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J,//?. 2 tf-o'fqS^ -i<3o/ REPORT OF THE THIRDTTEETING OF THE BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE; HELD AT CAMBRIDGE IN 1833. LONDON: JOHN MURRAY, ALBEMARLE STREET. 1834. LONDON: PRINTED BY RICHARD TAYLOR, RED LION COURT, FLEET STREET. PREFACE. The Transactions of the British Association consist of three parts ; first, of Reports on the State of Science drawn up at the instance of the Association ; secondly, of Miscel- laneous Communications to the Meetings ; and thirdly, of Recommendations by the Committees, having for their ob- jects to mark out certain points for scientific inquiry. It is proper to remark, that some of the Reports here printed are to be considered in the light of first parts of the intended survey of the sciences reviewed in them, the continuation being postponed to a future Meeting. Thus, the Report on Hydraulics, by Mr. G. Rennie, will be completed in a second part, to be presented to the Meeting at Edinburgh ; the Report on the mathematical theory of the same science, by the Rev. Mr. Challis, which is here restricted to problems on the common theory of Fluids, will be further extended to the theories which have recently been advanced respecting the internal constitution of Fluids and the state of their caloric, to account for certain phseno- mena of their equilibrium and motion ; and the Report on Analytical Science, by the Rev. Mr. Peacock, which in the present volume includes Algebra, and the application of Algebra to Geometry, is intended to be hereafter concluded by a review of the Differential and Integral Calculus and the theory of Series. In like manner, to the Report on Botany, by Dr. Lindley, which embraces only the physiological part of the science, that which Mr. Bentham has under- taken on the State and Progress of Systematic Botany will be supplemental ; and to the present Report, by Dr. Charles Henry, on one branch of Animal Physiology, a more general review of the progress of that science will be added by the Rev. Dr. Clark. With respect to the next part of the Transactions, which includes the communications made to the Sections, two a 2 IV PKEFACE. rules have been adopted ; the first is, to print no oral com- munications unless furnished or revised by the Author him- self. In the former volume this rule was slightly deviated from, for the purpose of showing in what manner the Meetings were conducted. But however valuable a part of the proceedings of the Meetings the verbal communications and discussions maybe, it is evidently impossible to publish a safe and satisfactory report of them from any minutes which can be taken. The second rule is, not to print any of the miscellaneous communications at length ; but either abstracts of them, or notices* only, the object of the rule being to keep the Transactions within the bounds which the Association has prescribed to itself, and to prevent any interference with the publications of other societies. In the present volume, there is one paper printed at lengthf , which contains the results of certain experiments instituted expressly at the request of the Association. The Recommendations of various subjects for scientific inquiry agreed upon at Cambridge have been here incor- porated with those adopted at former Meetings, and the Suggestions which are contained in the Reports on the state of science, published in the present and preceding volume, have likewise been added; so as to present a general view of the desiderata in science to which attention has been invited. To this part of the volume are also appended those direc- tions for the use of observers which have proceeded from Committees appointed to promote particular investigations. To the Transactions is prefixed a brief outline of the General Proceedings of the Cambridge Meeting, a fuller Rc; port of them having been rendered unnecessary by the ac- count which has already issued from the University press. The observations, however, delivered by the Rev. Mr. Whe- WELL on the state of science as it is exhibited in the first volume of the Reports of the Association, not having been before published, are printed at length. • The notices of Communications will be found in the general account of the Proceedings of the Sections, p. 353. f " Experiments on the Quantity of Rain which falls at different Heights in the Atmosphere." CONTENTS Page. Proceedings of the Meeting ix TRANSACTIONS. Report on the State of Knowledge respecting Mineral Veins. By John Taylor, F.R.S., Treasurer of the Geological Society and of the British Association for the Advancement of Science, &c. 1 On the Principal Questions at present debated in the Philosophy of Botany. By John Lindley, Ph. D., F.R.S., Professor of Botany in the University of London 27 Report on the Physiology of the Nervous System. By William Charles Henry, M.D., Physician to the Manchester Royal In- firmary .59 Report on the present State of our Knowledge respecting the Strength of Materials. By Peter Barlow, F.R.S., Corr. Memb. Inst. France, &c. &c 93 Report on the State of our Knowledge respecting the Magnetism of the Earth. By S. Hunter Christie, M.A., F.R.S., M.C.P.S., Corr. Memb. Philom. Soc. Paris, Hon. Memb. Yorkshire Phil. Soc. ; of the Royal Military Academy ; and Member of Trinity College Cambridge 105 Report on the present State of the Analytical Theory of Hydro- statics and Hydrodynamics. By the Rev. J. Challis, late Fel- low of Trinity College Cambridge 131 Report on the Progress and present State of our Knowledge of Hydraulics as a Branch of Engineering. By George Rennie, F.R.S., &c. &c 153 Report on the recent Progress and present State of certain Branches of Analysis. By George Peacock, M.A., F.R.S., F.G.S., F.Z.S., F.R.A.S., F.C.P.S., FeUow and Tutor of Trinity Col- lege Cambridge 185 TRANSACTIONS OF THE SECTIONS. I. Mathematics and Physics. Professor CErsted on the Compressibility of Water 353 W. R. Hamilton on some Results of the View of a Characteristic Function in Optics 360 The Rev. H. Lloyd on Conical Refraction 370 VI CONTENTS. Page. Sir John F. W. Herschel on the Absorption of Light by coloured Media, vieM'ed in connexion with the undulatory Theory .... 373 The Rev. Baden Powell on the Dispersive Powers of the Media of the Eye, in connexion with its Achromatism 374 R. Potter, Jun., on the power of Glass of Antimony to reflect Light 377 on a Phsenomenon in the Interference of Light hitherto undescribed 378 Sir John F. W. Herschel's Explanation of the Principle and Con- struction of the Actinometer 379 M. Melloni's Account of some recent Experiments on Radiant Heat 381 John Prideaux on Thermo-Electricity 384 W. Snow Harris on some new Phaenomena of Electrical Attrac- tion 386 The Rev. John G. MacVicar on Electricity 390 The Rev. J. Power's Inquiry into the Cause of Endosmose and Exosmose 391 Michael Faraday on Electro -chemical Decomposition 393 Dr. Turner's Experiments on Atomic Weights 399 Prof. Johnston's Notice of a Method of analysing Carbonaceous Iron 400 R. Potter, Jun. A Communication respecting an Arch of the Aurora Borealis 401 John Phillips's Report of Experiments on the Quantities of Rain falling at different Elevations above the Surface of the Ground at York 401 II. Philosophical Instruments and Mechanical Arts. The Rev. Wm. Scoresby on a peculiar Source of Error in Experi- ments with the Dipping Needle 412 The Rev. W. H. Miller on the Construction of a new Barometer 414 W. L. Wharton on a Barometer with an enlarged Scale 414 W. S. Harris on the Construction of a new Wheel Barometer . . 414 J. Newman on a new Method of constructing a Portable Barometer 417 The Rev. James Cumming on an Instrument for measuring the total heating Effect of the Sun's Rays for a given time 418 on some Electro-magnetic Instruments 418 Andrew Ure on the Thermostat, or Heat-governor 419 Thomas Davison on a Reflecting Telescope 420 W. L. Wharton on a Steam-engine for pumping Water 421 E. J. Dent on the Application of a glass Balance-siDring to Chro- nometers 421 E. HoDGKiNsoN on the EflFect of Impact on Beams 421 on the direct tensile Strength of Cast Iron 423 J. I. Hawkins's Investigation of the Principle of Mr. Saxton's loco- motive differential Pulley, &c 424 John Taylor's Account of the Depths of Mines 427 J. Owen on Naval Architecture 430 CONTENTS. VU Page. III. Natural History — Anatomy — Physiology. Professor Agardh on the originary Structure of the Flower, and the mutual Dependency of its Parts 433 Professor Daubeny's Notice of Researches on the Action of Light upon Plants 436 Walter Adam on some symmetrical Relations of the Bones of the Megatherium 437 R Harlan on some new species of Fossil Saurians found in Ame- rica 440 The Rev. L. Jenyns's Remarks on Genera and Subgenera, &c. . . 440 J. Macartney on some parts of the Natural History of the Com- mon Toad 441 J. Blackwall's Observations relative to the Structure and Func- tions of Spiders 444 W. Yarrell on the Reproduction of the Eel 44G C. WiLLCox on the Naturalization in England of the Mytilus cre- natus, a native of India, and the Acematichcerus Heros, a native of Africa 448 J. Macartney's Abstract of Observations on the Structure and Functions of the Nervous System 449 H. Carlile's Abstract of Observations on the Motions and Sounds of the Heart 454 H. Earle on the Mechanism and Physiology of the Urethra .... 460 Burt on the Nomenclature of Clouds 460 G. H. Fielding on the peculiar Atmospherical Phaenomena as ob- served at Hull during April and May 1833, in relation to the prevalence of Influenza 461 IV. History of Science. Francis Baily's short Account of some MS. Letters (addressed to Mr. Abraham Sharp, relative to the Publication of Mr. Flam- steed's Historia Calestis,) laid on the table for the inspection of the Members of the Association 462 Recommendations of the British Association for the Advancement of Science 467 Recommendations of the Committees 469 Appendix 484 Prospectus of the Objects and Plan of the Statistical Society of London 492 Objects and Rules of the Association 497 Index 501 o to o Cft^ CO CO »o =.J — CO CM O >r; C O O J3 ;. -a M OJ o u o iX! CCS CO <: w P5 H CO QO _ O _ X o w W a; *>» i" '^ B ^ P S; G t- 5j 3 3 QJ c/^ Ph ^ >- S s S O . a s "" 1-3 O ;«■ H 125 a o 1-a -pq iS H Q c <u t; s 03 ° 5 2 5 S S s bo 53 ,^»-HOOOOO00O QulC^O^O COCOCO»-i ^ — «o ■^ ^ Q 00 f-i 5 00 «i« < ^ 3 ►J •- 03 <! 1/3 *J 4J O = *J CQ E o CJ rt &, o ••- 3 < u OS Q C8 fe H «o «o k, (U a> t3 ^ s ei QC ■±3 =4- o CO pi3 ^ •O O <u ■n 4-1 a ;ri n H S c2 J3 1 s JJ X c V C3 !U ^ n .* bn 60 c CS c «0 ira o "3 > ca to c Cm o lO ^ o rt 3) J3 en T3 c J3 a c 3 o £ *-• c n •rl QJ J2 CO Q) 13 a 3 ■*-» C5 P. CO .a ti-, o u C 0) 'o <§ o o o ^ § ~i 1 -^ o CO o 05 w (M rt s -^• ^ THIRD REPORT. PROCEEDINGS OF THE MEETING. 1833. The third Meeting of the British Association commenced its sittings at Cambridge on Monday, the 24th of June, 1833. It was attended by more than nine hundred Members, and was honoured with the presence of several foreign philosophers. The extent of accommodation provided by the University, and by the societies of which it consists, corresponded with the magnitude of the Meeting. The public schools, with two adjoining halls, were allotted to the use of the Sections and Committees, and the Senate-house was appropriated to the reception of the general assemblies ; a large proportion of the visitors were lodged within the walls of the Colleges, and the great halls of the two principal foundations were opened in hospitality to a concourse of guests collected from all parts by a common interest in scientific pursuits. GENERAL MEETING. On Monday evening, at eight o'clock, the Members assem- bled in the Senate-house : and a public discussion took place on the phenomena and theory of the Aurora Borealis. On Tuesday, at 1 p. m. a General Meeting was held in the Senate-house ; the President of the preceding year, (the Rev. Dr. Buckland,) resigned his office. In the course of his speech*, he congratulated the Meeting on the proof af- forded by the Report recently published, that the Association was pursuing a course of peculiar utility to science, whilst at the * A fuller account of the speeches delivered at the Meeting -will be found annexed to the lithographed signatures, &c., published at Cambridge. 1833. b X THIRD KEPORT — 1833. same time it had fully redeemed its pledge of not interfering with the province of other Scientific Societies. The President (the Rev. Professor Sedgwick,) stated, in his opening speech, that it was the desire of the Vice-Chancellor and the Heads of Colleges that everything should be done on the present occasion to emulate, as far as circumstances per- mitted, the splendid reception which had been given to the Association by the sister University of Oxford. He dwelt on the advantages which such a Meeting brought with it to the places in which it was held, by inducing scientific foreigners to visit them, and expressed the delight with which he hailed such visits, as an omen that the great barriers which for a length of time had served man for man, had now been broken down. He described the character of the Reports which the Association has published ; and added that he attached so much value to these expositions of the state of science, that he had requested one of the Secretaries, (the Rev, William Whewell,) to present to the Meeting a fuller analysis of their contents. The President concluded his speech with the fol- lowing gratifying announcement : " There is a philosopher," he said, " sitting among us whose hair is blanched by time, but possessing an intellect still in its healthiest vigour, — a man whose whole life has been devoted to the cause of truth, — my vener- able friend Dr. Dalton. Without any powerful apparatus for making philosophical' experiments, with an apparatus, indeed, which many might think almost contemptible, and with very limited external means for employing his great natural powers, he has gone straight forward in his distinguished course, and obtained for himself in those branches of knowledge which he has cultivated, a name not perhaps equalled by that of any- other living philosopher in the world. From the hour he came from his mother's womb the God of nature laid his hand upon him, and ordained him for the ministration of high philosophy. But his natural talents, great as they are, and his almost intuitive skill in tracing the relations of material phsenomena, would have been of comparatively little value to himself and to society, had there not been superadded to them a beautiful moral simpUcity and singleness of heart, which made him go on steadily in the way he saw before him, without turning to the right hand or to the left, and taught him to do homage to no authority before that of truth. Fixing his eye on the most extensive views of science, he has been not only a successful experimenter, but a philosopher of the highest order; his experiments have never had an insulated character, but have been always made as contributions towards some important PROCEEDINGS OF THE MEETING. XI end, as among the steps towards some lofty generalization. And with a most happy prescience of the points to which the rays of scattered observations were converging, he has more than once seen light while to other eyes all was yet in darkness ; out of seeming confusion has elicited order ; and has thus reached the high distinction of being one of the greatest legis- lators of chemical science. "It is my delightful privilege this day to announce (on the authority of a Minister of the Crown who sits near me,*) that His Majesty, King William the Fourth, wishing to manifest his attachment to science, and his regard for a character like that of Dr. Dalton, has graciously conferi'ed on him, out of the funds of the Civil List, a substantial mark of his royal favour." The Rev. William Whewell, being called vipon by the Pi'esident, delivered the following address : — " The British Association for the Advancement of Science meets at present under different circumstances from those which accompanied its former Meetings. The publication of the volume containing the Reports applied for by the Meeting at York, in 1831, and read before the Meeting at Oxford last year, must affect its proceedings during our sittings on the present occasion ; and thus we are now to look for the operation of one part of the machinery which its founders have endea- voured to put in action. Entertaining the views which sug- gested to them the scheme and plan of the Association, they must needs hope that such an event as this publication will exercise a beneficial influence upon its future career. " This hope is derived, they trust, from no visionary or presumptuous notions of what institutions and associations can effect. Let none suppose that we ascribe to assembled num- bers and conjoined labours extravagant powers and privileges in the promotion of science ; — that we believe in the omnipo- tence of a parliament of the scientific world. We know that the progress of discovery can no more be suddenly accelerated by a word of command uttered by a multitude, than by a single voice. There is, as was long ago said, no royal road to knowledge — no possibility of shortening the way, because he who wishes to travel along it is the most powerful one ; and just as little is there any mode of making it shorter, because they who press forward are many. We must all start from our actual position, and we cannot accelerate our advance by * The Right Honourable T. Spring Rice. b2 xii . THIIID REPORT — 1838. any method of giving to each man his mile of the march. Yet something we may do : we may take care that those who come ready and willing for the road, shall start from the proper point and in the proper direction ; — shall not scramble over broken ground, when there is a causeway parallel to their path, nor set off confidently from an advanced point when the first steps of the road are still doubtful ; — shall not waste their powers in struggling forwards where movement is not progress, and shall have pointed out to them all glimmerings of Hght, through the dense and deep screen which divides us from the next bright region of philosophical truth. We cannot create, we cannot even direct, the powers of discovery ; but we may perhaps aid them to direct themselves ; we may perhaps enable them to feel how many of us are ready to admire their success, and willing, so far as it is possible for intellects of a common pitch, to minister to their exertions. " It was conceived that an exposition of the recent progress, the present condition, the most pressing requirements of the principal branches of science at the present moment, might answer some of the purposes I have attempted to describe. Several such expositions have accordingly been presented to the Association by persons selected for the task, most of them eminent for their own contributions to the department which they had to review ; and these are now accessible to Members of the Association and to the public. It appears to be suitable to the design of this body, and likely to further its aims, that some one should endeavour to point out the bearing which the statements thus brought before it may and ought to have upon its future proceedings, and especially upon the laboiu's of the Meeting now begun. I am well persuaded that if the President had taken this ofiice upon himself, the striking and important views which it may naturally suggest would have been pre- sented in a manner worthy of the occasion : he has been influenced by various causes to wish to devolve it upon me, and I have considered that I should show my respect for the Asso- ciation better by attempting the task, however imperfectly, than by pleading my inferior fitness for it. " The particular questions which require consideration, and the researches which most require prosecution, in the sciences to which the Reports now before you refer, will be offered to the notice of the Sections of the Association which the subjects respectively concern, at their separate sittings. It is conceived that the most obvious and promising chance of removing deficiencies and solving difficulties in each subject, is to be found in drawing to them the notice of persons who have paid PROCEEDINGS OF THE MEETING. Xlll a continued and especial attention to the subject. The con- sideration of these points will therefore properly form a part of the business of the Sectional Meetings ; and all Members of the Association, according to their own peculiar pursuits and means, will thus have the opportunity of supplying any wanting knowledge, and of throwing light upon any existing perplexity. " But besides this special examination of the suggestions which your Reports contain, there are some more general reflexions to which they natui-ally give rise, which may perhaps be properly brought forward upon this first General Assembly of the present Meeting ; and which, if they are well founded, may preside over and influence the aims and exertions of many of us, both during our present discussions and in our future attempts to further the ends of science. " There is here neither time nor occasion for any but the most rapid survey of the subjects to which your Reports refer, in the point of view in which the Reports place them before you. Astronomy, which stands first on the list, is not only the queen of sciences, but, in a stricter sense of the term, the only perfect science ; — the only branch of human knowledge in which particulars are completely subjugated to generals, eflPects to causes ; — in which the long observation of the past has been, by human reason, twined into a chain which binds in its links the remotest events of the future ; — in which we are able fully and clearly to interpret Nature's oracles, so that by that which we have tried we receive a prophecy of that which is untried. The rules of all our leading facts have been made out by observations of which the science began with the earliest dawn of history ; the grand law of causation by which they are all bound together has been enunciated for 150 years; and we have in this case an example of a science in that elevated state of flourishing maturity, in which all that remains is to determine with the extreme of accuracy the con- sequences of its rules by the profoundest combinations of mathematics, the magnitude of its data by the minutest scru- pulousness of observation ; in which, further, its claims are so fully acknowledged, that the public wealth of every nation pre- tending to civilization, the most consummate productions of labour and skill, and the loftiest and most powerful intellects which appear among men, are gladly and emulously assigned to the task of adding to its completeness. In this condition of the science, it will readily be understood that Professor Airy, your Reporter upon it, has had to mark his desiderata, in no cases but those where some further developement of calcula- Xiv THIRD REPORT — 1833. tion, some furthei- delicacy of observation, some further accu- mulation of exact facts, are requisite ; though in every branch of the subject the labour of calculation, the delicacy of obser- vation, and the accumulation of exact facts, have already gone so far that the mere statement of what has been done can hardly be made credible or conceivable to a person unfamiliar with the study. " One article, indeed, in his list of recommendations to future labourers, read at the last Meeting of the Association, may ap- pear capable of being accomplished by more limited labour than the rest, — the determination of the mass of Jupiter by obser- vations of the elongations of his satellites. And undoubtedly, many persons were surprised when they found that on this, so obvious a subject of interest, no measures had been obtained since those which Pound took at the request of Newton. Yet in this case, if an accuracy and certainty worthy of the present condition of Astronomy were to be aimed at, the requisite ob- servations could not be few nor the calculation easy, when it is considered in how complex a manner the satellites disturb each other's motions. But the Meeting will learn with pleasure that the task which he thus pointed out to others, he has him- self in the intervening time executed in the most complete manner. He has weighed the mass of Jupiter in the way he thus recommended ; and it may show the wonderful perfection of such astronomical measures to state, that he has proved with certainty, that this mass is more than 322 and less than 323 times the mass of the terrestrial globe on which we stand. " Such is Astronomy : but in proceeding to other sciences, our condition and our task are of a far different kind. Instead of developing our theories, we have to establish them ; instead of determining our data and rules with the last accuracy, we have to obtain first approximations to them. This, indeed, may be asserted of the next subject on the list, though that is, in its principles, a branch of Physical Astronomy ; for that alone of all the branches of Physical Astronomy had been al- most or altogether neglected by men of science. I speak of the science of the Tides. Mr. Lubbock terminated his Report on this subject, by lamenting in Laplace's words this unmerited neglect. He himself in England, and Laplace in France, were indeed the only mathematicians who had applied themselves to do some portion of what M^as to be done with respect to this subject. Since our Meeting last year, Mr. Dessiou has, under Mr. Lubbock's direction, compared the tides of London, Sheer- ness, Portsmouth, Plymouth, Brest, and St. Helena ; and the comparison has brought to light very remarkable agreements . PROCEEDINGS OF THE MEETING. XY in the law which regulates the time of high water, agreements both with each other and with theory ; and has at the same time brought into view some anomalies which will give a strong impulse to the curiosity with which we shall examine the re- cords of future observations at some of these places and at many others. I may perhaps here take the Uberty of mention- ing my own attempts since our last Meeting, to contribute something bearing on this department. It appeared to me that our knowledge of one particular branch of this subject, the motion of the tide-wave in all parts of the ocean, was in such a condition, that by collecting and arranging our existing mate- rials, we should probably be enabled to procure abundant and valuable additions to them. This, therefore, I attempted to do ; and I have embodied the result of this attempt in an ' Essay towards a First Approximation to a Map of Cotidal Lines,' which is now just printed in the Philosophical Transac- tions of the Royal Society. If the time of the Meeting allows, I would willingly place before you the views at which we have now arrived, and the direction of our labours which these suggest. *' In the case of the science of Tides, we have no doubt about the general theory to which the phaenomena are to be referred, the law of universal gravitation ; though we still desiderate a clear application of the theory to the details. In another sub- ject which comes under our review, the science of Light, the prominent point of interest is the selection of the general theory. Sir David Brewster, the author of our Report on this subject, has spoken of ' the two rival theories of light,' which are, as you ai'e aware, that which makes light to consist in material particles emitted by a luminous body, and that which makes it to consist in undulations pi'opagated through a sta- tionary ether. The rivalry of these theories, so far as they can now be said to be rivals, has been by no means barren of interest and instruction during the year which is just elapsed. The discussions on the undulatory theory in our scientific journals have been animated, and cannot, I think, be considered as having left the subject where they found it. The claims of the undulatory theory, it will be recollected, do not depend only on its explaining the facts which it was originally intended to explain ; but on this ; — that the suppositions adopted in order to account for one set of facts, fall in most wonderfully with the suppositions requisite to explain a class of facts en- tirely different ; in the same manner as in the doctrine of gra- vitation, the law of force which is derived from the revolutions of the planets in their orbits, accounts for the apparently re- XVi THIRD REPORT — 1833. mote facts of the precession of the equinoxes and the tides. To all this there is nothing corresponding in the history of the theory of emission ; and no one, I think, well acquainted with the subject, would now assert, that if this latter theory had been as much cultivated as the other, it might have had a simi- lar brilliant fortune in these respects. " But if the undulatory theory be true, there must be solu- tions to all the apparent difficulties and contradictions which may occur in particular cases ; and moreover the doctrine will probably gain general acceptance, in proportion as these solu- tions are propounded and understood, and as prophecies of untried results are dehvered and fulfilled. In the way of such prophecies few things have been more remarkable than the prediction, that under particular circumstances a ray of light must be refracted into a conical pencil, deduced from the theory by Professor Hamilton of Dublin, and afterwards verified ex- perimentally by Professor Lloyd. In the way of special diffi- culties, Mr. Potter proposed an ingenious experiment which appeared to him inconsistent with the theory. Professor Airy, from a mathematical examination of this case, asserted that the facts, which are indeed difficult to observe, must be somewhat different from what they appeai-ed to Mr. Potter ; and having myself been present at Professor Airy's experiments, I can venture to say, that the appearances agree exactly with the results which he has deduced from the theory. Another gen- tleman, Mr. Barton, proposed other difficulties founded upon the calculation of certain experiments of Biot and Newton ; and Professor Powell of Oxford has pointed out that the data so referred to cannot safely be made the basis of such calcula- tions, for mathematical reasons. There is indeed here, also, one question of fact concerning an experiment stated in New- ton's Optics : In a part of the image of an aperture where Newton's statement places a dark line, in which Mr. Barton has followed him, Professors Airy, Powell, and others, have been able to see only a bright space, as the theory would require. Probably the experiments giving the two different results have not been made under precisely the same circumstances ; and the admirers of Newton are the persons who will least of all consider his immoveable fame as exposed to any shock by these discussions. " Perhaps, while the undulationist will conceive that his opinions have gained no small accession of evidence by this ex- emplification of what they will account for, those who think the advocates of the theory have advanced its claims too far, will be in some degree concihated by having a distinct acknow- PROCEEDINGS OF THE MEETING. XVll ledgement, as during these discussions they have had, of what it does not pretend to explain. The whole doctrine of the absorption of light is at present out of the pale of its calcula- tions ; and if the theory is ever extended to these phaenomena, it must be by supplementary suppositions concerning the ether and its undulations, of which we have at present not the slight- est conception. " There are various of the Physical subjects to which your Reports I'efer, which it is less necessary to notice in a general sketch like the present. The recent discoveries in Thermo- electricity, of which Professor Gumming has presented you with a review, and the investigations concerning Radiant Heat which have been arranged and stated by Professor Powell, are subjects of great interest and promise ; and they are gradually advancing, by the accumulation of facts bound together by subordinate rules, into that condition in which we may hope to see them subjugated to general and philosophical theories. But with regard to this prospect, the subjects I have mentioned are only the fragments of sciences, on which we cannot hope to theorize successfully except by considering them with refer- ence to their whole ; — Thermo-electricity with reference to the whole doctrine of electricity ; Radiant Heat with reference to the whole doctrine of heat. " If the subjects just mentioned be but parts of sciences, thei'e is another on which you have a Report before you, which, though treated as one science, is in reality a collection of several sciences, each of great extent. I speak of Meteorology, which is reported on by Professor Forbes. There is perhaps no por- tion of human knowledge more capable of being advanced by our conjoined exertions than this : some of the requisite ob- servations demand practice and skill ; but others are easily made, when the observer is once imbued with sound elemen- tary notions ; and in all departments of the subject little can be done without a great accumulation of facts and a patient in- quiry after their rules. Some such contributions we may look for at our present Meeting. Professor Forbes has spoken of the possibility of constructing maps of the sky by which we may trace the daily and hourly condition of the atmosphere over large tracts of the earth. If, indeed, we could make a stratigraphical analysis of the aerial shell of the earth, as the geologist has done of its solid crust, this would be a vast step for Meteorology. This, however, must needs be a difficult task : in addition to the complexity of these superincumbent masses, time enters here as a new element of variety : the strata of the geologist continue fixed and permanent : those of the meteoro- Xviii THIRD REPORT — 1833. legist change from one moment to another. Another difficulty is this ; that while we want to determine what takes place in the whole depth of the aerial ocean, our observations are neces- sarily made almost solely at its bottom. Our access to the heights of the atmosphere is more limited, in comparison with what we wish to observe, than our access to the depths of the earth. " Geology, indeed, is a most signal and animating instance of what may be effected by continued labours governed by common views. Mr. Conybeare's Report upon this science gives you a view of what has been done in it during the last twenty years ; and his ' Section of Europe from the North of Scotland to the Adriatic,' which is annexed to the Report, conveys the general views with regard to the structure of Central Europe, at which geologists have now arrived. To point out any more recent additions to its progress or its prospects is an undertaking more suitable to the geologists by profession, than to the pre- sent sketch. And all who take an interest in the subject will rejoice that the constitution and practice of the Geological So- ciety very happily provide, by the annual addresses of its Pre- sidents, against any arrear in the incorporation of fresh acquisi- tions with its accumulated treasures. " The science of Mineralogy, on which I had the honour of offering a Report to the Association, was formerly looked upon as a subordinate portion of Geology. It may, however, now be most usefully considered as a science co-ordinate and closely allied with Chemistry, and the most important questions for examination in the one science belong almost equally to the other. Mr. Johnston, in his Report on Chemical Science, has, as the subject required, dwelt upon the questions of isomor- phism and plesiomorphism, which I had noticed as of great im- portance to Mineralogy. Dr. Turner and Prof. Miller, who at the last Meeting undertook to inquire into this subject, have examined a number of cases, and obtained some valuable facts ; but the progress of our knowledge here necessarily requires time, since the most delicate chemical analysis and the exact measurement of 30 or 40 crystals are wanted for the satisfac- tory estabhshment of the properties of each species *. In Che- • Perhaps I shall not have a more favourable occasion than the present of correcting a statement in my Report, which is not perfectly accurate, on a point which has been a subject of controversy between Sir David Brewster and Mr. Brooke. I have noticed (p. 338.) the sulphato-tricarbonate of lead of Mr. Brooke, as a mineral which at first appeared to contradict Sir David Brewster's general law of the connexion of crystalline form with optical structure, in as much as it appeared to be of the rhombohedral system, and was found to have PROCEEDINGS OF THE MEETING. XIX mistry, besides the great subject of isomorphism to which I have referred, there are some other yet undecided questions, as for instance those concerning the existence and relations of the sulpho-salts and chloro-saUs ; and these are not small points, for they affect the whole aspect of chemical theory, and thus show us how erroneously we should judge, if we were to consider this science as otherwise than in its infancy. " In every science. Notation and Nomenclature are questions subordinate to calculation and theory. The Notation of Cry- stallography is such as to answer the purposes of calculation, whether we take that of Mohs, Weiss, or Nauman. It appears very desirable that the Notation of Chemistry also should be so constructed as to answer the same purpose. Dr. Turner in the last edition of his Chemistry, and Mr. Johnston in his Report, have used a notation which has this advantage, which that commonly employed by the continental Chemists does not possess. " I have elsewhere stated to the Association how little hope there appears at present to be of purifying and systematizing our mineralogical nomenclature. The changes of theory in Chemistry to which I have already referred, must necessarily superinduce a change of its nomenclature, in the same manner in which the existing nomenclature was introduced by the pre- valent theory ; and the new views have in fact been connected with such a change by those who have propounded them. It will be for the Chemical Section of the Association to consider how far these questions of Nomenclature and Notation can be discussed with advantage at the present Meeting. " The Reports presented at the last Meeting had a reference, for the most part, to physical rather than physiological science. The latter department of human knowledge will be more pro- minently the subject of some of the Reports which are to come before us on the present occasion. There is, however, one of two axes of double refraction ; and which was afterwards found to confirm the law, the apparently rhombohedral forms being found by Mr. Haidinger to be not simple but compound. It seems, however, that the solution of the difficulty (for no one now will doubt that it lias a solution,) is somewhat different. There appear to have been included under this name two different kinds of crystals belonging to different systems of crystallization. Some which Mr. Brooke found to be rhombohedral, Sir David Brewster found to have a single optical axis with no trace of composition ; others were prismatic with two axes ; and thus Mr. Brooke's original determinations were probably correct. The high reputa- tion of the parties in this controversy does not need this explanation ; but pro- bably those who look with pleasure at the manner in which the apparent excep- tions to laws of nature gradually disappear, may not think a moment or two lost in placing the matter on its proper footing. XX THIRD REPORT — 1833. last year's Reports which refers to one of the widest questions of Physiology ; that of Dr. Prichard on the History of the Human Species, and its subdivision into races. The other lines of research which tend in the same direction will probably be brought before the Association in successive years, and thus give us a view of the extent of knowledge which is accessible to us on this subject. " In addition to these particular notices of the aspect under which various sciences present themselves to us as resulting from the Reports of last years, there is a reflexion which may I think be collected from the general consideration of these sciences, and which is important to us, since it bears upon the manner in which science is to be promoted by combined labour such as that which it is a main object of this Association to stimulate and organize. The reflexion to which I refer is this ; — that a combination of theory with facts, of general views with experimental industry, is requisite, even in subordinate contributors to science. It has of late been common to assert that facts alone are valuable in science ; that theory, so far as it is valuable, is contained in the facts ; and, so far as it is not contained in the facts, can merely mislead and preoccupy men. But this antithesis between theory and facts has probably in its turn contributed to delude and perplex ; to make men's ob- servations and speculations useless and fruitless. For it is only through some view or other of the connexion and relation of facts, that we know what circumstances we ought to notice and record ; and every labourer in the field of science, however humble, must direct his labours by some theoretical views, original or adopted. Or if the word theory be unconquerably obnoxious, as to some it appears to be, it will probably still be conceded, that it is the rviles of facts, as well as facts themselves, with which it is our business to acquaint ourselves. That the recollection of this may not be useless, we may collect from the contrast which Professor Airy in his Report has drawn between the astronomers of our own and of other countries. "In En- gland," he says, (p. 184,) " an observer conceives that he has done everything when he has made an observation," " In foreign observatories," he adds, " the exhibition of results and the comparison of results with theory, are considered as de- serving more of an astronomer's attention, and demanding greater exercise of his intellect, than the mere observation of a body on the wire of a telescope." We may, indeed, perceive in some measure the reason which has led to the neglect of theory with us. For a long period astronomical theory was greatly a-head of observation, and this deficiency was mainly PROCEEDINGS OF THE MEETING. XNl supplied by the perseverance and accuracy of English ob- servers. It was natural that the value and reputation which our observations thus acquired for the time, should lead us to think too disrespectfully, in comparison, of the other depart- ments of the science. Nor is the lesson thus taught us con- fined to Astronomy ; for, though we may not be able in other respects to compare our facts with the results of a vast and yet certain theory, we ought never to forget that facts can only become portions of knowledge as they become classed and con- nected ; that they can only constitute truth when they are in- cluded in general propositions. Without some attention to this consideration, we may notice daily the changes of the winds and skies, and make a journal of the weather, which shall have no more value than a journal of our dreams would have ; but if we can once obtain fixed measures of what we notice, and connect our measures by probable or certain rules, it is no longer a vacant employment to gaze at the clouds, or an un- profitable stringing together of expletives to remark on the weather ; the caprices of the atmosphere become steady dispo- sitions, and we are on the road to meteorological science. " It may be added — as a further reason why no observer should be content without arranging his observations, in what- ever part of Physics, and without endeavouring at least to classify and connect them — that when this is not done at first, it will most likely never be done. The circumstances of the observation can hardly ever be properly understood or inter- preted by others ; the suggestions which the observations themselves supply, for change of plan or details, cannot in any other way be properly appreciated and acted on. And even the mere multitude of unanalysed observations may drive future students of the subject into a despair of rendering them useful. Among the other desiderata in Astronomy which Professor Airy mentions, he observes, " Bradley's observations of stars," made in 1750, " were nearly useless till Bessel undertook to re- duce them" in 1818. "In like manner Bradley's and Mas- kelyne's observations of the sun are still nearly useless," and they and many more must continue so till they are reduced. This could not have happened if they had been reduced and compared with theory at the time ; and it cannot but grieve us to see so much skill, labour and zeal thus wasted. The per- petual reference or attempt to refer observations, however nu- merous, to the most probable known rules, can alone obviate similar evils. " It may appear to many, that by thus recommending theory we incur the danger of encouraging theoi-eticul speculations XXii THIRD REPORT — 1833. to the detriment of observation. To do this would be indeed to render an ill service to science : but we conceive that our purpose cannot so far be misunderstood. Without here at- tempting any nice or technical distinctions between theory and hypothesis, it may be sufficient to observe that all deductions from theory for any other pupose than that of comparison with observation are frivolous and useless exercises of ingenuity, so far as the interests of physical science are concerned. Specu- lators, if of active and inventive minds, will form theories whether we wish it or no. These theories may be useful or may be otherwise — we have examples of both results. If the theories merely stimulate the examination of facts, and are modified as and when the facts suggest modification, they may be erroneous, but they will still be beneficial ; — they may die, but they will not have lived in vain. If, on the other hand, our theory be supposed to have a truth of a superior kind to the facts ; to be certain independently of its exemplification in par- ticular cases ; — if, when exceptions to our propositions occui', instead of modifying the theory, we explain away the facts, — our theory then becomes our tyrant, and all who work under its bidding do the work of slaves, they themselves deriving no benefit from the result of their labours. For the sake of ex- ample we may point out the Geological Society as a body which, labouring in the former spirit, has ennobled and eni-iched itself by its exertions : if any body of men should employ themselves in the way last described, they must soon expend the small stock of a priori plausibility with which they must of course begin the world. " To exemplify the distinction for a moment longer, let it be recollected that we have at the present time two rival theories of the history of the earth which prevail in the minds of geo- logists ; — one, which asserts that the changes of which we trace the evidence in the earth's materials have been produced by causes such as are still acting at the surface ; another, which considers that the elevation of mountain chains and the transi- tion from the organized world of one formation to that of the next, have been produced by events which, compared with the present course of things, may be called catastrophes and con- vulsions. Who does not see that all that those theories have hitherto done, has been, to lead geologists to study more ex- actly the laws of permanence and of change in the existing organic and inorganic world, on the one hand; and on the other, the relations of mountain chains to each other, and to the phaenomena which their strata present ? And who doubts, that, as the amount of the full evidence may finally be, (which PROCEEDINGS OF THE MEETING. XXIII may, indeed, perhaps require many generations to accumulate,) geologists will give their assent to the one or the other of these views, or to some intermediate opinion to which both may gradually converge? " On the other hand — to take an example from a science with which I have had a professional concern- — the theory that cry- stalline bodies are composed of ultimate molecules which have a definite and constant geometrical form, may properly and philosophically be adopted, so far as we can, by means of it, reduce to rules the actually occurring secondary faces of such substances. But if we assume the doctrine of such an atomic composition, and then form imaginary arrangements of these atoms, and enunciate these as explanations of dimorphism, or plesiomorphism, or any other apparent exception to the general principle, we proceed, as appears to me, unphilosophi- cally. Let us collect and classify the facts of dimorphism and plesiomorphism, and see what rules they follow, and we may then hope to discern whether our atomic theory of crystalline molecules is tenable, and what modifications of it these cases, uncontemplated in its original formation, now demand. " I will not now attempt to draw forth other lessons which the Report of last year may supply for our future guidance ; although such offer themselves, and will undoubtedly affect the spirit of our proceedings during this Meeting. But there is a reflexion belonging to what I may call the morals of science, which seems to me to lie on the face of this Report, and which I cannot prevail upon myself to pass over. In looking steadily at the past history and present state of physical knowledge, we cannot, I think, avoid being struck with this thought, — How little is done and how much remains to do ; — and again, not- withstanding this, how much we owe to the great philosophers who have preceded us. It is sometimes advanced as a charge against the studies of modern science, that they give men an overweening opinion of their own acquirements, of the supe- riority of the present generation, and of the intellectual power and progress of man ; — that they make men confident and con- temptuous, vain and proud. That they never do this, would be much to say of these or of any other studies ; but, assuredly, those must read the history of science with strange preposses- sions who find in it an aliment for such feelings. What is the picture which we have had presented to us ? Among all the attempts of man to systematize and complete his knowledge, there is one science, Astronomy, in which he may be considered to have been successful ; he has there attained a general and certain theory : for this success, the labour of the most highly- Xxiv THIRD REPORT — 1833. gifted portion of the species for 5000 years has been requisite. There is another science, Optics, in which we are, perhaps, in the act of obtaining the same success, with regard to a part of the phaenomena. But all the rest of the prospect is compara- tively darkness and chaos ; hmited rules, imperfectly known, imperfectly verified, connected by no known cause, are all that we can discern. Even in those sciences which are considered as having been most successful, as Chemistry, every few years changes the aspect under which the theory presents the facts to our minds, while no theory, as yet, has advanced beyond the mere horn-book of calculation. What is there here of which man can be proud, or from which he can find reason to be pre- sumptuous ? And even if the Discoverers to whom these sciences owe such progress as they have made — the great men of the present