the demonstration appearing assensum con- stringere, non rem. In a word, his conception of the matter is still imperfect. But between this state- of mind and that which is produced by the contemplation of any elementary proposition, there is no- fixed or definite boundary. Every one who has really studied mathematics must remember cases in which, after long and patient thought, the reason of the truth of a propo- sition, with the demonstration of which he may have been acquainted for years, has- seemed to dawn on him ; the propo- sition thenceforth becoming., as it were, a part of his own mind a matter about which he is no more capable of doubting than about the primary conceptions of form and magnitude. The mind thus brought into nearer, if not immediate, contact with necessary truth is conscious of its own- development ; and herein, I believe, resides the special benefit to be derived from the study of mathematics, a benefit, that is,, distinct from the exercise of patience and attention which it undoubtedly requires, but which is required also in other pursuits. The study of mathematics is especially valuable, not because it gives the Student practice in ratiocination but because it enlarges the sphere of his intuition, by giving him distinct and conscious possession of truths which lay hid in his conceptions of figure, number, and the like* But THE COURSE OF MATHEMATICAL STUDIES* 419 In order to tins kind of mental development, it is necessary not only that the Student should master the successive steps of the demonstrations set "before him and retain them in his memory, "but that his mind should become imbued with their spirit and essence. His real progress therefore is not to be measured simply by the extent of ground over which he has passed : it varies also according to the degree in which he has approached towards a complete intuition into the results which he is able to prove. I believe that this principle ought to be our guide in ex- amining the merits and defects of a course of mathematical study intended to form part of a liberal education. But the connexion of natural philosophy with mathematics must, to a greater or less extent, modify the conclusions to which it would lead us. II. It would be impossible to trace in detail the conse- quences which appear to follow from this way of considering the subject. They may be classed under two heads, the choice of subjects, and the choice of methods. With respect to the former, my impression has long been that a good deal might be omitted which now enters into our course of reading, not only without impairing its utility, but with positive advantage. A more rigorous subordination of details to fundamental principles would not only save the Student's time, but would make the principles themselves be more clearly apprehended. Everything received into our course ought to justify its admission there, either by its own importance or by its connexion with something more valuable than itself. Mere exercises of industry and in- genuity, long numerical calculations, complicated processes of algebraical reduction, tricks of transformation for the evaluation of integrals and the solution of differential equations, and the like, may all be accounted comparatively useless. These things may be impressed on the memory but will hardly long remain, there, and meanwhile are felt to be rather a burden than an acquisition. So in mixed mathematics, many of the approxi- mate formula* in optics, the less important astronomical cor- rections, detailed descriptions of philosophical instruments, &c., might all be advantageously laid aside. Not that these things are npt worth knowing, but that they do not properly belong to such a course of mathematics as we are considering in which the chief end proposed is a clear insight into fundamental prin- 272 420 THE COURSE OF MATHEMATICAL STUDIES. ciples. In general it may "be said that formulae of approxima- tion are unsuited to the end we have in view ; they give little or nothing on which the mind can rest : their value resides in the practical application which is to be made of them, but which the Student never makes. It is the predominance of approxi- mate results which renders the lunar and planetary theories unsatisfactory portions of the Student's course. They are, how- ever, by no means to be omitted or curtailed, and with respect to the latter, the evil might be lessened, though not without some inconvenience, by giving more prominence to the general theory of the variation of parameters as we find it in the M6- canique Analytigue and in some of Poisson's memoirs. Sir W. Hamilton's essays in the Philosophical Transactions, and those of Jacobi in Oelle's Journal^ might, perhaps, give some ad- ditional materials for the formation of a course of study on this part of natural philosophy. III. Secondly, as to the choice of methods, and especially as to the preference to be given to geometry or to analysis. Ever since the introduction of the modern analysis into Univer- sity reading, there have been complaints of its having super- seded the older methods and traditions of the Cambridge system. Those who favoured its- progress affirmed, and most truly, that by its aid the Student advances faster, and goes farther, than he could do without it ; he gains in fact more knowledge of the subjects set before him. But this argument had little weight with those who held that not the knowledge but the process of acquiring it the training and discipline of the mind was the thing chiefly to be thought of. It has been said that if information merely is the end in view, mathematics have less claim on our attention than many other things, and that most of the arguments in their favour cease to be applicable if geometry is discarded or disparaged. Of late these views seem to have gained ground in the Univer- sity : their influence may be traced in the recent legislation on the subject of mathematical honours. The principle on which this re-action against the newer methods is chiefly based, namely, that the mind of the Student ought to be as much as possible conversant with fundamental conceptions is, I think, perfectly correct. But it does not follow that analytical methods ought to be discouraged. Demonstrations may be geometrical, and THE COURSE OF MATHEMATICAL STUDIES. 