and the past — if they might be elate and confident in the exercises of their intellectual powers, who are ive, that we should ape their mental attitudes ? — we, who can but with pain and eiFort keep a firm hold of the views which they have disclosed ? But it has not been so ; they, the really great in the world of intellect, have never had their characters marked with admiration of themselves and contempt of others. Their genuine nobility has ever been superior to those ignoble and low-born tempers. Their views of their own powers and achieve- ments have been sober and modest, because they have ever felt how near their predecessors had advanced to what they had done, and what patience and labour their own small progress had cost. Knowledge, like wealth, is not likely to make us proud or vain, except when it comes suddenly and unlearned ; and in such a case, it is little to be hoped that we shall use well, or increase, our ill-understood possession. " Perhaps some of the appearance of overweening estimation of ourselves and our generation which has been charged against science, has arisen from the natural exultation which men feel at witnessing the successes of art. I need not here dwell upon the distinction of science and art ; of knowledge, and the ap- plication of knowledge to the uses of life ; of theory and practice. In the success of the mechanical arts there is much that we look at with an admiration mingled with some feeling of triumph ; and this feeling is here natural and blameless. For what is all such art but a struggle, — a perpetual conflict with the inertness of matter and its unfitness for our purposes? And when, in this conflict, we gain some point, it is impossible we should not feel some of the exultation of victory. In all stages of civilization this temper prevails : from the naked in- habitant of the islands of the ocean, who by means of a piece PROCEEDINGS OF THE MEETING. XXV of board glides through the furious and apparently deadly line of breakers, to the traveller vi'ho starts along a rail-road with a rapidity that dazzles the eye, this triumphant joy in suc- cessful art is universally felt. But we shall have no difficulty in distinguishing this feeling from the calm pleasure which we receive from the contemplation of truth. And when we con- sider how small an advance of speculative science is implied in each successful step of art, we shall be in no danger of im- bibing, from the mere high spirits produced by difficulty over- come, any extravagant estimate of what man has done or can do, any perverse conception of the true scale of his aims and hopes. " Still, it would little become us here to be unjust to prac- tical science. Practice has always been the origin and stimulus of theory : Art has ever been the mother of Science ; the comely and busy mother of a daughter of a far loftier and serener beauty. And so it is likely still to be : there are no subjects in which we may look more hopefully to an advance in sound theoretical views, than those in which the demands of practice make men willing to experiment on an expensive scale, with keenness and perseverance ; and reward every addition of our knowledge with an addition to our power. And even they — for undoubtedly there are many such — who require no such bribe as an inducement to their own exertions, may still be glad that such a fund should exist, as a means of engaging and recompensing subordinate labourers. " I will not detain you longer by endeavouring to follow more into detail the application of these observations to the proceedings of the General and Sectional Meetings during the present week. But I may remark that some subjects, circum- stanced exactly as I have described, will be brought under your notice by the Reports which we have reason to hope for on the present occasion. Thus, the state of our knowledge of the laws of the motion of fluids is universally important, since the motion of boats of all kinds, hydraulic machinery, the tide^, the flowing of rivers, all depend upon it. Mr. Stevenson and Mr. Rennie have undertaken to give us an account of different branches of this subject as connected with practice ; and Mr. Challis will report to us on the present state of the analytical theory. In like manner the subject of the strength of materials, which the multiplied uses of iron, stone and wood, make so inter- esting, will be brought before you by Mr. Barlow. These were two of the portions of mechanics the earliest speculated upon, and in them the latest speculators have as yet advanced little beyond the views of the earliest. XXvi THIRD REPORT— 183,S. " I mention these as specimens only of the points to which we may more particularly direct our attention. I will only observe, in addition, that if some studies, as for instance those of Natural History and Physiology, appear hitherto to have occupied less space in our proceedings than their importance And interest might justly demand, this has occurred because the Reports on other subjects appeared more easy to obtain in the first instance ; and the balance will I trust be restored at the present Meeting. I need not add anything further on this subject. Among an assembly of persons such as are now met in this place, there can be no doubt that the most important and profound questions of science in its existing state will be those which will most naturally occur in our assemblies and discussions. It merely remains for me to congratulate the As- sociation upon the circumstances under which it is assembled ; and to express my persuasion that all of us, acting under the elevating and yet sobering thought of being engaged in the great cause of the advancement of true science, and cherishing the views and feelings which such a situation inspires, shall derive satisfaction and benefit from the occasions of the present week." Mr. Whewell having concluded his Address, the Meeting adjourned, after electing by a general vote the candidates who had been approved by the Council and by the General Com- mittee. At eight P.M., the Members having reassembled in the Senate- house, Mr. Taylor read a Report on the state of our know- ledge respecting Mineral Veins, which was followed by a general discussion on the nature and origin of veins. On Wednesday at one p.m., the Chairmen of the Sections hav- ing read the minutes of their proceedings to the Meeting, the Rev. G. Peacock delivered a brief abstract of his Report on the state of the Theory of Algebra. Professor Lindley read a Report on the state of Physiological Botany ; and Mr. G. Ren- nie on the state of Practical Hydraulics. Auditors were ap- pointed to examine the accounts. On Thursday, at one p.m., the auditors reported the state of the accounts. The Chairmen of the Sections read the mi- nutes of their proceedings. Professor Christie read a Report on the present state of our knowledge respecting the Magnetism of the Earth. A summary of the contents of a Report on the state of knowledge as to the Strength of Materials, by Pro- PKOCEEDINGS OF THE MEETING. XXvni fessoi' Bavlow, was given, in the absence of the Author, by the -Rev. W. Whewell. In the evening, Mr. Whewell delivered a Lecture in the Senate-house, on the manner in which observations of the Tide may be usefully made to serve as a groundwork for general views ; either by observing the time of high water at different places on the same day, in order to determine the motion of the summit of the tide-wave ; or by continuing the observations for a considerable time, and comparing them with the moon's transit to obtain the semi-menstrual inequality. He observed, that it appears from Mr. Lubbock's recent researches on the subject, that the tides of Portsmouth and Brest agree very closely in the law of this inequality, and that the tides of Ply- mouth and London also agree ; but that there is an anomaly which cannot at present be explained in the comparison of Brest with Plymouth. Professor Parish explained to the Meeting the advantages which he conceived would be derived from ap- plying the power of steam to carriages on undulating roads in preference to level rail-ways. On Friday, at one p.m., the Chairmen of the Sections having read the minutes of their proceedings, the Rev. J. Challis made a Report on the progress of the Theory of Fluids. The Pre- sident stated the appropriation * to certain scientific objects of a portion of the funds of the Association to the amount of 600/. Mr. Babbage, at the President's request, explained his views respecting the advantages which wovild accrue to science from such a collection of numerical facts as he had formerly recommended under the title of " Constants of Nature and Art." The President announced, that it had been resolved by the General Committee, that the Meeting of 1834 should take place at Edinburgh in the early part of the month of Sep- tember ; he read the names of the Officers and Members of the Council appointed for the ensuing year. The thanks of the Meeting were then voted to the Vice- Chancellor and the other authorities of the University, to the retiring Officers and Members of the Council, to the President, the Secretaries for Cambridge, the Local Committee of Manage- ment, and the General Secretary. The President, in his concluding Address to the Meeting, explained an irregularity which had occurred in the formation of a new Section. In addition to the five Sections into which the Meeting had been divided by the authority of the General * For a particular account of these appropriations, see p. xxxvi. c2 XXviii THIRD REPORT 1833. Committee, he stated that another had come into operation, the object of whicli was to promote statistical inquiries. It had originated with some distinguished philosophers, but could not be regarded as a legitimate branch of the Association till it had received the recognition of the governing body ; there could be little doubt, however, that the new Section would obtain the sanction of the General Committee, with some limitation per- haps of the specific objects of inquiry. On this subject he made the following observations : — " Some remarks may be expected from me in reference to the objects of this Section, as several Members may perhaps think them ill fitted to a Society formed only for the promotion of natural science. To set, as far as I am able, these doubts at rest, I will explain what I understand by science, and what I think the proper objects of the Association. By science, then, I understand the consideration of all subjects, whether of a pure or mixed nature, capable of being reduced to measurement and calculation. All things comprehended under the categories of space, time and number properly belong to our investigations ; and all phaenomena capable of being brought under the sem- blance of a law are legitimate objects of our inquiries. But there ai'e many important subjects of human contemplation which come under none of these heads, being separated from them by new- elements ; for they bear upon the passions, affections and feel- ings of our moral nature. Most important parts of our nature such elements indeed are ; and God forbid that I should call upon any man to extinguish them ; but they enter not among the objects of the Association. The sciences of morals and politics are elevated far above the speculations of oiu- philosophy. Can, then, statistical inquiries be made compatible with our objects, and taken into the bosom of our Society? I think they unquestionably may, so far as they have to do with matters of fact, with mere abstractions, and with numerical results. Considered in that light they give what may be called the raw material to political economy and political philosophy ; and by their help the lasting foundations of those sciences may be per- haps ultimately laid. These inquiries are, however, it is import- ant to observe, most intimately connected with moral phaeno- mena and economical speculations, — they touch the mainsprings of passion and feeling, — they blend themselves with the generali- zations of political science ; but when we enter on these higher generahzations, that moment they are dissevered from the ob- jects of the Association, and must be abandoned by it, if it means not to desert the secure ground which it has now taken. " Should any one affirm (what, indeed, no one is prepared PROCEEDINGS OF THE MEETING. XXIX to deny,) that all truth has one common essence, and should he then go on to ask why truths of different degrees should be thus dissevered from each other, the reply would not be dif- ficult. In physical truth, whatever may be our difference of opinion, there is an ultimate appeal to experiment and ob- servation, against which passion and prejudice have not a single plea to urge. But in moral and political reasoning, we have ever to do with questions, in which the waywardness of man's will and the turbulence of man's passions are among the strongest elements. The consequence it is not for me to tell. Look around you, and you will then see the whole framework of society put in movement by the worst passions of our na- ture; you will see love turned into hate, deliberation into dis- cord, and men, instead of mitigating the evils which are about them, tearing and mangling each other, and deforming the moral aspect of the world. And let not the Members of the Association indulge a fancy, that they are themselves exempt from the common evils of humanity. There is that within us, which, if put into a flame, may consume our whole fabric, — may produce an explosion, capable at once of destroying all the principles by which we are held together, and of dissi- pating our body in the air. Our Meetings have been essen- tially harmonious, only because we have kept within our proper boundaries, confined ourselves to the laws of nature, and steered clear of all questions in the decision of which bad passions could have any play. But if we transgress our pro- per boundaries, go into provinces not belonging to us, and open a door of communication with the dreary wild of politics, that instant will the foul Daemon of discord find his way into our Eden of Philosophy. " In every condition of society there is some bright spot on which the eye loves to rest. In the turbulent republics of ancient Greece, where men seemed in an almost ceaseless war- fare of mind and'body, they had their seasons of solemnity, when hostile nations made a truce with their bitter feelings, as- sembled together, for a time, in harmony, and joined in a great festival ; which, however differing from what we now see in its magnitude and forms of celebration, was consecrated, like our present Meeting, to the honour of national genius. What- ever have been the bitter feelings which have so often disgraced the civil history of mankind, I dare to hope that they will never find their resting-place within the threshold where this Associa- tion meets ; that peace and good will, though banished from every other corner of the land, will ever find an honoured seat amongst us ; and that the congregated philosophers of the empire, throwing aside bad passion and party animosity, will, XXX. THIRD REPORT 1833. year by year, come to their philosophical Olympia, to witness a noble ceremonial, to meet in a pacific combat, and share in the glorious privilege of pushing on the triumphal car of Truth. " The last duty I have to perform this morning would be a painful one indeed, were our Assembly to be broken up into elements which were not again to be reunited. The Association is not, however, dissolved ; its meeting is only adjourned to an- other year; and it has been a matter of great joy to me to an- nounce to you, that the Committee has elected for your next President a distinguished soldier and philosopher ; and that it will be your privilege to reassemble in one of the fairest capitals of the world, — in a city v/hich has nursed a race of literary and philosophic giants, — ^^in a land filled with natural beauties, and wedded to the imagination and the memory by a thousand en- dearing associations. " There is a solemnity in parting words, which may, I think, justify me (especially after what has been so well said this morn- ing by the Marquis of Nortliampton,) in passing the limits I have so far carefully prescribed to myself, and in treading for a moment on more hallowed ground. In the first place, I would entreat you to remember that you ought above all things to re- joice in the moral influence of an Association like the present. Facts, which are the first objects of our pursuit, are of compa- ratively small value till they are combined together so as to lead to some philosophic inference. Physical experiments, con- sidered merely by themselves, and apart from the rest of nature, are no better than stones lying scattered on the ground, which require to be chiselled and cemented before they can be made into a building fit for the habitation of man. The true value of an experiment is, that it is subordinate to some law, — that it is a step toward the knowledge of some general truth. Without, at least, a glinnnering of such truth, physical knowledge has no true nobility. But there is in the intellect of man an appetency for the discovery of general truth, and by this appetency, in subordination to the capacities of his mind, has he been led on to the discovery of general laws ; and thus has his soul been fitted to reflect back upon the world a portion of the counsels of his Creator. If I have said that physical phsenomena, unless con- nected with the ideas of order and of law, are of little worth, I may further say, that an intellectual grasp of material laws of the highest oi'der has no moral worth, except it be combined with another movement of the mind, raising it to the perception of an intelligent First Cause. It is by help of this last movement that nature's language is comprehended ; that her laws become pregnant with meaning ; that material phaenomena are instinct with life ; that all moral and material changes become linked PROCEEDINGS OF THE MEETING. XXXI together ; and that Truth, under whatever forms she may pre- sent herself, seems to have but one essential substance. "I have before spoken of the distinctions between moral and physical science; and I need not repeat what I have said, unless it be once more solemnly to adjure you not to leave the straight path by which you are advancing, — not to desert the cause for which you have so well combined together. But let no one misunderstand my meaning. If I have said that bad passions mingle themselves with moral and political sciences, and that the conclusions of these sciences are made obscure from the want of our comprehending all the elements with which we have to deal, I have only spoken the truth ; but still I hold that moral and political science is of a higher order than the physical. The latter has sometimes, in the estimation of man, been placed on a higher level than it deserves, only from the circumstance of its being so well defined, and grounded in the evidence of ex- periments appealing to the senses. Its progress is marked by indices the eye can follow ; and the boundaries of its conquests are traced by landmarks which stand high in the horizon of man's history. But with all these accompaniments, the moral and political sciences entirely swallow up the physical in impor- tance. For what are they but an interpretation of the governing laws of intellectual nature, having a relation in time pi'esent to the social happiness of millions, and bearing in their end on the destinies of immortal beings ? " Gentlemen, if I look forward with delight to our meeting again at Edinburgh, it is a delight chastised by a far different feeling, to which, had not these been parting words, I should not have ventured to give an utterance. It is not possible we should all again meet together. Some of those whose voices have been lifted up during this great Meeting, whose eyes have brightened at the presence of their friends, and whose hearts have beat high during the intellectual commu- nion of the week, before another year may not be numbered with the living. Nay, by that law of nature to which every living man must in his turn yield obedience, it is certain that before another festival, the cold hand of death will rest on the head of some who are present in this assembly. If a thought like this gives a tone of grave solemnity to words of parting, it svu-ely ought to teach us, during our common rejoicings at the triumphal progress of science, a personal lesson of deep humility. By the laws of nature, before we can meet again, many of those bright faces which during the past week I have seen around me may be laid low, for the hand of death may have been upon them ; but wherever we reassemble, God grant that all our attainntents in science may tend to our moral improvement; and XXxii THIRD REPORT 1833. may we all meet at last in the presence of that Almighty Being, whose will is the rule of all law, and whose bosom is the centre of all power!" SECTIONAL MEETINGS. The Sections assembled daily at eleven a.m., and occasion- ally also at half-past eight p.m., at their respective places of meeting, in the Schools, the Astronomical Lecture-room, and the Hall of Caius College. On Saturday, the Section of Na- tural History made an excursion to the Fens. Abstracts of most of the Communications which were made to the Sections will be found in a subsequent part of the volume. In addition to the communications of which abstracts are there given, notices of the following transactions appear on the minutes : — M. Quetelet described the observations which he had made on Falling Stars. It was suggested that such observations might be available in certain cases for determining differences of longitude. Mr. Potter communicated some calculations of the height of the Aurora BoreaHs, seen on the 21st of March 1833. Mr. Hopkins gave an abstract of a paper on the Vibration of Air in Cyhndrical Tubes of definite length. Dr. Ritchie made some remarks on the Sensibility of the Eye, and the errors to which it is subject. Mr. Barton gave a view of his opinions on the Propagation of Heat in solid bodies. A letter was received from Mr. Frend regarding certain points in the Theory of the Tides. The Rev.W. Scoresby described a Celestial Compass invent- ed by Col. Graydon. Mr. R. Murphy read some remarks on the utihty of observ- ing the Magnetic Dip in Mines. M. Quetelet gave an account of some observations made by himself and M. Necker de Saussure, which corroborate the statements of M. Kuppfer, respecting theinequahty of magne- tic intensity at the top and the base of mountains. Professor Christie stated his views relative to the cause of the Magnetism of the Earth. Mr. A. Trevelyan read a paper on certain Vibrations of Heated Metals. Mr. Brunei exhibited and explained a Model in illustration of his method of constructing Bridges without centering. PROCEEDINGS OF THE MEETING. XXXUI A notice of some experiments relative to Isomorphism, by Dr. Turner and Professor Miller, was read. Dr. Daubeny made a communication on the Gases given off from the surfaces of the water in certain thermal springs. The Rev. W. V. Harcourt exhibited specimens of Metal taken out of the crevices at the bottom of a mould in which a large bronze figure had been cast by Mr. Chantrey ; together with fragments of the Bronze employed in the casting, from which the former specimens differed considerably in colour, frangi- bility, &c. Mr. Lowe gave an account of various chemical products found in the retorts and flues of Gas Works. Mr. Pearsall made a communication on the bleaching powers of Oxygen. Mr. J. Taylor described the character of the Ecton Mine, and the occurrence of the copper ore in connected cavities which had been explored to a depth of 225 fathoms without reaching the termination of them. Dr. Buckland described the manner in which fibrous Lime- stone occurs in the Isle of Purbeck and other situations. Mr. Murchison stated, and illustrated by Maps and Sections, the principal results of his inquiries into the sedimentary de- posits which occupy the western parts of Shropshire and Here- fordshire, and are prolonged in a S.W. direction through the counties of Radnor, Brecknock, and Caermarthen, and the in- trusive igneous rocks which occur in certain parts of the di- strict. He mentioned the occurrence of freshwater Limestone in a detached Coal-field of Shropshire. Professor Sedgwick described the leading features in the Geology of North Wales, the lines of elevation, the relation of the trap rocks to the slate system, the cleavage of the slate ; pointed out the relations of this tract to that examined by Mr. Murchison ; and drew a general parallel between the slate formations of Wales and Cumberland. Mr. J. Taylor having read to the Section the concluding part of his Report on Veins, in the discussion which followed, M. Dufrenoy entered into a consideration of some phaenomenaof the igneous rocks of Britanny and Central France, viewed with reference to the connexion between them and the metalliferous veins of those districts, and remarked on the occurrence in Central France of mineral veins, only in the narrow zone at the junction of the unstratified and stratified rocks. He also made some remarks on the association of dolomite and gypsum, with the igneous rocks of the Alps and the Pyrenees. Professor Sedgwick gave a general account of the Red Sand- stones connected with the Coal-measures of Scotland, and the XXxiv THIRD REPORT — 1833. Isle of Arran, with the view of showing that they are perfectly distinct from the similar rocks connected with the Magnesian Limest*)ne. JMr. Hartop exhibited a Map and Sections to illustrate the series of Coal Strata in South Yorkshire, and their direction and varying dip in the valley of the Dun, and to the north and south of that river; described the characters of the strata, and the in- fluence of certain great dislocations on the quality of the coal. Mr. Greenough exhibited a Map of Western Europe, on which the relative levels of land and water were represented by means of colours, instead of engraving. Mr. Greenough was requested to permit a map on this plan to be published. The Rev. J. Hailstone communicated some notices relating to Mineral Veins. Sections of the Well in the Dock Yard at Portsmouth, and of the Well in the Victualling Yard at Weevil, were communi- cated by the Rev. Mr. Leggat and Mr. Blackburn, on the pai't of the Portsmouth Philosophical Society; and a letter from Mr. Goodrich, explanatary of the Sections, was read. Mr. Mantell exhibited a perfect Femur of the Iguanodon, and explained its distinctive anatomical characters. Mr. W. C. Trevelyan exhibited specimens of Coprolites, and remains of Fishes, from the Edinburgh Coal-field. Mr. Fox exhibited specimens of Fishes from the Magnesian Limestone and Marl-slate of Durham. Mr. Gray made some remarks on the occurrence of Water in the Valves of Bivalve Shells, and exhibited a specimen of Spondylus varius, in which water was contained in both the valves. Mr. Ogilby gave an account of his views respecting the classification of Ruminating Quadrupeds, which he proposed to found upon the presence or absence of horns on the female sex ; the peculiar form of the upper lip ; and the presence or absence of the subocular and submaxillary glands. He showed the ap- plication of these views to the division of hollow-horned rumi- nating animals without horns in the female sear, which he dis- tributed into five new genera. The Rev. W. Scoresby communicated some observations on the adaptation of the Structure of the Cetacea to their habits of life and residence in the Ocean ; and suggested the use which might be made of the peculiar forms of the Whalebone in their classification. Lieutenant Colonel Sykes exhibited a specimen of the Short- tailed Manis, and communicated some observations on its mode of progression. Mr. Brayley communicated a memoir on the laws regulating PROCEEDINGS OF THE MEETING. XXXV the distribution of the powers of producing Light and Heat among Animals. Mr. H. Strickland made some remarks on the Vipera Chersea, showing its specific difterence from the common Viper. The subject of the use of the Pith in Plants, was discussed by Professor Burnett, Professor Henslow, Mr. Curtis, and Mr. Gray. Dr. Roupell exhibited some Drawings representing the effects of irritant Poisons upon the hving membrane of the in- testinal canal of Men and Animals. Mr. Fisher communicated some observations on the physical condition of the Brain during sleep. Mr. Brooke made some remarks on the physiology of the Eye and the Ear. Dr. Marshall Hall gave an abstract of his views respecting the reflex function of the Medulla oblongat^i and Medulla spi- nalis. COMMITTEES. The General Committee met daily at ten a.m., and at other hours by adjournment, in the Hall of Trinity Hall. The Com- mittees of Sciences met as soon after ten as the business of the General Coimnittee permitted, in the rooms of their respective Sections. The General Committee made the necessary arrange- ments for the conduct of the Meeting ; formed the Sectional Committees of Sciences ; determined the place and time of the next Meeting ; appointed the new Officers and Council ; and passed the following Resolutions : — 1. That the thanks of the Association be given to the Societies and Institutions from which it has received invitations, — in Bris- tol, Birmingham, Liverpool, Newcastle and Edinburgh. 2. That Members of the Association whose subscription shall have been due for two years, and who shall not pay it on proper notice, shall cease to be Members, powder being left to the Com- mittee or Council to reinstate them on reasonable grounds within one year, on payment of their arrears. 3. That the number of Deputies which provincial Institutions shall be entitled to send to the Meetings as Members of the General Committee, shall be two from each Institution. 4. That the following instructions be given to each of the Com- mittees of Sciences : — To select those points of science, which, on a review of the former Recommendations of the Committees, or those contained XXXvi THIRD REPORT — 18,33. in the Reports published by the Association, or from sugges- tions made at the present Meeting, they may think most fit to be advanced by an application of the funds of the Society, either in compensation for labour, or in defraying the expense of apparatus, or otherwise. The Committee are requested to confine their selections to definite as well as important objects ; to state their reasons for the selection, and where they may think proper, to designate individuals to undertake the desired investigations ; they are to transmit their Recommendations through their Secretaries to the General Committee. The Committees of Sciences having complied with these in- structions, the following Resolutions were passed by the General Committee : 1. That a sum not exceeding 200/. be devoted to the dis- cussion of observations of the Tides, and the formation of Tide Tables, under the superintendence of Mr. Baily, Mr. Lubbock, Rev. G. Peacock, and Rev. W. Whewell. 2. That a sum not exceeding 50/. be appropriated to the construction of a Telescopic Lens, or Lenses, out of Rock Salt, under the direction of Sir David Brewster. S. That Dr. Dalton and Dr. Prout be requested to institute experiments on the specific gravities of Oxygen, Hydrogen, and Carbonic Acid ; and that a sum not exceeding 50/. be appropri- ated to defray the expense of any apparatus which may be re- quired. 4. That a series of experiments on the effects of long con- tinued Heat be instituted at some iron furnace, or in any other suitable situation ; and that a sum not exceeding 50/. be placed at the disposal of a Sub-Committee, consisting of Professor Daubeny, Rev. W. V. Harcourt, Professor Sedgwick, and Pro- fessor Turner, to meet any expense which may be incurred *. 5. That measurements should be made, and the necessary data procured, to determine the question of the permanence or change of the relative Level of Sea and Land on the coasts of Great Britain and Ireland ; and that for this purpose a sum not exceeding 100/. be placed at the disposal of a Sub-Com- mittee, consisting of Mr. Greenough, Mr. Lubbock, Mr. G. Rennie, Professor Sedgwick, Mr. Stevenson, and Rev. W. Whewell ; — the measurements to be so executed, as to furnish the means of reference in future times, not only as to the re- lative levels of the land and sea, but also as to waste or exten- sion of the land. • These experiments have been instituted by Mr. Harcourt, in Yorkshire, at the Low Moor Iron Works, the property of Messrs. Hird and Co., and at the Elsecar Furnace, belonging to Earl Fitzwilliam. PROCEEDINGS OF THE MEETING. XXXVU 6. That the effects of Poisons on the Animal Economy should be investigated and illustrated by graphic representations ; and that a sum not exceeding 251. be appropriated for this object. Dr. Roupell, and Dr. Hodgkin were requested to undertake this investigation. 7. That the sensibilities of the Nerves of the Brain should be investigated ; and that a sum not exceeding 251. should be appropriated to this object. Dr. Marshall Hall and Mr. S. D. Broughton were requested to undertake these experiments. 8. That a sum not exceeding 100/. be appropriated towards the execution of the plan proposed by Professor Babbage, for collecting and arranging the Constants of Nature and Art*. 9. That a representation be submitted to Government on the part of the British Association, stating that it would tend greatly to the advancement of astronomy, and the art of navigation, if the observations of the sun, moon and planets, made by Bradley, Maskelyne and Pond, were reduced; and that a deputation -j- be appointed to wait upon the Lords of the Treasuiy with a re- quest, that public provision may be made for the accomplish- ment of this great national object. Proposals for the formation of a Statistical Section were ap- proved. It was resolved, that the inquiries of this Section should be restricted to those classes of facts relating to communities of men which are capable of being expressed by numbers, and which promise, when sufficiently multiplied, to indicate general laws. A Committee of Statistical Science was formed %. The Re- commendations § of the several Committees of Science were re- vised and approved. TRUSTEES OF THE ASSOCIATION. Charles Babbage, F.R.S. Lucasian Professor of Mathe- matics, Cambridge. R. I. Murchison, F.R.S. V.P.G.S. &c. John Taylor, F.R.S. Treas. G.S. &c. • For an abstract of Mr. Babbage 's plan, see the Appendix. + The deputation consisted of Professor Airy, Mr. Baily, Mr. D. Gilbert and Sir John Herschel. The application was immediately complied with by the Go- vernment. \ For an account of the proceedings of this Committee, see the Appendix. § These Recommendations will be found marked with an asterisk in the col- lection of Recommendations and Suggestions printed in the latter part of the volume. XXXviii THIRD REPORT — 1833. OFFICERS. President. — Rev. Adam Sedgwick, F.R.S. G.S. and Wood- wardian Professor of Geology, Cambridge. Vice-Presidents. — G. B. Airy, F.G.S. Plumian Professor of Astronomy, Cambridge. John Dalton, D.C.L. F.R.S. Instit. Reg. Sc. Paris. Corresp. President elect.— lAeni. Gen. Sir T. M. Brisbane, K.C.B. F.R.S. L. & E. President of the Royal Soc. Edinb. Inst. Reg. Sc. Paris. Corresp. Vice-Presidents elect. — Sir David Brewster, K.G.H. LL.D. F.R.S. L. & E. Rev. J. Robinson, D.D. Astronomer Royal at Armagh. Treasurer.— John Taylor, F.R.S. Treas. G.S. General Secretary. — Rev. W. V. Harcourt, F.R.S. G.S. Assistant Secretary. — John Phillips, F.R.S. G.S. Professor of Geology in King's College, London. Secretaries for Oxford. — Charles Daubeny, M.D. L.S. Professor of Botany. Rev. B. Powell, F.R.S. Professor of Geometry. Secretaries for Cambridge. — Rev. J. S. Henslow, G.S. Professor of Botany. Rev. W. Whewell, F.R.S. Secretaries for Edinburgh. — John Robison, Sec. James D. Forbes, F.R.S. L. & E. F.G.S. Professor of Natural Philosophy. Secretary for Dublin. — Rev. Thomas Luby. COUNCIL. Rev, W. Buckland, D.D, F.R.S. Professor of Geol. and Min. Oxford. W. Clift, F.R.S. Rev. T.Chalmers, D.D. Professor of Divinity, Edinbvn-gh. S. H. Christie, F.R.S. Professor of Ma- thematics at Woolwich. Earl Fitzwilliam, F.R.S. G.S. G. B. Greenough, F.R.S. Pres. of the Geol. Society. T. Hodg- kin, M.D. London. W. R. Hamilton, Astronomer Royal for Ireland. W. J. Hooker, F.R.S. Professor of Botany, Glasgow. Robert Jameson, F.R.S. Professor of Natural Hi- story, Edinburgh. John Lindley, F.R.S. Professor of Botany in the University of London. J. W. Lubbock, Treas. R.S. Rev. B. Lloyd, D.D. Treas. Prov. of Trin. Coll. Dublin. R. I. Murchison, F.R.S. &c. Patrick Neill, M.D. F.R.S.E. Edinburgh. George Rennie, F.R.S. Rev. W. Ritchie, LL.D. F.R.S. Professor of Nat. Philosoohy in the University of Lon- don. J. S. Traill, M.D. W. Yarrell, F.L.S. &c. Ex officio members, — The Trustees and Officers of the Association. ,S'ecre/fm>.s.— Edward Turner, M.D. F.R.S. Sec. G.S. Rev. James Yates, F.L.S. G.S. F.R.S. Savilian F.L.S. &c.- R.S.E. PROCEEDINGS OF THE MEETING. XXXIX COMMITTEES OF SCIENCES. I. Mathematics and General Physics. Chairman. — Sir D. Brewster, F.R.S. &c. Deputy Chairman. — Rev. G. Peacock, F.R.S. Secretary. — Professor Forbes. Viscount Aclare, F.R.S. Professor Airy. Professor Bab- bage. Francis Baily, V.P.R.S. John Barton, F.R.S. Rev. J. Bowstead. Sir T. M. Brisbane, F.R.S. Professor Christie. Rev, H. Coddington, F.R.S. E. J. Cooper. Dr. Corrie, F.R.S. G. DoUond, F.R.S. Lieut. Drummond. Davies Gilbert, D.C.L. F.R.S. Rev. R. Greswell, F.R.S. Pro- fessor W. R. Hamilton. Hon. C. Harris, F.G.S. G. Harvey, F.R.S. Sir John F. W. Herschel, F.R.S. E. Hodgkinson. W. Hopkins. John Hymers. Rev. Professor T. Jarratt. Rev. Dr. Lardner, F.R.S. Rev. Dr. Lloyd. Professor Lloyd. J. W. Lubbock, Treas. R.S. R. Murphy, F.R.S. Phil- pott. R. Pottei-, jun. Professor Powell. Professor Quetelet. Professor Rigaud. Rev. Dr. Robinson. Rev. R. Walker, F.R.S. W. L. Wharton. C. Wheatstone. Rev. W. Whewell, F.R.S. Rev. R. Willis, F.R.S. II. Chemistry, Mineralogy, Sj-c. Chairman.— 3. Dalton, D.C.L. F.R.S. Deputy Chairman. — ^Rev. Professor Cumming. Secretary. — Professor Miller. Professor Daniell. Professor Daubeny. M. Faraday, D.C.L. Rev. W. Vernon Harcourt, F.R.S. W. Snow Harris, F.R.S. W. Hatfeild, F.G.S. J. F. W. Johnston, A.M. Rev. D. Lardner, LL.D. F.R.S. Rev. B. Lloyd, LL.D. T. J. Pearsall. Dr. Prout, F.R.S. Professor W. Ritchie. Rev. W. Scoresby, F.R.S. W. Sturgeon. Professor Turner. III. Geology and Geography. Chairman. — G. B. Greenough, F.R.S. Pres. G.S. Deputy Chairmen. — Rev. Dr. Buckland, F.R.S. G.S. R.L Murchison, F.R.S. V.P.G.S. Secretaries.— W. Lonsdale, F.G.S. John Phillips, F.R.S. G.S. Dr. Boase. James Bryce, jun. F.G.S. Joseph Carn^j F.R.S. G.S. Major Clerke, C.B. F.R.S. M. Dufrenoy. Sir Philip Malpas de Grey Egerton, F.R.S. G.S. Dr. Fitton, F.R.S. G.S. Rev. J. Hailstone, F.R.S. G.S. Professor Harlan. G. Mantell, F.R.S. G.S. Lieut. Murphy, R. E. Marquis of xl THIRD REPORT — 1833. Northampton, F.R.S. G.S. Rev. Professor Sedgwick. Colonel Silvertop, F.G.S, W. Smith. John Taylor, F.R.S. Treas. G.S. W. C. Trevelyan, F.G.S. H. T. M. Witham, F.G.S. Rev. J. Yates, F.G.S. IV. Natural History. Chairman.— Kev. W. L. P. Garnons, F.L.S. Deputy Chairman. — Rev. L. Jenyns, F.L.S. Secretaries. — C. C. Babington, F.L.S. D. Don, F.L.S. Professor Agardh. G. Bentham, Sec. Hort. Soc. F.L.S. J. Blackwall, F.L.S. W. J. Burchell. Professor Burnett. W.Christy, F.L.S. Allan Cunningham, F.L.S. J. Curtis,F.L.S. E. Forster, F.R.S. Treas. L.S. G. T. Fox, F.L.S. J. £. Gray, F.R.S. Rev. Professor Henslow. Rev. Dr. Jermyn. Rev. W. Kirby, F.R.S. L.S. Professor Lindley. W. Ogilby, F.L.S. Dr. J. C. Prichard, F.R.S. J. F. Royle, F.L.S. J. Sabine, F.R.S. L.S. P. J. Selby, F.L.S. J. F. Stephens, F.L.S. H.Strickland. Colonel Sykes, F.R.S. L.S. Richard Taylor, F.L.S. G.S. W. G. Werscow. J. O. Westwood, F.L.S. W. Yarrell, F.L.S. V. Anatomy, Medicine, ^c. Chairman. — Dr. Haviland. Deputy Chairman. — Dr. Clark. Secretaries. — Dr. Bond. Mr. Paget. Dr. Alderson. S. D. Broughton, F.R.S. W. Clift, F.R.S. G.S. Dr. Dugard. H. Earle, F.R.S. Dr. Marshall Hall, F.R.S. Dr. Hewett. Dr. Malcavey. Dr. Macartney. Pro- fessor Mayo. Dr. Paris, F.R.S. Dr. Prout, F.R.S. Dr. Roget, F.R.S. G.S. Dr. Thackeray. Dr. D. Thorp. VI. Statistics. Chairman. — Professor Babbage. Secretary. — J. E. Drinkwater, M.A. H. Elphinstone, F.R.S. W. Empson, M.A. Earl Fitz- william, F.R.S. H. Hallam, F.R.S. E. Halswell, F.R.S. Rev. Professor Jones. Sir C. Lemon, Bart. F.R.S. J. W. Lubbock, Treas. R.S. Professor Malthus. Capt. Pringle. M. Quetelet. Rev. E. Stanley, F.L.S. G.S. Colonel Sykes, F.R.S. F.L.S. G.S. Richard Taylor, F.L.S. G.S. [ 1 ] TRANSACTIONS. Report on the State of Knowledge respecting Mineral Veins. By John Taylor, F.R.S., Treasurer of the Geological So- ciety and of the British Associatioii for the Advancement of Science, Sfc. 8fc. X HAVE found it very difficult to execute the task proposed to me in a manner satisfactory to myself, as we have at this time no digested account of the views entertained by geologists of the present day upon this interesting subject. The most per- fect treatise is that of Werner, which deserves much attention for the observation of facts which it displays; but as it was written to propound a theory, and as that theory depended upon views of the structure of the crust of the earth which modern geology has at least thrown much doubt upon, so his work cannot be taken as an outline of our present state of knowledge. Since his time but little has been attempted respecting vein formations; and the subject has been, I think, rather neglected by geologists, who have advanced other branches of the science with extraordinary skill, industry and success. Detached pa- pers have, indeed, appeared by English authors, among which that on the veins of Cornwall, by Mr. Joseph Carne, holds a distinguished place. As some proof that the subject of veins has not been much attended to, I would remark, that in the Second Series of the Transactions of the Geological Society of London, consisting now of the first and second volumes complete, and two Parts of the third volume, no paper expressly on veins is to be found. In the First Series there are two papers, one by the late Mr. W. Phillips, giving an outline of facts more generally obsei'ved with respect to veins in Cornwall, from observations made principally in the year 1800. Another is by Dr. Berger, on the physical structure of Devon and Cornwall, from observa- tions made in 1809. The writer adopts the Wernerian theory, and mentions cases which he thinks confirmatory of its truth. In the four volumes of the Transactions of the Royal Geo- logical Society of Cornwall, we shall find this subject more 1833. B 2 THIRD REPORT — 1833. attended to, and there are several communications relating to it : among the authors are Dr. Boase, Mr. Carne, Dr. Davey, Mr. K. W. Fox, and Mr. John Hawkins. One of the papers by Mr. Carne is that to which I have before alluded. One of the most recent works by foreign writers is that of the late M. Schmidt of Siegen. He was an experienced prac- tical miner, and wrote chiefly with a view to his art, describing the various derangements in mineral veins, and tracing the best rules to be observed in pursuing researches in difiicult circum- stances. He adopts the Wernerian theory of formations, and refers to the author of it as the great master of the subject. Though no general theory has of late been produced in re- gular form, yet with the great attention that has been given to geology by so many eminent men, an extended field of observa- tion has taken place, leading to a very general change of opi- nion on most important points ; many conjectures respecting the formation of veins have sprung up, and which, when the facts are more investigated, and they shall have been recorded and classified, may form the groundwork for a more enlarged and rational theory, by which their phaenomena and structure may be explained, and the causes of their formation, the manner of filling up, and the circumstances of the varied derangements and dislocations, may be traced and be better understood. The subject is of threefold importance : first, as it relates to science, wherein a better knowledge of veins generally must very materially contribute to sound investigations as to the structure of the rocks that inclose them: secondly, as it is much owing to the pursuit of the minerals which are deposited in veins that we have acquired and may yet extend our knowledge of geology in general : thirdly, in relation to the question some- times proposed as to the usefulness of geological science, the most ready answer may be given, if it be considered that this inquiry will relate to subjects of practical utility, in which man- kind are universally and largely interested. Before I proceed to any account of the opinions as to the formation of veins, I would offer some definition descriptive of their character and structure, that in proceeding with our sub- ject we may clearly understand what is meant to be treated on. Werner lays it down, " That veins are particular mineral re- positories, of a flat or tabular shape, which in general traverse the strata of mountains, and are filled with mineral matter dif- fering more or less from the nature of the rocks in which they occur. . " Veins cross the strata, and have a direction different from theirs. Other mineral repositories, such as particular strata or REPORT ON MINERAL VEINS. 9 beds, of whatever thickness they occur, have, on the contrary, a similar direction with the strata of the rock, and instead of crossing, run parallel with them : this forms the characteristic difference." Playfair says : " Veins are of various kinds, and may in ge- neral be defined, separations in the continuity of a rock, of a determinate width, but extending indefinitely in length and depth, and filled with mineral substances different from the rock itself. The mineral veins, strictly so called, are those filled with sparry or crystallized substances, and containing the me- tallic ores." Mr. Carne says : " By a true vein I understand the mineral contents of a vertical or inclined fissure, nearly straight, and of indefinite length and depth. These contents are generally, but not always, different from the strata or the rocks which the vein intersects. True veins have regular walls, and sometimes a thin layer of clay between the wall and the vein ; small branches are also frequently found to diverge from them on both sides." Mr. Carne mentions other veins, which he distinguishes from the true ones as being shorter, crooked, and irregular in size ; he considers these to have formed in a different manner : but this will be discussed hereafter. These definitions seem to me to be sufficient for our pur- pose ; but it may be advantageous here to introduce some further description of circumstances connected with veins, and to explain the terms usually employed to describe them. Being tabular masses, generally of no great width, any one will, whether vertical or inclined, present at its intersection with the surface a line nearly straight : this may be from north to south, or from east to west, or in any intermediate course. This is usually called the direction ; by miners frequently the run of the vein, or the course of the vein, and is denoted by the points of the compass it may cross. The length, as Werner states, is indefinite, it being doubtful whether any vein has been pursued to a perfect termination. The tabular mass, again, may be either vertical to the plane of the earth's surface, or may deviate from this position by in- clining to one side or the other of the perpendicular. This deviation is called the inclination of the vein ; by the Cornish miners the underlie. It is measured by the angle made with the perpendicular ; and as the dip will be to one side of the direction, the latter being known, the other is easily expressed. The depth to which veins descend into the earth is unknown, as well as the length, and for the same reason. b2 4 THIRD REPORT — 1833. The only dimension we can ascertain is that across from one side to the other of the tabular mass, and is measured from one wall to the other, which is the term used in England for the cheeks or sides presented by the inclosing rock. This dimen- sion is called the width, or frequently the size of the vein. The width varies considerably in the same vein. In Europe a vein containing ore is considered to be a wide one if it ex- ceeds five or six feet. In Mexico the width of veins is gene- rally greater. In metalliferous veins the deposits of ore are extremely irre- gular, forming masses of very diversified form and extent, and are separated from each other by intervening masses of vein- stone or matrix, either entirely devoid of ore, or more or less mixed with it. It is rare to find a vein entirely filled with ore in any part. In this respect they differ from most beds, where, as in those of coal, the whole is a uniform mass. The layer of clay, which, as Mr. Carne says, is frequent in such veins, will deserve particular notice when we consider their general structure and the theories of their formation : this is called Saal-bande by the Germans, and flookan by the Cornish miners. The clearest idea of a vein will be obtained by imagining a crack or fissiu'e in the rocks, running in nearly a straight line, extending to great and unknown length and depth, and filled with various substances. I do not intend by this description to convey any theoretic opinion as to the manner in which such fissures may have been formed, or as to the mode of their being furnished with their present contents. These are subjects on which the greatest diversity of opinion has existed in former times, and this diver- sity is continued to the present period. It is the main business of this Report to state these opinions, and to describe our pre- sent state of knowledge of this difficult subject. I feel great distrust of my power to do it justice; but I am encouraged by the idea that a feeble sketch may induce abler hands to pursue the design, and throw more and more light upon this interesting branch of geology. It would be of little use to go into details of the conjectures of ancient authors, or into the mysteries with which this sub- ject was enveloped in the age of alchemy. The earliest writer who is worthy to be consulted is Agricola (whose proper name was Bauer) : he resided in the Saxon Erz- gebirge, and died in the middle of the sixteenth century. He has been called the father of mineralogy, and of the science of REPORT ON MINERAL VEINS. 5 mining. He had the rare merit of emerging from the mists and clouds of an absurd school of philosophy, which had till then obscured the objects which it pretended to illustrate ; and he first subjected them to inquiries prompted by sound reason and just views of nature. His writings were numerous, and in such pure Latin that they are said to be entitled to a place among the classics. He treats of veins in a work called Bermannus, but more particularly in the third book of his great work De Re MetaUica. Agricola being held to be the first who has written anything certain on the formation of veins, and his theory of the manner of their being filled up having, with some modifications, been for a long period generally received, and in part even adopted by Werner, I shall commence from his time the notice of the opinions promulgated by various writers antecedent to Werner and Hutton. Some have maintained. That veins and their branchings are to be considered as the branches and twigs of an immense trunk which exists in the interior of the globe : That from the bowels of the earth metallic particles issued forth in the form of vapours and exhalations through the rents, in the same manner as sap rises and circulates in vegetables. This speculation was proposed by Von Oppel, captain-ge- neral of the Saxon mines, who wrote in 1749. He was a skilful miner and an accurate observer; and it is singular that this opi- nion is not consistent with most that he has elsewhere said on the subject, which generally rather agreed with the views which were adopted by Werner and others. Henkel, who wrote in the early part of the seventeenth cen- tury, and who has been held to be the father of mineralogical chemistry, first attributed the formation of the contents of veins to peculiar exhalations : he supposed the basis of each metal and mineral to have existed in the substance of the rock, and to have been developed by a peculiar process of nature. Becher about the same time supported very similar views. Stahl, who commented upon the writings of Becher, had ad- vanced a somewhat similar opinion; but he afterwards rejected this theory, and considered veins, as well as the substances of which they are composed, as having been formed at the same time with the earth itself. Zimmerman, chief commissioner of mines in Saxony, who died in 1747, had an idea that the variety of minerals contained in veins had been produced by a transformation of the sub- stance of the rock. Charpentier, in 1778, supported nearly similar opinions, and 6 THIRD REPORT — 1833. combated strenuously against the theory which considers veins to have been rents that were afterwards filled up by different mineral substances. This is the theory, however, which, from the time of Agricola to the present day, has been most generally received, namely, that veins were Jissiires which have been since filled up by de- grees tvith mineral matters. The causes of such fissures, and the mode of their contents being deposited, have been variously stated, and have given rise to much conjecture; and allowing for these differences, the main proposition has been supported by many writers. Among these I would name Agricola; Balthazar Rosier, an eminent miner of Freyberg, who died in 1673; Hoffman, a commissioner of mines at the same place, in 1746; Von Oppel, before mentioned, who, though he had indulged in other speculations, distinctly lays down in his Introduction to Subterranean Geometry , (Dres- den, 1749,) that veins were formerly fissures, open in their su- perior part, and that they traverse and intersect the strata. Bergman entertained opinions very similar, which were also supported by Delius, an author on mining, of considerable ce- lebrity, who wrote about 1770. Gerhard, in his Essay on the History of the Mineral Ki?igr dom, (Berlin, 1781,) gives a collection of interesting facts con- cerning veins, and considers them to have originally been rents, which were afterwards filled up with mineral substances. To this list may be added Lasius, in his Observations on the Mountains oftheHartz,in 1787; and Linnaeus is stated "to have wondered at the nature of that force which split the rocks into those cracks ; and adds, that probably the cause is very familiar, — that they were formed moist, and cracked in drying*." In England we have testimony to the same opinion from Dr. Pryce, who wrote his Mineralogia Cormibiensis in 1778. He says, "When solid bodies were separated from fluid, certain cracks, chinks and fissures in various directions were formed, and as the matter of each stratum became more compact and dense by the desertion of moisture, each stratum within itself had its fissures likewise, which, for the most part, being in- fluenced by peculiar distinct laws, were either perpendicular, oblique," &c. He afterwards adds, that those very fissures are the wombs or receptacles of all metals, and most minerals. He assigns the derangements of veins to the effect of fracture by violence, and quotes subsidence as one of the probable causes of such dislo- cations. He says there can be no doubt that many alterations * Hill. REPORT ON MINERAL VEINS. V have happened to various parts of the earth before, at, and after the Flood, from inundations, earthquakes, and the dis- solvent powers of subterranean fire and ivater, which variety of causes and circumstances must infalUbly have produced many irregularities in the disposition and situation of circumjacent strata and lodes *. He describes twelve kinds of lodes or veins in Cornwall, naming them from their chief contents. But the most remark- able observation of Dr. Pryce is respecting the relative age of veins, of which he seems to have given the first intimation. Werner, long after, states this as a discovery of his own, and as an essential part of his theory. His translator, however, (Dr. Anderson,) does Pryce justice, and remarks that his ob- servations must have been unknown to Werner, who showed much anxiety in all cases to confer on every writer the merit which was due to him. Dr. Anderson quotes the passage as one of much importance. " Because the cross gossans or cross flookans run through all veins of opposite directions, without the least interruption from them, but, on the contrary, do apparently disjoint and dislocate all of them, it seems reasonable to conclude, that the east and west veins were antecedent to cross veins, and that some great event, long after the Creation, occasioned those transverse clefts and openings. But how or when this should come to pass, we cannot presume to form any adequate ideaf ." Kirwan supports the doctrine that some veins were originally open, as appears from the rounded stones and petrifactions found in them. Thus, in the granitic mountain of Pangel in Silesia there is a vein filled with globular basalt. So also in veins of wacken, in Joachimstahl in Bohemia, trees and their branches have been found. But he deems it improbable that all veins were originally open to day, and filled from above. He inclines to the theory of veins being filled by the percolation of solutions of the me- tals and earths. Having now taken a cursory view of the opinions held before Werner published his Theory of Veins, and seen something of the state of knowledge relating to this subject, we may bear in mind the materials which he had to work with, and take into account his well-known views as to the origin of rocks from aqueous deposition, and we shall comprehend the system which he developed, with respect to veins, in the only work, I believe, which proceeded from his own hand, and which was published * ' Lode ' is the term used in Cornwall for a metalliferous vein. + Miiwialoffia Cornubicnsis, ji. 101. '' 8 THIRD REPORT — 1838. at Freyberg in 1791. Werner adopts, in the first place, the proposition that the spaces now occupied by veins were origi- nally rents formed in the substance of rocks, and states that this is not a new opinion. He claims the merit of having ascertained in a more positive manner the causes which have produced these rents, and of having brought forward better proofs of it than had formerly been done. He admits that rents may be produced by many different causes, but he assigns the greater part to subsidence. He lays it down, that when the mass of materials of which the rocks were formed by precipitation in the humid way, and which was at first soft and moveable, began to sink and dry, fissures must of necessity have been formed, chiefly in those places where mountain chains and high land existed. He adds, that rents and fissures are still forming from time to time in mountains which have a close resemblance to those spaces now occupied by veins, and that this happens in rainy seasons and from earthquakes. He adduces as a proof of his assertions, that veins, in respect of their form, situation and position, bear a strong resemblance to rents and fissures which are formed in rocks and in the earth ; that is to say, both have the same tabular figure, and the deviations which they make from their general direction are few in number and very inconsiderable ; and he remarks, that all the veins of a mining district, more particularly when they are of the same formation, have a similar direction, which shows them to have been produced by the same general cause. But what Werner claimed as altogether new, and what he challenges as his own particular discovery is, 1. To have determined and described in a more particular manner the internal structure of veins, as well as the formation of the different substances of which they are composed, and to have settled the relative age of each. 2. To have given the most accurate observations and most perfect knowledge of the meetings and intersections of veins, and to have made these observations subservient to the deter- mining their relative ages. 3. To have determined the different vein formations, parti- cularly metalliferous veins, as well as their age. 4. To have been the first who entertained the idea that the spaces which veins occupy were filled by precipitations from the solutions, which at the same time formed by other precipi- tations the beds of mountains, and to have furnished proofs of this : and, REPORT ON MINERAL VEINS. 9 5. To have determined the essential differences that are found between the structure of veins and that of beds. Werner illustrates his propositions by many observations, which his intimate acquaintance with the extensive mining di- stricts in which he was engaged gave him the power of observing and recording ; and it must be conceded, at least, that his state- ment of facts, and his arrangement of them, give him a manifest superiority over most writers upon this subject. Every one who has had opportunity to see much of these storehouses of nature will be struck with the accuracy of most of his descrip- tions, whether they admit the theory by which they are ex- plained, or not. He allows that the enrichment of veins, or their being filled with ores or metals, may have taken place by, 1. a. A particular filling up from above. b. By particular internal canals. c. By infiltration across the mass of the vein. 2. A metallic vein may be increased by the junction of a new metalliferous vein, 3. Though rarely, the richness of a vein may be the effect of an elective attraction or affinity of the neighbouring rock. The mode assigned by Werner for the formation of the spaces now occupied by veins is still further demonstrated, in his opinion, by the relation which veins have to one another ; as, Their intersecting one another. Their shifting one another. Their splitting one another into branches. Their joining and accompanying one another. Their cutting oft" one another. All these peculiarities, he remarks, are produced by the ef- fects of a new fissure upon one that is older. Subsidence having been the cause of fissures he thinks is proved by the difference in the level in the parts of the same stratum or bed in which a vein is inclosed ; and this throwing up or down, as the miners term it, bears a proportion to the size of the vein. The interior structure of many veins is quoted to show that the fissures had been originally open, and which had been af- terwards filled by degrees. Such veins are composed of beds, arranged in a direction pa- rallel to their sides ; their crystallizations are supposed to show these beds to have been deposited successively on each other, and that those next the walls have been first formed. A cir- cumstance much relied on, also, is the existence of rolled masses or water-borne stones, fragments of the adjacent rock, some- 10 THIRD REPORT — 1833. times forming a breccia, remains or impressions of organic bo- dies, coal and rock salt substances of recent formation, and other matters, which should appear to have come in from above. This theory obtained considerable attention, and was very generally adopted from the time of its being made known ; and it has, I believe, many adherents at this day, particularly among miners or those who have much opportunity of actual observa- tion. Hutton's Theory of the Earth was published afterwards, in 1795; and as his views regarding the operations employed in the formation of the structure of the rocks differed entirely from those who assigned to them an aqueous origin, so it will readily be supposed that he would promulgate a new explana- tion of the formation of veins. According to Playfair, this theory embraced the following propositions : — It allowed that veins are of a formation subsequent to the hardening and consolidation of the strata which they traverse, and that the crystallized and sparry structure of the substances contained in them shows that these substances must have con- creted from a fluid state. It assumes that this fluidity was simple like that of fusion by heat, and not compound like that of solution in a menstruum. It is inferred that this is so from the acknowledged insolu- bility of the substances that fill the veins in any one menstruum, and from the total disappearance of the solvent, if there was any, it being argued that nothing but heat could have escaped from the cavities. It is further maintained, that as the metals generally appear in veins in the form of sulphurets, the combination to which their composition is owing could only have taken place by the action of heat. And, furthei", that metals being also found na- tive, to suppose that they could have been precipitated pure and uncombined from any menstruum, is to trespass against all analogy, and to maintain a physical impossibility. It is therefore inferred, that the materials which fill the mi- neral veins were melted by heat, and forcibly injected in that state into the clefts and fissures of the strata. The fissures must have ai'isen, not merely from the shrinking of the strata while they acquired hardness and solidity, but from the violence done to them when they were heaved up and elevated in the manner which the theory has laid down. Slips or heaves of veins, and of the strata inclosing them, are to be explained from the same violence which has been exerted. REPORT ON MINERAL VEINS. 11 It is admitted as interesting to remark, that in the midst of the signs of disturbance which prevail in the bowels of the earth, there reigns a certain symmetry and order, which indi- cates a force of incredible magnitude, but slow and gradual in its eftects. Further, that as a long period was required for the elevation of the strata, the rents made in them are not all of the same date, nor the veins all of the same formation. A vein that forces the other out of its place, and preserves its own direction, is evidently the more recent of the two. The parallel coats lining the walls or sides of the vein, which are attributed by Werner and others to aqueous deposition, are ascribed to successive injections of melted matter. Veins have been considered as traversing only the stratified parts of the globe. They do, however, occasionally intersect the unstratified parts, particularly the granite ; the same vein often continuing its course across rocks of both kinds without suffering material change. It is asserted that all the countries most remarkable for their mines are primary, and that Derbyshire is the most considera- ble exception to this rule that is known. This preference which the metals appear to give to the pri- mary strata, is considered as consistent with Dr. Hutton's theory ; and particularly as these strata, being the lowest, have also the most direct communication with those regions from which the mineral veins derive all their riches. In arguing further upon this theory, it is assumed that no- thing of the substances which fill the veins is to be found any- where at the surface ; and that, contrary to the allegation of some that mineral veins are less rich as they go further down, it is stated that this is not generally so, and that the mines in Derbyshire and Cornwall are richest in depth, as they would be if filled with melted matter from below. Again, it is said that if veins were filled from above, and by water, the materials ought to be disposed in horizontal layers across the vein; and that this opinion is sufficiently refuted by the fact that rarely any metallic ore is found out of the vein, or in the rock on either side of it, and least of all where the vein is richest. The foregoing seem to be the most important allegations in support of the Huttonian theory ; and I have taken them nearly in the order in which they are given in Professor Playfair's il- lustrations of this celebrated system. There is yet another doctrine regarding the formation of veins, which, though it is not of modern date, and has had but IS THIRD REPORT — 1833. few supporters among writers upon the subject, has yet claims to be considered, and particularly as it has of late been urged upon our notice, and by some whose observations have been made in districts where veins of vai'ious order are abundant. This theory is, in short. That veins were formed at the same time with the rocks themselves ; that the whole was a contem- poraneous creation ; and that there have been neither fissures subsequent to the consolidation of the mass, nor filling up from above or below, or disturbances to produce the heaves or shifts which we see. When this hypothesis was first proposed I do not know, but that it was long since we may infer, as Agricola regards the opinion which supposes veins such as we now see them to have been formed at the same time with our globe, to be at variance with fact, and he calls it the opinion of the vulgar. The same hypothesis was indeed supported by Stahl ; but he seems to have adopted it rather on account of the difficulties attendant on any other explanation that had been proposed, than for any good reason that he had to give. Such are, however, but assertions, to be received with doubt by any one who inquires freely and without prejudice. Partial evidence may appear for some such formations ; but it is another affair to attribute all veins to such an origin, and thus to sweep away at once the difficulty of explaining many complicated ap- pearances. The doctrine of a contemporaneous formation of veins has lately found an advocate in Dr. Boase, in his paper on the geology of Cornwall. After commenting on the division into different orders, which Mr. Carne had indicated as to veins, according to certain appearances in their direction and the character of the substances with which they are filled, he says he cannot detect any characters which are not common to all the Cornish veins ; and since some of them are generally ac- knowledged to be contemporaneous with the rock, he concludes that they have all the same origin. Dr. Boase, however, candidly sets out by stating that he had purposely refrained from making inquiries at the mines con- cerning the phasnomena of veins, and that bis experience is therefore principally confined to jhose which occur in cliffs, quarries, and natural sections that are exposed to open view. Lest this admission should create surprise, he remarks that such sources of information are invaluable as the only ones easily available to exercise the senses on the nature of veins ; for, unless to those much accustomed to descend into mines, they may as well be visited blindfold. REPORT ON MINERAL VEINS. 13 He remarks, however, as to the veins of Cornwall, that their great irregularity in size and in form, their fi'equent ramifica- tions, their similarity of composition and intimate connexion with the rocks which they traverse, and, above all, the large masses of slate which they envelop, are all circumstances to disprove their origin from fissures, and to support their con- temporaneous origin. Dr. Boase suggests that veins follow the arrangement of the joints of the rocks, and that it may thus be explained why the different series of veins cross each other, and why the veins of each series are respectively parallel. And he thinks that thus we may suppose how veins which are crossed may seem to abut or terminate against those that are opposed thereto ; having, when in the same line, that pecu- liar appearance that has been attributed to intersection, and the appearance of being heaved when on the opposite sides of the cross vein, they are not on the same line, but occur in the parallel joints of distant layers. The latter occurrence, he remarks, although very common, is not however universal ; for, in some instances, the part of the vein supposed to have been intersected has never been found. As Mr. Carne had observed, that when contemporaneous veins meet each other in a ci'oss direction, they do not exhibit the heaves and interruptions of true veins, bvit usually unite. Dr. Boase says that this statement is opposed to his obser- vations, and that the phaenomenon of intersection is common to all kinds of veins. Further, he expresses a doubt whether heaves in veins are not after all rather apparent than real, but explains that he does not mean to assert that they do not ex- hibit these phaenomena, but that this arrangement, as in the case of small veins, only gives the appearance of being moved from the original positions. I have now stated the opinions which, as far as I know, have been generally received on the subject of the formation of veins, from which it wiU appear that there are three leading hypo- theses. 1st. That which supposes them to have been open fissures, caused by disruption, and occasioned principally by subsidence of parts of the rocks, which fissures were afterwards filled up with various matters by deposits from aqueous solution, chiefly from above. Modifications of this theory are. That such rents in the earth may have been caused in other ways, such as earthquakes, or certain great convulsions, as well as by subsidence : That they may have been filled by the infiltration of sola- i'4 THIRD REPORT 1833. tions, which deposited the substances with which they were charged in the veins, or by the process of subhmation from below. The second theory allows that veins were formed subse- quently to the consolidation of the rocks ; but the cause prin- cipally assigned for such fissures is the violence done to the strata by the elevation or upheaving of other rocks from below. And it is an essential part of this theory that the materials which fill the veins were forcibly injected upwards in a state of complete fusion by heat. The third theory is that denying any subsequent processes which might either cause rents and fissures, or might fill them with matter which differs from the rocks which inclose them : the whole formation was contemporaneous with the rocks them- selves, the mineral substances which we find in veins having separated and arranged themselves into the forms in which we now see them to exist. The advocates of these theories have each zealously asserted the truth of his own system, and refused to admit of causes or explanations which appeared to militate against it ; and thus a boundary has been set, as it appears to me, to that freedom of inquiry which is so desirable in such cases, and a limit drawn round the reasoning faculties of man upon evidence which may come before him. It will appear, from what has already been said, that veins have very different characters and appearances ; and this might be made more clear, if it were here the proper place to enlarge upon the subject and point out the distinctions. For our pur- pose, however, it may be sufficient to remark upon two or three principal varieties. First, then, are those which have beyond all comparison been most explored and examined, on account of the stores which they contain, — the metalliferous veins. As these have been penetrated in all directions to the greatest ex- tent that human power and ingenuity have been able to effect, so their structure is better known and more accurately ob- sei'ved. Similar to these, and occurring with them, and therefore well known, are others, which, though baiTen of metals, are yet often called true veins ; and these, as well as the first, come pretty fully under the view of the miner. Next there are veins, regular in their structure to a great extent, filled with matter which has the character of being de- rived from igneous origin, such as are usually called dykes of trap, whinstone, &c., &c. ; to which would be added by most REPiORT ON MINERAL VEINS. 15 geologists of the present day, the veins of granite, porphyry, quartz, &c. Some of these have been examined below the svirface, where they pass through coal-fields, or other deposits of useful mine- rals, but containing in themselves nothing to reward the toU of exploring them : little has been seen of their contents and con- figuration, and our knowledge of them is more limited. Lastly, there are tortuous and irregular veins or ramifications in most rocks, extending to limited distances, as far as our ob- servations permit us to judge, seldom offering a valuable return for any effort to explore them, and of which, therefore, our knowledge is but superficial. Such veins, according to Mr. Carne, have been usually di- stinguished from true veins by their shortness, crookedness, and irregularity of size, as well as by the similarity of the con- stituent parts of the substances which they contain to those of the adjoining rocks, with which they are generally so closely connected as to appear a part of the same mass. Two other distinctive marks may be added ; one is, that when they cross they do not exhibit the heaves of true veins, but usually unite ; the other is, that when there is an apparent heave it is easy to perceive that what appear to be separate parts of the same vein are different veins terminating at the cross vein. Such may be, probably, of contemporaneous formation ; and there may be deposits of ore also which it would be difficult to refer the structure of to any other hypothesis, particulai-ly such as contain ores so intimately mixed with the rocks as to form a constituent part of them. I would suggest, that if from any one of these classes we were to form a judgement as to the whole, error would probably be the consequence, or, at any rate, the view would be a narrow and contracted one, and our decisions would be defective in many important respects. To have conducted the inquiry in this manner seems to me to have been the error in many who have preceded us in for- warding the state of knowledge on vein formations. Nor do I mean to detract from the great merit of many of them on this accovmt ; the field of observation is too vast to become fully acquainted with it ; it extends over the most rugged parts of the earth's surface, and its boundaries are not reached in the deep recesses of its bowels. It is no wonder that in the earlier stages of such inquiries men should be strongly impressed with what lay immediately before them, and should view with dis- trust what they might only learn from description. Such impressions may be traced in looking at the authors of 16 THIRD REPORT — 1833. the systems which we have reviewed. Werner expressly tells us, that we are indebted to miners for the theories which he deemed most worthy of acceptation, and he names as such Agricola, Rosier, Henkel, HoiFman, Von Oppel, Charpentier, and Trebra. We may add his own name and that of Dr. Pryce, in our own country, as intimately acquainted with mining. Now all such men would be more acquainted with the metalliferous veins and such as accompany them; and from these they would derive much evidence in favour of the opinions which they ad- vocated ; at least, partaking, as I probably do, in the same pre- judices, so it would appear to me, if by the labour of other inquirers I did not know that there were other facts requiring a different explanation. Again, Dr. Hutton and his commentators had largely ob- served veins which may fairly be attributed to injection; they had found dykes of trap passing through coal-beds, and con- verting them into cinder. Such evidence of the effects of heat and of a filling up by matter in fusion is not to be resisted ; but when we look at what is said of the metalliferous veins by some of the writers on this side of the question, we observe great want of practical knowledge and many errors, arising out of the attexnpt to make all bend to a single method of solving the problem. For the third hypothesis of contemporaneous formation there is this to be said, — that some veins exist which seem to admit of no other explanation ; and that this being allowed to such as will have but one theory, this is at once the easiest, because it gets rid of many difficulties without further trovible ; but we can hardly be satisfied to adopt it as universal upon experience that has been principally confined to sections in quarries and in cliffs, or to such as are exposed to open view. Our present state of knowledge as to the formation of veins should therefore, in my opinion, be allowed to admit that most of the causes which have been stated have operated at various periods and through a long succession of time, some prevailing at one epoch, and some at another, modified by circumstances which we can but imperfectly comprehend or explain. In this view we may allow of a classification of veins accord- ing to their probable mode of origin ; and such a classification has been thought of by some of our ablest geologists of the present day, and was indeed propounded in one of our sections at Oxford last year by our present learned President, who ex- pressed his opinion that there were three different sets of veins : — 1. Those which have been plainly mere fissures or cracks, and which have been subsequently filled ; 2. Those of injec- REPORT ON MINERAL VEINS. 17 tion ; 3. The contemporaneous veins, which might more aptly be termed veins of segregation. Here I might close this Report, which is already much too tedious, were it not that I may be expected to notice briefly some of the facts adduced by the advocates of the respective theories, and, by comparing them, show how far they are enti- tled to be considered as objections on one side, or as proofs on the other, with the confidence which has been assigned to them. Werner and Hutton agree in allowing that rents took place subsequently to the consolidation of the rocks, or at the time of their consolidation. They differ as to the cause of the rents : Werner ascribes it to subsidence, or to sinking and shrinking of the solid materials of our globe ; Hutton, to violent upheaving of matter from below, breaking up the superinjacent strata. Either of these causes seems adequate to the effect, and in either case corresponding strata might be found having different levels of position on opposite sides of the fissure, as is constantly the case. This by miners in the North of England is called the throw of the vein ; and it is clear that one side may as well be thrown up as the other thrown down. Mr. Fox and Dr. Boase urge the great irregularity of the width of veins, the difficulty of supposing the sides to be supported, and some other objec- tions to the hypothesis of open fissures. Irregularity of width is but a comparative term ; and taking into consideration the immense extent of their dimensions in length and depth, it amounts in my opinion to but little. The other objections are in a great degree anticipated and answered by Werner ; and, after all, difficulties can hardly be urged against the positive testimony of some veins having been open, which is afforded by the substances found in them, such as rolled pebbles, petrifactions, &c. The parallelism of veins of one formation is insisted upon by Werner as a proof of his view of the subject ; and I confess that there appears to me to be considerable difficulty in explaining this, on the supposition that fissures were caused by a mass protruded upwards through strata already foi'med. From such a cause one should expect not to have a number of cracks pa- rallel to each other, but rather to see them radiating from the centre of the greatest disturbance. In the metalhferous veins we may certainly observe this parallelism to a great extent. Mr. Carne has beautifully illustrated this in Cornwall, and has shown how the productive veins generally have an east and west course ; how, as they differ in their contents, they diff'er also in their direction, each class being, however, parallel in 18^3. c 18 THIRD REPORT — 1833. itself; ami how tliese facts illustrate relative ages of forma- tion. This tendency to an east and west direction of the metallife- rous veins may be observed not only in Cornwall but in the stratified parts of England, in the mining districts of Europe, and in the range of the great veins of Mexico. Mr. Robert Fox, having discovered galvanic action to ensue by the connexion of an apparatus, constructed to detect it, with portions of metalHferous veins, suggests whether some analogies may not be traced between electro-magnetic ciu'rents and the directions of veins : nothing upon which any hypothesis can be built seems, however, as yet to have been proposed ; and it may be doubted whether, when this test is apphed to masses of ore, the experiment is not liable to many objections. A principal one seems to be, that by the very act by which we gain access to the vein, we lay it open to atmospheric action, and conse- quently to decomposition. Chemical agency commences, and with it, very naturally, galvanic influences are excited. Veins containing ores little subject to decomposition have, I apprehend, been found to give httle or no indications of this nature. It may, however, be that this general direction of metallife- rous veins may not obtain as to veins of injection ; and in that case we shall have additional reason to admit more causes than one to have been in operation. This is a matter deserving ex- tensive observation. Other veins have been stated to cross the metalliferous veins : they are generally filled in a different manner. If they contain any ores, they are frequently of difterent metals from those in the former. They pass through or traverse the other veins, cutting them through, and sviffering a disturbance to take place in their linear direction, or what the miners significantly term a heave. This fact is relied upon as proving that veins are of different ages, as first asserted by Pryce, much insisted upon by Werner, and allowed by Hutton and Playfjiir. Those who dispute this inference, therefore, are the advo- cates for the sole operation of contemporaneous causes : they object that rules which have been proposed for ascertaining the exact tendency of such disturbances having been found to be subject to exceptions, tlie proof of dislocation is wanting, or that dislocation has taken place without motion. The latter proposition, at any rate, appears to me to be very difficult to understand ; and I think if any part of this intricate subject is clear and intelligible, it is that the relative age of veins is made REPORT ON MINERAL VEINS. 19 out by these fiicts, even although we may not yet be able to apply rules for every case, — a subject which has been con- sidei'ed as highly important in its practical application to the art of mining. The greatest controversy, however, relates to the mode in which veins have been filled. Here, again, we must remark, how the opinions of observers have been influenced by the facts cominff under their immediate observation. Werner, and the mining authors on whom he relies, drew their inferences from metalliferous veins. Hutton and his fol- lowers regarded chiefly those of another class ; and this great author and his commentator Professor Playfair were evidently ill informed as to metalliferous veins. That certain veins have been filled by injection from below, and with matter in igneous fusion, seems to be rendered certain by evidence, which is clearer than most we possess on such sub- jects, and must be admitted at once. Thus, when we see a trap dyke traversing a bed of coal and charring the combusti- ble matter, and afl^ecting the rock itself with visible efifects of great heat, we must assent to the cause assigned ; and when we see matter of igneous origin not only filling the veins, but over- flowing on the surface, or insinuating itself between adjacent beds, the case is plainer than most that occur in geological re- search. But though one class of theorists have proposed this as the universal cause of the filling up of veins, ought we to admit this to be true, when we find so many in which no similar appear- ances are to be traced? Why, for instance, if the ores were forced from below, did the power which injected them just limit itself to raising them within a short distance of the surface,— for where shall we find an instance of their being protruded above it ? If the metallic contents of veins were injected from below, we ought to be able to trace something like the direction of the currents in which the matter flowed ; we ought to see some continuity in the operation, and some connexion between the masses of ore which occur in veins ; whereas the contrai'y of each is notoriously evident to every observer. It would seem also to be very probable, if the enrichment was from below, and the matter was forced in from those regions whence their treasures are supposed to be derived, that by a nearer approach to the depths of the earth we should find the riches more abundant. Professor Playfair admits this inference, and disposes of the difficulty by arguing that it is so ; and says, that though mines c2 so THIRD REPORT 1833. in Mexico and Peru are said to be less rich as they descend further, those of Derbyshire and Cornwall exhibit the very contrary. He is unfortunate in this allegation, and the facts will not bear him out, as every one of common experience must know ; and thus, as I have before observed, we have hypotheses sup- ported by a limited knowledge of the facts. The theory of the filling up of veins by precipitation from aqueous solutions, is defective in not being able to show what menstruum could render such substances soluble in water ; and this difficulty must remain an important one, unless en- larged knowledge should hereafter afford the means of ex- plaining it. But when we are told that the supposition is absurd, that water cannot arrange its deposits in planes highly inclined, that no appearance of stalactites is to be found in veins, nor can we see in them any substance like those on the earth's surface, which aqueous action has removed, — it must be recollected that we know silex is soluble in water at high temperatures ; that crystals do arrange themselves on the sides of vessels in planes highly inclined; that stalactites of chalcedony, of quartz, and of iron pyrites, have been found deep in the veins in Cornwall, and that much of the substance of the surrounding rocks, and such as we see on the surface, and adjoining and inclosing the veins themselves, is found in them, occupying much of their space, previously having been worn down into fragments, into loose sand, and into clay or mud, the latter of which is so common that, as I have before observed, it is relied on by the miner as a distinguishing character of regular veins*. The action of water may, I think, be as fairly assumed as that of fire ; and we may consider what their joint powers might be, when compelled, as it were, to act together, under circum- stances that immense pressure might produce. But in examining the contents of veins, we are, I think, likely to be struck, not only by the appearance of a complication of causes, but by evidence of their succession, admitting the pro- bability not only of different agents having been employed, but of their having done their work separately as well as conjointly, • Mr. Weavei' describes the contents of the great vein of Bolanos in Mexico thus : " The chief mass of this vein may be said to consist of the detritus of the adjacent rocks, more or less consolidated, and generally hard ; nay, in places, it is actually composed of a conglomerate. Proper vein-stones, such as fluov or calc spar, are, comparatively speaking, casualties. In this basis the finer delicate silver ores and native silver are dispersed, in common with the harder and coarser ores of blende, iron, and copper, besides lead ores." REPORT ON MINERAL VEINS. 21 —of having operated at different periods, and of one having produced effects for which another was inadequate. As we cannot easily conceive how the metahic ores can have been deposited from solution in water, and appearances are much against their having been injected in a state of fusion, there is another supposition which, though not free from diffi- culties, has yet probability enough in its favour to have gained it many supporters, — which is, that these and some other sub- stances have been raised from below by sublimation. This is not a new opinion, for though the older writers expressed it in an indistinct manner, and spoke of metallic vapours and exha- lations, — and thus we shall find it proposed by Becher, Stahl, Henkel, and others, — yet their meaning evidently was, that sub- stances had been volatilized by heat, and assumed their places in veins by condensation, or by combining with other materials. We know for certain that some of the metallic sulphurets may be so volatilized, and will reassume their form and be produced in a crystallized state ; and so far nothing is assumed beyond our knowledge : but as we find these sulphurets, which compose by far the greater part of the metdllic contents of veins, in insulated masses, surrounded on all sides by other substances, which we can hardly conjecture to have been sublimed, we en- counter much difficulty in explaining how the process can have taken place ; and it becomes even more difficult when we see how very much these different classes of substances are incor- porated, and how they completely, in most instances, envelop and inclose each other. The hypothesis of filling up by sublimation would also seem to require that the deepest portions of veins should be richer, especially considering the very small extent to which after all they have been perforated ; but yet, shallow as our workings into the earth really have been, there is much appearance of their having in many instances gone below the richest deposits of the metals. This seems to have been the case in some of the deepest mines in Mexico, and in several in our own country. It is im- possible, indeed, to say that greater deposits may not exist still lower down ; and though veins have not been traced to their termination, they have in many instances been pursued until the indications of metallic produce have become faint and hope- less. And these unfavourable appearances have increased very commonly with increasing depth, which is as much, perhaps, as we are likely to know about it, as the operations of the miner are thus arrested, and the inducement to further experiment is taken away. 22 THIRD REPORT — 1833. The agency of sublimation has lately been advocated by Pro- fessor Necker of Geneva, in a paper read before the Geological Society of London * ; and he has extended an ingenious hypo- thesis of Dr. Bovi^, who would bring under a general law the relation of metalliferous veins and deposits to those crystalline rocks which, by the majority of modern geologists, are consi- dered to have been produced by fire ; and thus to lead to the inference that the metals were deposited in the former by sub- limation from the latter. M. Necker inquires, 1. Whether there is near each of the known metalliferous deposits any unstratified rock 'i 2. If none is to be found in the immediate vicinity, is there no evidence which would lead to the belief that an unstratified rock may extend under the metalliferous district ? 3. Do there exist metalliferous deposits entirely disconnected from unstratified rocks ? Professor Necker answers these questions by showing that in various countries there are such relations as he supposes, and admits, in reply to the last, that there are cases where the depo- sits seem to be unconnected with any trace of unstratified rock. If metalliferous deposits are commonly in crystalline rocks which are attributed to igneous origin, it must be allowed also that there are others abundantly rich where no apparent con- nexion is to be traced. M. Necker mentions the mountain limestone as such ; but he does not seem aware of the extent of those deposits, which, with the beds of grit and shale which alternate with it, present numberless regular veins abounding with certain ores. As this fact is indisputable, it seems necessary to show not only that unstratified rocks may be under them — which there is little doubt about, — but that there should be some connexion between the veins which contain the metals and similar chan- nels or passages in the rocks below. No such evidence, I be- lieve, at present exists ; and I am not aware of any veins having yet been found to penetrate from the stratified rocks into those upon which they rest. This supposition must therefore, like many others, be taken as a mere probability to account for some appearances in certain places, but not to explain all the phaenomena. There is one point which, before I conclude, I would endea- vour to press on the attention and consideration of future ob- servers, because, in the first place, it does not appear to have been much regarded by writers on the subject ; and next, be- « March 28th, 1832. REPORT ON MINERAL VEINS. 23 cause, though it seems to offer objections to some received theories, it may, when better understood, assist in developing the truth. This is the relation that the contents of a vein bear to the nature of the rock in which the fissure is situated. Thus in the older rocks, we see the same vein intersecting clay-slate and granite: it is itself continuous, and there is no doubt of its identity ; and yet the contents of the part inclosed by the one rock shall differ very much from what is found in the other. In Cornwall, a vein that has been productive of copper ore in the clay-slate, passing into the granite becomes richer, or, what is more remarkable, furnishes ores of the same metal differently mineralized. If we pursue it further into the granite, the produce of metal frequently is found to diminish. Veins in some cases cut through the elvan courses, as well as the clay-slate inclosing these porphyries : the ores are rich and abundant in the latter ; in other instances they fail altogether. Less striking differences in the structure of the rock seem to aff'ect the contents of the veins ; and appearances as to the tex- ture and formation of the strata are often regarded by miners with mox'e anxiety ihan the indications presented by the vein itself; and a change of ground is relied upon with an assu- rance, derived from experience, as a more certain basis to au- gur upon, for better or for worse, than almost any other which the difficult art of mining has to offer. Numberless facts might be collected and adduced to show that this is not mere speculation ; but it will nowhere appear more clearly than if we examine the various beds of limestone grit, &c., in the great lead mines in the North of England. Here we shall find a series of stratified rocks, and that por- tion of the series which has been most productive of lead ores, occupying a thickness of nearly 280 yards. It is divided into 55 distinct beds, which are accurately described in Mr. West- garth Forster's section, each having its name known to the miners of the country. Nine of these beds are of limestone, about 18 are of gritstone or siliceous sandstone, and the re- mainder are plate or black shale, with thin beds of imperfect coal. Now the lead veins pass through all these beds, and have been worked more or less into all of them; and it has thus been proved, that though the fissure is common to all, yet lead ore is only found abundantly in particular beds, and those very much the same, if we examine the immense number of mines which are working in this district. Where the veins pass through the shale, little or no ore is to 24 THIRD REPORT — 1833. be found in them ; where they are inclosed by the gritstones, there they become more productive ; but it is in one of the beds of limestone, and one only, that the great deposit of lead ore is to be found. In the great mining field of Alstone Moor, this bed is called the great limestone, and yet its thickness is only about 23 yards out of the 280 which the series of lead measures occupy; and notwithstanding this, four-fifths of all the lead ore found in the district is derived from such parts of the veins as are inclosed by this particular stratum. The veins equally passing through the other beds, and traced by innumerable workings through them, are yet only rich in metallic treasure where they repose in this favoured stratum *. Though perhaps few cases are so striking as this, yet it is evident that the same thing takes place to a certain extent with all the metals, in all rocks and in all countries. If it is a fact and correctly stated, it must be considered in reference to the theories propounded to us, and it seems directly opposed to the doctrine of forcible injection; but it may admit of probable explanation by calling in certain affinities, either by the advocates of precipitation from water, or by those who may contend that sublimed vapours might be attracted to particular spots. * To illustrate the comparative bearing of the different beds in the manor of Alstone Moor, Mr. Thomas Dickinson, the Moor Master for Greenwich Hospi- tal, extracted for me an exact account of the ore produced from each bed in all the mines of the manor in the year 1822, which gave the following results: — Limestone Beds. — Great limestone 20,827 bings. Little limestone 287 Four-fathom limestone 91 Scar limeston e 90 Tyne bottom limestone 393 ■ 21,688 Gritstone Beds. — High slate sill 107 Lower slate sill 289 Firestone 262 Pattinson's sill 259 High coal sill 327 Low coal sill 154 Tuft 306 Quarry hazel 44 Nattrass Gill hazel 21 , Six-fathom hazel 576 Slaty hazel 18 Hazel imder scar limestone 2 2,365 Whole produce of the mines of the manor 24,053 bings. REPORT ON MINERAL VEINS. 