421 yet in a high degree artificial ; and first principles may be lost sight of in a maze of triangles, -no less than in a maze of equa- tions. Though in mathematical investigations there is no royal road, yet there is a natural one, that, namely, which enables the Student, as far as possible, to grasp the natural relations which exist among the objects of his contemplation. If this route be followed, it matters but little whether the reasoning be expressed by one set or kind of symbols or by another in plain words in short hand or algebraically. To change the notation is merely to translate from one language into another. It is common to find persons in Cambridge and elsewhere who insist upon it that geometry is geometry, and analysis analysis; but it may be doubted whether this notion of an absolute separation between the two things is not the result of a want of familiarity with either. It seems to be supposed that if a mathematician treats a problem geometrically, he has to think about -it for himself, whereas if he treats it symbolically, the symbols think for him. Perhaps it may be said that the fact of there being any tendency towards so childish a notion is in itself evidence of the mischief produced by the use of sym- bols ; and certainly if symbols were never used, the notion could not exist. But neither could it exist if they were rightly used and rightly understood. The phrases which I believe may now and then be heard from some of our younger analysts, such as fi< the irrefragable a?," and " putting it into the mill," for ex- pressing the conditions of a problem symbolically, show perhaps that those who use them have but a half understanding of what they are doing. But this evil is not to be remedied by discou- raging the tise of symbols. That our methods should be geo- metrical is not by any means essential ; they ought to be natural, and it has been too hastily supposed that they will necessarily "be so if symbols are excluded: whereas it is not by precise adherence to any particular mode of expression that we are to bring the Student to a familiar apprehension of the principles of what he is engaged on. This is to be accomplished rather by a "melange heureux de synth&se et d' analyse," to use the words of a great master in the art of which he speaks, than by imposing either on teacher of students any unnecessary re- straints. Let us consider the question more generally* When the conditions of a problem have been stated* the solution may 422 THE COURSE OF MATHEMATICAL STUDIED be evolved from them by innumerable sets of combinations. It does not often occur that even a practised mathematician divines the simplest and the pest. His choice among the routes which he may follow is determined by an infinity of circum- stances, and more especially by the way in which the conditions have been expressed. " Words shoot back on the understanding of the wisest," and so do symbols ; and if the conditions of the problem, whether geometrical or mechanical, or, if we will, logical, are expressed by means of algebraical symbols, he will, in all probability, not deal with them as he would have done had they been expressed in common language : the reason of which is, that the combinations and inferences which are the most obvious when one mode of expression is employed, cease to be so when it is replaced by another. Hence and from, other causes arises a variety of forms of demonstration, often, it is true, perplexing, and yet, if attentively considered, full of in- struction. For as to a mind which has attained to a perfect mastery of the subject, and by which,. therefore, the connexion of the data of the problem, with its solution is perceived as by intuition, all the demonstrations appear to be in their essence identical, different modes merely of presenting the same con- ceptions, so contrariwise the comparison of the different demon- strations by which a given result has been established, tends to make us recognize the grounds of their essential unity. It is not by merely fixing in the memory the successive steps of a single mode of demonstration, or even by studying several, if we allow them to remain in the mind as distinct and hetero- geneous processes of thought, that we are to acquire a complete insight into the subject in hand, but by a more discursive method, by inquiring perpetually into the grounds and reason of what we are doing, by interpreting our symbols and follow- ing the train of geometrical or physical conceptions to which their interpretation leads, and again by retracing our steps and passing from general considerations or purely geometrical rea- soning to the technical language of symbols. Every change of form should be suggestive of a new aspect of the subject, and it is thus that the simplest way of considering it is to be dis- covered. In confirmation of some of the opinions which I have been endeavouring to express, I may refer to Poinsot's admirable THE COURSE OF MATHEMATICAL STUDIES. 