25 That metallic ores are found to repose in rocks which seem congenial to them, and that their combinations are modified by changes in the rocks, will not I think be disputed by practised miners, or by those who have most narrowly searched into the hidden recesses of the earth. Facts must be observed and compared, effects must be traced to probable causes, and difficulties must be explained or can- didly admitted, if we would enlarge and generalize our know- ledge of vein formations. There are obstacles to the progress of this knowledge ; for, as Dr. Boase has remarked, it is not easy for a person unaccustomed to it to use his eyes with much ad- vantage, in the places where the study can best be pursued. It is the miners' business, however, not only to see clearly, but to consider all the inti-icate appearances that veins exhibit ; and I would exhort them not to be satisfied merely with the obser- vations their art may seem to require, but to extend them to a larger view of the subject, and to contribute, as many of their eminent predecessors have done, to the common stock of general science. If the imperfect view which I have thus endeavoured to give of prevalent opinions should assist in such endeavours, or should stimulate any persons in undertaking a further pursuit of the subject, it would be to me a source of great gratification; as the desire of promoting such inquiries must be my apology for attempting the task which I have undertaken. [ ~^7 ] On the Principal Questions at present debated in the Pliiloso- j)hy of Botany. By John Lindley, Ph.D., F.R.S., 8fc., Professor of Botany in the University of London. If we compare the state of Botany at the end of the last cen- tury with its present condition, we shall find that it has become so changed as scarcely to be recognised for the same science. Improvements in the construction of the microscope, the disco- veries in vegetable chemistry, the exchange of artificial methods of arrangement for an extended and universal contemplation of natural affinities, the reduction of all classes of phagnomena to general principles, and, above all things, the adoption of the philosophical views of Gothe, together with the recognition of an universal unity of design throughout the vegetable world, are undoubtedly the principal causes to which this change is to be ascribed. As the general nature of recent discoveries, and a sufficient- ly definite notion of the present state of botanical science, may be collected from the introductory works which have appeared in this country within the last three years, it is presumed that the object of the British Association will be attained if the present Report is confined to the most interesting only of those subjects upon which botanists have been recently occupied, and to an indication of the points to which it is more particularly desirable that inquiries should now be directed. I have also excluded everything that relates to mere systematic botany, in the hope that some one will take that subject as the basis of a separate Report. Elementary Organs. — This country has, till lately, been re- markably barren of discoveries in vegetable anatomy, since the time of Grew, who was one of the fathers of that branch of science. Whatever progress has been made in the determina- tion of the exact nature of those minute organs, by the united powers of which the functions of vegetation are sustained, it has been chiefly in foreign countries that it has taken place : the names of Mirbel, Moldenhauer, Kieser, Link and Amici, stand alone during the period when their works were published ; and it has only been within a very few years that those of Brown, Valentine, Griffith and Slack have entered into com- petition with the anatomists of Germany and France. By the researches of these and other patient inquirers, mc 28 THIRD REPORT — 1833. have already reduced our knowledge of the exact internal struc- ture of plants to a state of very considerable precision; although it must be confessed that vegetable anatomy is still the field where the greatest discoveries may be expected. It is now generally agreed that the old opinions, that the tis- sue of plants is either a membrane doubled together in endless folds, or a congeries of cavities formed in solidifiable mucus by the extrication of gaseous matter, are equally erroneous, and that it really consists of distinct sacs or cells, pressed to- gether and adhering to each other by the sides where they are in contact. It is considered that this is proved by the following circum- stances. 1. By the action of some powerful solvent, such as nitric acid, the cells may be artificially separated from each other. 2. In parts which become succulent, the cells separate spontaneously, as in the receptacle of the strawberry, the berry of the privet, &c. 3. When the parts are young, their tissue may be easily separated by pressure in water. 4. It is con- formable to what has been observed in the growth of plants. Amici found that the new tubes of Chara appear like young buds from the points or axillae of pre-existing tubes ; an observation that has been confirmed by Mr. Henry Slack *. It has been distinctly proved by M. Mirbelf , that the same thing occurs in the case of Marchantia polymorplia. That learned botanist, in the course of his inquiries into the structure of this remarkable plant, may be said to have been present at the birth of its cel- lular tissue ; and he found that in all cases one tube or utricle generated another, so that sometimes the young masses of tis- sue had the appearance of knotted or branched cords. He satis- fied himself, by a beautifully connected series of observations, that new parts are not formed by the adhesion of vesicles origi- nally distinct, as many have asserted, but by the generative power of one first utricle, which engenders others endowed with the same property. It appears that when first formed the sacs are completely closed up, so that there is no communication between the one and the other, excepting through the highly permeable mem- brane of which they are composed. This, indeed, is not con- formable to the observations of those who have described and represented pores or passages of considerable magnitude pierced in the sides of the sacs; but it has been satisfactorily shown by Dutrochet, that the spaces supposed by such observers to be * Transactions of the Society of Arts, vol. xlix. f " Recherches Anatomiques et Physiologiques sur le Marchantia po/t/mor- pha," in Nouv. Ann, du Museum, vol. i. p. 93. REPORT ON THE PHILOSOPHY OF BOTANY. 29 pores are nothing more than grains of amylaceous matter stick- ing to the sides of the sacs ; for he found that by immersing the latter in hot nitric acid, the supposed pores became opake, and by afterwards moistening them with a weak solution of caustic potash, they recovered their transparency : we also find that the supposed pores are readily detached from the sides of the sacs to which they adhere ; and I think it may be added, that our microscopes are now alone sufficient to show what they are. The question as to the perceptible porosity of vegetable tis- sue may therefore be considered, I think, disposed of as a general fact; for the objection that Dr. Mohl has taken to this explanation *, — namely, that in a transverse section we ought to find such grains projecting from the sides of the cells Hke little eminences, — cannot surely be entitled to much weight, if we op- pose to this negative observation the positive evidence already- mentioned, and especially if we consider that it is next to im- possible for the keenest knife to make a section of svich delicate parts without carrying away such particles upon its edge. There are, nevertheless, cases in which the point is still open to in- vestigation. Thus Mirbel, in his second memoir on the Marchantia f , positively declares that the curious cells which line the anther of the common gourd, are continuous membranes till just be- fore the expansion of the flower, when they very suddenly en- large, and their sides divide into the narrow ribands or threads which give their name to what we call fibrous cells. In this, and the multitudes of similar cases with which Purkinje has made us acquainted, there can be no doubt that the sides of the cells consist ultimately of nothing but openwork; but still it seems certain that during the principal part of their existence they were completely closed up. It is also probable that in other cases the sides of the cells or vessels ultimately give way and slit ; but this rending seems to be a phaenomenon attendant upon the cessation of the ordi- nary functions of tissue, and independent of their original con- struction. In coniferous plants the wood is in a great measure com- posed of closed tubes, tapering to each end, the sides of which are marked with circles, containing a smaller circle in their cen- tre. These circles have long been considered undoubted pores^ and it does not appear possible to prove them otherwise by any of the tests already mentioned. * Ueher die P or en des Pflanzen- Zell'jewehes, p. 11. Tubingen, 1828. t Archives de Botaniqiie, vol. i. 30 TIITRD REPORT — 1833. I have endeavoured to show * that they are glands of a pecu- liar figure, which stick to the sides of the tubes ; and I have ascertained that the large round holes that are certainly found in coniferous tissue are caused by the dropping or rubbing off of such supposed glands. But a very different opinion is en- tertained by Dr. Mohlf, whose observations have been con- firmed by Dr. Unger;}:. In the opinion of the former of these botanists the supposed glands of coniferous tissue are circular spaces where the membrane of the tube becomes abruptly ex- tremely thin ; and it is said that transverse slices of coniferous wood, made at an angle of forty-five degrees, demonstrate the fact. Dr. Mohl is also of opinion, as has been already said, that the porous appearances above mentioned, and ascribed to the adhesion of amylaceous matter to the sides, are of a similar nature. It has been shown by Mr. Griffiths, that in the kind of tissue called the dotted duct, the suspicion of Du Petit Thouars that this form of tissue is composed of short cylindrical cells placed end to end, and opening into each other, is correct ; their com- munication, however, is not by means of an organic perfo- ration, but is produced by the absorption and rupture of the ends which come in contact. Mr. Slack has also stated, in a very good paper upon Vegetable Tissue §, that in other cases the vessels of plants open into each other where they come in contact ; as, for example, at the conical extremities, where ducts join each other ; but he represents this to be owing to the obliteration of their membrane at that point ; the internal fibre, of which they are in part composed, remaining like a grating stretched across the opening where the enveloping membrane has disappeared. In a short paper, published in the Journal of the Royal In- stitution in December 1831, I have endeavoured to show that membrane and fibre are to be considered the organic elements of vegetable tissue, contrary to the more usual opinion that membrane only is its basis : this was attempted to be proved, not only by the fact that the simple cells of the testa of Mau~ randia, &c., are apparently formed by a fibre twisted spirally in the inside of their membrane, but also by the elastic spires I had discovered on the outside of the seed of Collomia, in which it is plain that no membrane whatever is generated. * Introduction to Botany, p. 16. t. 2. f. 7. t Ueher die Poren des Pjiemzen-Zellgeivehes. X Botanische Zeittmg, October 7, 1832. § Transactions of the Societij of Arts, vol. xlix. REPORT ON THE PHILOSOPHY OP BOTANY. 31 It would, however, appear from the researches of Mirhel*, that the presence of a twisted fibre within a cell is not always the cause of the spiral or fibrous character so common in tissvie. He finds, as has been already stated, that the cells that line the anther of a gourd are at first membranous and closed, and that they continue in this state till just before the bursting of the anther, when they suddenly divide in such a way as to assume the appearance of delicate threads, curved in almost elliptical rings, which adhere to the shell of the anther by one end; these rings ai-e placed parallel with each other in each cell, to which they give an appearance like that of a little gallery with two rows of pilasters, the connecting arches of which remain after the destruction of the roof and walls. He also watched the development of the curious bodies called elaiers in the Mar- chantia, which he describes to the following efi^ect. At first they are long slender tubes, pointed at each end ; at a subse- quent stage their walls thicken, and become less transparent, and are marked all round through their entire length with two parallel, very close, spiral streaks ; later still the tubes enlai'ge, and their streaks become slits, which divide the walls all round, from one end to the other, into two filaments ; and, finally, the circumvolutions of the filaments separate, assume the appear- ance of a corkscrew, acquire a rust colour, and the elater is complete. These elaters he considers organically identical with the spiral vessel, and hence he concludes that every description of vessel is a cell, differing from ordinary cells in being larger. Upon the general accuracy' of these observations I am dis- posed to place great confidence ; and I would even add, that the theory of pierced or open cellular tissue being produced by the spontaneous rending of its membrane, is apparently con- nected with an observation of my ownf, that in some plants simple vegetable membrane will tear more readily in one direc- tion than another. It is nevertheless to be observed, that the theory of fibre being one of the organic elements of tissue does not seem to have occurred to the experienced physiologist to whose observations I am referring, and that some of the ap- pearances he mentions at a stage pi-eceding transformation are very like those of the development of an internal fibre. The opinion of the organic identity of all the forms of tissue has also been maintained by Mr. Slack, in the paper already referred to, and by Dr. Mohl, in his memoir on the comparative anatomy of the stem of Cycadece^ Coniferce, and Tree Ferns. » Archives de Botantque, vol, i. t Introduction to Botany, p. 2. S3 THIRD REPORT — 1833. The latter considers that the dotted tubes of Cycadece un- doubtedly pass directly into the vessels called by the Germans vasa scalarlformia \ but my own observations do not confirm this statement. ' Circulation. — Whether or not plants have a circulation ana- logous to that of animals, is a topic that was more open to con- jecture at a time when the real structure of the former was un- known, than it can be at the present day. Knowing, as we now do, that a tree is more analogous to a Polype than to a simple animal ; that it is a congeries of vital systems, acting indeed in concert, but to a great degree independent of each other, and that it has myriads of seats of life, we cannot expect that in such productions anything absolutely similar to the mo- tion of the blood of animals from and to one common point should be found. The idea of circulation existing in plants must therefore be abandoned ; but that a motion of some kind is constantly going on in their fluids was sufficiently proved by the well-known facts of the flow of the sap, the bleeding of the vine, the immense loss plants sustain by evaporation, and by similar phaenomena. The motion was for the first time beheld by Amici, the Professor at Modena, who discovered it in the Char a. He found that in this plant the cylindrical cells of the stem are filled with fluid, in which are suspended grains of green matter of irregular form and size. These grains were distinctly seen to ascend one side of each tube, and descend the other, after the manner of a jack-chain, and to be continually in action, in the same manner, as long as the cell retained its life ; the motion of the grains was evidently due to the ascend- ing and descending current in the fluid contained within the tube-like cell. It could not be ascertained that any kind of communication existed between the cells, but each was seen to have a motion of its own. The observations of Amici have been verified in this country chiefly upon species of Nitella ; and from the investigations of Mr. Solly, Mr. Varley, and Mr. Slack*, the nature of the phee- nomenon has been determined with considerable precision. Among other things, it has been ascertained that in Nitella the currents have always a certain relation to the axis of growth, the ascending current vmiformly passing along the side of the cell most remote fi-om the axis, and the descending current along the side next the axis. Similar motions have been seen in several other plants. In the cells of Hydrocharis Morsus-Ranae the fluid has been ob- » Transactions of the Society of Arts, vol. xlix. REPORT ON THE PHILOSOPHY OF LJOTANY. 3^ served to move round and round their sides in a rotatory man- ner, which, however, has not been seen to follow any particular law. In the joints of the hairs of Tradescmitia virginica several currents of a similar nature exist ; and in the hair of the corolla of a species of Pentstemon, Mr. Slack has observed several currents taking various directions, some continuing to the summit of the hair, whilst others turn and descend in va- rious places, two currents frequently uniting in one channel. It may hence, possibly, be assumed that in the cells of plants, when filled with fluid, there is a very general rotatory move- ment, which is confined to each particular cell. This, it is ob- vious, can form no part of the general circulation of the system, which must often occur with great rapidity, and which must take place from the roots to the extremities. The rotatory motion may perhaps be considered a sort of motion of di- gestion, and connected with the chemical changes which matter undergoes in the cells from the united action of light, heat, and air. What has been supposed to be a discovery of the universal motion of sap has been made by Professor Schultz of Berlin, who remarked two torrents, one of which was progressive, and the other retrogressive, in what he calls the vital vessels (ap- parently the woody fibre) in the veins of Cheliclonium majus, and in the stipulae of Ficus elastica. His observations have been repeated by a Commission of the Institute, composed of MM. Mirbel and Cassini, who have reported* that they have also seen the motion described by Professor Schultz ; and I have myself witnessed it as is repre- sented by those observers. But it appears probable, from se- veral circumstances, that the motion that has been seen has either been owing merely to the vessels in which it was re- marked having been cut through, and emptying themselves of their contents, as Mr. Slack has suggested, or else was nothing ])ut the common rotatorymotion imperfectly observed. Sfri.'cfure of the Axis. — From the period when INl. Desfon- taines first demonstrated the existence of two totally distinct modes of increase in the diameter of the stems of plants, it has been received as a certain fact that monocotyledonous plants increase by addition to the centre of their stem, and dicotyle- donous by addition to the circumference. Nothing has yet arisen to throw any doubt upon the exactness of this notion in regard to dicotyledonous plants ; but Dr. Hugo Mohl has endeavoured to showf that monocotyledonous stems are not * Aniinlea ih.i Sr.'ienres, vol. xxii. p. 80. i Molil, "])i> I'aliiiavum Stnictura," in Marlhis's Genrra ct Species Palmarum. IS;}.']. D 34f THIRD REPORT — 1833. formed in the manner that has been supposed. According to him, the new matter from which the wood results is not a mere addition of new matter to the centre, but consists of bundles of wood, which, originating at the base of the leaves, take first a direction towards the centre, and then a course outwards towards the circumference, forming a curve ; so that the stem of a Palm is, in fact, a mass of woody arcs intersecting each other, and having their extremities next the circumference of the trimk. I regret that I have not been able to consult Dr. von Martius's splendid work on Palms since this Report was commenced, and that I am therefore unable to state upon what evidence Dr. Mohl has rested his theory. The same writer has stated* that Cycadece — that singular tribe, which is placed, as it were, on the boundary line between cellular and vascular plants, — are not in a great measure desti- tute of vessels as is commonly supposed, but, on the contrary, are composed exclusively of spiral vessels and their modifica- tions, without any mixture of woody fibre. I have already ad- verted to this hypothesis in speaking of the same author's state- ment, that the dotted tubes of Cycadece are a slight modi- fication of vasa scalariformia. Dr. Mohl is also of opinion that Cycadeae are not exogenous in their mode of growth, as seems to be indicated by their appearance when cut, and by their dicotyledonous embryo, but that they are more like Palms in their manner of forming their wood, which is essen- tially endogenous. He asserts that the stem of Cycadece, in regard to its anatomical condition, must be considered inter- mediate between that of Tree Ferns and Coniferce, just as their leaves and fructification undoubtedly are. He states that in Cycadece a body of wood is gradually formed of the fibres con- nected with the central and terminal bud ; that so long as this original wood is soft, and capable of giving way to the fibres that are continually passing downwards, no second cylinder of wood is formed ; but in time the original wood becomes hard- ened, and then the new fibres find their way outward and down- ward, collecting into a second cylinder on the outside of the original wood. It is obvious that this explanation is not so sa- tisfactory as could be desired ; for, in the first place, such a distinction between Cycadece and Exogence as that which Dr. Mohl states to exist, is verbal rather than real, since he admits that the second cylinder of wood is formed externally to the first ; and secondly, it is obvious that if that structure which is represented in the 21st plate of the third volume of the Hortus * Uebcr den Bau des Cycadeen Stammex und sein Verhaltniss xu den Slamni der Coniferen und Banmfarrn. 4to. Munich, 1832. REPORT ON THE PHILOSOPHY OF BOTANY. 35 Malabaricus be correct, where the stem of Cycas c'lrcinal'is is shown to have several concentric zones, precisely as in other exogenous trees, it must follow that Di\ Mohl's explanation would be still more inadmissible ; accordingly, this author dis- credits the fact of the stem of Cycas circinalis having numer- ous concentric zones. It is, however, certain, from the speci- mens brought to England by Dr. WalUch, that the structure of this Cycas is really such as is shown in the Hortus Malaba- ricus. It is nevertheless extremely well worth further inquiry whether there is not some important but as yet vmdiscovered peculiarity in the mode of forming their stem by Cycadecs; for it must be confessed that growth by a single terminal bud, after the manner of Palms, is not what we should expect to meet with in exogenous trees. Pi-ofessor Schultz of Berlin has indicated* the existence of a group of plants, the structure of whose stems he considers at variance with all the forms at present recognised ; and to this group he refers Cycadece : but the assemblage of orders which he collects under what he calls the same plan of growth is so extremely incongruous as to lead to no other conclusion than that subordinate modifications of internal structure are of no general importance, but are merely indicative of individual pe- culiarities. Dr. Mohl further states, that Cryptogamic plants of the highest degree of organization, such as Ferns, Lycopodiacece, Alarsileacece, and Mosses, in all which a distinct axis is found, have a mode of growth neither exogenous nor endogenous, but altogether of a peculiar nature. In these plants, when once the lower part of the stem is formed it becomes incapable of any further alteration, but hardens, and the stem continues to grow only by its point, which lengthens merely by the progres- sive development of the parts already formed, without sending downwards any fibrous or woody bundles, as both in exoge- nous and endogenous plants. M. Lestiboudois, the Professor of Botany at Lille, distin- guishes Monocotyledons from Dicotyledons, upon principles different from those generally adniitted. According to this writer, dicotyledonous trees have two systems, one, the central, consisting of the medullary sheath and the wood ; the other, the cortical, composing the bark. These two systems increase separately, so that in Dicotyledons there are two surfaces of increase, that of the central system, which adds to its outside, and that of the cortical system, which adds to its inside : but * NuiurHchpn Sijstem des Pflanxenreichs nach seiner inncrcn Organization. 8vo. Berlin, 1^32." 36 THIRD REPORT— 18i]o. in the stem of Monocotyledons there is only one surfoce of in- crease, namely, that on the inside ; and hence he concludes that such plants have only a cortical system, and consist of bark alone. It must be obvious that there are too many ana- tomical objections to this theory to render it deserving of any other than this incidental notice*. The cause of the formation of wood has always been a sub- ject upon which physiologists have been unable to agree ; and if the opinions held by the writers of the last century have been disproved, it cannot be added that those of the present day are by any means settled. It is now, indeed, admitted on all hands that wood is a deposit in some way connected with the action of leaves ; for it has been proved beyond all question that the quantity of wood that is formed is in direct proportion to the number of leaves that are evolved, and to their healthy action, and that where no leaves are formed, neither is wood deposited. But it is a subject of dispute whether wood is actually or- ganized matter generated by the leaves, and sent downwards by them, or whether it is a mere secretion, which is deposited in the course of its descent from the leaves to the roots. The former opinion has been maintained in different forms by De la Hire, Darwin, Du Petit Thouars, Poiteau, and myself, and would perhaps have been more generally adopted if it had not been too much mixed up with hypothetical statements, to the reception of which there are in the opinion of many persons strong objections. For example, it has been assei'ted that the wood of trees is an aggregation of the roots of myriads of buds in a state of action, and that consequently a tree is an asso- ciation of individuals having a peculiar organic adhesion and a common system of growth, but each its own individual life. To this view it is no doubt very easy to raise objections, some of which it may be difficult, in the present state of our know- ledge, to answer ; and therefore it is better for the moment to leave this part of the proposition out of consideration, and to confine it to the simple statement that wood is organized matter, generated by the leaves, and sent downvt^ards by them. In support of this it is argued : 1 st, That an anatomical examination of a plant shows that the woody systems of the leaf and stem are continuous : Sndly, That this is not only the fact in exogenous plants, but in all endogenous and cellular plants that have been examined ; so that it may be considered a universal law : ordly. That in the early spring, and for some time after plants be- gin to grow, the woody matter is actually to be seen and traced • Achille Richai'd, Nouveaux EUmens de la Botaniqtte, 5me edit. p. 119. REPOllT ON THE PHILOSOPHY OF BOTANY. 37 descending in parallel tubes from the origin of the leaves, and from no other place : 4thly, That in all cases where obstacles are presented to the descent of such tubes, they turn aside, and afterwards resume their parallelism when the obstacle has been passed by : 5thly, That in endogenous plants, such as Palms, and in some exogenous trees, such as Lignum Vita, they cross and interlace each other in a manner which can only be accounted for by their passing downwards, the one over the other, as the leaves are developed : and, finally, That the perfect organization of the wood is incompatible with a mere deposit of secreted matter. To all which the following evidence has been added by M. Achille Richard. He states * that he saw in the pos- session of Du Petit Thouars a branch of Robinia Pseudacacia, on which Robinia hispida had been grafted. The stock had died, but the scion had continued to grow, and had emitted from its base a sort of plaster, formed of very distinct fibres, which surrounded the extremity of the branch to some distance, and formed a sort of sheath ; thus demonstrating incontestibly that fibres do descend from the base of the scion, to overlay the stock. To this several objections have been taken, the most im- portant of which are the following. If wood were really or- ganized matter, emanating from the leaves, it must necessarily happen that in grafted plants the stock ought in time to acquire the nature of the scion, because its wood would be formed en- tirely by the addition of new matter, said to be furnished by the leaves of the scion ; so far, however, is this from being the fact, that it is well known that in the oldest grafted trees there is no action whatever exercised by the scion upon the stock, but that, on the contrary, a distinct line of organic demarcation separates the wood of the one from the other, and the shoots emitted from the stock by wood said to have been generated by the leaves of the scion, are in all respects of the nature of the stock. Again, — if a ring of bark from a red-wooded tree is made to grow in the room of a similar ring of bark of a white- wooded tree^ as it easily may be made, the trunk will increase in diameter, but all the wood beneath the ring of red bark will be red, although it must have originated in the leaves of the tree which produces white wood. It is further urged, that in grafted plants the scion often overgrows the stock, increasing much the more rapidly in diameter, or that the reverse takes place, as when the Pavia littea is grafted upon the common Horse-chestnut, — and that these circumstances are inconsistent » yuufcaiu- E/emcns de la BoiaiiiqKC, 5mc edit. p. 105. 38 THIRD REPORT — 1833. with the supposition that the wood is organic matter engendered by leaves. To these statements there is nothing to object as mere facts, for they are true; but they certainly do not warrant the conclusions tliat have been drawn from them. One most important point is overlooked by those who employ these argu- ments, namely, that in all plants there are two distinct simul- taneous systems of growth, the ceUular and the fibro-vascular, of which the former is horizontal, and the latter vertical. The cellular gives origin to the pith, the medullary rays, and the principal part of the cortical integument ; the fibro-vascular, to the wood and a portion of the bark ; so that the axis of a plant may be not inaptly compared to a piece of linen, the cellular system being the woof, the fibro-vascular the warp. It has also been proved by Mr. Knight* and M. De Candollef that buds are exclusively generated by the cellular system, while roots are evolved from the fibro-vascular system. Now if these facts are rightly considered, they will be found to ofFer an obvious expla- nation of the pha?nomena produced by those botanists who think that wood cannot be matter generated in an organic state by the leaves. The character of wood is chiefly owing to the colour^ quantity, size, and distortions of the medullary rays, which be- long to the horizontal system; it is for this reason that there is so distinct a line drawn between the wood of the graft and stock, for the horizontal systems of each are constantly pressing together with nearly equal force, and uniting as the trunk in- creases in diameter. As buds from which new branches elon- gate are generated by cellular tissue, they also belong to the horizontal system ; and hence it is that the stock will ahvays produce branches like itself, notwithstanding the long super- position of new wood which has been taking place in it froni the scion. The case of a ring of red bark always forming red wood be- neath it, is precisely of the same nature. After the new bark has adhered to the mouths of the medullary rays of the stock, and so identified itself with the horizontal system, it is gradually pushed outwards by the descent of woody matter from above through it : but in giving way it is constantly generating red matter from its hoi-izontal system, through which the wood de- scends, which thus acquires a colour that does not properly belong to it. With regard to the instances of grafts over- growing their stock, or vice versa, it is obvious that these are susceptible of explanation upon the same principle. If the hori- zontal system of both stock and scion has an equal power of • Philosophical Transaciiuns, 1S05, p. 257. •{■ Physiohgic Vcnitcde, p. 158. REPORT ON THE PHILOSOPHY OF BOTANY. SSi lateral extension, the diameter of each will remain the same ; but if one grows more rapidly than jthe other, the diameters will necessarily be different : where the scion has a horizontal system that develops more rapidly than that of the stock, the latter will be the smaller, and vice versa. It is* however, to be observed, that in these cases plants are altogether in a morbid state, and will not live for any considerable time. Those who object to the theory of wood being generated by the action of leaves, either suppose — 1st, that liber is developed by alburnum, and wood by liber ; or, 2ndly, that " the woody and cortical layers originate laterally from the cambium furnished by preexisting layers, and nourished by the descending sap *." The first of these opinions appears to be that of M. Turpinf , as far as can be collected from a long memoir upon the grafting of plants and animals ; but I must fairly confess that I am not sure I have rightly understood his meaning, so much are his facts mixed up with gratuitous hypothesis and obscure specu- lations upon the action of what he calls globuline. The second is the opinion commonly entertained in France, and adopted by M. De Candolle in his latest published work. The objections to the views of M. Turpin need hardly be stated in a Report like this, where conciseness is so much an object. Those which especially bear upon the view taken by M. De Candolle are, that his theory is not applicable to all parts of the vegetable kingdom, but to exogenous plants only; that it is inconceivable how the highly organized parallel tubes of the wood, which can be traced anatomically from the leaves, and which are formed with great rapidity, can be a lateral de- posit from the liber and alburnum ; that they are manifestly formed long before it can be supposed that the leaves have commenced their office of elaborating the descending sap ; and, finally, that endogenous and cryptogamic plants, in which there is no secretion of cambium, nevertheless have wood. Such is the state of this subject at the time I am writing. To use the words of M. De Candolle, " The whole question may be reduced to this, — Either there descend from the top of a tree the rudiments of fibres, which are nourished and developed by the juices springing laterally from the body of wood and bark, or new layers are developed by preexisting layers, which are nourished by the descending juices formed in the leaves |." As this is one of the most curious points remaining to be settled among botanists, and as it is still as much open to dis- * De Candolle, Phyaiologie Vegetale, p. 165. + Sue Annales des Sciences, vols. xxiv. and xxv., particularly vol. x.w. p. •13. J De Candollu, Physiologie f'rgelale, p. 157. 40 THIRD REPORT 1833. cussion as ever, I have dwelt upon it at an unusual length, in the hope that some Member of the British Association may have leisvire to prosecute the inquiry. Perhaps there is no mode of proceeding to elucidate it which would be more likely to lead to positive results, than a very careful anatomical exami- nation of the progressive development of the Mangel Wiu'zel root, beginning with the dormant embryo, and concluding with the perfectly formed plant. Arrangemeni of Leaves. — It has for a long time been thought that the various modes in which leaves, and the organs which are the result of them, are arranged upon a stem might be re- duced to the spiral, and that all deviations from this law of arrangement are to be considered as caused by the breaking of spires into verticilli. In the Pine Apple, for instance, the Pine Cone, the Screw Pine, and many other plants, the spiral arrange- ment of the leaves is so obvious that it cannot be overlooked ; in trees with alternate leaves this same order of arrangement may be discovered if a line is drawn from the base of one leaf to that of another, always following the same direction; even in verticiilate plants we not unfrequently see that the whorls are dislocated by the praBternatural elongation of their axis, and then become converted into a spire ; and the same pha3nome- non is of common occurrence among the verticilli of leaves in the form of calyx, corolla, stamens, and carpella, which com- pose the flower. This will be the more distinctly apparent if we consider that, as M. Adolphe Brongniart has shown*, what we calLvvhorls in a flower often are not so, strictly speaking, but only a series of parts placed in close approximation, and at dif- ferent heights, upon the short branch that forms their axis. Dr. Alexander Braun has endeavoured f to prove mathema- tically that the spiral arrangement of the parts of plants is not only universal, but subject to laws of a very precise nature. His memoir is of considerable length, and would be wholly un- intelligible without the plates that illustrate it. It is therefore only possible on this occasion to mention the results. Setting out witli a contemplation of the manner in which the scales of a Pine Cone are placed, to which a long and ingenious method of analysis v/as applied, he found that several different series of spires are discoverable, between which there invariably exist peculiar arithmetical relations, v/hich are the expression of the various combinations of a certain number of elements disposed in a regular manner. All these spires depend upon the posi- * Annales des Sciences, vol. xxiii. p. 226. + Verglekhendr Vritersuchunj iibcr die Ordininfj der Schuppen an den Tan- nen~apfcn. Ito. IPjd. REPORT ON THE PHILOSOPHY OF BOTANY. 41^ tion of a fundamental series, from wliicli the otliers are devia- tions. The nature of the fundamental series is expressed by a fraction, of which the numerator indicates the whole number of turns required to complete one spire, and the denominator the number of scales or parts which constitute it: thus ^-^ in- dicates that eight turns are made round the axis before any scale or part is exactly vertical to that which was first formed, and the number of scales or parts that intervene before this coincidence takes place is twenty-one. It does not appear that this inquiry has as yet led to any practical application, although one might have expected that as the natural affinities of plants are determined, in a great de- gree, by the accordance that is observable in the relative posi- tion of their parts, the spires of which those parts are composed might have had something in common M'hich would be suscep- tible of being expressed by numbers. If any practical applica- tion can be made of Dr. Braun's fractions, it seems likely to be confined to the distinction of species. His observations seem, however, to have established the truth of the doctrine that, be- ginning with the cotyledons, the whole of the appendages of the axis of plants, — leaves, calyx, corolla, stamens, and car- pella, — form an uninterrupted spire, governed by laws which are almost constant. Structtire of Leaves. — The leaves of plants have been found by M. Adolphe Brongniart to be not merely expansions of the cellular integument of stems, traversed by veins originating * in the woody system, but to be organs in which the inteqnal parenchyma is arranged with beautiful uniformity, in the man- ner most conducive to the end of exposui'e to light and air, and of elaboration, for which the leaves are chiefly destined. In their usual structure leaves have been found by this observer either to consist of two principal layers, — of which the upper, into which the ascending sap is first introduced, is formed of compact cells, more or less perpendicular to the plane of the cu- ticle, and the under, into which the returning sap is propelled, is formed of very lax cavernous tissue, more or less parallel with the cuticle of the lower surface, — or else of two layers perpen- dicular to the cuticle, with a central parallel stratum. The observations of Drs. Mohl and Meyen generally confirm this ; but at the same time the latter instances several cases in which the texture of the leaf has been found to be nearly the same throughout. Dutrochet* states, in addition, that the interior of the leaf * Anitales des Sciences, vol. xxv. p. 215. 42 THIRD REPORT 1833. is divided completely by a number of partitions caused by the ribs and jirincipal veins, so that the air cavities have not actually a free communication in every direction through the parenchy- ma, but are to a certain extent cut off from each other. This is conformable to what M. Mirbel has described in Marchantia, who finds the leafy expansions of that plant separated by par- titions into chambers, between which he is of opinion that there is no other communication than what results from the per- meability of the tissue *. The statement of M. Adolphe Brongniart, that all leaves in- tended to exist in the air are furnished with a distinct cuticle on their two faces, while those which are developed under w^ater have no cuticle at ail, has not been disproved, unless in the case of Marchantia f , whose under surface can scarcely be said to have a distinct cuticle; but this jilant, which can only exist in humid shady places, is perhaps rather a proof of the accuracy of the theory of M. Brongniart than an exception to it. That the stomata in all cases open into internal cavities in the leaf, where the tissue is extremely lax and cavernous, ap- pears also exti'emely probable. It was especially found to be the case by M. Mirbel in his so often quoted remarks upon Marchaniia. With regard to the stomata themselves, no one appears yet to have confirmed the observation of Dr. Brown;}:, that their apparent orifice is closed up by a membrane. On the contrary, the observations of M. Mirbel on Marchantia, if they are to be taken as illustrative of the usual structure of those singular organs, go to establish the accuracy of the common opinion that the stomata are apertures in the cuticle. That most skil- ful physiologist, while watching the development of Marchantia, remarked the very birth of the stomata, which he describes as taking place thus : — The appearance of a little pit in the middle of four or five cells placed in a ring is a certain indication of the beginning of a stoma. The pit evidently increases by the enlargement and separation of the svirrounding cells. If the nascent stoma consists of five cells, of which one is surrounded by four others, then the central one is destroyed; but if it con- sists of three or four cells adjusted so as to form a disk, then the stoma is caused by the separation of their sides in the cen- tre, by which means a sort of star is created. It is true that * " Reclierclies Anatomiques et Pliysiologiques sur le Marchantia poli/ruor- pha," in Nouvcaux Annalcs du Museum, vol. i. p. 7. + Ibid. p. 93. J Suppl. prlmttm Pradromi Floros Novce IloUaiidicc, p. 3. REPORT ON THE PHILOSOPHY OF BOTANY. 45 the stomata of Marchantia are in some respects different from what are found upon flowering plants ; yet I think we can hardly doubt that the plan upon which they are all formed is essen- tially the same. Dutrochet also confirms * the statement of Amici, that the stomata are perforations; for he finds that when leaves are de- prived of their air by the aii'-pump, it is chiefly on the under side, where the greatest number of stomata is found, that little air bubbles make their appearance ; and that it is through the stomata that water rushes into the cavernous parenchyma to supply the loss occasioned by the abstraction of air. Anther, Sj-c. — Some curious remarks upon the nature of the tissue that lines the cells of the anther have been published by Dr. John E. Purkinje, Professor of Medicine at Breslau. His researches are chiefly directed to the determination of the na- ture of the tissue that is in immediate contact with the pollen ; and he has demonstrated in an elaborate Essay f, that the opi- nion emitted by Mirbel in 1808 J, that the cause of the dehis- cence of the anther is its lining, consisting of cellular tissue cut into slits and eminently hygrometrical, is substantially true. He shows tjiat this lining is composed of cellular tissue chiefly of the fibrous kind, which forms an infinite multitude of little springs, that when dry contract and pull back the valves of the anthers by a powerful accumulation of forces which are indivi- dually scarcely appreciable : so that the opening of the anther is not a mere act of chance, but the admirably contrived result of the maturity of the pollen, — an epoch at which the surround- ing tissue is necessarily exhausted of its fluid by the force of endosmosis exercised by each particular grain of pollen. That this exhaustion of the circumambient tissue by the en- dosmosis of the pollen is not a mere hypothesis, has been shown by Mirbel in a continuation of the beautiful memoir I have already so often referred to§. He finds that, on the one hand, a great abundance of fluid is directed into the utricles, in which the pollen is developed a little before the maturity of the latter, and that by a dislocation of those utricles the pollen loses all organic connexion with the lining of the anther ; and that, on the other hand, these utricles are dried up, lacerated, and disorganized, at the time when the pollen has acquired its full development. * Annales des Sciences, vol. xxv. p. 24?, t De Cellulis Anther arum fihrous. 4to. Wratislaviae, 1830. X " Observations sur iin Systcnie d'Anatomie Comparee de Vegetaiix, fondes sur rOrganization de la Fleur," in Mcmoire.s de I'lnstitut, 1808, "p. W.'A. § "Complement des ObsciTations sur le Marchantia jwlymorphu," in Ar- chives de Botanique, vol. i. 44 THIRD REPORT — 1833. The Origin of the Pollen, connected as it intimately is with the singular pha;nomena of vegetable sexuality, has naturally been of late an object of some inquiry. To the important dis- coveries of the younger Brongniart and of Dr. Robert Brown, M. Mirbellias added some observations*, detailed with that admirable clearness and precision which give so great a value to all his writings, and wiiich are the more interesting as they serve to explain what was before obscure, and to correct what appears to have been either inaccurately or imperfectly de- cribed. This he has been enabled to do by beginning his in- quiry at the very earliest period when the organization of the anther can be discovered : his subject was the common Gourd. At a very early time the whole tissue of the anther is of the most perfect uniformity, consisting of cellules, the transverse section of which represents nearly regular hexagons and penta- gons. In every cell, without even excepting those which com- pose the superficies of the anther, are found little loose bodies, so exceedingly minute that a magnifying power of 500 or 600 diameters is scarcely sufficient to examine them : they may be compared to transparent, nearly colourless vesicles, more or less round, and of unequal size. At a stage but little more ad- vanced, you may observe on each side of the medial line of a transverse section of a lobe of an anther, a collection of cellules rather larger than the remainder: it will afterwards be seen that it is here that the pollen is engendered; such cells are therefore called pollen-cells. In a bud, a line and a half or two lines in diameter, some remarkable alterations were found to have taken place ; the pollen-cells had enlarged and their gra- nules had so much increased in number, that they nearly filled the cells in opake masses. These granules and pollen-cells formed together a greyish mass, connected with the rest of the tissue by the intervention of a cellular membrane, which, not- withstanding its organic continuity with the surrounding parts, is at once distinguishable ; for while the cells of the surrounding parts elongate parallel to the plane of the surface, and to the plane of the base of the anther, those of the cellular membrane elongate from the centre to the circumference. In more ad- vanced anthers, the sides of the pollen-cells, from being thin and dry, had changed to a perceptible thickness, and their sub- stance, gorged with fluid, resembled a colourless jelly. When the buds were three or four lines long, an unexpected phaeno- menon presented itself. At first the thick and succulent walls of each pollen-cell dilated so as to leave an empty space between the inner face and the granules, not one of which sepiu'ated * " Complement des Observations," i^c, as above quoted. REPORT ON THE PHILOSOPHY OF BOTANY. '4S from the general mass, which showed that some power kept them united. Shortly after, four appendages, like knife-blades, developed at equal distances upon the inner face of the cell, and gradually projected their edges towards the centre, till at last they divided the granular mass into four little triangvdar bodies; when the appendages had completely united at their edges, they divided the cavity of the pollen-cells into four di- stinct boxes, which then began to rounden, and finally became little spherical masses. Each of these was the rudiment of a grain of pollen, subsequently acquired a membranous integu- ment, hardened, became yellow, and thus arrived at maturity. Wliat is perhaps most important in these observations is the demonstration of the original organic continuity of all the parts of the anther, against the statement of M. Adolphe Brongniart, and also against what appears to be the opinion of Dr. Brown, as far as can be collected from the manner in which he speaks of the evolution of the pollen in Tradescantia rirginica *. Although it is not directly shown by these observations whe- ther the perfect grain of pollen has one or two integuments, — a question that may still be said to be unsettled, — it nevertheless appears from other instances that M. Mii'bel admits the exist- ence of an outer not distensible coat, and of an inner highly extensible lining. A curious paper upon this point f has been published by a Saxon botanist named Fritzsche. By means of a mixture of two parts by weight of concentrated sulphuric acid, and five parts of water, he found that the grains of pollen can be rendered so transparent as to reveal their internal struc- ture, and that the whole process of the emission of the pollen- tubes can be distinctly traced. He describes the universal pre- sence of two coatings to the grains of pollen ; and he also finds that the pollen contains a quantity of oily particles in addition to the moving corpuscles, — a fact which has also been noticed by Dr. Brown. Although the generalizations in this work are less satisfactory than could be desired, it must nevertheless be considered a most valuable collection of facts, and as containing the best arrangement that has as yet appeared of the various forms un- der which the pollen is seen. Fert/lhation. — The road which some years since was so happily opened by Amici to the discovery of the exact manner in which vegetable fertilization takes effect, is every day be- coming more and more direct. The doubts of those vvho could not discern the tubes that are projected into the style by the * Observations upon Orchidon? n?id Asclrpiacfea?, p. 21. f Beitrage ?:ur Kenntnks dcs Pollen. 4to. Berlin, 1S33. 46 THIRD REPORT — 1833. pollen, have been removed; the important demonstration by Dr. Brown of the universal presence of a passage through the integuments of the ovulum at the point of the nucleus has been extended and confirmed by M. Mirbel in a paper of the high- est interest * ; the fact that it is at the point of the nucleus (where this passage exists,) that the nascent embryo makes it appear- ance, is now undisputed; the passage of the contents of the pollen down the pollen-tubes, and the curious discovery of a power of motion in the granules that are thus emitted, are also recognised : it now only remains to be proved that the pollen- tubes come in contact with the nucleus, and the whole secret of fertilization is revealed. A few remarkable contributions to this part of the subject have lately been made. Some plants have the passage or foramen in their ovulum so remote from any part through which the pollen-tubes can be supposed to convey their influence, as to have thrown consider- able difficulty in the way of the supposition that actual contact between the point of the nucleus and the fertilizing tissue is indispensable. The manner in which, notwithstanding the apparent difficulty of such contact taking place, this happens in Statice Armeria, was long since made out by Dr. Brown, in whose possession I several years ago saw drawings illustrating this phsenomenon ; it has since been explained by M. Mirbel. Another case, pre- senting similar apparent difficulties, occurs in Helianthemutn. In plants of that genus the foramen is at that end of the ovulum which is most remote from the hilum ; and although the ovula themselves are elevated upon cords much longer than are usually met with, yet there are no obvious means of their coming in con- tact with any part through which the matter projected into the pollen-tubes can be supposed to descend. It has, however, been ascertained by M. Adolphe Brongniartf, that at the time when the stigma is covered with pollen, and fertilization has taken effect, there is a bundle of threads, originating from the base of the style, which hang down in the cavity of the ovarium, and, floating there, are abundantly sufficient to convey the in- fluence of the pollen to the points of the nuclei. So again in AsclepiadecE. In this tribe, from the peculiar conformation of the parts, and from the grains of pollen being all shut up in a sort of bag, out of which there seemed to be no escape, it was supposed that this tribe must at least form an exception to the general rule. But before the month of November 1828:}:, the * Nouvelles Piecherehes sur la Sfnicture dc VOoule Vegetal et sur ses Deve- loppements. Also Additions aiix 'Nouvelles Rec/irrclies,' ^c. •f- Annales des Sciences, vol. xxiv. p. 123. X Linntpa, vol. iv. p. 94. REPORT ON THE PHILOSOPHY OF BOTANY. 47 celebrated Prussian traveller and botanist Ehrenberg had dis- covered that the grains of pollen of Asclepiadece acquire a sort of tails which are all directed to a suture of their sac on the side next the stigma, and which at the period of fertilization are lengthened and emitted ; but he did not discover that these tails are only formed subsequently to the commencement of a new vital action connected with fertilization, and he thought that they were of a different nature from the pollen-tubes of other plants ; he particularly observed in Asclepias syriaca that the tails become exceedingly long and hang down. In 1831 the subject was resumed by Dr. Brown* in this country, and by M. Adolphe Brongniartf in France, at times so nearly identical, that it really seems to me impossible to say with which the discovery about to be mentioned originated : it will therefore be only justice if the Essays referred to are spoken of collectively instead of separately. These two distinguished botanists ascertained that the production of tails by the grains of pollen was a phaenomenon connected with the action of ferti- lization ; they confirmed the existence of the suture described by Ehrenberg ; they found that the true stigma of Asclepia- dece is at the lower part of the discoid head of the style, and so placed as to be within reach of the suture through which the pollen-tubes or tails are emitted ; they remarked that the latter insinuate themselves below the head of the style, and follow its surface vmtil they reached the stigma, into the tissue of which they buried themselves so perceptibly that they were enabled to trace them, occasionally, almost into the cavity of the ovarium ; and thus they established the highly important fact, that this family, which was thought to be one of those in which it was impossible to suppose that fertilization takes place by actual contact between the pollen and the stigma, offers the most beautiful of all examples of the exactness of the theory, that it is at least owing to the projection of pollen-tubes into the sub- stance of the stigma. In the more essential parts these two observers are agreed : they, however, differ in some of the de- tails ; as, for instance, in the texture of the part of the style which I have here called stigma, and into which the pollen- tubes are introduced. M. Brongniart both describes and figures it as much more lax than the contiguous tissue, while on the other hand Dr. Brown declares that he has in no case been able to observe " the slightest appearance of secretion, or any dif- * Observations on the Organs and Mode of Fecundation of Orchideae and Asclepiadece. London, October 1831. f Annft/fis des Sciences for October and November 1831 ; from observations made in July, August and September of that yoar. 48 THIRD REPORT — ISoJ. ferences whatever in texture between tliat part and the general surface of the stigma" (meaning what I have described as the discoid head of the style) : but this is not the place for entering into the discussion of these subordinate points. Orchidece are another tribe in which similar difficulties have been found in reconciling structure with the necessity of con- tact between the pollen and stigma in order to effect impregna- tion. Indeed it seems in these plants as if every possible pre- caution had been taken by nature to prevent such contact. Nevertheless it is represented by M. Adolphe Brongniart, in a paper read before the Academy of Sciences of Paris in July 1831 *, that contact is as necessary in these plants as in others, and that in the emission of pollen-tubes they do not differ from other plants. These statements have been followed up by Dr. Brown f, in itn elaborate Essay upon the subject, in which the results that are arrived at by our learned countryman are es- sentially to the same effect. To these tliere is at present no- thing equally positive to oppose ; but as the indirect observa- tions of Mr. Bauer %, and the general structure of the order, are very much at variance with the probability of actual contact being necessary, and especially as Dr. Brown is obliged to have recourse to the supposition that tlie pollen of many of these plants must be actually carried by insects from the boxes in which it is naturally locked up, — it must be considered, I think, that the mode of fertilization in Orchidece is still far from being determined. I must particularly remark that the very proble- matical agency of insects, to m hich Dr. Brown has recourse in order to make out his case, seems to be singvilarly at variance with his supposition § that the insect forms, which in Oplirys are so striking, and which he finds resemble the msects of the countries in which the plants are found, are intended rather to repel than i^^ attract. It may be true, as Dr. Brown observes, that there is less necessity for the agency of insects in such flowers as the European OpJinjdecs ; but what other means than the assistance of insects can be supposed to extricate the pollen from the cells in the insect flowers of such plants as ReiioufJiera Arac/iniies, the whole genus Oncidiian, Teiramicra rigldn, several species of Epidendrinn, Cipnbidlum tenuijolium, Vanda peduncidans, and a host of others ? * Annales des Sciences, vol. xxiv. p. 113. t Observations upon the Organs and Mode of Fecundation of Orcliidese and Asclepiadese. X Illustrations of the Genera and Species of Orchideovs Plants. Part II. " Fructification," tabb. 5. 12. 13. ] k § " Proceedings of tlie Linnean Society," June o, 1832, as given in the Lon- don and Edinburgh Philosopliical iMagcuiitc and Journal. REPORT ON THE PHILOSOPHY OF BOTANY. 49 Origin of Organs. — There is no part of vegetable physiology SO obscure as that which relates to the origin of organs. We find a degree of simplicity that is perfectly astonishing in the fundamental structure of the whole vegetable kingdom ; we are able to prove by rigorous demonstration that every one of the appendages of the axis is a modification of a leaf, to which there is a constant tendency to revert; we see that in some cases a part which usually performs one function assumes another, as in the Alstromerias, whose leaves by a twist of their petiole turn their under surface upwards : but we are entirely ignorant of the causes to which these changes are owing. An impor- tant step in elucidating the subject has been lately taken by M. Mirbel, in his memoir upon the structure of Marchantia po~ lymorplia. The young bulbs by which this plant is multiplied are originally so homogeneous in structure, that there is no apparent character in their organization to show which of their faces is destined to become the upper surface, and which the under. For the purpose of ascertaining whether there existed any natural but invisible predisposition in the two faces to un- dergo the changes which subsequently become so apparent, and by means of which their respective functions are performed, or whether the tendency is given by some cause posterior to their first creation, the following experiments were instituted. Five bulbs were sown upon powdered sandstone, and it was found that the face which touched the sandstone produced roots, and the opposite face formed stomata. It Avas, however, possible that the five bulbs might have all accidentally fallen upon the face which was predisposed to emit roots ; other experiments of the same kind were therefore tried, first with eighty and afterwards with hundreds of little bulbs, — and the result was the same as with the five. This proved that either face was originally adapted for producing either roots or sto- mata, and that the tendency was determined merely by the po- sition in which the surfaces were placed. The next point to ascertain was, whether the tendency once given could be after- wards altered ; some little bulbs, that had been growing for twenty-four hours only, had emitted roots ; they were turned, so that the upper surface touched the soil, and the under was exposed to light. In twenty-four hours more the two faces had both produced roots ; that which had originall)'^ been the under surface went on pushing out new roots ; that which had originally been the upper surface had also produced roots : but in a few days the sides of the young plants began to rise from the soil, became erect, turned over, and finally recovered 1833. E 50 THIRD REPORT — 1833. in this way their original position, and the face which had ori- ginally been the uppermost, immediately became covered with stomata. It, therefore, appears that the impulse once given, the predisposition to assume particular appearances or func- tions is absolutely fixed, and will not change in the ordinary course of nature. This is a fact of very high interest for those who are occupied in researches into the causes of what is called vegetable metamorphosis, an expression which has been justly criticised as giving a false idea of the subject to which it relates. Morphology. — When those who first seized upon the im- portant but neglected facts out of which the modern theory of morphology has been constructed, asserted that all the appen- dages of the axis of a plant are metamorphosed leaves, more was certainly stated than the evidence would justify ; for we cannot say that an organ is a metamorphosed leaf, which in point of fact has never been a leaf. What was meant, and that which is supported by the most conclusive evidence, is, that every appendage of the axis, whether leaf, bractea, sepal, petal, stamen, or pistillum, is originally constructed of the same ele- ments, arranged upon a common plan, and varying in their manner of development, not on account of any original differ- ence in structure, but on account of special and local predis- posing causes : of this the leaf is taken as the type, because it is the organ which is most usually the result of the develop- ment of those elements,^ — is that to which the other organs generally revert, when from any accidental disturbing cause they do not assume the appearance to which they were originally predisposed, — and, moreover, is that in which we have the most complete state of organization. This is not a place for the discussion of the details upon which the theory of morphology is founded ; it is sufficient to state that it has become the basis of all philosophical views of structure, and an inseparable part of the science of botany. Its practical importance will be elucidated by the following circum- stance. Fourteen or fifteen years ago I was led to take a view of the structure of Reseda very different from that usually assigned to the genus ; and when a few years afterwards that view was published, it attracted a good deal of attention, and gained some converts among the botanists of Germany and France. It was afterwards objected to by Dr. Brown upon several grovmds ; but I am not aware that they were considered sufficiently valid to produce any change in the opinions of those who had adopted my hypothesis. Lately, however, Professor REPORT ON THE PIIILOSOPIIV OF BOTANY. 51 Henslow has satisfactorily proved*, in part by the aid of a monstrosity in the common Mignonette, and in part by a severe appHcation of morphological rules, that my hypothesis must necessarily be false ; and I am glad to have this opportunity of expressing my full concurrence in his opinion. It has long been known that the ligulate and tubular corollas of CompositcB are anatomically almost identical, and that their difterence consists only in the five petals of the tubular corolla all separating regularly for a short distance from their apex, while the five petals of the ligulate corolla adhere up to their very points, except on the side next the axis of inflorescence, where two of them are altogether distinct except at their base. M. Leopold von Buch explains this circumstance in the follow- ing manner. He states that these ligulate corollas when vmex- panded bear at their point a little, white, and very viscid body or gland, which is a peculiar secretion that dries up when it comes in contact with the atmosphere. The adhesion of this gland is too powerful to be overcome by the force of the style and stamens pressing against it from within. The corollas, which are gradually curved outwards by the growth of those in the centre of the inflorescence, at the same time bend down the style, which consequently presses up against the line of union of the two petals nearest the axis : although the style cannot overcome the adhesion of the viscid gland at the point of the corolla, it is able in time to destroy the union of the two inte- rior petals, which finally give way and allow the stamens and style to escape. As soon as this takes place, the corolla can no longer remain erect, but falls back toAvards the circumference of the capitulum, and thus contributes to the radiating character of this sort of inflorescence. When the viscid body is either not at all, or very imperfectly produced at the point of the co- rolla, as sometimes happens in the genus Hieracium, especially H. bifurcum, tubular corollas are produced instead of ligulate ones. The ovulum is the organ where the greatest difficulty has occurred in reducing the structure to anything analogous to that of other parts. It is true that Du Petit Thouars regarded it as analogous to a leaf bud ; but his view appears to have been purely hypothetical, for I am not aware that he had any distinct evidence of the fact. Some years ago M. Turpin, in showing the great similarity that exists between the convolute bracteas of certain Marcgraviacece and the exterior envelope of the ovulum, took the first step towards proving that the hypothesis * Transactions of the Philosophical Society of Cambridge, vol, v. Part I. E 2 52 THIRD REPORT — 1833. of Du Petit Thouars was susceptible of demonstration ; it was more distinctly shown by the interesting discovery of Professor Henslow, that the leaves of Malaxis paludosa had on their margins what no doubt must be considered buds, but what in structure are an intermediate state between buds and ovula ; and it has been recently asserted by Engelmann*, still, how- ever, without the production of any proof, that "ovula are buds of.a higher kind, their integuments leaves, and their funiculus the axis, all which, in cases of retrograde metamorphosis, are in fact converted into stem and gi-een leaves." The nearest ap- proach to a demonstration that has yet been afforded of ovula being buds is in a valuable paper by Professor Henslow, just printed in the Transactions of the Philosophical SocietT/ of Cam- bridge]-, in which it is shown that in the Mignonette the ovula are in fact transformed occasionally into leaves, either solitary or rolled together round an axis, of which the nucleus is the termination. M. Dumortier has endeavoured to prove | that the embryo itself is essentially the same as a single internodium of the stem with its vital point or rudimentary bud attached to it. Although the author's demonstration is a failure, and his paper a series of confused and illogical reasoning, yet there can be little doubt that the hypothesis itself is a close approximation to the truth. Dr. George Engelmann has recently attempted § to classify the aberrations from normal structure, which throw so much light upon the real origin and nature of the organs of plants. He has collected a very considerable number of cases under the following heads. 1 . Retrograde metamorphosis (Regressus), when organs assume the state of some of those on the outside of them, as when carpella change to stamens or petals, hypo- gynous scales to stamens, stamens to petals or sepals, sepals to ordinary leaves, irregular structure to regular, and the like. 2. Foliaceous metamorphosis {Virescentia), when all the parts of a flower assume more or less completely the state of leaves. 3. Disunion {Disjunctio), when the parts that usually cohere are separated, as the carpella of a syncarpous pistillum, the filaments of monadelphous stamens, the petals of a monopeta- lous corolla, &c. 4. Dislocation {Apostasis) ; in this case the whorls of the flower are broken up by the extension of the axis, 5. Viviparousness {Diaphysis), when the axis is not only elongated, but continues to grow and form new parts, as in those * De Antholysi Prodromus, p. 61. t vol. v. Part I. I Nova Acta Academice Naturee Curiosorum, vol. xvi. p. 245. ^ De Antholysi Prodromus. REPORT ON THE PHILOSOPHY OF BOTANY. S$ instances where one flower grows from within another. And finally, 6. Proliferousness {Ecblastesis), when buds are deve- loped in the axillae of the floral organs, so as to convert a sim- ple flower into a mass of inflorescence. A very considerable number of instances are adduced in illustration of these divi- sions, and the work will be found highly useful as a collection of curious or important facts. The doctrines of morphology, and the evidence in support of them, may now be considered so far settled as to require but little further illustration for the present. This is, however, only true of flowering plants : in the whole division of flower- less plants there has been scarcely any attempt to discover the analogy of organs, and to reduce their structure to a correspond- ing state of identification. I some time since * endeavoured to excite attention to this subject, by hazarding some speculations which had at least the merit of novelty to recommend them ; but I cannot discover that any one has since turned his atten- tion to the inquiry, although it must be confessed that the com- parative anatomy of flowerless plants is among the most inter- esting topics still remaining for discussion, and that it is rather discreditable to Cryptogamic botanists that the elucidation of so very curious a matter should be postponed to the compara- tively unimportant business of distinguishing or dividing genera and species. Gradual Development. — The theory of the gradual deve- lopment of the highest class of organic bodies, in consequence of a combination and complication of the phaenomena attendant upon the development of the lowest classes, has acquired so great a degree of probability among animals, that it has become a question of no small interest whether traces of the same, or a similar law, cannot be found among plants. In an inquiry of such a nature, it seems obvious that attention should in the first instance be dii'ected to a search after positive and incontestable facts, and that mere hypotheses should in the beginning be to- tally rejected. The only circumstances that occur to me as bearing directly upon this point are the following. It has been ascertained by M. Mirbel, in his memoir on the Marchantia, that the sporule of that very simple plant is a single vesicle, which, when it begins to grow, produces other vesicles on its surface, which go on propagating in the same manner, every new vesicle engendering others ; and that different modifica- tions of this process produce the different parts that the per- fect plant finally develops. * Outlines of the First Principles of Botany, p. 533, &c. Introduction to the Natural System of Dot any, p. 313, &c. 54 THIRD REPORT — '1833. The same principle of growth appears to obtain in Confervas, and probably is found in other vegetables of the lowest grade. This is analogous to what takes place in the formation of the embryo of Vasciilares. In the opinion of Dr. Brown and of Mirbel, the first rudiment of a plant far more comphcated than Marchaniia, consists also of a vesicle, but suspended by a thread to the summit of the cavity of the ovulum ; and the dif- ference between the one case and the other is, that while in the Marchaniia the original vesicle, " as soon as it is formed, pos- sesses all the conditions requisite for developing a complete plant on the surface of the soil ; on the other hand, that of flowering plants must, on pain of death, commence its deve- lopment in the interior of the ovulum, and cannot continue it further until it has produced the rudiments of root, stem, and cotyledons*. Beyond this I do not think that any attempt has been made to elucidate the question. Irritability. — I3r. Dutrochet has published f the result of some experiments v,ith the air-pump upon the pneumatic system of plants. Independently of confirming the fact, already gene^ rally known, of plants having the means of containing a large quantity of air, he arrived at the unexpected result, that the sleep of plants and their irritability are certainly dependent upon the presence of air within them. A sensitive plant, left in the vacuum of an air-pump for eighteen hours, indicated no sign whatever of the accustomed collapse of its leaflets on the ap- proach of night, nor when it was restored to the air could it be stimulated by the smartest shocks ; but in time it recovered its irritabihty. When flowers that usually close at night were placed in a vacuum while expanded, they would not close ; and when flowers already closed were placed in the same situation, they would not unfold at the return of morning ; whence Dr. Dutrochet infers that the internal air of plants is indispensably necessary to the exercise of their alternate motions of sleeping and waking, and in general to the existence of the faculty they possess of indicating by their movements the influence of ex- ternal exciting causes. Action of Coloured Light. — Professor Morren, of Ghent, lias mentioned j the result of some experiments upon the action of the coloured rays upon germination ; and he has fovmd that while those rays in which the illuminating power is the most feeble were, as might have been expected, the most favourable to germination, their power of decomposing carbonic acid, and * Archives de Botanique, vol. i. t ■•'innalcs des Sciences, vol.xxv. p. 243. J Annates des Sciences, vol. xxvii, p. 201, REPORT ON THE PHILOSOPHY OF BOTANY. 55 producing a green deposit in the parenchyma, is in proportion to their illuminating property ; that no decomposed rays effect this so rapidly as white light; and that the yellow ray possesses the greening power in the highest degree, the orange in a very slight degree, and violet, red and purple not at all. Colours. — Nothing can be named in the whole range of bo- tany upon which information is so much wanted as the cause of the various colours of plants. It was, indeed, long since sus- pected by Lamarck that the autumnal colouring of leaves and fruits was a morbid condition of those parts ; and it has subse- quently been ascertained that all colours are owing to the pre- sence of a substance, called chromiile by De CandoUe, which fills the parenchyma, assuming different tints. Green has also been clearly made out to be connected with exposure to light, and has been considered to be in all probability owing to the deposition of the carbon left upon the decomposition of car- bonic acid. Some botanists have also observed the connexion of red colour with acidity ; but still we had scarcely any positive knowledge of the cause of the production of any colour except green, till M. Macaire of Geneva* remarked, that just before leaves begin to change colour in the autumn, they cease parting with oxygen in the day, although they go on absorbing it at night ; whence he concluded that their chromule is oxygenated, by which a yellow colour is first caused, and then a red, — for he found that in all cases a change to red is preceded by a change to yellow. He also ascertained that the chromule of the red bracteae and calyx of Salvia splendens is chemically the same as that of autumnal leaves. Coupling this with the fact that petals do not part with oxygen, it would seem as if their colour, if yellow or red, may also be owing to a kind of oxygenation. But according to M, Theodore de Saussure f , coloured fruits part with their oxygen ; so that, if this be true, red and yellow cannot always be ascribed to such a cause. M. De CandoUe J has some excellent observations upon this subject in his recent admirable digest of the laws of vegetable physiology ; in which he concludes, from the inquiries hitherto instituted, that all co- lours depend upon the degree of oxygenation. When oxygen is in excess, the colour seems to tend to yellow or red ; and when it is deficient, or when the chromule is more carbonized, which is the same thing, it has a tendency to blue. Local ad- ditions of alkaline matters are also called in aid of an explanation of the various shades of colour that flowers and fruits present. * Mcmoirrs de la Socictc Physique de Geneve, vol. iv. p. 50. t Ibid. vol. i. p. 284. % Phjsiologie Vegetale, p. 906. 56 THIRD KEPORT — 1833. Dr. Dutrocliet is of opinion * that the whitish spots we some- times see in leaves, and the paler tint that generally character- izes the under side of the same organs, are owing to the presence of air beneath the cuticle. He finds that the arrow-head shaped blotch on the upper side of the leaf of Tnfolium pratense, and the whitish spots on Pulmonaria officinalis, disappear when the leaves are plunged in water beneath the exhausted receiver of the air-pump, and that the lower surface of leaves acquii'es the same depth of colour as the upper under similar circumstances. This he ascribes to the air naturally found in the leaves being abstracted, and its place supplied with water ; a conclusion which agrees with what might be inferred from the anatomical structure of the parts in question. Excretions. — It has long been known that some plants are incapable of growing, or at least of remaining in a healthy state, in soil in which the same species has previously been cultivated. For instance, a new apple orchard cannot be made to succeed on the site of an old apple orchard, unless some years inter- vene between the destruction of the one and the planting of the other : in gardens, no quantity of manui'e will enable one kind of fruit-tree to flourish on a spot from which another tree of the same species has been recently removed ; and all farmers practically evince, by the rotation of their crops, their expe- rience of the existence of this law. Exhaustion of the soil is evidently not the cause of this, for abundant manuring will not supersede the necessity of the usual rotation. The celebrated Duhamel long ago remarked, that the Elm parts by its roots with an unctuous dark-coloured substance ; and, according to De CandoUe, both Humboldt and Plenck suspected that some poisonous matter is secreted by roots ; but it is to M. Macaire, who at the instance of the first of these three botanists undertook to inquire experimentally into the subject, that we owe the discovery of the suspicion above al- luded to being well founded. He ascertained f that all plants part with a kind of faecal matter by their roots, that the nature of such excretions varies with species or large natural orders : in Cichoracecs and Papaveracece he found that the matter was analogous to opium, and in Leguminosce to gum ; in Graminece it consists of alkaline and earthy alkalies and carbonates, and in Euphorbiacecs of an acrid gum-resinous substance. These excretions are evidently thrown off by the roots on account of their presence in the system being deleterious ; and it was found by experiment, that plants artificially poisoned parted with the * Aiinales des Sciences, vol. xxv. p. 216. f De CandoUe, Physiologic Vegetale, p. 249. REPORT ON THE PHILOSOPHY OF BOTANY. 57 poisonous matter by their roots. For instance, a plant of Mer- curialis had its roots divided into two parcels, of which one was immersed in the neck of a bottle filled with a weak solution of acetate of lead, and the other parcel was plunged into the neck of a corresponding bottle filled with pure water. In a few days the pure water had become sensibly impregnated with acetate of lead. This, coupled with the well known fact that plants, although they generate poisonous secretions, yet cannot absorb them by their roots without death, as, for instance, is the case with Atropa Belladonna, seems to prove that the necessity of the rotation of crops is more dependent upon the soil being poisoned than upon its being exhausted. This is a part of vegetable physiology of vast importance to an agricultural country like England, and may possibly cause a total revolution in our system of husbandry. All that M. Macaire can be said as yet to have done, is to have discovered the fact and to have pointed out certain strong examples of it ; but if the discovery is to be converted to a practically useful purpose, we require positive information upon the following points : — 1. The nature of the faecal excretions of every plant culti- vated by the farmer. 2. The nature of the same excretions of the common weeds of agriculture. 3. The degree in which such excretions are poisonous to the plants that yield them, or to others. 4. The most ready means of decomposing those excretions by manures or other means. It would be superfluous to point out what the application would be of such information as thi;?, ; but I cannot forbear ex- pressing a hope that a question upon which so many deep inter- ests are involved may be among the first to occupy the atten- tion of the chemists of the British Association. [ 59 1 Report on the Physiology of the Nervous System. By Wil- liam Charles Henry, M.D., Physician to the Manchester Royal Infirmary, Introduction. — Tme science of Physiology has for its object to ascertain, to analyse, and to classify the qualities and actions which are peculiar to living bodies. These vital properties re- side exclusively in organized matter, which is characterized by a molecular arrangement, not producible by ordinary physical attractions and laws. Matter thus organized consists essen- tially of solids, so disposed into an irregular network of laminae and filaments, as to leave spaces occupied by fluids of various natures. 'Texture' or 'tissue' is the anatomical term by which such assemblages are distinguished. Of these the cellular, or tela cellulosa, is most elementary, being the sole constituent of several, and a partial component of all tissues and systems. Thus the membranes and vessels consist entirely of condensed cellular substance ; and even muscle and nerve are resolvable, by microscopic analysis, into globules deposited in attenuated cellular element. But though the phenomena, which are designated as vital, are never found apart from organization, and have even by some naturalists been regarded as identical with it, yet in the order of succession vital actions seem necessarily to stand to organized structures in the relation of antecedents ; for the production of even the most rudimentary forms and textures implies the previous operation of combining tendencies or 'vital affinities'. The origin and early development of these vital tendencies, and of organized structures, are beyond the pale of exact or even of approximative knowledge. But it is matter of certainty, that life is the product only of life ; that every new plant or animal proceeds from some pre-existent being of the same form and character ; and thus that the image of the great Epicurean poet, " Quasi cursores vital lampada tradunt," pos- sesses a compass and force of illustration which, as a supporter of the doctrine of fortuitous production, he could not have him- self contemplated. The popular notions I'especting life are obscure and indeter- minate ; nor are the opinions even of philosophers characterized by much greater distinctness or mutual accordance. Like other complex terms, ' life ' can obviously be defined only by an cnume- 60 THIRD REPORT — 1833. ration of the phenomena which it associates. This enumera- tion will comprehend a greater or a smaller number of particu- lars, according to the station in the scale of living beings which is occupied by the object of survey. In its simplest manifesta- tion, the principle of life may be resolved into the functions of nutrition, secretion and absorption. It consists, according to Cuvier, of the faculty possessed by certain combinations of matter, of existing for a certain time and under a determinate form, by attracting unceasingly into their composition a part of surrounding substances, and by restoring portions of their own substance to the elements. This definition comprehends all the essential phenomena of vegetable life. Nutritive matter is drawn from the soil by the spreading fibres of the root, through the instrumentality of spongioles or minute turgid bodies at their extremities, which act, according to Dutrochet, by a power which he has called 'endosmosis.* The same agency raises the nutrient fluid through the lymphatic tubes to the leaves, where it seems to undergo a kind of respiratory process, and becomes fit for assimilation. These changes, and the subsequent pro- pulsion of the sap to the different parts and textures, plainly indicate independent fibrillary movements, which are repre- sented in animal life by what Bichat has termed ' the pheno- mena of organic contractility'. The power residing in each part of detecting in the circulating fluid, and of appropriating, matters fitted to renovate its specific structure, is designated in the same system by the term ' organic sensibility'. Ascending from the vegetable to the animal kingdom, the term ' life' advances greatly in comprehensiveness. The exist- ence of a plant is limited to that portion of space in which acci- dent or design has inserted its germ; while animals are for the most part gifted with the faculties of changing their place, and of receiving from the external world various impressions. Along with the general nutritive functions, the higher attri- butes of locomotion and sensation are therefore comprised in the extended compass of meaning which the term ' life' acquires with the prefix ' animal'. The nutritive functions, too, emerging from their original simplicity, are accomplished by a more com- plex mechanism, and by agencies further removed from those which govern the inanimate world. Locomotion is effected either by means of a contractile tissue, or of distinct muscular fibres. These fibres have been said to consist of globules resembling, and equal in magnitude to, those of the blood, disposed in lines, in the elementally cellulosity , which by an extension of the analogy is compared to serum. But the latest microscopical observations of Dr. Hodgkin are opposed REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 61 to this globular constitution of the contractile fibre. " Innu- merable very minute but clear and fine parallel lines or striae may be distinctly perceived, transversely marking the fibrillas." Irritability, or the faculty of contracting on the application of a stimulant, is a property inherent in the living fibre. It is an essential element of all vital operations, except of those which have their seat in the nervous system, such as sensation, voli- tion, the intellectual states, and moral affections. All the phe- nomena of life, in the higher animals, may then be ultimately resolved into the single or combined action of these two ele- mentary properties, — irritability and nervous influence, each residing in its appropriate texture and system. These preliminary remarks are designed to unfold the prin- ciples to be followed in classifying the vital functions. In ge- neral or comparative physiology, a strictly scientific ari'angement would contemplate first the phenomena of the most elementary life, and would successively trace the more perfect development of those simple actions and their gradual transition into more complex processes, as well as the new functions, superadded in the ascending scale of endowment. But such a mode of classification is wholly inapplicable to the particular physiology of man and of the more perfect animals, viewed by itself and without reference to inferior orders of beings ; for the nutri- tive functions of this class, which correspond with the elemen- tary actions of the simplest vegetable life, are effected by a complex system of vessels and surfaces, deriving their vital powers from contractile fibres, and controlled, if not wholly governed, by nervous influence. It is then manifest, that in the higher physiology the general lav/s of contractility and ' in- nervation' must precede the description of the several functions, which all depend on their single or imited agency. The parti- cular functions will afterwards be classed, as they stand in more immediate relation to one or other of the two essential princi- ples of life. In the present state of physiological knowledge, it is impos- sible to determine absolutely, and without an opening to con- troversy, whether the functions of muscle or those of nerve are entitled to precedency. If each were equally independent of the other in the performance of their several offices, the question of priority would resolve itself into one of simple convenience. The actions of the nervous system, if contemplated for the short interval of time during which they are capable of persisting without renovation of tissue, are entirely independent of the contractile fibre. But it is certain that the cooperation of nerve is required in most, if not in all, the actions of the mus- 62 THIRD REPORT — 1833. cular system. Tims the voluntary muscles in all their natural and sympathetic contractions receive the stimulant impulse of volition through the medium of nerve ; and though the mode, in which the motive impression is communicated to the invo- luntary muscles, is still matter of controversy, there seems suffi- cient evidence* to sanction the conclusion that nerve is in this case also the channel of transmission ; — " that the immediate antecedent of the contraction of the muscular fibre is univer- sally a change in the ultimate nervous filament distributed to that fibre." If this be correct, the physiological history of muscle cannot be rendered complete without reference to that of nerve. In the higher manifestations of life, nervous matter is in- vested with the most eminently vital attributes. It is the ex- clusive seat of the various modes of sensation, and of all the intellectual operations ; or, rather, it is the point of transition, where the physical conditions of the organs, which are induced by external objects, pass into states of mind, becoming per- ceptions ; and where the mental act of volition first impresses a change on living matter. These two offices of conducting motive impressions from the central seat of the will to the mus"- cles, and of propagating sensations from the surface of the body and the external organs of sense to the sensorium commune, have been of late years shown to reside in distinct portions of nervous substance. The honour of this discovery, doubtless the most important accession to physiological knowledge since the time of Harvey, belongs exclusively to Sir Charles Bell. It constitutes, more- over, only a part of the new truths, which his researches have unveiled, regarding the general laws of nervous action, and the offices of individual nerves. His successive experiments on function, guided always by strong anatomical analogies in struc- ture, in origin, or in distribution, have led to the entire remo- deUing of nervous physiology, and to the formation of a system of arrangement, based on essential affinities and on parity of intimate composition, instead of on apparent sequence or prox- imity of origin. Among the continental anatomists, MM. Ma- gendie and Flourens have contributed most largely to our knowledge of this part of physiology ; the former by repeating and confirming the experiments of Bell, as well as by various original inquiries ; the latter by his important researches into the vital offices of the brain and its appendages. Much light, * See " A Critical and Experimental Enquiry into the Relations subsisting between Nerve and Muscle," in the 37th vol. of the JUdinbure/h Medical and Surgical Journal. RKPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 63 too, has been thrown on the functions of several of the ence- phalic nerves, and especially of those supplying the face and its connected cavities, by Mr. Herbert Mayo, who has analysed their anatomical composition, and pursued their course with singular precision, and has thus been enabled to correct some errors of detail in the system of Sir Charles Bell. Nervous System. — In man, and in other vertebrated animals, the nervous system consists of the cerebrum, cerebellum, me- dulla oblongata, medulla spinalis, and of the encephalic, spinal, and ganglionic nerves. It seems most natural to observe this order of anatomical sequence in recording what is known of nervous functions. Cerebrum, or Brain-proper. — The physiology of the brain has received of late years very considerable accessions, and its vital offices, viewed as an entire organ, have now probably been ascertained with sufficient precision. Some portion of this newly acquired knowledge has been gathered from experiments on living animals, but the greater and more valuable part has flowed from the study of comparative development. In this latter field of inquiry, Tiedemann's elaborate history of the pro- gressive evolution of the human brain during the period of foetal existence, with reference to the comparative structure of that organ in the lower animals, merits an early and detailed notice. It had been discovered by Harvey, that the foetus in the human species, as well as in inferior animals, is not a pre- cise facsimile of the adult, but that it commences from a form infinitely more simple, and passes through several successive stages of organization before reaching its perfect development. In the circulatory system, these changes have been minutely observed and faithfully recorded*. Tiedemann has traced a similar progression in the brain and nervous system, and has moreover established an exact parallel between the temporary states of the foetal brain in the periods of advancing gestation, and the jiermanent development of that organ at successive points of the animal scale. The first part of his work is simply descriptive of the nervous system of the embryo at each suc- cessive month of foetal life. It constitutes the anatomical grovmd- work upon which are raised the general laws of cerebral forma- tion, and the higher philosophy of the science. In the second part, Tiedemann has established, by examples drawn from all the grand divisions of the animal kingdom, the universality of the law of formation, as traced in the nervous system of the • See an excellent Essay on the Development of the Vascular System in the Foetus of Vertebrated Animals, by Dr, Allen Thomson. 64 THIRD REPORT — IS33. human foetus, and the existence of one and the same funda- mental type in the hrain of man and of the inferior animals. The facts which have been unfolded by the industry of Tiede- mann, besides leading to the universal law of nervous develop- ment, throw important light upon nervous function : for it is observed that the successive increments of nervous matter, and especially of brain, mark successive advances in the scale of being ; and, in general, that the development of the higher in- stincts and faculties keeps pace with that of brain. Thus, in the zoophyta, and in all living beings destitute of nerves, no- thing that resembles an instinct or voluntary act is discovera- ble. In fishes the hemispheres of the brain are small, and marked with few furrows or eminences. In birds they are much more voluminous, more raised and vaulted than in rep- tiles ; yet no convolutions or anfractuosities can be perceived on any point of their surface, nor are they divided into lobes. The brain of the mammalia approaches by successive steps to that of man. That of the rodentia is at the lowest point of organization. Thus the hemispheres in the mouse, rat, and squirrel are smooth and without convolutions. In the carnivo- rous and ruminating tribes, the hemispheres are much larger and marked by numerous convolutions. In the ape tribe the brain is still more capacious and more convex ; it covers the cerebellum, and is divided into anterior, middles, and posterior lobes. It is in. man that the brain attains its greatest magni- tude and most elaborate organization. Sommerring has proved that the volume of the brain, referred to that of the spinal mar- row as a standard of comparison, is greater in man than in any other animal. Various attempts have been made of late years, chiefly by the French physiologists, to ascertain the functions of the brain by actual experiment. It will appear from a detailed survey of their labours, that little more than a few general facts respecting the function of its larger masses and great na- tural divisions have flowed from this mode of research. The offices of the smaller parts of cerebral substance cannot with any certainty be derived from the phenomena that have been hitherto observed to follow the removal of those parts, since the most practised vivisectors have obtained conflicting results. Nor is it difficult, after having performed or witnessed such experiments, to point out many unavoidable sources of fallacy. In operations on living animals, and especially on so delicate an organ as the brain, it is scarcely possible for the most skilful manipulator to preserve exact anatomical boundaries, to restrain haemorrhage, or prevent the extension to contiguous parts, of REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. G5 the morbid actions consequent upon such serious injuries, and to distinguish the secondary and varying phenomena, induced by the pressure of extravasated blood, or the spread of an in- flammatory process, from those which are essential and pri- mary. The ablation of small and completely insulated portions of brain must, then, be classed among the " agenda" of experi- mental physiology. The most decisive researches, that have been hitherto insti- tuted on the functions of the brain, are those of M. Flourens. His mode of o^ierating was to remove cautiously successive thin slices of cerebral matter, and to note the corresponding changes of function. He commenced with the hemispheres of the brain, which he found might be thus cut away, including the corpora striata and thalami optici, without apparently occasioning any pain to the animal, and without exciting convulsive motions. Entire removal of the cerebrum induces a state resembling coma ; the animal appears plunged in a profound sleep, being wholly lost to external impressions, and incapable of originating mo- tion ; it is deprived, too, according to Flourens, of every mode of sensation. Hence the cerebrum is inferred to be the organ in which reside the faculties of perception, volition and memory. Though not itself sensible, in the ordinary acceptation of the word, — that is, capable, on contact or injury, of propagating sen- sation, — yet it is the point where impressions made on the ex- ternal organs of sense become objects of perception. This ab- sence of general sensibihty observed in the brain has also been experimentally demonstrated in the nerves dedicated to the func- tions of sight, of smell and of hearing, and constitutes, perhaps, one of the most remarkable phenomena that have been disclosed by interrogating living nature. Flourens appears, however, to have failed in proving that all the sensations demand for their perception the integrity of the brain. He has himself stated that an animal deprived of that organ, when violently struck, " has the air of awakening from sleep," and that if pushed for- wards, it continues to advance after the impelling force must have been wholly expended. Cuvier has therefore concluded, in his Report to the Academy of Sciences upon M. Flourens' paper, that the cerebral lobes are the receptacle in which the impressions made on the organs of sight and hearing only, be- come perceptible by the animal, and that probably there too all the sensations assume a distinct form, and leave durable im- pressions, — that the lobes are, in short, the abode of memory. The lobes, too, would seem to be the part in which those mo- tions which flow from spontaneous acts of the mind have their origin. But a power of effecting regular and combined move- 1833. p 66 THIRD REPORT — 1833. ments, on external stimulation, evidently survives the destruc- tion of the cerebral hemispheres. A very elaborate series of experiments on the functions of the brain in general, and especially on those of its anterior por- tion, have been since performed by M. Bouillaud *. That ob- server concurs with Flourens in vievping the cerebral lobes as the seat of the remembrance of those sensations which are fur- nished to us by sight and hearing, as well as of all the intel- lectual operations to which these sensations may be subjected, such as comparison, judgment and reasoning. But he proves that the ordinary tactual sensibility does not require for its manifestation the presence of the brain. For animals entirely deprived of brain were awakened by being struck, and gave evident indications of suffering when exposed to any cause of physical pain. Bouillaud observes, too, that the iris continues obedient to the stimulus of light, after ablation of the hemi- spheres, and on this ground calls in question the loss of vision asserted by Flourens. Nor are the lobes (he contends,) the only receptacle of intelligence, of instincts and of volition : for to admit this proposition of Flourens would be to grant that an animal which retains the power of locomotion, which makes every effort to escape from irritation, which preserves its appro- priate attitude, and executes the same movements after as be- fore mutilation, may perform all those actions without the agency of the will or of instinct. Another doctrine of Flourens, which has been experimentally refuted by Bouillaud, is, " that the cerebral lobes concur as a whole in the full and entire exercise of their functions ; that when one sense is lost, all are lost ; when one faculty disappears, all disappear ;" in short, that a certain amount of cerebral matter may be cut away without ap- parent injui'y, but that when this limit is passed, all voluntary acts and all perceptions perish simultaneously. Bouillaud, on the contrary, has described several experiments which show that animals, from whom the anterior or frontal part of the brain had been removed, preserved sight and hearing, though de- prived of the knowledge of external objects, and of the power of seeking their food. The second part of M. Bouillaud's researches is entirely de- voted to the functions of the anterior lobes of the brain. These were either removed by the scalpel, or destroyed by the actual cautery, in dogs, rabbits and pigeons. Animals thus mutilated feel, see, hear and smell ; are easily alarmed, and execute a number of voluntary acts, but cease to recognise the persons * Magendie, Journal de Physiolorjie, torn. x. p. 361 REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 67 or objects which surround them. They no longer seek food, or perform any action announcing a comljination of ideas. Thus the most docile and intelhgent dogs lost all power of compre- hending signs or words which were before familiar to them, became indifferent to menaces or caresses, were no longer amenable to authority, and retained no remembrance of places, of things, or of persons. They saw distinctly food presented to them, but had ceased to associate with its external qualities all perception of its relations to themselves as an object of de- sire. The anterior or frontal part of the brain is hence inferred to be the seat of several intellectual faculties. Its removal oc- casions a state resembling idiotism, characterized by loss of the power of discriminating external objects, which, however, co- exists with the faculties of sensation. It will be unnecessary to describe fully in this place the ex- periments of Professor Rolando of Turin, performed in 1809, and published in Magendie's Journal, tom. iii., 1823, since the more important of his facts have reference, not to the brain-proper, but to the cerebellum. His paper certainly contains some cu- rious anticipations of phenomena, since more accurately ob- served by Flourens and Magendie ; yet as regards the brain, properly so called, his results are vague and inconclusive. Accident, rather than a well matured design, seems to have directed what parts of the brain he should remove ; and from having comprehended in the same injury totally distinct anato- mical divisions, he has rendered it impossible to arrive at the precise function of any one part. Thus we are told that injury of the thalami optici and tubercula quadrigemina in a dog was followed by violent muscular contractions. Now all subsequent experimenters agree, that irritation of the thalami is incapable of inducing convulsive motions ; and Flourens has proved that this property has its beginning in the tubercula, — an important fact, which Rolando, with a little more precision in anatomical manipulation, could scarcely have failed to discover. Magendie has described* some curious experiments on the corpora striata, which, though closely analogous in their results to those on the cerebellum, have their proper place in this section. Removal of one corpus striatum was followed by no remarkable change ; but when both had been cut away, the animal rushed violently forwards, never deviating from a recti- linear course, and striking against any objects in its way. In his lecture of February 7, 1828, Magendie, in the presence of his class, removed both corpora striata from a rabbit. The anunal * Journal de Physiologie, tom. iii. p. 376. 