423 tract on the motion of a rigid body. He has there shown, with great felicity both of thought and of expression, that the art of combining symbols is by no means the whole of mathematical analysis ; that we must join to it the art of interpretation, and that in many cases the essence and meaning of a result are scarcely more obvious in the equations which express it than in the original " mise en equation." It did not belong to his purpose to point out that on the other hand geometrical are not necessarily natural methods of demonstration ; but that there is a real distinction cannot, I think, be questioned. If it were not so, if the Student felt that by studying a subject geometrically he acquired more real insight into it than he could else have got, geometry would be more popular in the University than it now is. In truth, the difficulty of remembering many geometrical demonstrations is in itself a proof of their artificial character ; for that which the mind has once completely grasped it does not easily forget. What we want is the introduction of a freer and more liberal method*, and especially the abandonment of the notion that anything is gained by a rigorous separation of geometry and analysis. It is this which for the most part makes our geometry pedantic, and many of our analytical text books dry and sterile. If it be asked how such a change could he brought about, I am inclined to think it could only result from a change in the opinions of those on whom the character of our studies chiefly depends, the Professors, Tutors, and Examiners. It could hardly be made the subject of direct legislation. IV. The ,same remark would apply to the subject more especially suggested by the seventh query, namely, the proper limits of an undergraduate course of mathematics. " In every branch of mathematics there are parts which from their abstruse- ought not to be introduced into the degree examination; * What may be called tlxe new geometry seems to be little studied in the Uni- versity,* yet the method of which it makes so much, use, namely, the generation and transformation of figures by ideal motion, is more natural and philosophical than the (so to speak) rigid geometry to which our attention has been confined. It has been well said that the differential calculus is the symbolical expression of the law of continuity, and probably the principles of the calculus would be better understood if notions connected with this law -were introduced at an earlier period of our course. Seo, on the impossibility of severing our conceptions of space from those of time and motion, Trendelenburg's Logiscke 424 THE COURSE OF MATHEMATICAL STUDIES, but it would be found difficult to trace any precise and perma- nent line of demarcation by which these might be separated from the rest. In the progress of every science its methods tend to become simpler ; and to refer especially to one of the subjects mentioned in the query, namely. Electricity, I may remind the Commissioners of the great simplification which the theory of induced electricity has recently received. Professor William Thomson's theory of electrical images has made, so to speak, elementary, problems which previously required a " very refined and laborious analysis." This theory, if electricity is to be studied at all, would now almost of necessity form a portion of the undergraduate course. It is, however, sometimes doubted whether not only electricity but also the cognate theories of heat, light, and magnetism ought not be excluded from the degree examination. I confess to being unwilling that the ter- minus of our mathematical studies should be made to recede, and am disposed to believe that with the changes suggested in the earlier part of these remarks, sufficient time would be found for these subjects to be, up to a certain point, satisfactorily studied. They now engage so much of the attention of scientific men that it seems particularly desirable that the highest class of Students should leave the University in a condition to follow their progress and development. A young man will not wil- lingly forget what he learned at Cambridge if he finds that it enables him to understand the discoveries and researches which are now going on; on the contrary, he will probably always retain, at least, an interest in scientific matters. This advantage would in many cases, perhaps in most, be lost if the subjects in question were not studied until after the B.A. degree. Few even of our best Students could then be induced to devote them- selves to fresh mathematical studies, notwithstanding the influ- ence of any system of prizes either for mere proficiency or for original research. Of all such prizes 4 * it may, I think, be said that they could not be made to form a natural element of the system of University education. I do by no means deny, or even doubt of, their utility ; but we must remember that a University may be considered in two * The remark does not apply to the Smith's Prizes, the examination for these prizes being in effect a kind of sequel to the degree examination, and, therefore, not requiring a distinct course of reading. THE COURSE OF MATHEMATICAL STUDIES. 425 points of view, -as a seat of learning and as a place of education. Much may be done by means of prizes to encourage learned men in the pursuit of the kind of knowledge to which they have especially dedicated themselves ; but as things are, and perhaps as they ought to be, even a liberal education must end about the age at which the Bachelor's degree is commonly taken. And it may further be affirmed, with much show of reason, that those whom after that epoch circumstances still permit " inter silvas Acadcmi quaerere verum," may with advantage, so far as the symmetrical development of the mind is concerned, turn from mathematical to other studies. V. With respect to the subjects mentioned in the seventh query, I have already made some remarks on the lunar and planetary theories, as well as on electricity and the kindred branches of physics. There remain, therefore, only the calculus of variations and definite and elliptic integrals. Of these sub- jects the first seems not unsuited for University reading. It involves important principles and admits of important applica- tions. From its connexion with the theory of conditions of integrability it forms a natural sequel to the integral calculus ; and, on the other hand, if the planetary theory were to be studied in the manner which I have suggested, some previous acquaintance with its principles would be indispensable. In favour of definite integrals there is not so much to be said, many of them depending for their evaluation on particular artifices, which