68 THIRD REPORT — 1833. attempted to rush forwards, and, if restrained, appeared rest- less, continuing in the attitude of incipient progression. One thalamus opticus was then cut away from the same animal. The direction of its motion was immediately changed from a straight to a curved line. It continued for some time to run round in circles, turning towards the injured side. When the other thalamus was removed, the animal ceased its motions and re- mained perfectly tranquil, with the head inclined backwards. These experiments, it may be observed, furnish no support to the opinions of MM. Foville and Pinel Grandchamps, who have assigned the anterior lobes and corpora striata as the parts presiding over the movements of the inferior extremities, and the posterior lobes and thalami as regulating the superior. Cerebelhim. — It may be regarded as nearly established by modern researches, that the cerebellum is more or less directly connected with the function of locomotion. The precise nature and extent of its control over the actions of the voluntary muscles are, however, far from being clearly determined. In the higher animals, the mental act of volition probably has its commencing point, as productive of a physical change, in the brain-proper ; though it must be confessed that some of the experiments of Flourens, and all of those of Bouillaud, indicate the persistence of many instinctive, and even of some automatic motions, after destruction of the brain. But there does appear sufficient evidence to prove that those volitions which have motion as their effect, whatever be their origin, whether in the cerebrum, cerebellum, or medulla oblongata*, require for their accomplishment the cooperation of the cerebellum. This evi- dence has been mainly supplied by the same inquirers whose researches on the cerebrum have been already analysed. In the order of time, though not of importance, the experi- ments of Professor Rolando stand foremost. Injuries of the cerebellum, he observed, were always followed by diminished motive power ; and this partial loss of power was always in direct proportion to the amount of injui-y. A turtle survived upwards of two months the entire removal of the cerebellum, continuing sensible to the slightest stimulus ; but when irritants were applied, it was totally unable to move from its place. M. Flourens has since arrived at similar, but more definitive results. He removed in succession thin slices from the cere- bellum. After the first two layers had been cut away, a slight weakness and want of harmony and system in the automatic movements were noticed. When more cerebellic substance had * Flourens, Mhnoires de I' Academie, torn. ix. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 69 been removed, great general agitation became apparent. The pigeon which was the subject of operation retained, as at first, the senses of sight and hearing, but was capable of executing only irregular unconnected muscidar efforts. It lost by degrees the power of flying, of walking, and even of standing. Removal of the whole cerebellum was followed by the entire disappear- ance of motive power. The animal, if laid upon its back, tried in vain to turn round ; it perceived and was apprehensive of blows, with which it was menaced, heard sounds, seemed aware of danger, and made attempts to escape, though ineffectually, — in short, while it preserved, uninjured, sensation and the ex- ercise of volition, it had lost all power of rendering its muscles obedient to the will. The cerebellum is hence supposed by Flourens to be invested with the office of " balancing, regu- lating or combining separate sets of muscles and limbs, so as to bring about those complex movements depending on simvd- taneous and conspiring efforts of many muscles, which are ne- cessary to the different kinds of progressive motion." Bouil- laud, who has successfully disputed several of the opinions of Flourens respecting the functions of the cerebrum, fully concurs with him as to those of the cerebellum. Yet, it must be admitted, that there exists also conflicting experimental testimony on this subject. M. Fodera* states that he has found the removal of a part of the cerebellum to be followed, in all cases, either by motion backwards, or by that position of the body which precedes retrograde movement. The head is thrown back, the hind legs separated, and the fore legs extended forwards, and pressed firmly against the ground. More complete destruction of the cerebellum occa- sions the animal to fall on its side ; but the head is still inclined rigidly backwards, and the anterior extremities agitated with convulsive movements, tending to cause retrograde motion of the body. Injuries of one side of the cerebellum were observed to produce paralysis of the same side of the body ; as might, indeed, have been anticipated from the direct course, without decussation, of the restiform columns which ascend to form the cei'ebellum. Magendie has described f precisely the same re- sults. A duck, whose cerebellum had been destroyed, could swim only backwards. In the course of his experimental lec- tures, Magendie, having removed the cerebellum in several rab- bits, demonstrated to his class the phenomena of retrograde movement, exactly as they have been recorded by Fodera. It is, then, impossible to regard the conclusions of Flourens as * Journal de Physique, July 1823. f Ibid. toin. iii. p. 157. 70 THIRD REPORT — 1883. fully established, opposed as they are by those of so skilful an experimenter as Magendie. Indeed, while Flourens conceives the cei'ebellum to preside over motion, MM. Foville and Pinel Grandchamps attribute to it the directly opposite function of sensation : and this doctrine seems to derive some support from anatomical disposition ; for it has been proved by Tiedemann that the cerebellum is nothing more than an expansion or pro- longation of the corpora restiformia, and posterior columns of the spinal medulla, which columns have been shown by Sir Charles Bell to have the office of conveying sensations. But it is not the less true that all recent experiments, even those of Fodera and Magendie, point to some connexion between the cerebellum and the power of voluntary motion. In the present state of our knowledge it would be unsafe to contend for more than the probable existence of some such general relation. This, then, is all that seems deserving of confidence respect- ing the functions of the cerebellum itself. But there are some singular phenomena which, though residing in other structures more or less near to the cerebellum, are so analogous to those already described as to call for notice in this place. Magendie has described * the results of injury to the crura cerebelli of a rabbit. Complete division of the right crus was followed by rapid and incessant rotation of the body upon its own axis, from left to right. This singular motion having continued two hours, Magendie placed the rabbit in a basket containing hay. On visiting it the following day he was surprised to find the animal still turning round as before, and completely enveloped in hay. The eyes were rigidly fixed in different lines; that of the injured side being directed forwards and downwards, that of the other side backwards and upwards. If both crura were divided, no motion followed. Magendie hence concluded that these ner- vous cords are the conductors of impulsive forces which coun- terbalance one another, and that from the equilibrium of these two forces result the power of standing, and even of maintaining a state of rest, and of executing the diflTerent voluntary motions. The inquiry naturally presented itself, whether these forces are inherent in the crura themselves, or emanate from the cere- bellum or some other source. To determine this question, portions of substance were removed from both sides of the cerebellum, but unequally, so as to leave intact | on the left side and ^ only on the right. The animal rolled towards the right side, and its eyes were fixed in the manner already de- scribed. But the left crus being divided, the animal rolled to * Journal tie Physiologie, torn. iv. 399. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 71 the left side. Hence it appears that section of the crus lias more influence over the lateral rotation of the hotly than injury of the cerebellum itself; and that the impulsive force does not belong (at least exclusively) to the cerebellum. Wlien the cere- bellum was divided precisely in the median line, the animal seemed suspended between two opposing forces, sometimes in- clining towards one side, as if about to fall, and again thrown suddenly back to the opposite side. Its eyes were singularly agitated, and seemed about to start from the orbits. Similar movements followed division of the continuous fibres in the pons Varolii. Serres has described a case of similar rotatory motion occurring in the human subject. A shoemaker ha- bituated to excess in alcoholic liquors, after great intemperance was seized with an irresistible disposition to turn round upon his own axis, and continued to move so till death ensued. On inspecting the brain, one of the crura cerebelli was found much diseased, and this was the only alteration of structure visible in any part of the nervous system. M, Flourens has published in a recent volume of the Me- moir es de VAcad^mie des Sciences* a description of some striking abnormal motions which followed the division of the semicircular canals of the ears of birds. Though these organs have no anatomical relation to the cerebrum or cerebellum, the altered motions resulting from their division are so analogous to those observed by Magendie after lesions of the corpora striata and crura, that they may be most conveniently described in the same section. Two of the semicircular canals are ver- tical, and one horizontal. Division of the horizontal canals on each side occasioned a rapid horizontal movement of the head from right to left, and back again, and loss of the power of maintaining an equilibrium, except when standing, or when perfectly motionless. There was also the same singular rota- tion of the animal round its own axis which follows injury of the crura cerebelli. Section of the inferior vertical canal on both sides produced violent vertical movements of the head, with loss of equilibrium in walking or flying. There was in this case no rotation of the body upon itself, but the bird fell back- wards, and remained lying on its back. When the superior vertical canals were divided, the same phenomena were ob- served as in section of the inferior, except that the bird fell forward on its head, instead of backward. All the canals, both vertical and horizontal, having been divided, in another pigeon, violent and iwegular motions in all directions ensued. When, * torn. ix. p. 454. 12 THIRD REPORT — 1833. however, the bony canals were so cautiously divided as to leave their internal membranous investment uninjured, these ab- normal motions were not produced. It is, therefore, in these membranes, or rather in the expansion of the acoustic nerve which overspreads them, that the cause of this phenomenon must reside. No explanation is proposed by Flourens of the control thus exercised by a nerve supposed to minister exclu- sively to the sense of hearing, over actions so entirely opposite in character. It is remarkable that the irregular movements should observe the same direction in their course as the canals, by the section of which they are induced. Thus the direction of the inferior vertical canal is posterior, that of the superior is anterior, corresponding perfectly with the directions of the abnormal motions. Medulla Oblongata. — The medulla oblongata, or "bulbe rachidien," is reducible into six columns, or three pairs, viz. two anterior or pyramidal, which partially decussate, two mid- dle or olivary, and two posterior or restiform, which proceed forwards without crossing. It is continuous in structure with the spinal marrow, and enjoys, by virtue of this relation, the same function of propagating motion and sensation. But it is distin- guished from the spinal medulla by special and higher attributes, being endowed with the faculty of originating motions, as well as with that of regulating and conducting them. The medulla oblongata, with the cerebrum and cerebellum, constitute, in short, according to Flourens *, those portions of the nervous system which exercise their functions "spontaneously or primordial- ly," and which originate and preside over the vital actions of the subordinate parts. To this latter order of parts, which re- quire an exciting or regulating influence, belongs the spinal medulla. In the superior class, Flourens seems to assign even a higher place to the medulla oblongata than to the cerebrum or cerebellum. For the cerebrum, he observes, may act without the cerebellum; and this latter organ continues to regulate the motions of the body after removal of the cerebrum. But the functions of neither cerebrum nor cerebellum survive the destruc- tion of the medulla oblongata, which seems to be the common bond and central knot combining all the individual parts of the nervous system into one whole. The medulla oblongata was regarded by Legallois as the mainspring or "premier mobile" of the inspiratory movements. He repeated before a Commission of the Institute of France the leading experiments on which his opinion rested f. In a rabbit * Memoires de I'Academie des Sciences, torn. ix. p. 478. t CEuvres de Legallois, torn. i. p. 247. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 73 five or six clays old, the larynx was detached from the os hy- oides and the glottis exposed to view. The brain and cere- bellum were then extracted without arresting the inspirations, which were marked by four simultaneous motions,' — a gaping of the lips, an opening of the glottis, the elevation of the ribs, and the contraction of the diaphragm. Legallois next removed the medulla oblongata, when all these motions ceased together. In a second rabbit, instead of extracting at once the entire me- dulla, it was cut away in successive thin slices. The four in- spiratory movements continued after the removal of the three first slices, but ceased after the fourth. It was found that the fourth had reached the origin of the eighth pair of nerves. If, instead of destroying the part in which this motive influence resides, it be simply prevented from communicating with the muscles which are subservient to inspiration, a similar effect ought to be produced. Now it is obvious that the medulla oblongata must transmit its influence to the muscles which raise the ribs, through the medium of the intercostal nerves, and therefore of the spinal marrow, and to the diaphragm through the phrenic nerves, and to these through the spinal marrow. In another rabbit, therefore, the medulla spinalis was cut across about the level of the seventh cervical vertebra. The effect of this operation was to arrest the elevation of the ribs, the other three inspiratory motions still continuing. A second section was made near the first cervical vertebra, and consequently above the origin of the phrenic, with the effect of suspending the contraction of the diaphragm. The par vagum was next divided in the neck, and the opening of the glottis ceased. There remained then, of the four inspiratory move^ ments, only the gaping of the lips, which, however, was suflS- cient to attest that the medulla oblongata still retained the power of producing them all. This power had ceased to call forth the other three motions, only because it no longer had communication with their organs. M. Flourens, in a recent memoir already referred to *, has confirmed and extended the views first announced by Legal- lois. He has distinctly traced the comparative action of the medulla spinalis and oblongata, on respiration, in the four classes of vertebrated animals. In birds, he found that all the lumbar and the posterior dorsal medulla might be destroyed without impeding the respiratory function, though it was arrested by removal of the costal medulla. In the mammalia the costal also * Memoir cs de V Academie, torn. ix. 1830. 74 THIRD REPORT — 1833. might be removed, for though the raisjng of the ribs ceased, the action of the diaphragm continued as long as the origin of the phrenic nerve remained uninjured. In frogs, all the spinal medulla may be destroyed, except the portion, whence spring the nerves supplying the hyoideal apparatus. Every part of the spinal marrow may be removed in fishes without affecting re- spiration ; for all the nerves distributed to the respiratory organs of fishes have their origin in the medulla oblongata. It is hence apparent that the spinal marrow exercises only a variable and relative action on the I'espiratory function, in the different classes of vertebrated animals. In descending from the higher to the lower points of this scale, the spinal marrow is seen pro- gressively to disengage itself from cooperation in these move- ments, while the medulla oblongata tends more and more to concentrate them in itself, till in fishes the proper functions of the two meduUge show themselves completely distinct, the spinal ministering to locomotion and sensation, and the oblongata to re- spiration. The medulla oblongata is, then, the "premier mo- teur " or the exciting and regulating principle of the inspiratory movements in all classes of vertebrated animals ; besides par- ticipating, by virtue of its continuity with the spinal marrow, in the proper functions of that organ. From a second series of experiments, M. Flourens concludes that there exists a point in the nervous centres at which the section of those centres produces the sudden annihilation of all the inspiratory move- ments ; and that this point corresponds with the origin of the eighth pair of nerves, commencing immediately above, and ending a little below, that origin, — a result precisely agx'eeing with that obtained by Legallois. Spinal Marrow. — It is apparent, that the functions of the three grand divisions of the nervous system, already described, have not yet been distinctly and fully ascertained. Our know- ledge of those, which next fall under survey, is more definite and substantial. The vital offices of the spinal medulla — re- garded by Legallois as the mainspring of life, and as alone re- gulating the actions of the heart and nobler organs, — are now reduced to conveying to the muscles the motive impulse of voli- tion, and to propagating to the sensorium commune, impressions made on the external senses. It is not invested with the power possessed by the cerebrum and cerebellum, and perhaps by the medulla oblongata, of spontaneously originating muscular mo- tions. It is mainly, if not exclusively, a conductor; a medium of communication between the brain and the external instru- ments of locomotion and sensation. Flourens, indeed, conjee- REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 75 tures that it also has the office of associating the partial con- tractions of individual muscles into "mouvemens d'ensemble," necessary to the regular motions of the limbs. Before recording what is known of the spinal cord itself, it will be proper to advert to some recent experiments of Magen- die on the serous fluid in which it is immersed. It would appear that a quantity of liquid, varying from two to five ounces in the human subject, is always interposed between the arachnoid tunic and the pia mater, or proper membrane of the cord. The intermembranous bag, occupied by this fluid, communicates with the ventricular cavities at the calamus scrip- torius by a round aperture, often large and patent in hydroce- phalic subjects. Magendie has therefore named this serous liquid ' cerebro-spinal '. In living animals, it issues in a stream from a puncture of the arachnoid. Its removal occasions great nervous agitation, and symptoms resembling those of canine madness. The sudden increase of its quantity induces coma. Its presence seems essential to the undisturbed and natural ex- ercise of the nervous functions ; and this influence pi-obably is dependent upon its pressure, temperature and chemical con- stitution, since any variation of these conditions is followed by the phenomena of nervous disorder. The great medullary cord is divided by a double furrow into two lateral halves ; and each of these is again subdivided by the insertions of the ligamenta dentata into two columns, one pos- terior and one anterior. It has been long known that section of any part of the spinal marrow excludes from intercourse with the brain all those parts of the body, which derive their nerves from the cylinder of medulla below the point of injury. The muscles, so supplied, are no longer obedient to the control of the will, and the tegumentary membranes similarly situated en- tirely lose their sensibility. This interruption of the relations which subsist between the central seat of volition and sensation, and the rest of the body, whether due to direct injury of the great nervous masses or communicating nerves, or produced by the pressure of extravasated fluids, by morbid growths, or by various poisonous matters, constitutes the condition known by the name ' pai-alysis'. In cases of this kind it is frequently ob- served that the powers of sensation and locomotion are simulta- neously impaired or destroyed. But examples are not want- ing, even in the earliest clinical records, of the total loss of one of those faculties with perfect integrity of the other. Such facts naturally suggested the belief that the power of propagating sensations, and that of conveying motive impressions, resided in distinct portions of the nervous system. This opinion, how- 76 THIRD REPORT — 1833. ever, remained mere matter of conjecture until a recent period, when it was unequivocally established by Sir Charles Bell. From the original experiments of that most distinguished phy- siologist, repeated and confirmed by Magendie, it follows that the faculty of conducting sensations resides exclusively in the two posterior columns of the medulla, while that of communica- ting to the muscular system the motive stimulus impressed by volition is the attribute of the two anterior columns. The same limitation of function is found in the nervous roots which spring from these separate columns. Thus each spinal nerve is fur- nished Avitli a double series of roots, one set of which have their origin in the anterior medullary column, and one in the pos- terior. The spinal nerves are, in consequence of this anatomi- cal composition, nerves of twofold function, containing in the same sheath distinct continuous filaments from both columns, and exercising, in the parts to which they are distributed, the double oflice of conductors of motion and sensation. It will afterwards appear, in our history of individual nerves, that all those which spring from the brain, except the fifth and eighth pairs, possess only a single function. Sufficient experimental proof of the foregoing propositions has been furnished by Sir Charles Bell and by M. Magendie. Thus, division of the posterior roots of the spinal nerves is uni- formly followed by total absence of feeling in the parts of the body to which the injured nerves are distributed, while their motive power remains undiminished. Magendie has further observed, that if the medullary canal be laid open, and the two posterior cords be touched or pricked slightly, there is instant expression of intense sufi^ering; whereas, if the same or a greater amount of irritation be applied to the anterior columns, there are scarcely any signs of excited sensibility. The central parts of the medulla seem also nearly impassable *. They may be touched, and even lacei*ated, according to Magendie, without exciting pain, if precautions are taken to avoid the surrounding medullary substance. In general, the properties of the spinal marrow, and especially its sensibility, seem to reside mainly on its surface ; for slight contact, even of the vascular membranes covering the posterior columns, caused acute pain. The first experiment of Sir C. Bell consisted in laying open the &pinal canal of a living rabbit, and dividing the posterior roots of the nerves that supply the lower limbs. The animal was able to crawl. In his second trial he first stunned the rabbit, and then exposed the spinal marrow. On irritating the * Annales de Chimie et de Physique, torn, xxiii. p. 436. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 77 posterior roots, no motion was induced in any part of the mus- cular frame ; but on grasping the anterior roots, each touch of the forceps was followed by a corresponding contraction of the muscles supplied by the irritated nerve. Magendie has de- scribed* the following experiments, which he has since declared were made without any knowledge of the prior ones of Sir C. Bell. The subjects chosen for the operation were puppies about six weeks old; for in these it was easy to cut with a sharp scalpel through the vertebrae and to expose the medulla. In the first, the posterior roots of the lumbar and sacral nerves were divided, and the wound closed : violent pressure, and even prick- ing with a shai'p instrument, awakened no sensation in the limb supplied by the nerves which had been cut ; but its motive power was uninjured. A second and a third trial gave the same re- sults. Magendie then divided in another animal, though with some difficulty, the anterior roots of the same nerves on one side. The hind limb became flaccid and entirely motionless, though it preserved its sensibility. Both the anterior and pos- terior roots were cut in the same subject with destruction of motion and sensation. In a second paperf Magendie has re- lated the following additional facts. The introduction of nux vomica into the animal economy is well known to give rise to violent tetanic convulsions of the whole muscular system. This property was made available as a test of the functions of the separate orders of nervous I'oots. It was found that, while all the other muscles of the body were agitated, when under the influence of this poison, by violent spasmodic contractions, the limb, supplied by nerves whose anterior roots had been pre- viously divided, remained supple and motionless. But when the posterior roots only had been cut, the tetanic spasms were universal. It would seem, however, that the seats of the two faculties of conducting motion and sensation are not strictly insulated by exact anatomical lines, but that they rather pass into each other with rapidly decreasing intensity. Thus irri- tation of the anterior roots, when connected with the medvdla, gives birth, along with motive phenomena, to some evidences of sensibility ; and, vice versa, stimuli applied to the posterior roots, also undivided, occasion slight muscular contractions. In this last case it is, indeed, probable that the irritation tra- velled from the posterior roots upwards to the brain in the ac- customed channel, and gave rise to a perception of pain, which prompted the muscular effort. Indeed, after division of the posterior nervous roots, ordinary stimulants, applied to the • Journal de Physiologie, torn. ii. p. 276. August 1822. f Ibhl torn. ii. p. 366. 78 THIRD REPORT — 1833. ends not connected with the medulla, produced no apparent eftects ; though the galvanic fluid directed upon either order of roots gave rise to muscular contractions. These were more complete and energetic when the anterior roots were the sub- jects of the experiment. Besides the evidence thus obtained by direct experiments on living animals, several important facts have been gathered from the pathology of the nervous system in man. These consist of cases of insulated paralysis of either motion or feeling, referred to the changes in structure obsei'ved after death. Sir Charles Bell has himself recorded several examples of this kind strongly confirming his experimental results ; and others of similar ten- dency are scattered through the successive volumes of Magen- die's Journal*. But it must be admitted, that evidence of this kind is seldom distinct and conclusive. The structural changes, induced by disease, are rarely so circumscribed in seat and extent as to represent adequately the operations of the scalpel; and often when they are thus isolated within anatomical bound- ing lines, they affect, by pressure, or by the spread of the same morbid process, in a degree too slight to leave decided traces, the functions of contiguous parts, thus clouding the judgments of the best pathologists, and invalidating their inferences. There is, however, a very I'emarkable case described by Pro- fessor Royer CoUard, to which these objections do not apply. Sprevale, an invalided soldier, was upwards of seventeen years the subject of medical observation in the Maison de Sante of Charenton. This individual remained for the last seven years of his life with the legs and thighs permanently crossed, and totally incapable of motion, though retaining their sensibility. On opening after death the spinal canal, there was found the pultaceous softening {ramolUssement) of the whole anterior part of the medulla, and of almost the whole of the fibrous cords which form it. The anterior roots of the spinal nerves had also lost their accustomed consistency; while the posterior sur- face of the spinal cord, and its investing membrane, were healthy. Several of the cases observed by Sir Charles Bell furnish also unequivocal proof of the soundness of the views developed by experiment. There exist, indeed, few truths in physiology established on so wide and solid a basis of experimental research and pa- thological observation, as those deduced by Sir Charles Bell, the original discoverer, and by Magendie, his successor in the path of inquiry, respecting the offices of the spinal medulla. * See in particular Dr. Rullier's case, torn. iii. p. 1 73 ; and Dr. KorefTs, torn. iv. p. 376. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 79 This organ may now be regarded as mainly, if not solely, a medium of intercourse between the external world and the brain, and again between the brain and the voluntary muscles, its two anterior columns being subservient to motion, its two posterior to sensation. In the present state of our knowledge it would be fruitless to try to penetrate into the minute philosophy of these actions : but it seems probable, from recent discoveries on the ultimate anatomy of tissue, that these actions are mole- cular, having their place in the globular elements, into which all living textures are resolvable by microscopic analysis ; — that the physical changes, e. g. impressed by external objects on the delicate net-work of nerve which invests the tegumentary mem- branes and open cavities, are propagated thence, from particle to particle, along the continuous filaments, to their origins in the posterior spinal columns, and thence to the central point, where they become objects of perception; — and that the motive sti- mulus of volition is similarly transmitted down the anterior co- lumns and nerves, to the organs of locomotion. Indeed, it is a legitimate inference from Sir Charles Bell's discoveries, that a simple nervous filament, or medullary column, can only propa- gate an impression in one line of direction, viz. either towards or from the central seat of perception and of will ; and this cu- rious law of nervous actions would seem to point at some in- sensible molecular motion as their essential condition. It remains to investigate the arguments which have been supposed to prove the residence in the spinal marrow of the power of originating and controlling the actions of the heart. This question has been matter of eager controversy, from its bearing vipon the general relations of nerve and muscle. With- out prejudging this latter topic, it may simplify its future con- sideration, and will at the same time be more consistent with strict arrangement, to state here merely the facts which have reference to the spinal medulla. The woi'k of Legallois, entitled ^^ Experiences sur le Prin- cipe de la Vie, notamment sur celni des Mouvemens du Cceur et si/r le Siege de ce Principe* ," was the first remarkable essay on the relations between the heart and the spinal cord. It will, however, be sufficient to allude in general terms to the conclu- sions of Legallois, since they have been entirely subverted by the subsequent researches of Dr. Wilson Philip and M. Flou- rens. Legallois's main doctrine was, that the principle which animates each part of the body resides in that part of the spinal medulla whence its nerves have their origin; and that it is also • CEuvres de Legallois, torn. i. pp. 97, 99, &c. 80 THIRD REPORT — 1853. from the spinal cord that the heart derives the principle of its life and its motion*. The experimental proof supposed to establish these propositions consisted in destroying in different rabbits portions of the cervical, dorsal and lumbar medulla. Cessation of the heart's action Avas affirmed to be the constant result of the operation ; but even in some of Legallois's own ex- periments f, the motions of the heart continued after consider- able injury had been inflicted on the spinal cord, and especially on its lower divisions. Still more unequivocal is the evidence that has been advanced by Dr. Wilson Philip, in his Inquiry into the Laivs of the Vital Functions. His experiments, which were very numerous and judiciously varied, show that the cir- culation continues long after entire removal of the spinal mar- row, and that by artificially maintaining respiration, the motions of the heart may be almost indefinitely prolonged. Flourens, in the 10th vol. of the Mem. de lAcademieX, ^^^^ lately con- firmed Dr. Philip's views : he has shown that the circulation is entirely independent of the spinal marrow. The influence ap- parently exerted is only secondary, being due to the suspension of the respiratory movements. Thus all those portions of the spinal marrow which can be destroyed in the different classes of animals without arresting respiration, may be removed with- out affecting the cii'culation. In fishes and frogs the entire spinal cord may be destroyed without checking the heart's mo- tions, because in these classes the medulla oblongata presides exclusively over the respiratory function. Nerves. — The classification of nerves, which is most conve- nient to the physiologist, is based upon their vital properties or functions. Such an arrangement wovdd distribute them into — 1, nerves of motion; 2, nerves both of motion and sensation; 3, the nerves ministering to the senses of sight, smell and hear- ing ; and 4, the ganglionic system, or, according to Bichat, nerves of organic life. Sir Charles Bell has added a fifth class, comprising nerves which he supposes are dedicated to the respiratory motions. But it will afterwards appear, that the existence of an exclusive system of respiratory nerves is not supported by sufficient evidence. The first class of nerves exercising the single office of con- veying motion comprehends the 3rd, 4th, 6th, portio dura of the 7th, the 9th, and perhaps two divisions of the 8th, viz. the glossopharyngeal and spinal accessory. Mr. H. Mayo's expe- riments detailed in his Anatomical and Physiological Commen- taries, No. 1 1 . (and Journal de Physique, tom. iii.) throw much * p. 259. t pp. 100^ 101, 105. + p. 625. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 81 light on the functions of several of these nerves. The motions of the iris, he shows, require the integrity of the third pair, division of these nerves being always followed by full dilatation of the pupils, which cease to be obedient to the stimulus of light. If the divided end of the nerve communicating with the eye be pinched by the forceps, the iris contracts. Hence it is apparent that dimi- nution of the aperture of the pupil is the result of action, and dilatation of the pupil the result of relaxation, of the iris. Flou- rens has shown that complete extirpation of the tubercula quad- rigemina also paralyses the iris, and that irritation of those bo- dies excites its contractions. The same effect is noticed by Mayo to arise from division or irritation of the optic nerve. He divided the optic nerves within the cranium of a pigeon immediately after decapitation. When the end of the nerve connected with the ball of the eye was seized in the forceps, no action ensued; but when the end attached to the brain was irritated, the iris immediately contracted. These several experiments clearly in- dicate the dependence of the iris upon the optic nerve, upon the tubercula from which one root of that nerve springs, and upon the third pair. The stimulus of light impinges upon the retina, is conveyed along the optic nerve through the tubercle to the sensorivmi, whence the motive impression is propagated to the iris by the third encephalic nerve. It is not so easy to define the precise mode of action of the pathetici, or fourth pair of nerves. Sir Charles Bell * supposes that they are destined " to provide for the insensible and in- stinctive rolling of the eyeball, and to associate this motion of the eyeball with the winking motions of the eyelids." He even conjectures that "the influence of the fourth nerve is, on cer- tain occasions, to cause a relaxation of the muscle to which it goes." It is certain, however, from its exclusive distribution to the superior oblique muscle, that the fourth is a nerve of motion. The sixth nerve is also a nerve of voluntary motion, and is sent to the rectus externus of the eyeball. Sir Charles Bell has placed the portio dura of the seventh pair among his respiratory nerves. There is, however, no doubt that it is simply a motive nerve, and that it is indeed the only nerve of motion, which supplies all the muscles of the face, except those of the lower jaw and palate. Division of this nerve occasions no expression of pain, according to Bell; but Mayo's experience is opposed to this absence of sensibility f. "The motion of the nostril of the same side instantly ceased, * Natural System of Nerves, p. 3.58. t See Mr. H. Mayo's Anatomical and Pliijsioloffical Commentaries, Part I.; and Outlines of Human PLy^ioloijij, 2nd edit., p. 3;34, 1 S.Jo. G 82 TIIIUD REPORT — 1833. after its section in an ass *, and that side of the face remained at rest and placid during the highest excitement of the other parts of the respiratory organs." These and similar observa- tions are all consistent with the opinion, that the seventh is simply a nerve of voluntary motion. It will afterwards appear that it has no claim to any further endowment. Mr. Herbert Mayo infers from his experiments, that the three divisions of the eighth pair are all nerves both of motion and sensation. Thus the glossopharyngeus is a nerve of motion to the pharynx, and perhaps of sensibility to the tongue. He observed that " on irritating the glossopharyngeal nerve in an animal recently killed, the muscular fibres about the pharynx acted, but not those of the tongue f." Irritation of the spinal accessory produced both muscular contractions and pain. The par vagum, he conceives, bestows sensibility on the membrane of the larynx, besides conveying the motive stimulus to its muscles. This nerve has been the subject of experiment from the earliest times, and Legallois has minutely described the results obtained by successive inquirers X- These were singu- larly discordant, and gave origin to the most opposite theories of the mode of action of the par vagum. In the greater number of experiments, section of this nerve was followed, after a longer or shorter interval, by death. Piccolhomini contended that the division of the nerve was fatal from its arresting the move- ments of the heart, and after him Willis supported the same doctrine. By Haller, on the contrary, the cause of death was sought in disturbance of the digestive fvmctions. Bichat and Dupuytren seem to have been the first to obtain a glimpse of the true seat of injury. The former remarked that the respiration became very laborious after section of the nerve, and Dupuytren distinctly traced death to asphyxia. Legallois has established by numerous experiments the accuracy of this last view. He has shown that in very young animals death is the immediate consequence of the operation of cutting either the par vagum or its recurrent branch, and that the suddenness of the effect is due to the narrowness of the aperture of the glottis in early age. In adult animals, the asphyxia is induced by the effusion of serous fluids and ropy discoloured mucus into the bronchial tubes and air-cells. More recently. Dr. Wilson Philip has prac- tised the section of the par vagum with an especial reference to its influence upon digestion. He divided the nerve below the origin of the inferior laryngeal branch, as in this case the « pp. 106, 107. t Outlines of Human Physiology, 2nd edit., p. 337. + (Euvres, p. 1 54 et seq. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 83 dyspnoea is much less considerable than when the wound is in- flicted on the higher portion*. It was found, in all these trials, that food introduced into the stomach after the operation re- mained wholly undigested. Hence Dr. Philip infers the de- pendence of secretion upon nervous influence, a conclusion, it has been remarked by Dr. Alison, not logically deducible from the experimental dataf. The par vagum cannot then, it is obvious, be included in the class of nerves subservient solely to motion ; and it is even doubtful whether the other two divisions of the eighth pair are not also endowed with sensibility. Respecting the function of the ninth, or lingvial, there is, however, no place for hesitation. It has been experimentally proved by Mr. Mayo to supply the muscles of the tongue ; though he also asserts that pinching it with the forceps excited pain. Three of these nerves, the third, sixth, and ninth, arise, it was first remarked by Sir Charles Bell, from a tract of medullary matter continuous with the an- terior column of the spinal marrow: and hence their exclusive oflice of conducting motive impressions. II. There are thirty-two pairs of nerves of similar anatomical origin and composition, which possess the twofold office of com- municating motion and sensation. Of these, all excepting one (the fifth pair of the cerebral nerves) spring from the spinal marrow. These thirty-one pairs are precisely analogous in formation, being all constituted of two distinct series of roots, one from the anterior column, and one from the posterior column of the spinal marrow. The posterior funiculi collected together form a ganglion, seated just before this root is joined by the anterior root. It has been already stated that the power of propagating sensation resides in the posterior column, and in the nervous roots arising from it, and that the motive faculty has its seat in the anterior column and roots. The evidence, also, supplied by Bell and Magendie, that the spinal nerves are hence nerves of double office, has been fully detailed. It re- mains, then, to establish the title of the fifth pair of cerebral nerves to be included in the same class with the spinal nerves. The analogy in structure and mode of origin between the fifth pair and the nerves of the spine has been long matter of observation. Prochaska has thus distinctly noticed it in a pas- sage of his Essay De Structurd Nervorum, published in 1779, first pointed out to me by my friend Dr. Holme : " Quare omnium cerebri nervorum, solum quintum par post ortum suum * Experimental Inquirt/, 3rd edit., p. 109. t Dr. Alison, Journal of Science, vol. ix. p. 106. g2 84 THIRD REPORT — 1833. more nervorum spinalium, ganglion semilunare dictum, facere debet? sub quo peculiaris funiculorum fasciculus ad tevtium quinti paris ramum, maxillarem inferiorem dictum, properat, insalutato ganglio semilunari, ad similitudinem radicum ante- riorum nervorum spinalium ?" Siimmerring has also pointed out with equal clearness the resemblance in distribution be- tween the smaller root of the fifth and the anterior roots of the spinal nerves. But Sir Charles Bell was the first to establish the identity of their functions, and to arrange them prominently in the same natural division. His experiment consisted in exposing the fifth pair at its root, in an ass, the moment the animal was killed. " On irritating the nerve, the muscles of the jaw acted, and the jaw was closed with a snap. On dividing the root of the nerve in a living animal, the jaw fell relaxed." In another experiment the superior maxillary branch of the fifth nerve was exposed. " Touching this nerve gave acute pain ; the side of the lip was observed to hang low, and it was dragged to the other side." Sir Charles Bell concluded that the fifth nerve and its branches are endowed with the attri- butes of motion and sensation. This, though correct as regards the nerve itself, viewed as a whole, is strictly true only of the lowest of its three divisions, viz. the inferior maxillary. The ophthalmic and the superior maxillary, the subject of the last experiment, are nerves simply of sensation. Mr. Herbert Mayo in the Essay already referred to, has pointed out this error, and has defined with minute precision the relative offices of the fifth and seventh nerves. By a careful dissection of the fifth nerve he found that the anterior branch, or smaller root, which goes, as Prochaska was aware, entirely to the inferior maxillary, is distributed exclusively to the circumflexus palati, the ptery- goids, and temporal and masseter muscles. He observed that sec- tion of the supra and infra orbitar branches, and of the inferior maxillary, near the foramina, whence they emerge, induces loss of sensation in the corresponding parts of the face. It may then be regarded as fully proved that the trigeminus or fifth pair is the nerve which bestows sensation on the face and its appen- dages, and motion only on the muscles connected with the lower jaw. The other muscles of the face derive their motive power from the portio dura of the seventh nerve. M. Magendie has also published several memoirs on the functions of the fifth pair. In these he attempts to prove that the olfactory nerve is not the nerve of smell ; that the op- tic is but partially the nerve of vision ; and that the auditory is not the principal nerve of hearing. It is in the fifth pair that he supposes all these distinct and special endowments to reside. IlEPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 85 But the experimental proof will be found to be singularly in- conclusive. The olfactory nerves were entirely destroyed in a dog. After the operation it continued sensible to strong odours, as of ammonia, acetic acid, or essential oil of lavender ; and the introduction of a probe into the nasal cavity excited the same motions and pain as in an unmutilated dog. The fifth pair was then divided in several young animals, the olfactory being left entire. All signs of the perception of strongly odorous sub- stances, as sneezing, rubbing the nose, or turning away the head, entirely disappeared. From these facts Magendie infers that the seat of the sensations of smell is in the fifth, and not in the first pair of nerves. It is obvious that Magendie has con- founded two modes of sensation, which are essentially distinct in their nature and in their organic seat, viz. the true percep- tions of smell, and the common sensibility of the nasal passages. The phenomena, which he observed to cease after the section of the fifth nerve, are the results of simple irritation of the pi- tuitary membrane, and are manifestly wholly unconnected with the sense of smelUng, since they are producible by all powerful chemical agents, even though inodorous, as, for example, by sulphuric acid. No proof has been given that the true olfac- tory perceptions do not survive the destruction of the fifth pair. Indeed, in a subsequent paper, Magendie confesses that the loss of sensibility in the nasal membrane, after section of the fifth, does not prove the residence of the sense of smell in the branches of that nerve ; but merely that the olfactory nerve re- quires, for its perfect action, the cooperation of the fifth pair, and that it possesses only a special sensibility to odorous parti- cles. There is even less ground for supposing that the fifth pair is in any degree subservient to the senses of sight and hearing. After cutting this nerve on one side, the flame of a torch was suddenly brought near the eye, without inducing contraction of the pupil ; but the direct light of the sun caused the animal to close its eyelids. Thus the sensibility of the retina, though somewhat impaired, was not destroyed by division of the fifth pair. But section of the optic nerves was immediately followed by total blindness. In another rabbit Magendie divided the fifth pair on one side, and the optic nerve on the other. The animal, he states, was completely deprived of sight, though the eye, in which the fifth pair only had been cut, remained suscep- tible to the action of the solar rays. No evidence, however, is offered to show that the animal was entirely blind : on the con- trary, the only change observed, on approaching a torch to an vninjiired eye, was contraction of the iris ; and this we are told 86 THIRD REPORT 1833. was actually observed in the eye of the side, on which the fifth nerve had been divided. Magendie has assigned another singular function to the fifth pair, viz. to preside over the nutrition of the eye. Twenty-four hours after section of this nerve, incipient opacity of the cornea was observed, which gradually increased till the cornea became as white as alabaster. There was also great vascularity of the conjunctiva extending to the iris, with secretion of pus, and for- mation of false membranes in the anterior chamber. About the eighth day, the cornea began to detach itself from the sclerotica, the centre ulcerated, and the humours of the eye finally escaped, leaving only a small tubercle in the orbit. In this experiment, the nerve had been divided in the temporal fossa, but when cut immediately after leaving the pons Varolii, the morbid changes were less marked, the movements of the globe of the eye were preserved, the inflammation was limited to the superior part of the eye, and the opacity occupied only a small segment of the circumference of the cornea. After division of the nerve near its origin in the medulla, no traces of disease were discoverable in the eye till the seventh day, and these symptoms never be- came very prominent. Several cases have been since recorded of structural disease of this nerve in the human subject, with the concomitant symptoms. That of Laine, described by Serres in the 4th vol. of Magendie's Journal, furnishes strong support to the views of Magendie *, A different explanation of this fact and of others which have a tendency to refer secretion and nutrition to the control of the nervous system has been proposed by Dr. Alison. Mucous surfaces are protected from the contact of air and foreign bo- dies by a copious secretion, which is evidently regulated in amount by their sensibility, since it is increased by any unusual iri'itation. This is especially true of the membrane of the eye. Now section of the fifth pair is known to paralyse the sensibi- lity of that organ, and the contact of air or other irritating body upon the insensible membrane, instead of inducing an aug- mented mucous discharge, will excite the inflammatory process described by Magendie. The disorder of the digestive func- tion f, which followed division of the par vagum in the experi- ments of Dr. Wilson Philip, and the ulceration of the coats of the bladder after injury of the lower part of the spinal marrow, are attributed by Dr. Alison to the same cause. The class of nerves which comprehends the fifth pair and * See also a case of destruction of the olfactory nerves, lom. v. •f Outlines of Phyniolngy, p. 71. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 87 the thirty-one pairs of spinal nerves, becomes, after the vniion of their roots, invested with a twofold endowment, and conti- nues so throughout their entire course and final distribution to the muscular tissue. It would appear, indeed, from a later paper of Sir Charles Bell*, that nerves of sensation, as well as of motion, are necessary to the perfect action of the voluntary muscles. "Between the brain and the muscles there is a circle of nerves ; one nerve conveys the influence from the brain to the muscle, another gives the sense of the condition of the muscle to the brain." In the case of the spinal nerves this circle of intercourse is at least probable ; but proof of its ne- cessity must be obtained, from observing the habitudes of those encephalic nerves, which minister exclusively to motion. Now it is found, on minute dissection, that the muscles of the eye- ball, which are supplied by the third, fourth and sixth motive nerves, also receive sensitive filaments from the ophthalmic branch of the fifth ; and that the muscles of the face, to which the portio dura is distributed, are also furnished with branches of sensation from the fifth. Sir Charles Bell has further shown that the muscles of the lower jaw, to which the motive im- pression is propagated by the muscular branch of the inferior maxillary, draw nervous supplies also from the ganglionic or sensitive branch of that division of the fifth pair. This com- plicated provision has its origin, he supposes, in its being " ne- cessary to the governance of the muscular frame that there should be consciousness of the state or degree of action of the muscles." III. The olfactory, auditory and optic nerves are gifted with a special sensibility to the otajects of the external senses, to which they respectively minister, Magendie seems to have been the first to prove, experimentally, that they do not also share the common or tactile sensibility. He exposed the olfac- tory nerves, and found that, like the hemispheres of the brain from which they spring, they are insensible to pressure, prick- ing, or even laceration. Strong ammonia was dropped upon them without eliciting any signs of feeling. The optic nerve, and its expansion on the retina, participate with the olfactory in this insensibility to stimulants. This was proved by Ma- gendie in the human subject as well as in animals. In perform- ing the operation of depressing the opaque lens, he repeatedly touched the retina in two diflPerent individuals without awaken- ing the slightest sensation. The portio mollis, or acoustic nerve, was also touched, pressed, and even torn without causing pain. • Philosophical Transactions, 1826, p. 163. 88 THIRD REPORT 18S3. IV. The functions of the ganglia, of the great sympatlietic nerve, and its intricate plexuses and anastomotic connexions, are matter, at present, of conjectui'e. Dr. Johnstone, in an Essay on the Use of the Ganglions, published in 1771, has described a few inconclusive experiments on the cardiac nerves. He supposes that " ganglions are the instruments by which the motions of the heart and intestines are rendered uniformly in- voluntary," — a notion which Sir Charles Bell has shown to be to- tally unsound. The best history of opinions, to which indeed our knowledge reduces itself, will be found in the physiological section of Lobstein's work, De Nervi Sympathetici Fabrica, Usu, et Morbis*. In the earliest of his communications to the Royal Society, as well as in his last work on the Nervous System ■!-, Sir Charles Bell has maintained the existence of a separate class of nerves, subservient to the regular and the associated actions of respira- tion. The origins of these nerves J "are in a line or series, and from a distinct column of the spinal marrow. Behind the corpus olivare, and anterior to that process, which descends from the cerebellum, called sometimes the corpus restiforme, a convex strip of medullary matter may be observed. From this tract of medullary matter, on the side of the medulla oblongata, arise, in succession from above downwards, the portio dura of the seventh nerve, the glossopharyngeus nerve, the nerve of the par vagum, the nervus ad par vagum accessorius, and, as I imagine, the phrenic and the external respiratory nerves." The fourth pair is also received into the same class. This doctrine of an exclusive system of respiratory nerves, associated in function by virtue of an anatomical relation of their roots, has not, as Sir Charles Bell seems himself aware §, received the concurrence of many intelligent physiologists of this counti'y or of the Continent. Mr. Herbert Mayo, in the ad- mirable Essay already referred to, was the first to point out the true relations of the fifth and seventh nerves. He has shown that the muscles of the face, excepting those already enumer- ated, which elevate the lower jaw, receive their motive nerves exclusively from the seventh, and consequently that this nerve must govern all their motions, voluntary as well as respiratory. But Dr. Alison, in his very elaborate paper || " On the Physiolo- gical Principle of Sympathy," has cast considerable doubts on * Paris 1823. t 4to, 1830. J The Nervous System of the Human Body, p. 129. 'Ito, 1830. § Op. cit., p. 11;'). II Transacliom of Oie Medko-chirurykal Socidy of Edinburgh, 1826, vol. ii. p. 165. TJEPOKT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 89 the soundness of this part of Sir Charles Bell's arrangement, as respects not only the individual nerves thus classed together, but even the general principle on which the entire system rests. The reasoning of Dr. Alison consists, first, in referring the phenomena of natural and excited respiration to the compre- hensive order of sympathetic actions. In these " the pheno- menon observed is, that on an irritation or stimulus being applied to one part of the body, the voluntary muscles of an- other, and often distant part, are thrown into action." Now it has been long since fully established by Dr. Whytt, that these associations in function cannot be referred to any con- nexions, either in origin or in course, of the nerves supplying remote organs so sympathizing; and that a sensation is the necessary antecedent of the resulting muscular aetion. Thus it is known that these actions cease in the state of coma ; are not excited when the mind is strongly impressed by any other sensation or thought ; and that the same muscular con- tractions may be induced by the irritation of different parts of the body, provided the same sensation be excited. Dr. Alison has, however, failed to show* that the essential acts of inspira- tion, viz. the contractions of the diaphragm and intercostals, require the intervention of a sensation. Their continuance in the state of coma, and in the experiments of Legallois and Flourens after the entire removal of the brain, and their di- .stinct reference by these two inquirers to the medulla oblon- gata, which has never been supposed to be the seat of sensa- tion, prove them to be independent of the will and of perception. But this is true only of the essential, not of the associated respiratory phenomena. Dr. Alison proceeds to show that there is equal reason for classing almost all the nerves of the brain, and many more of the spinal nerves, with those exclusively named respiratory by ♦Sir Charles Bell. Thus the lingual nerve governs an infinite number of motions strictly associated with respiration : the in- ferior maxillary " moves the muscles of the lower jaw in the action of sucking, — an action clearly instinctive when first per- formed by the infant, frequently repeated voluntarily during life, and always in connexion with the act of respiration." Again, the sensitive branches of the fifth pair cooperate in the act of sneezing. But if these nerves be admitted into the system, the fundamental principle of that system, viz. origin in a line or series, is at once violated. Nor is this connexion in origin more than matter of conjecture, as regards two of the * p. 1 76, and note. 90 THIRD REPORT 1833. most important of the nerves, classed by Sir Charles Bell himself as respiratory, — the phrenic and the external respiratory. These two nerves branch from the cervical or regular double-rooted series. Moreover, the circumstance of rising in linear suc- cession is not found to associate nerves in function. " Be- tween the roots of the phrenic nerve and those of the inter- costals, there intervene in the same series the origins of the three lowest cervical nerves, and the first dorsal, which go chiefly to the axillary plexus and to the arm, and which are not respiratory nerves." In recapitulation, the following facts are among the most important that have been fully ascertained in the physiology of the nervous system. 1. One universal type has been followed in the formation of the nervous system in vertebrated animals. The brain of the human foetus is gradually evolved in the successive months of uterine existence ; and these stages of progressive develop- ment strictly correspond with permanent states of the adult brain at inferior degrees of the animal scale. 2. These successive increments of cerebral matter are found to be accompanied by parallel advances in the manifestation of the higher instincts and of the mental faculties. 3. That the brain is the material organ of all intellectual states and operations, is proved by observation on comparative development, as well as by experiments on living animals, and by the study of human pathology. But there does not exist any conclusive evidence for referring separate faculties, or moral affections, to distinct portions of brain. 4. Certain irregular movements are produced by injuries of the corpora striata, thalami optici, crura cerebelli, and semi- circular canals of the internal ear. 5. The tubercula quadrigemina preside over the motions of the iris, and their integrity seems essential even to the func- tions of the retina. They are also, according to Flourens, the points, at which irritation first begins to excite pain and mus- cular contractions. 6. The cerebellum appears to exercise some degree of con- ti'ol over the instruments of locomotion ; but the precise na- ture and amount of this influence cannot be distinctly defined. 7. The cerebrum, cerebellum and medulla oblongata possess the faculty of acting primordially, or spontaneously, without requiring foreign excitation. The spinal cord and the nerves are not endowed with spontaneity of action, and are therefore termed subordinate parts. REPORT ON THE PHYSIOLOGY OF THE NERVOUS SYSTEM. 91 8. The medulla oblongata exercises the office of originating and regulating the motions essential to the act of respiration. By virtue of its continuity with the spinal marrow, it also par- ticipates in the functions of that division of nervous matter. 9. The function of the spinal cord is simply that of a con- ductor of motive impulses, from the brain to the nerves supply- ing the muscles, and of sensitive impressions from the surface of the body to the sensorium commune. These two vital offices reside in distinct portions of the spinal medulla, — the propagation of motion in its anterior columns, the transmission of sensations in its posterior columns. There is no necessary dependence of the motions of the heart, and the other invo- luntary muscles, on the spinal marrow. 10. The nerves are comprehended in the four following classes : — I. Nerves simply of motion ; II. Of motion and sen- sation ; III. Of three of the senses ; IV. The ganglionic sy- stem. I. The nerves of motion are the third, fourth, sixth, portio dura of the seventh, and the ninth. It is not ascertained whe- ther the glossopharyngeal and spinal accessory nerves belong to this or to the second class. 11. The function of ministering both to motion and sensation is possessed by the fifth pair of cerebral nerves, and by the spinal nerves, which agree precisely in anatomical composition. The par vagum, however, which is one of the irregular nerves, has also a twofold endowment. III. This division comprises the first and second pairs, and the portio mollis of the seventh pair. These nerves are insen- sible to ordinary stimulants, and possess an exclusive sensibility to their respective objects, — viz. odorous matter, light, and aei-ial undulations. IV. The system of the great sympathetic nerve, and its as- sociated plexuses and gangUa. [ 93 ] Report on the present State of our Knowledge respecting the Strength of Materials. By Peter BarloWj Esq., F.R.S., Corr. Memb. Inst. France, 8fc. Sfc. The theory of the strength of materials, considered merely as a branch of mechanical or physical science, must be admitted to hold only a very subordinate rank ; but in a covmtry in which machinery and works of every description are carried to a great extent, it certainly becomes a subject of much practical im- portance ; and it was no doubt viewing it in this light which led the Committee of the British Association, at their last Meeting, to do me the honour to request me to furnish them with a communication on the subject. In drawing my attention to this inquiry, the Committee have subdivided it into the following heads : — 1. Whether, from the experiments of different authors, we have arrived at any general principles ? 2. What those principles are ? 3. How modified in their application to dif- ferent substances ? And what are the differences of opinion which at present prevail on those subjects ? To these questions, without a formal division of the Essay, I shall endeavour to reply in the following pages, by drawing a concise sketch of the experimental and theoretical researches which have been undertaken with reference to these inquiries. The subject of the sti*ength of materials, from its great prac- tical importance, has engaged the attention of several able men, both theoretical and practical, and much useful informa- tion has been thereby obtained. As far as relates to the me- chanical effects of different strains, everything that can be desired has been effected ; but the uncertain nature of mate- rials generally, will not admit of our drawing from experiment such determinate data as could be wished. Two trees of the same wood, grown in the same field, having pieces selected from the same parts, will frequently differ from each other very considerably in strength, when submitted to precisely the same strain. The like may be said of two bars of iron from the same ore, the same furnace, and from the same rollers, and even of different parts of the same bar ; and so likewise of two ropes, two cables, &c. We must not, therefore, in ques- tions of this kind, expect to arrive at data so fixed and deter- minate as in many other practical cases ; but still, within cer- tain limits, much important information has been obtained for 94 THIRD REPORT IS.'JS. the guidance of practical men ; and by tabulating such results in a subsequent part of this article, I shall endeavour to answer the leading questions of the Committee of the British Associa- tion, as far, at least, as relates to experimental results. In re- ference to theory, it must also be admitted that some uncer- tainty still remains ; but this likewise is in a great measure to be referred to the nature of the materials, which is such as to offer resistances by no means consistent with any fixed and determinate laws. Hence some authors have assumed the fibres or crystals composing a body to be perfectly incompressible, and others as perfectly elastic ; whereas it is known that they are strictly neither one nor the other, the law of resistance being differ- ently modified in nearly every different substance ; and as it is requisite theoretically to assume some determinate law of action, it necessarily follows that some doubt must also hang over this branch of the subject. It is, however, fortunate that whatever may be the uncertainty on these points, the relative strength of different beams or bolts of the same material, of similar forms and submitted to similar strains, is not thereby affected ; so that whatever may be the law which the fibres or particles of a body observe in their resistance to compression or extension, still, from the result of a well conducted series of experiments, the absolute resisting force of beams of similar forms, of the same materials, of any dimensions, submitted to similar strains, may, as far as the mean strength can be depended upon, be satisfactorily deduced. An examination of these different views taken of the subject by different writers will, it is hoped, be found to furnish a reply to the other queries of the Committee. The first writer who endeavoured to connect this inquiry with geometry, and thereby to submit it to calculations, was the ve- nerable Galileo, in his Dialogues, published in 1633. He there considers solid bodies as being made up of numerous small fibres placed parallel to each other, and their resistance to se- paration to a force applied parallel to their length, to be pro- portional to their transverse area, — an assumption at once ob- vious and indisputable, abstracting from the defects and irre- gularities of the materials themselves. He next inquired in what manner these fibres would resist a force applied perpen- dicularly to their length : and here he assumed that they were wholly incompressible ; that the fibres under every degree of tension resisted with the same force, and, consequently, that when a beam was fixed solidly in a horizontal position, with one end in a wall or other immoveable mass, the resistance of the integrant fibres was equal to the sum of their direct resistances REPORT ON THE STRENGTH 6f MATERIALS. 95. multiplied by the distance of the centre of gravity of their sec- tion from the lowest point ; abovit which point, according to this hypothesis, the motion must necessarily take place. The fallacy of these assumptions was noticed, but not cor- rected, by several subsequent authors. Leibnitz objected to the doctrine of the fibres resisting eqvially under all degrees of tension, but admitted their incompressibility, thereby still making the motion take place about the lowest point of the sec- tion; but he assumed for the law of resistance to extension that it was always proportional to the quantity of extension. Ac- cordingly as the one or the other of these hypotheses was adopted, the computed transverse resistance of a beam, as de- pending on the absolute strength of its fibres, varied in the I'atio of 3 to 2 ; and many fanciful conclusions have been drawn by different authors relative to the strength of differently formed beams, founded upon the one or the other of these assumptions, which, however, it will be unnecessary to refer to more parti- cularly in this article. We have seen that each of these distinguished philosophers supposed the incompressibility of the fibres ; but James Ber- noulli rejected this part of Leibnitz's hypothesis, and considered the fibres as both compressible and extensible, and that the resistance to each force was proportional to the degree of ex- tension or compression. Consequently, the motion instead of taking place, as hitherto considered, about the lowest point of the section, was now necessarily about a point vi^ithin it ; and his conclusion was, that whatever be the position of the axis of motion, or, as it is now commonly called, the neutral axis, the same force applied to the same arm of a lever will always pro- duce the same effect, whether all the fibres act by extension or by compression, or whether only a part of them be extended, and a part compressed. Dr. Robison, in an elaborate article on this subject, also assumes the compressibility and exten- sibility of the fibres, and as a consequence, assumes the centre of compression as a fulcrum, about which the forces to exten- sion are exerted, and the resistance of both forces to be directly proportional to the degree of compression or extension to which they are exposed ; that is, he assumed each force, although not necessarily offering equal power of resistance, to be indivi- dually subject to the law of action appertaining to perfectly elastic bodies. In carrying on the experiments which laid the foundation of my Essay on the Strength of Timber, Sfc, in 1817, I v/as led by several circumstances I had observed to doubt whether, in the case of timber, this assumption of perfect elas- ticity was admissible. And as some of the specimens used in 96 thiiId report — 1833. my experiments showed very distinctly after the fracture the line about which the fracture took place, I thought of availing myself of this datum, and that which gave the strength of direct cohesion, in order to deduce the law of resistance from actual experiment, instead of using any assumed law whatever. The result of this investigation implied that the resistance was nearly as first assumed by Galileo, and although very dif- ferent from what I had anticipated, yet, as an experimental re- sult, I felt bound to abide by it, attributing the discrepance to the imperfect elastic properties of the material. Mr. Hodgkin- son, however, in a very ingenious paper read at the Manchester Philosophical Society in 1822, has pointed out an error in_ my investigation, by my having assumed the momentum of the forces on each side the neutral axis as equal to each other, instead of the forces themselves ; consequently the above de- duction in favour of the Galilean hypothesis fails. This paper did not come to my knowledge till the third edition of my Essay was nearly printed off, and the correction could not then be made ; but being made, it proves that the law of actual resistance approaches much nearer to that of perfect elasticity than from the nature of the materials there could be any reason to expect ; so that in cases where the position of the neutral axis is known, and also its resistance to direct cohesion, a tolerably close ap- proximation may be made to the transverse strength of a beam of any form, by assuming the resistance to extension to be pro- portional to the quantity of extension, and the centre of com- pression as the fulcrum about which that resistance is exerted. But I have before observed, and beg again to repeat, that by far the most satisfactory data will always be obtained by ex- periments on beams of the like form (however small the scale,) and of the same material as those to be employed, because then the law of resistance forms no part of the inquiry, and does not necessarily enter into the calculation, the ultimate strengths being dependent on the dimensions only, whatever may be the absolute or relative resistance of the fibres to the two forces we have been considering. At present I have only considered the resistance of a beam to a transverse strain ; but there is another mode of application, in which, again, the law of resistance necessarily enters, and has led to many curious and even mysterious conclusions. This is when a force of compression is applied parallel to the length. In the case of short blocks, the resistance of the material to a crushing force is all that is necessary to be known ; and in the Philosophical Transactions for 1818 we have a highly valuable table of experimental results on a great variety of materials, by REPORT ON THE STRENGTH OF MATERIALS. 97 George Rennie, Esq., which contains nearly all the information on this subject that can be desired. But when a beam is of con- siderable length in comparison with its section, it is no longer the crushing force that is to be considered : the beam will bend and be ultimately destroyed by an operation very similar to that which breaks it transversely ; and the investigation of these circumstances has called forth the efforts of Euler, Lagrange, and some other distinguished mathematicians. When a cylindric body considered as an aggregate of pa- rallel fibres is pressed vertically in the direction of its length, it is difficult to fix on data to determine the point of flexure, since no reason can be assigned why it should bend in one way rather than in another ; still, however, we know that practically such bending will take place. And it is made to appear, by the investigations of Euler and Lagrange, that with a certain weight this ought theoretically to be the case, but that with a less weight no such an effect is produced, — an apparent interruption of the law of continuity not easily explained, which exhibits itself, however, analytically, by the expression for the ordinate of greatest inflection being imaginary till the weight or pressure amounts to a certain quantity. Another mysterious result from these investigations is, that while the column has any definite di- mensions, and is loaded with a certain weight, inflection as above stated takes place ; but if the column be supposed infinitely thin, then it will not bend till the weight is infinitely great. These investigations of two such distinguished geometers are highly interesting as analytical processes, but the hypothesis on which they are founded, namely, that of the perfect elas- ticity of the materials, is inconsistent with the nature of bodies employed in practice : they form, therefore, rather an exercise of analytical skill than of useful practical deductions. There is, however, one useful result to be drawn from these processes, which is, that the weight under which a given column begins to bend is directly as its absolute elasticity ; so that, having de- termined experimentally the weight which a column of given elasticity will support safely, or that at which inflection would commence, we may determine the weight which another column of the same dimensions, but of different elasticity, may be charged with without danger. M. Gerard, a member of the Institute of France, aware of the little practical information to be drawn from investigations wholly hypothetical, has given the detail of a great number of actual experimental results connected with this subject on oak and fir beams of considerable dimensions, carried on at the ex- 1833. H 98 THIRD REPORT — 1883. pense of the French Government, from which he has drawn the following empirical formulae, viz.^ 1. In oak bea,„s Ul = ■1784451 (/+ -03). «■ bo V'o 2. In fir beams ?/! = 8161 128 a /r^ o where P = half the weight in kilogrammes, a the less, and h the greater sides of the section, /half the length of the column, and b the versed sine of inflection, the dimensions being all in metres *. How far these formulae are to be trusted in practical con- structions is, however, I consider, rather doubtful, because they are drawn from a number of results which differ very greatly from each other ; and in one case in particular the result, as referred to the deflection of beams, has been satisfactorily shown to be erroneous by Baron Charles Dupin, in vol. x. of the Jour- nal de VE'cole Polytechnique, as also by a carefully conducted series of experiments in my Essay on the Strength of Timber, ^•c. I conceive it, therefore, to be very desirable that a set of experiments on this application of a straining force on vertical columns should be undertaken, and it is, perhaps the only branch of the inquiry connected with the strength of materials in which there is a marked deficiency of practical data ; at the same time it is one in which both timber and iron are being con- stantly employed. We see every day in the metropolis houses of immense height and weight being built, the whole fronts of which, from the first floors, are supported entirely by iron or wooden columns ; and all this is done without any practical rule that can be depended upon for determining whether or not these columns are equal to the duty they have to perform. I say this with a full knowledge that Mr, Tredgold has fur- nished an approximate rule for this purpose ; but the principle on which it is founded has no substantial basis. The extra- ordinary skill which Mr. Tredgold possessed in every branch of this subject, and the great ingenuity he has displayed in in- vestigating and simplifying every calculation connected with architectural and mechanical construction, certainly entitle his opinion to high consideration ; but still on a subject of such high importance, it would be much more satisfactory to be pos- sessed of actual experimental data. The supposition he ad- vanced was made entirely as a matter of necessity, and I am * See Traite Analytique de la Rhistance des Solidei. REPORT ON THE STRENGTH OF MATERIALS. 99 confident that no one would have been more happy than him- self to have been enabled to substitute fact for hypothesis, had he possessed the means of adopting the former. But unfor- tunately such a series of experiments are too expensive and laborious to be undertaken by an individual situated as he was, having a family to maintain by his industry, and whose close and unremitting application to these and similar inquiries, in all probability shortened his valuable life*. At present I have referred principally to experiments made with a view of determining the ultimate strength of materials ; and with data thus obtained practical men have been enabled to pursue their operations with safety, by keeping sufficiently within the limits of the ultimate strain the materials would bear, or rather with which they would just break, some working to a third, others to a fourth, &c., of the ultimate strength, according to the nature of the construction, or the opinion of the con- structor. But it is to be observed, that although we may thus ensure perfect safety as far as relates to absolute strength, there are many cases in which a certain degree of deflection would be very injurious. It is therefore highly necessary to attend also to this subject, particularly as the deflection of beams and their ultimate strength depend upon different principles, or are at least subject to different laws. Hence most writers of late date give two series of values, one exhibiting the absolute or relative strength, and the other the absolute or relative elasticities. These values will of course be found to differ in different au- thors, on account of the uncertainty in the strength of the ma- terials already referred to, but amongst recent experiments the difference is not important : they will also be found differently expressed, in consequence of some authors deducing these numbers from experiments differently made. Some, for ex- ample, have drawn their formulae for absolute strength from experiments made on beams fixed at one end and loaded at the other, using the whole length ; some, again, from experiments on beams supported at each end and loaded in the middle, using the half length. Some take the length in feet, and the section in inches ; others all the dimension in inches ; and a similar variety occurs in estimating the elasticity. Also, in the latter case, some authors employ what is denominated the mo- dulus of elasticity, in which latter case the weight of the beam * Mr. Tredgold's Principles of Carpentry, and his Treatise on the Strength of Iron, ought to be in the possession of every practical builder; besides which two works, he published many separate articles on the same subject in different numbers of the Philosophical Magaziiie. H 2 100 THIRD REPORT — 1833. itself, and consequently its specific gravity, enters. These va- rieties of expressions, however, are not to be understood as arising from any difference of opinion amongst the authors from whom they proceed, but merely as different modes of expressing the same principles : indeed, in reply to that inquiry of the Committee with reference to this point, I may, I think, venture to say there is not at present any difference of opinion on any of the leading principles connected with the strength of mate- rials, with the excejDtion of such as are dependent entirely upon the imperfect nature of the materials themselves, and which, as we have seen, will give rise to different results in the hands of the same experimenter and under circumstances in every re- spect similar. As I distinguish the doctrine of the absolute resistance or strength of materials, which is founded on experiment, from that which relates to the amount and resolution of the forces or strains to which they are exposed, which is geometrical ; and as I confine myself to the former subject only in this Essay, it is not, I conceive, necessary to extend the preceding remarks to any greater length. I shall therefore conclude by giving a table of the absolute and relative values of the ultimate strength and elasticity of various species of timber and other materials, selected from those results in which I conceive the greatest reliance may be placed. Formula; relating to the ultimate Strength of Materials in cases of Transverse Strain. — Let I, b, d, denote the length, breadth and depth in inches in any beam, w the experimental w breaking weight in pounds, then will j— r-^ = S be a constant quantity for the same material, and for the same manner of ap- plying the straining force ; but this constant is different in dif- ferent modes of application. Or, making S constant in all cases for the same material, the above expression must be prefixed by a coefficient, according to the mode of fixing and straining. 1. When the beam is fixed at one end, and loaded at the other, , bcT^-^' 2. When fixed the same, but uniformly loaded, 1 lie _ ^ 3. When supported at both ends, and loaded in the middle, 1 Itv _ ^ 4 ^ ZTrf^^ ~ ^' REPORT ON THE STRENGTH OF MATERIALS. 101 4. Supported the same, and uniformly loaded, 1 8 Iw „ ^ bd^-^' Fixed at b oth ends. and loaded in the middle, 1 6 Iw ^ Fixed the same, bui . uniformly loaded, 1 12 Iw ^, ^ bd^~ 7. Supported at the ends, and loaded at a point not in the middle. Then, n m being the division of the beam at the point of application, 11711 I HI _ „ Some authors state the coefficients for cases 5 and 6 as -rj- 1 . ^ and jp, but both theory and practice have shovpn these numbers to be erroneous. By means of these formulse, and the value of S, given in the following Table, the strength of any given beam, or the beam requisite to bear a given load, may be computed. This column, however, it must be remembered, gives the ultimate strength, and not more than one third of this ought to be depended upon for any permanent construction. Formulce relating to the Deflection of Beams in cases of Transverse Strains. — Retaining the same notation, but repre- senting the constant by E, and the deflection in inches by 8, we shall have. Case 1. 2. 3. 32., Pw _ Case 4. 5 Piv 8 ^ bdn~ ^ 1 ^ bdH-^- 12 Pw 1 ^ bdH- ^• 5. 2 Pw 3 ^ bdH"^ 1 Ptv „ 1 ^ bdH-^- 6. 5 Pio ^ 12 ^ bdH-^ Hence, again, from the column marked E in the following Ttble, the deflection a given load will produce in any cate may be computed; or, the deflection being fixed, the dimension of the beam may be found. Some authors, instead of this measure of 102 THIRD REPORT — 1833. 73 elasticity, deduce it immediately from the formula . .^ ^ = E, substituting for w the height in inches of a column of the ma- terial, having the section of the beam for its base, which is equal to the weight w, and this is then denominated the modulus of elasticity. It is useful in showing the relation between the weight and elasticity of different materials, and is accordingly introduced into the following Table. The above formulae embrace all those cases most commonly employed in practice. There are, of course, other strains con- nected with this inquiry, as in the case of torsion in the axles and shafts of wheels, mills, &c., the tension of bars in suspen- sion bridges, and those arising from internal pressure in cylin- ders, as in guns, water-pipes, hydraulic presses, &c. ; but these fall rather under the head of the resolution of forces than that of direct strength. It may just be observed, that the equation due to the latter strain is t{c — n) ■= n R, where t is the thickness of metal in inches, c the cohesive power in pounds of a square inch rod of the given material, n the pressure on a square inch of the fluid in pounds, and R the in- terior radius of the cylinder in inches. Our column marked C will apply to this case, but here again not more than one third the tabular value can be depended upon in practice. KEPORT ON THE STRENGTH OF MATERIALS. 103 Table of the Mean Strength and Elasticity of various Materials, as deduced from the most accurate Experiments. Names of Materials. Woods. Acacia Ash Beech Birch, Common , American Black Box Bullet Tree Cabacully Deal, Christiana , , Memel .... Elm Fir, New England — .Riga — , Mar Forest ..., Green heart , Larch, Scotch Locust Tree Mahogany Norway Spars Oak, English | j^°'" , African , Adriatic , Canadian ... , Dantzic Pear Tree Poon Pine, Pitch , Red Teak , Tonquin Bean . Iron. from. Iron, Cast < , , Malleable , Wire Speci. fie Gra- vity. 710 760 700 700 750 1000 1030 900 680 590 540 550 750 700 1000 540 950 637 580 700 900 980 990 872 760 646 600 660 660 750 1050 7200 c. Mean strength of cohesion on an inch sec- tion. lbs. 17000 11500 20000 11000 11000 5780 12000 12600 12000 7000 20580 8000 12000 9000 15000 14400 14000 12000 14500 9800 14000 10500 10000 15000 7760 16300\ 36000 ] 60000 80000 Constants for trans- verse strains. 1800 2026 1560 1900 1500 2650 2500 1550 1730 1030 1100 1130 1140 2700 1120 3400 1470 1200 2260 2000 1380 1760 1450 2200 1630 1340 2460 2700 8100 9000 Constants for deflec- tions. 4609000 6580000 5417000 6570000 5700000 10512000 7437000 6350000 6420000 2803000 5967000 5314000 3400000 10620000 4200000 767000 5830000 3490000 Modulus of Elasticity. 3739000 4988000 4457000 5406000 3388000 5878000 4759000 5378000 6268000 3007000 6249000 4080000 2797000 6118000 4480000 4649000 5789000 2872000 700000047020000 9500000 3880000 8590000 4760000 6760000 5000000 7360000 9660000 10620000 55830000 2257000 5674000 3607000 6488000 4364000 6423000 7417000 5826000 69120000 5530000 91440000 6770000 Ilemarks, of English growth. ditto. ditto. ditto. American. Berhice. ditto. English. Scotland. Berbice. America, South. Results very va- riable. East Indies. East Indies. Berbice. [Mean of English f and Foreign. [ 105 ] Report on the State of our Knowledge respecting the Magnetism of the Earth. By S. Hunter Christie, Esq., M.A., F.R.S. M.C.P.S., Corr. Memh. Philom. Sac. Paris, Hon. Memb. Yorkshire Phil. Soc.; of the Royal Military Academy ; and Member of Tritiity College, Cambridge. Had the discovery of the loadstone's du-ective power been made by a philosopher who at the same time pointed out its import- ance to the purposes of navigation, we might expect that his name would have been handed down to posterity as one of the greatest benefactors of mankind. The discovery was, however, most likely made by one so engaged in maritime enterprise that, in his eyes, this application constituted its whole value ; and it is not improbable that, being for some time kept secret, it may have been the principal cause of the success of many enterprises attributed to the superior skill and bravery of the leaders. The knowledge of this property of the magnet, though gradually diffused, would long be guarded with jealousy by those who justly viewed it as of the highest advantage in their predatory or commercial excursions ; and this is, perhaps, the cause of the obscurity in which the subject is veiled. If the discovery is European, there is no people, from the character of their early enterprises, and, I may add, from the nature of the rocks of their country, more likely to have made it than the early Nor- wegians ; and as there is reason for believing that they were acquainted with the directive property of the loadstone at least half a century earlier than its use is supposed to have been known in other parts of Europe, it may be but justice to allow them the honour of having been the discoverers. Whether the discovery was made in Asia or in Europe, in the North or in the South, I am not, however, now called upon to decide, but to point out the consequences which have followed that disco- very by unveiling gradually phgenomena, though less striking, yet equally interesting, and some even more difficult of expla- nation. These phasnomena are, the variation of the magnetic needle, with its annual and diurnal changes ; the dip of the needle ; and the intensity of the magnetic force of the earth ; which are, how- ever, all comprised under two heads, — The Direction and the Ijitensity of the terrestrial magnetic force. 106 THIRD REPORT — 1833. I. The Direction of the Terrestrial Magnetic Force. 1. The Variation of the Needle. — For some centuries after the directive property of the loadstone was discovered, it was generally supposed that the needle pointed correctly towards the pole of the heavens. It has however been said, on the authority of a letter by Peter Adsiger, that the variation of the needle was known as early as 1269; and if we fully admit the authenticity of this letter, we must allow that the writer was at that date not only aware of the fact, but that he had observed the extent of the deviation of the needle from the meridian*. It is possible that such an observation as this may have been made at this early period by an individual de- voting his time to the examination of magnetical phaenomena; * This curious and highly interesting letter, dated the 8th of August 1269, is contained in a volume of manuscripts in the Library of the University of Leyden, and we are indebted to Cavallo for having published extracts from it. The variation is thus referred to : " Take notice that the magnet (stone), as well as the needle that has been touched (rubbed) by it, does not point exactly to the poles ; but that part of it which is reckoned to point to the south declines a little to the west, and that part which looks towards the north inclines as much to the east. The exact quantity of this declination I have found, after numer- ous experiments, to be five degrees. However, this declination is no obstacle to our guidance, because we make the needle itself decline from the true south by nearly one point and an half towards the west. A point, then, contains five degrees." (Letter of Peter Adsiger, Cavallo On Magnetism, London 1 800, p. 317.) It is certainly extraordinary, if so clear an account of the deviation of the needle from the meridian as this, was communicated to any one by the person who had himself observed that deviation, that for more than two centuries afterwards we should have no record of a second observation of the fact. This alone would throw doubt on the authenticity of the letter, and the estimate given of the variation may appear to confirm these doubts ; for, according to the period of change which best agrees with the observations during more than two hundred years, the variation, if observed, would have been found to be westerly instead of easterly in 1269. It may however be urged, that as the whole period of change has not yet elapsed since observations were made, we are not in pos- session of a sufficient number of facts to authorize us to draw conclusions re- specting the variation at such an early date ; and also, that if the letter be spu- rious, or the original date have been altered to that which it bears, this or the fabrication can only have been for the purpose of founding claims in consequence of the contents of this letter ; and as no such claims have been advanced, there appears no motive either for fabrication or alteration. In a preceding part of the letter the author gives methods for finding the poles of a loadstone ; and certainly the direction of the axis could not be determined to within five degrees by either of these ; so that, as regards the loadstone, we may, I think, conclude that the author did not make the observation. As a matter of curious history connected with magnetism, it is desirable that either the authenticity of this letter should be clearly established, or reasons given for doubting it, by those who have an opportunity of cousulting the original. REPORT ON THE MAGNETISM OF THE EARTH. 107 and as it is probable that for some time subsequent to the dis- covery of the directive property of the needle the deviation in Europe was not of sufficient magnitude to have been easily de- tected by means of the rude instruments then in use, it may very likely be owing to this circumstance that we have not earlier records of the variation*. That Columbus, the most scientific navigator of his age, when he commenced his career of discovery, and undertook to show the western route to India, was not aware of it, is clear, since the discovery during his first voyage has been attributed to him. However, although Co- lumbus may have noticed that the needle did not in every situa- tion point due north, and Adsiger, long before him, may even have rudely obtained the amount of its deviation, the first ob- servations of the variation on which any reliance can be placed appear to have been made about the middle of the sixteenth century, and shortly afterwards it was well known that the va- riation is not the same in all places f . 