must uselessly burden the Student's memory ; and if in an examination he attempts to determine the value of one with which he is not already acquainted, he will often only waste time and ingenuity .to no purpose. Still certain definite inte- grals must be known, and the theory of definite integrals of periodic functions is especially important from its connexion with Fourier's theorem and the development of discontinuous functions. The difficulties of this theory have, I think, been sufficiently removed to justify its introduction into our course ; and it is not to be forgotten that no other step in the recent progress of analysis has exercised so great an influence on ma- thematical physics. The general theory of elliptic integrals is too extensive and too abstruse for University reading, and the study of isolated propositions almost useless. But Abel's theo- rem ought to be studied as the fcatural development of the theory 426 THE COURSE OF MATHEMATICAL STUDIES. of symmetrical functions, nor is there any difficulty in the de- monstration by which an intelligent student can be embarrassed. Likewise Abel's method for the division of the complete elliptic function might with advantage replace Gauss's solution of the binomial equation. It includes this solution as a particular case, and from its generality is far more intelligible: Boscovich's doctrine, that the more generally a subject is considered the more easily is it understood, being for the most part true. These instances 3 in which portions of the theory of the com- parison of transcendents serve to complete and illustrate the theory of equations, tend to show that a course of mathematical study cannot well be made to adhere precisely to any definite classification of the different branches of mathematics. VI. One of the obstacles which hinder our mathematical studies from being quite what they ought to be is touched on in the tenth query*. It is there asked whether the number of problems proposed in the Senate-House examination is not, regard being had to the time allowed for solving them, greater than it should be. It may be answered that it is necessary to set before the candidates for high honours more problems than any one is supposed capable of solving in the given time, in order by the variety of subjects to provide sufficient employ- ment for each, and at the same time to leave a certain freedom of choice ; and, further, that if no more problems were proposed than the ablest questionists might be presumed capable of solv- ing, the number would still be too great for those of inferior ability. All this is true, and it is therefore much less easy to point out a remedy than to perceive the evils which result from the present state of things, not only in the problem papers but throughout the examination. He who for a season can remem- ber a great deal, and who while he remembers it can reproduce * The tenth query ig : I s it your opinion, or the contrary, that the problems proposed in the examinations bear too large a proportion to the questions derived either immediately or by a very simple deduction from the books which are com- monly read ? Are they for the most part so proposed, as to admit of their being readily apprehended and their solution effected by a. Student who thoroughly understands the principles and applications of the branches of Mathematics upon which they depend, or are they not unfrequently involved in such a form as to require for their solution a peculiar tact distinct from accurate and philosophical knowledge ? Are you not disposed to think the number of problems proposed to be solved in the time allowed (as many, generally, as seven, or eight in one Hour of time) is greater than the best prepared Student can be expected to complete 1 THE COURSE OF MATHEMATICAL STUDIES. 427 it rapidly and well, generally attains to more honour than he quite deserves ; and though the nndue influence of mere memory is somewhat diminished by papers consisting of original pro- blems, yet at any rate the student is trained to be quick and ready rather than wise and thoughtful. In devising problems it is difficult to avoid mere puzzles things to be solved only by some happy guess ; and if, on the other hand, examiners were to confine themselves to tolerably obvious deductions from known propositions, the problem papers would cease to be a counterpoise to the rest of the examination. If the candidates for high honours were, as in the Smith's prize examination, examined apart from the rest, it might be possible, so far as they were concerned, to diminish these difficulties by varying the modes of examination. In- a select examination there could be no excessive inconvenience in giving almost unlimited time for the consideration of the questions proposed, and in these questions the candidate might be required to state accurately, and in detail, the grounds of his views on fundamental prin- ciples in analysis, geometry, ox physics. Again, in such an examination books might perhaps be introduced, and if so, in- teresting questions might be proposed of a kind now inadmis- sible ; to mention one class only the candidate might be re- quired to form an opinion on any controverted point, to examine for instance the correctness of Sir James Ivory's doctrine, that in certain cases the ordinary condition of fluid equilibrium is insufficient, and to state his reasons for adopting or rejecting it. It must, however, be remembered that after all possible im- provements the complaint that schools " lack profoundness and dwell too much on seeming," will always be more or less just. A course of study, of which the most obvious purpose is to pre- pare the student for an eV/Sei^?, can never be quite what a course of study ought to be, and might be made if higher ends could always be kept in view. OAMBK1DGB I PIOTXBD BY 0. J. OLAY, M.A, AT THE UNIVKBS1TY PBEBS. CAMBRIDGE, Mv. 1863. BOOKS PUBLISHED BY DEIGHTON, BELL, & CO. 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