2. Change in the Direction of the Needle. — When it was first determined by observation, about 1541, that the needle did not point to the pole of the earth, it was found that this vari- ation from the meridian, at Paris, was about 7° or 8° towards the east. In 1550 it was observed 8° or 9° east; and in 1580, 11 1° east. Norman appears to have been the first who observed the variation with any degree of accuracy in Lon- don. He states that he observed it to be 11° 15' east J, but he was not aware that it does not remain constant in the same place §. In 1580, Burough found the variation at Limehouse to be 11^° or 11^° east||, and his observations appear to be • Another reason why the variation was not earlier observed may be that the natural magnet was first used for the purposes of navigation, and its directive line was that which pointed to the pole star. As it was therefore considered that the natural magnet indicated the direction of the meridian, and it was found that a needle touched by it had the directive power, when the needle was introduced it was assumed that this also pointed in the meridian. t The Netv Attractive, by Robert Norman, chap. ix. London 1596. X Ibid. No date is given for this observation ; but from the circumstance of Burough referring to Norman's book in the preface to his Discourse of the Va- riation of the Compasse, dated 1581, (the copy of this to which I have access was printed in 1596, but the Bodleian Library contains one printed in 1581,) it would appear that there must have been an earlier edition of Norman's book than that of 1596, and that his observations must have been made before 1581. Bond, Philosophical Transactions, vol. viii. p. 6066, gives 1576 as the date of Norman's observations. § " And although this variation of the needle be found in travaile to be divers and changeable, yet at anie land or fixed place assigned, it remaineth alwaies one, still permanent and abiding." New Attractive, chap. ix. II The mean of his observations, which do not differ 20', is 11° 19' east. 108 THIRD REPORT — 1833. entitled to much confidence ; but he was of the same opinion as Norman with respect to the constancy of the variation*. Gunter, in 1612, found the variation in London to be 5° 36' east; and Gellibrand, in 1633, observed it 4° 4' east. Dr. Wal- lis considers Gellibrand to have been the discoverer of " the variation of the variation f ; " but if Gunter had any confidence in his own observations and those of Burough, he must have been aware of the change in the variation. In 1630, Petit found the variation at Paris to be 4^° east, but suspected, at the time, that the earlier observations there had been incorrect; and it was not until 1660, when he found the variation to be only 10' east, that he was satisfied of the change of the varia- tion. About ten years later, Azaut, at Rome, where the va- riation had been observed 8° east, found it to be more than 2° west; and Hevelius, who at Dantzick in 1642 had found it to be 3° 5' west, now found it to be 7° 20' west. 3. Diurnal Change in the Variation. — This was discovered in 1722 by Graham, to whose talents and mechanical skill science is so deeply indebted. He found that with several needles, on the construction of which much care had been be- stowed, the variation was not always the same ; and at length determined that the variation was different at different hours in the day, the greatest westerly variation occurring between noon and four hours after, and the least about six or seven o'clock in the evening f. Wargentin at Stockholm in 1750, and Canton in London from 1756 to 1759, moi-e particularly observed this phsenomenon; and the latter determined that the time of minimum westerly variation in London was between eight and nine in the morning, and the time of maximum be- tween one and two in the afternoon. Canton likewise deter- mined in 1759, that the daily variation was different at different times in the year, the maximum change occurring about the end of June, and the minimum in December §. Cassini, during more than five years and a half, namely, from May 1783 to January 1789, carefully observed, at particular hours, the direction of a needle suspended in the Observatory at Paris ; and although he does not correctly state the course of the daily variation, overlooking altogether the second maximum west, and the pro- gress of the needle towards the east in the early part of the * " For considering it remayneth alwaies constant without alteration in every severall place, there is hope it may be reduced into method and rule." Dis- course, chap. X. f Philosophical Transactions, 1701, vol. xxii. p. 10-36. + Ibid. 1724, vol. xxxiii. p. 96. § Ibid. 1759, vol. xli. p. JWS. REPORT ON THE MAGNETISM OF THE EARTH. 109 morning *, yet his observations and remarks are of great value as pointing out the annual oscillations of the needle f. Since this, the diui-nal variation has been very generally observed, but by no one with greater care and perseverance than by the late Colonel Beaufoy ;}:. In order to determine whether the course of the diurnal va- riation is influenced by the elevation of the place of observation, the zealous and indefatigable De Saussure undertook a series of observations on the Col du Geant, nearly 11,300 feet above the level of the sea. This series, after incurring much personal inconvenience and even risk in that region of snow and of storms, he completed ; and he has compared the results with observa- tions which he made immediately before and after at Chamouni and Geneva. From this comparison it appears that the course of the diurnal variation was nearly the same on one of the highest mountains, in a deep and narrow valley at its foot, and in the middle of a plain or of a large valley. The times of the maxima, east and west, are in each case nearly those previously determined by Canton, these maxima occurring rather later on the Col du Geant than at the other stations. Excluding in all cases the results where extraordinary causes appear to have operated, the extent of the diurnal variation at Chamouni ex- ceeds that at Geneva and also that on the Col, the two latter being very nearly the same. The observations, however, are, as Saussure very justly remarks, nuich too limited to give cor- rect means §. 5. The Dip of the Magnetic Needle. — Norman having found with different needles, and with one in particular on the con- struction of which he had bestowed much pains, that although perfectly balanced on the centre previously to being touched by the magnet, after this operation the north end always de- clined below the horizon, devised an instrument by which he • Journal de Physique, Mai 1792, torn. xl. p. 345. f Ibid. p. 348. Many of the results of Colonel Beaufoy 's observations are published in the Edinburgh PhilosophicalJoumal, vols. i. ii. iii. iv. and vii. § Saussure, Voyayes dans les Alpes, torn. iv. p. 302 au p. 312. does not give the mean results, I insert them here. As Saussure Time of absolute maximum. Time of second maximum. Geneva Cliamouni ... Col du Geant East. h m 7 56 A.M. 7 34 8 09 West. h m 1 09 P.M. 1 41 2 00 East. 6 26 P.M. 7 44 5 51 West. h m 11 17 P.M 10 46 10 17 Extent of Elevation diurnal above the change. sea. 15 42 17 06 15 43 Feet. 1305 3453 11274 110 THIRD REPORT — 1833. could determine the inclination of the needle to the plane of the horizon*. The figure given of the instrument is sufficiently I'ude, but the principles of its construction, as stated by Nor- man, are correct. With this instrument he found the inclina- tion of the needle to the horizon in London to be about 71° 50', but gives no date to the observation, though Bond assigns 1576 as the time f . Although in a theoretical point of view it would be desirable to have so early a record of the dip, particularly as subsequent observations lead us to suppose that the dip attained its maximum after this time, yet, considering the uncertainty attending such observations, even with the present improved instruments, we cannot place much confidence in this result, however we may rely upon the author having used every pre- caution in his power to ensure accuracy. Having determined the dip of the needle in London, Norman states that this de- clining of the needle will be found to be different at different places on the earth X, though he does not take a correct view of the subject, for he considers that the needle will always be directed towards a fixed point. 5. Variation of the Dip. — Subsequent observations by Bond, Graham, Cavendish, and Gilpin, and the more recent ones in our own time, have shown that the inclination of the needle to the horizon at the same place, like the angle which it makes with the meridian, is subject to change ; but the diurnal oscil- lations of the direction are of too minute a character to have been ascertained with the imperfect instruments which we possess. This is an outline of the phsenomena hitherto observed, de- pending upon the direction of the forces acting upon the needle. Various attempts have been made to account for those obser- vable at fixed points on the earth's surface at different periods, and also to connect those depending on the different positions of the places of observation, but hitherto with only very partial success. It is not my intention to enter into a detailed history of these attempts, but I may briefly notice some of the most remarkable. To Gilbert we are indebted not only for the first clear views of the principles of magnetism, but of their application to the phsenomenon of the directive power of the needle ; and indeed we may say that, with the exception of the recent discoveries, all that has been done since, in magnetism, has for its foundation the principles which he established by experiment §. He con- * New Attractive, chap. iii. iv. t Philosophical Transactions, 1673, vol. viii. p. 6066. X New Attractive, chap. vii. § Gilbert, De Magtiete, ^c, Lond. 1600.. REPORT ON THE MAGNETISM OF THE EARTH. Ill sidered that the earth acts upon a magnetized bai% and upon iron, like a magnet, the directive power of the needle being due to the action of magnetism of a contrary kind to that at the end of the needle directed towards the pole of the earth. He applied the term "pole" to the ends of the needle directed towards the poles of the earth, according to the view he had taken of terrestrial magnetism, designating the end pointing towards the north, as the south pole of the needle, and that point- ing towards the south, as its north pole*. It is to be regretted that some English philosophers, guided by less correct views, have since his time applied these terms in the reverse sense, which occasionally introduces some ambiguity, though now they are used in this country, as on the Continent, in the sense ori- ginally given to them by Gilbert. In 1668 Bond published a Table of computed variations in London, for every year, from that time to the year 1716 f. The variations in this Table agree nearly with those afterwards ob- served for about twenty- five years, beyond which time they differ very widely ; and I only notice this Table as the first em- pirical attempt at the solution of a problem which is, as yet, unsolved. Bond afterwards proposed to account for the change in the variation and dip of the needle by the motion of two magnetic poles about the poles of the earth. He professed not only to give the period of this motion, but to be able to point out its cause, and even proposed to determine the longitude by means of the dip J. He, however, did not make public either his methods or his views ; but with regard to the longitude, it is probable they were the same as those afterwards adopted by Churchman. Halley considered that the direction of the needle at different places on the earth's surface might be explained on the suppo- sition that the earth had four magnetic poles §, and that the change in the direction at the same place was due to the motion of two of these poles about the axis of the earth, the other two being fixed. He does not enter into any calculations to show the accordance of the phaenomena with such an hypothesis, but conjectures that the period of revolution of these poles is about 700 years li. Since this time, calculations have been made by various au- thors, both on the hypothesis of two magnetic poles and on that of four, with the view of comparing the results of these , * Gilbert, De Magnete, Sfc, lib. i. cap. iv. t Philosophical Transaction)!, 1668, vol. iii. p. 789. X Ibid. 1673, vol. viii. p. 6065. § Ibid. 1683, vol. xiii. p. 208. II /6irf. 1692, vol. xvii. p. 563. lis THIRD REPORT — 1833. hypotheses with actual observation. The most recent attempt of this kind is that by Professor Hansteen. He adopts Halley's hypothesis of four magnetical poles, but considers that they all revolve, and in different periods, the northern poles from west to east, and the southern ones from east to west. The results calculated on this hypothesis agree pretty nearly with the ob- servations with which they are compared ; but as considerable uncertainty attends magnetical observations, excepting those of the variation made at fixed observatories, and especially the early ones of the dip and variation, on which the periods of the poles and their intensities must so much depend, it would cer- tainly be premature to say that such an hypothesis satisfactorily explains the phaenomena of terrestrial magnetism. If we admit that the progressive changes which take place in the direction of the needle are due to the rotation of these poles, we must look to the oscillations of the same poles for the cause of the diurnal oscillation of the needle. Any hypothesis which by means of two or more magnetic poles will thus connect the phaenomena of magnetism, is of great advantage, however un- able we may be to give a reason for the particular positions of the poles, or for their revolution. Hansteen refers these to the agency of the sun and moon. Without assigning any cause either for the direction of the needle, or for the progressive change of that direction, attempts have been made to account for its diurnal oscillations. But before taking a review of these, it is necessary that I should state more particularly the precise nature of the phsenomenon. This I cannot do better than by referring to the results de- duced from Canton's observations*. From these it appears that in London, during the twenty-four hours, a double oscilla- tion of the needle takes place, the absolute maximum west happening about half-past one in the afternoon, and the abso- lute maximum east, that is, the minimum west, about nine in the morning; besides which there was another maximum east about nine in the evening, and a maximum west near midnight or very early in the morning, the two latter maxima being small compared with the absolute maxima. Colonel Beaufoy's very extensive series of observations, made when the variation was between 24° and 25° west, (Canton's having been made when it was 19°,) give nearly the same results, the absolute maxima happening somewhat earlier, and the second maxima west about eleven in the evening. Canton explained the westerly motion of the needle in the » Philosophical Transactions, 1759, p. 398, and 1827, pp. 333, 334. REPORT ON THE MAGNETISM OF THE EARTH. 1 Ifi latter part of the morning, and the subsequent easterly motion, by supposing that the heat of the sun acted upon the northern parts of the earth as upon a magnet, by weakening their in- fluence, but offered no explanation of the morning easterly mo- tion of the needle. Oersted's discovery of the influence of the closed voltaic circuit upon the magnetic needle, and the consequent discoveries of Davy, Ampere and Arago, immediately led to the considera- tion, whether all the phaenomena of terrestrial magnetism were not due to electric currents ; and the discovery of Seebeck, that electric currents are excited when metals having different powers of conducting heat are in contact, — which discovery with but few holds the rank to which it is eminently entitled, — pointed to a probable source for the existence of such currents. At the conclusion of a highly interesting paper on the develop- ment of electro-magnetism by heat, Professor Gumming re- marks that "magnetism, and that to a considerable extent, it appears, is excited by the unequal distribution of heat amongst metallic, and possibly amongst other bodies. Is it improbable that the diurnal variation of the needle, which follows the course of the sun, and therefore seems to depend upon heat, may result from the metals, and other substances which com- pose the surface of the earth, being unequally heated, and con- sequently suffering a change in their magnetic influence ? " And in the second part of a paper, detailing some thermo-magnetical experiments, read before the Royal Society of Edinbui-gh, Dr. Traill considers "that the disturbance of the equihbrium of the temperature of our planet, by the continual action of the sun's rays on its intertropical regions, and of the polar ices, must convert the earth into a vast thermo-magnetic apparatus : " and "that the disturbance of the equihbrium of temperature, even in stony strata, may elicit some degree of magnetism*." Previous to this, I had adopted the opinion that temperature, if not the only cause, is the principal one of the daily variation f. It did not, however, appear to me, that any of the experiments hitherto made bore directly on the subject, since the metals producing electric currents by their unequal conduction of heat were only in contact at particular parts, and in no case had such currents been excited by different metals having their surfaces symmetrically united throughout. I in consequence instituted a series of experiments with two metals so united, and found that electric currents were still excited on the • Transactions nf the Philosophieal Society of Cambridge, vol. ii. p. 64. + Pliihtxophtenl Transartions, 1823, p. 392. 18,'Jrj, 1 114 ' THIRD REPORT — 1833. application of heat, the phaenomena corresponding to magnetic polarization in a particular direction with reference to the place of greatest heat*. From these experiments I drew the con- clusion that one part of the earth, with the atmosphere, being more heated than another, two magnetic poles, or rather elec- tric currents producing effects referrible to such poles, would be formed on each side of the equator, poles of difterent names being opposed to each other on the contrary sides of the equa- tor; and that different points in the earth's equator becoming successively those of greatest heat, these poles would be carried round the axis of the earth, and would necessarily cause a de- viation in the horizontal needle f. On comparing experimentally the effects that would result from the revolution of such poles with the diurnal deviations at London, as observed by Canton and Beaufoy, also with those observed by Lieut. Hood at Fort Enterprise, and finally with the late Captain Foster's at Port Bowen, I found a close agreement in all cases in the general character of the pliEenomena, and that the times of the maxima east and west did not differ greatly in the several cases. The double oscillation of the needle, to which I have referred in Canton's and Beaufoy's observations, clearly resulted from this view of the subject. Some of the experiments to which I have referred showed that when heat was applied to a globe, the electric currents excited were such, that on contrary sides of the equator the deviations of the end of the needle of the same name as the latitude were at the same time always in the same direction, either both towards east or both towards west. No observations having at that time been made on the diurnal variation of the needle in a high southern latitude, I considered *' that the agreement of the theoretical results with such ob- servations would be almost decisive of the correctness of the theory." Captain Foster's observations at Cape Horn, South Shetland, and the Cape of Good Hope, show most decidedly that in the southern hemisphere the diurnal deviations of the south end of the needle correspond very precisely with those of the north end in the northern hemisphere ; and most fully bear me out in the view which I had taken. These valuable obser- vations have been placed in my hands by His Royal Highness the President, and the Council of the Royal Society, and I in- tend, when I have sufficient leisure, rigidly to compare them, and likewise those to which I have already referred in the northern hemisphere, with the diurnal deviations that would • "Theory of the Diurnal Variation of the Magnetic Needle," Philosoj)Mcal Tramactions, 1827, pp. 321, 326. I- Ibid. pp. 327, 328. REPORT ON THE MAGNETISM OF THE EARTH. 115 result at the corresponding places on the earth's surface, on the supposition that such electric currents as I have supposed are excited on contrary sides of the equator, in consequence of different parts on the earth's surface becoming successively the places of greatest heat, during its revolution upon its axis. Should there be found in these results that accordance w^hich I have reason to expect, there will, I think, be no doubt that the diurnal deviation of the needle is due to electric currents excited by the heat of the sun. I have already adverted to the hypotheses of two or more poles, by means of which attempts have been made to explain the phaenomena of terrestrial magnetism, and I may now re- mark, that if we admit the existence of such poles, we must be careful not to consider the magnetic meridians as great circles : they are unquestionably curves of double curvature. Nor must we consider these poles to be, like the poles of a magnet, cen- tres of force not far removed from the surface. If such centres of force exist for the whole surface of the earth, the experi- mental determinations of the magnetic force at different places, to which I shall shortly advert, at least show that they cannot be far removed from the centre of figure. In the delineation of magnetic charts, more attention has hitherto been paid to the Halleyan lines, or lines of equal varia- tion, than to any others ; and I am not disposed to undervalue charts where such lines alone are exhibited : to the navigator they are of the greatest value ; but they throw little light on the phae- nomena in general. If the meridians wei'e correctly represented, they would at least indicate clearly their points of convergence, if such in all cases exist ; but the lines that would be most likely to guide us to a true theory of terrestrial magnetism, are the nor- mals to the direction of the needle. If, as is highly probable, the direction of the needle is due to electric currents circulating either in the interior or near the surface of the earth, these normals would represent the intersection of the planes of the currents with the surface of the earth ; and, by their delineation, we should have exhibited in one view the course of the currents and the physical features by which that course may be modified, so that any striking correspondences which may exist, would be immediately seized, and lead to important conclusions. Changes of temperature I consider to be the principal cause of the diurnal changes in the direction of the needle : and if any connexion exist between these electric currents and climate, we are to expect that the curvature of these normal lines will be influenced by the forms, the extent and direction of the con- tinents or seas over which they pass, and also by the height, i2 11(> THIRD REPORT — 1833. direction and extent of chains of movmtains, and probably by their geological structure. These normal lines may, to a certain extent, agree with the lines of equal dip, which have already been delineated upon some charts. In Churchman's charts they are represented in the positions they would have on Euler's hypothesis of the earth having two magnetic poles. The only use, however, of such hypothetical representations is, that by comparison with actual observation they become tests of the correctness of the theory, or they may point out the modifications which it requires, in order that it may accord with observation. In Professor Hans- teen's chart the hnes of equal dip are projected from observa- tions reduced to the year 1780. Considering how very deficient we are, even now, in correct observations of the dip, I should not be disposed to place much reliance upon the accuracy of these lines, particularly where they cross great extents of sea aflbrding no points of land necessary for observations of the dip. Of these lines of equal dip the most important is the magnetic equator, or that line on the earth at which the dipping needle would be horizontal. The observations eivina^ this result can of course be but few, and are therefore very inadequate for the correct representation of this line. In order to obviate this difficulty, M. Morlet made use of all observations not very re- mote from the equator, determining the distance of that line from the place of observation by means of the law, that the tangent of the magnetic latitude is half the tangent of the dip, which is derived from the hypothesis of two magnetic poles near to the centre of the earth. By this means the position of the equator was determined throughout its whole extent; and a surprising agreement was fovind between the determinations of each point by means of different observations, which shows that, within certain limits near the equator, the hypothesis very correctly represents the observations. This line exhibits in- flections in its course which have been attributed, and probably with justice, to the physical constitution of the surface in their vicinity*. It has been considered also that a general resem- blance exists between the isothermal lines and the lines of equal dip on the surface of the earth f. All the lines to which I have here referred have been liitherto represented on a plane, either on the stereographical, the glo- bular, or Mercator's projection. Mr. Barlow has, however, very lately represented the lines of equal variation on a globe, from a great mass of the most recent documents connected with » Biot, Traite de Physique. f Hansteen, Edinhtrgh fhilosophical Journal, vol, iii. p. 127. REPORT ON THE MAGNETISM OF THE EARTH. 117 the variation, furnished to him by the Admiralty, the East India Company, and from other sources. If to the hues of equal .va- riation were added the magnetic meridians and their normals, the isodynamic lines, with those of equal dip, such a globe would form the most complete representation of facts connected with terrestrial magnetism that has ever been exhibited, and might indicate relations which have hitherto been overlooked. Having discovered that a peculiar polarity is imparted to iron by the simple act of rotation, I was led to consider whether the principal phsenomenon of terrestrial magnetism is not, in a great measvu-e, due to its rotation. The subsequent discovery by Arago, that analogous effects take place during the rotation of all metals, and Faraday's more recent discovery, that electrical currents are not only excited during the motion of metals, but that such currents are transmitted by them, render such an opinion not improbable. It is, however, to be remarked, that, in all these cases, motion alone is not the cause of the effects produced ; but that these effects are due to electricity induced in the body by its motion in the neighbourhood of a magnetized body. If, then, electrical currents aie excited in the earth in consequence of its rotation, we must look to some body exterior to the earth for the inducing cause. The magnetic influence attributed by Morichini and Mrs. Somerville to the violet ray, and the effect M'hich I found to be produced on a magnetized needle when vibrated in sunshine, and which appeared not to admit of explanation without attributing such influence to the sun's rays, might appear to point to the sun as the inducing body. The experiments, however, of Morichini and Mrs. So- merville, have not succeeded on repetition ; and in a recent re- petition of my own experiments, in a vacuum, by Mr. Snow Harris, the effects which I observed were not detected. I had found that the effects produced on an unmagnetized steel needle differed from those produced on a similar needle when magnet- ized, and therefore considered that the idea of these effects being independent of magnetism was precluded ; but Mr. Har- ris's results may possibly be considered to indicate that they were due solely to currents of air excited by the sun's rays. These circumstances render it doubtful whether the sun's rays possess any magnetic influence independent of their heating power ; but besides this, supposing such influence to exist, if electric currents were induced in the earth during its rotation, they would be nearly at right angles to the equator, and would therefore cause a magnetized needle to place itself nearly per- pendicular to the meridians, or parallel to the equator. Altiiough it Avould therefore appear that the rotation of the 118 THIRD REPORT — lSo3. earth is not the cause of its magnetism, yet it is highly pro- bable, from Mr. Faraday's experiments *, that, magnetism ex- isting in the earth independently of it, electrical currents may be produced, not only by the earth's rotation, but by the motion of the waters on its surface, and even by that of the atmosphere ; so that the direction and intensity of the magnetic forces would be modified by the influence of these currents. This subject is at present involved in obscurity : still, consi- dering how many have been the discoveries made within a few years, — all bearing more or less directly upon it, though none afford a complete explanation of the phaenomena, — it does not appear unreasonable to expect that we are not far removed from a point where a few steps shall place us beyond the mist in which we are now enveloped. Mr. Fox, having observed effects attributable to the electri- city of metalliferous veins, appears disposed to refer some of the phaenomena of terrestrial magnetism to electrical cur- rents existing in these veins -f ; but although we should not be jvarranted in denying the existence of these currents, indepen- dently of the wires made use of in Mr. Fox's experiments, or even their influence on the needle, yet I think we should be cautious in drawing conclusions from these experiments J. II. Intensity of the Terrestrial Magnetic Force. I have as yet said little on the intensity of the terrestrial mag- netic forces. Graham, after having discovered the daily varia- tion of the needle, suspected that the force which urges it varies not only in direction, but also in intensity. He made a great variety of observations with a dipping needle, but drew no ge- neral conclusion from his results. Indeed, with the instruments then in use, he was not likely to determine that which has al- most escaped detection with instruments of more accurate con- struction, for the diurnal variation of the whole magnetic force may perhaps still be considered doubtful. Later observations, particvilarly those of Professor Hansteen, have shown that the time of vibration of a horizontal needle varies during the day, from which it was inferred that the horizontal force also varies. Professor Hansteen, by this means, found that the horizontal intensity of terrestrial magnetism has a diurnal variation, de- • Philosophical Transactions, 1832, p. 176. f ^l>'^- 1830, .p 407. X Mr. Henwood informs me that he has repeated the experiments of Mr. Fox ill from forty to fifty places not before experimented on, and that he pro- poses greatly extending them. i\s far as he can yet see, he considers that his results go to confirm Mr. Fox's deductions, — I suppose with regard to the elec- tricity of metalliferous veins. IIEI'ORT ON THE MAGNETISM OF THE EARTH. 119 creasing, at Christiana, until ten or eleven o'clock in the morn- ing, when it attains its minimum, and then increases until four or five o'clock after noon, when it appeared to reach its maxi- mum*. By observing, at different times of the day, the direc- tion of a horizontal needle thrown nearly at right angles to the meridian, by the action of two powerful magnets, placed in the meridian, passing through its centre, after correcting the ob- servations for the effect of changes of temperature on the in- tensity of the force of the magnets, I found that at Woolwich the terrestrial horizontal intensity decreased until lO** 30" a.m., when it reached its minimum, and increasing from that time, attained its maximum about 1^ 30™ p.m. f . This agreement, in results obtained by totally independent methods, removes all doubt respecting the diurnal variation of the horizontal force. The difference in the time of the maximum in the two cases may be accounted for, independently of the difference in the variation at the two places of observation, by the circumstance that no correction for the effect of temperature on the time of vibration is made in Professor Hansteen's observation. As no such correction had hitherto been made, it must have been con- sidered that differences in the temperature at which observations were made had little influence on the intensity of the vibrating needle ; but in the communication containing these observations, I pointed out the necessity of such a correction J; and since then, in deducing the terrestrial intensity from the times of vi- • Edinburgh Philosophical Journal, vol. iv. p. 297. + Philosophical Transactions, 1825, pp. 50 & 57. An inconvenience attending the method which I employed is, that the observations require a correction for temperature which is not very readily applied, as will T)e seen by referring to my paper ; but this might in a great measure be obviated, by rendering the tempera- ture of the magnets employed always the same previous to observation. If, how- ever, in order to retain the needle in its position nearly at right angles to the me- ridian, torsion were applied instead of the repulsive forces of magnets, the correc- tion for temperature would be nearly reduced to that due to the eifects produced on the intensity of the needle itself by changes of temperature. But even this method is not without objection ; for the sensibility of the needle depending upon the number of circles of torsion requisite to bring it into the pi-oper posi- tion, if a wire were employed, unless very long, its elasticity would be impaired by more than two or three turns ; and it is doubtful whether a filament of glass of moderate length would bear more than this without fractiu-e. I had pro- posed to the late Captain Foster, previous to his last voyage, that he should de- termine the horizontal intensity at different stations, and also its diurnal changes by this method, and had a balance of torsion constructed for him for the purpose ; but as the instrument is extremely troublesome in its adjustments, I consider that the many other observations which he had to make did not allow him time for the extensive use of this instrument which he had proposed. It is, however, very desirable that it should be ascertained how far this method is applicable. X Philosophical l^ransaclions, 1825. l20 THIRD REPORT— 18S3. bration of a needle, it has been customary to apply a correction for differences in the temperatures at which the observations may have been made. The horizontal intensity varying during the day, it becomes a question whether this arises from a change alone in the direc- tion of the force, or whether this change of direction is not accompanied by a change in the intensity of the whole force. In a communication to the Philosophical Society of Cambridge *, I suggested that deviations, fi'om whatever cause, in the direc- tion of the horizontal needle, were referrible to the deviations which, under the same circumstances, would take place in the direction of the dipping needle. Adopting these views. Captain Foster infers, from observations made by him at Port Bowen, on the corresponding times of vibration of a dipping needle, supported on its axis and suspended horizontally, that the diur- nal change in the horizontal intensity is due principally, if not wholly, to a small change in the amount of the dip. The observa- tions, however, do not indicate that the force in the direction of the dip is constant. Captain Foster's obsei'vations at Spitzber- genf show, more decidedly, the diurnal variation of this force : there, its maximum intensity appears to have occun*ed at about 3^ 30"" A.M., and the minimum at 2^ 47™ p.m. ; its greatest change amountino; to -J^ of its mean value. The maximum horizontal intensity appears to have occurred a little after noon, and the minimum nearly an hour after midnight ; but there is consider-r able irregularity in the changes which it undergoes. It would, however, appear, from these observations, that the variations hi the absolute intensity were in opposition to those in the hori- zontal resolved part of it ; so that the principal cause of the latter variations must have been a change in the dip itself. Captain Foster considers " that the times of the day when these changes are the greatest and least, point clearly to the sun as the primary agent in the production of them ; and that this agency is such as to produce a constant inflection of the pole towards the sun during the twenty-four hours." This is in per- fect accordance with the conclusions I had previously drawn from the experiments on A'-hich I founded the theory of the di- urnal variation of the needle :j:, as I had shown that if the diur- nal variation of the needle arise from the cause which I have assigned for it, the dip ought to be a maximum, in northern la- titudes, nearly when the sun is on the south magnetic meridian, and a minimum when it has passed it about 130°. * Transactions of the Philosophical Society of Cambridffe, 1820. t Phitosophical Transactions, 1828. ' J Ibid. 1827, pp. 345, 349. REPOKT ON THE MAGNETISM OF THE EARTH. 121 Humboldt was the first who determined that the intensity of the whole magnetic force is different at different positions on the earth's surface. Having made observations on the times of vibration of the same dipping needle, at various stations in the vicinity of the equator, and approaching to the northern pole, he found that the intensity of the terrestrial force decreases in approaching the equator ; but no precise law, according to which the intensity depends upon the distance from the equator, can be determined from these observations. Numberless observa- tions have since been made in both hemispheres, with every precaution to ensure accuracy in the results, but they do not in general accord with the theoretical formulse with which they have been compared. On the hypothesis of two magnetic poles not far removed from the centre of the earth, if 8 represent the dip, A the mag- netic latitude of the place of observation, 1 the intensity of the force in the direction of the dip, and m a constant, then iind therefore. -v/ (4 - 3 sin2 8)' tan S = 2 tan K ; I = |V(3sin^A+l); or if i is the angular distance from the magnetic pole, or the complement of the latitude, I=|.^(3cos2i+l). By comparing his own observations with the first of these formulae, Captain Sabine came to the conclusion that they were " decisive against the supposed relation of the force to the ob- served dip, and equally so against any other relation whatso- ever, in which the respective phaenomena might be supposed to vary in correspondence with each other." Comparing them, however, with the last formula, he concludes that "the accord- ance of the experimental results with the general law proposed for their representation, cannot be contemplated as otherwise than most striking and remarkable." How the same set of observations should be in remarkable accordance with the one formvda and at variance with the other, when these formulae are dependent on each other, it is difficult to conceive ; but the conclusion drawn by Captain Sabine from his observations, at least shows the danger of relying upon any single set of obser- vations as confirmatory or subversive of theoretical views. I 122 THIRD RKPOKT — lS3o. have not yet compared with these results of theory the numer- ous observations made by Captain Foster, both in the northern and in the southern hemispheres ; but it is my intention to do this as soon as I can determine what correction ought to be made for the differences of temperature at the several stations: I do not, however, anticipate any very close accordance. In Captain Sabine's observations, the observed intensities, compared with those deduced from the preceding formulas, are in excess near the equator, and in defect near the pole ; and it is not improbable that, as Mr. Barlow has suggested, this in- crease of magnetic action near the equator above that which the theory gives, is due to the higher temperature in the equa- torial regions *. I am, however, disposed to assign even a more powerful influence than this to difference of temperature ; for I think it very possible, and indeed not improbable, that this may be the primary cavise of the polarity of the earth, although its influence may be much modified by other circumstances. At the conclusion of the paper on the diurnal variation f, to which I have already referred, I have suggested an experiment which I think might throw much light on this subject. I have pro- posed that a large copper sphere, of uniform thickness, should be filled with bismuth, the two metals being in perfect contact throvighout, and that experiments should be made with it simi- lar to those which I had made with one of smaller dimensions, but from which I was unable to obtain any very definite results, in consequence of the want of uniformity in the thickness of the copper and in the contact of the two metals. On heating the equator of such a sphere, the parts round the poles being cooled by caps of ice — which might not unaptly represent the polar ices, — we may expect that currents of electricity would be excited ; in which case the direction of those currents would decide whe- ther the experiment wei'e illustrative of the principal phaenome- non of terrestrial magnetism, or not. Should these currents of electricity be in the direction of the meridians, — which is impro- bable, since in this case opposing currents would meet at the poles, and there would be no means of discharge for them, — I think we might then conclude that the magnetism of the earth cannot be due to the difference in the temperature of its polar and equatorial regions ; but if, on the contrary, the currents should be in a direction parallel to the equator,^ — in which case their action upon a magnetized needle would be to urge it in the direction of the meridians, — I should then say that, in order to account for the terrestrial magnetic forces, and the diurnal * Edinburgh New Philosophical Journal, July 1 827. f Philosophical Transaclions, 1827, p. 354. REPORT On'tHE magnetism OF THE EARTH. 123 changes in their direction and intensity, it would only be re- quired to show, that electrical phaenomena may be excited, in such bodies as the earth and the atmosphere, by a disturbance in their temperature when in contact. As I consider that if such an experiment were carefully made it must give conclusive results, I would strongly suggest to the Council of the British Association the importance of having it made. It has been a question whether the intensity of terrestrial magnetism is the same at the surface of the sea and at heights above that surface to which we can attain. MM. Gay-Lussac and Biot, in their aerostatic ascent, could detect no difference at the height of 4000 metres *. Saussure had, however, con- cluded, from the observations which he made at Geneva, Cha- mouni, and on the Col du Geant, that the intensity was consi- derably less at the latter station than at either of the former, the difference in the levels being in the one case about 10,000 feet, in the other about 7800 f. M. Kupffer X also considers that his observations in the vi- cinity of Elbours, in which the difference of elevation of his two stations was 4500 feet, show clearly that the horizontal intensity decreases as we ascend above the surface ; and he accounts for this decrease not having been observed by MM. Biot and Gay- * Biot, Traite de Physique. f Voyages dans les Alpes, torn. iv. p. 313. — I take for granted that, admit- ting the accuracy of Saussure's observations, they warranted the conclusions he drew from them ; but some unaccountable errors must have crept in, either in transcribing or in printing them ; for not only the means which he deduces do not result from the observations, but the numbers which he employs contradict his conclusions. 1 transcribe the passage from the only edition I can consult, pub- lished at Neufchatel, 1796. " A' Geneve ces vingt oscillations employ erent 5m 2'; 4™ 50'; 5"*; 4™ 40'; dont la moyenne etoit 5™ 0*"4; le thermometre 6tant a 6 degres. A* Chamouni S"" 33' ; 5" 34' ; moyenne 5" 33''5 ; thermometre 12 deg. Au Col du Geant 5™ 30"-3 ; S-" 30'-5 ; 5" 3r-4; 5^ 34'-6, moyenne 5m 32"-45; thermometre 12-4 degres." " Or les forces magnetiques sont inversement comme les quarres des tems. Mais, a Geneve, le tems etoit ■')'" 0'-4 ou 300'.4, dont le quarre = 111155-56 ; a Chamouni 5" 33'-5 = 333'-5, dont le quarre = 111223. Au Geant 5" 32'-45 = 332'-45, dont le quarre = 11523-0025 ; d'ou il suivroit que la plus grande force etoit dans laplaine, et la plus petite sur la plus haute montagne, a pen pres d'une cinquieme : observation bien importante, si elle etoit confirmee par des experiences repetees, et faites a la meme temperature." The means of the above observations are 4™ 53* = 293', 5'" 33'-5 = 333'-5, andS"! 31'-7 = 331*-7; and the squares of these numbers are 85849, 111222-25, 110024-89. So that, according to this, the force was greatest at Geneva, and least at Chamouni. Taking Saussure's numbers, 300' -4, 333''-5, 332'-45, their squares are 90240-16, 111222-25, 110523-0025; so that still the general con- clusions are the same. t Voyage dans les Environs du Mont Elbronz. Rapport fait a I'Acadimie Jmpiriale des Sciences de St. Petersbourg, p. 88. 124 THIRD REPORT— 1838. Lussac, by its having been counteracted by the increase of in- tensity, arising from the diminution of temperature. Mr. Hen- wood informs me that he has made corresponding observations, consisting of two series, each of 3900 vibrations at each place ; on Cairn Brea Hill, 710 feet above the level of the sea ; at the surface of Dolcoath mine, 370 feet above the sea ; and at a depth of 1320 feet beneath the surface in Dolcoath mine, or 950 feet below the level of the sea ; and that, after clearing the results from the effects of temperature, the differences are so minute that he cannot yet venture to say he has detected any difference in the magnetic intensity at these stations. If, notwithstanding these results, we are to admit the correctness of M. Kupffer's conclusions, I think we must infer that the diminution of hori- zontal intensity at his higher station was due to an increase in the dip, which element would not probably be so much affected by a change of elevation in a comparatively level country, like Cornwall, as on the flank of such a mountain mass as Elbours. Before dismissing the subject of the terrestrial intensity, I should mention that attempts have been made to delineate on charts the course of isodynamic lines. Professor Hansteen has published a chart in which this is done for the year 1824. Of all observations, however, requisite for graphic exhibitions con- nected with tei'restrial magnetism, those on the authority of which such lines must be drawn are fewest in number and least satisfactory in their results ; we should, therefore, be very cau- tious in drawing conclusions from such delineations. Hitherto I have only referred to such changes in the direction of the magnetic force, and in its intensity, as appear to depend upon general causes ; but, besides these, sudden and sometimes considerable irregular changes occur. These have very gene- rally been attributed to the influence of the aurora borealis, whether visible or not at the place of observation ; and I think it not improbable that some may be due to a peculiar electrical state of the atmosphere, independent of that meteor. The in- fluence of the aurora borealis on the magnetic needle has, how- ever, been denied by some, principally because, during the occurrence of that meteor at Port Bowen, Captain Foster did not observe peculiar changes in the direction of the needle, al- though, from his proximity to the magnetic pole, the diurnal change sometimes amounted to 4° or 5° ; and, under such cir- cumstances, it was considered that these changes ought to have been particularly conspicuous. In a paper inserted in the se- cond volume of the Journal of the Royal Institution, I have, however, shown that Captain Foster's Port Bowen observations <lo not warrant the conclusions which have been drawn from REPORT ON THE MAGNETISM OF THE EARTH. 125 them, and have pointed out cu'cumstances which may, in this case, have rendered the effect of the aurora upon the horizon- tal needle less sensible than might have been expected. That changes in the direction and intensity of the terrestrial forces are simultaneous with the aurora borealis I feel no doubt, for I have seen the changes in the direction of the needle to accord so perfectly with the occurrence of this meteor, and to such an extent, that in my mind the connexion of the phgenomena be- came unquestionable*. As, however, the magnetic influence of the aurora boreahs has been doubted, I shall here point out the manner in which I consider the effects may be best ob- served. If the magnetic forces brought into action during an aurora are in the direction of the magnetic meridian, they will affect a dipping needle adjusted to the plane of that meridian, but the direction of an horizontal needle will remain unchanged : on the other hand, if the resultant of these forces makes an angle with the meridian, the direction of the horizontal needle will be changed, but the dipping needle may not be affected. In order to determine correctly the magnetic influence of the aurora by means of an horizontal needle, it is therefore necessary not only to have regard to those forces which influence its direction, but likewise to those which affect the horizontal intensity. The effects of the former are the objects of direct observation, but those of the latter are not so immediately observable. As, du- ring an aurora, the intensity may vai'y at every instant, — and it is these changes which are to be detected, — the method of deter- mining the intensity by the time of vibration of the needle can- not hei'e be applied, and other means must be adopted. The best method -appears to me to be that which I employed for determining the diurnal variation of the horizontal intensity, * the needle being retained nearly at right angles to the meridian by the repulsive force of a magnet, or by the torsion of a fine wire or thread of glass. For the purpose, then, of detecting in all cases the magnetic influence of the aurora, I consider that two horizontal needles should be employed ; one, adjusted in the meridian, for determining the changes which may take place in the direction of the horizontal force, and the other at right angles to the meridian, to determine the changes in the inten- sity of that force, arising principally from new forces in the plane of the meridian, and which would affect the direction of the dipping needle alone. Both these needles should be deli- * For the ohsei vations to which I here particularly refer, see the Journal of the Royal Inslitufiu/i, vol. ii. p. 272. 126 THIRD REPORT 1833. cately suspended, either by very fine wire, or by untwisted fibres of silk. In order to render the changes in the direction of the needle in the meridian more sensible, its directive force should be diminished by means of two magnets north and south of it, and having their axes in the meridian. These magnets should be made to approach the needle until it points about 30° on either side of the meridian, and they should be so ad- justed that the forces acting upon the needle will retain it i?i equiUbrio with its marked end at about 30° to the east and 30° to the west of north, and also at south. The needle is to be left with its marked end pointing south, for the purpose of ob- serving the changes occurring in its direction. If magnets are employed to retain the second needle nearly at right angles to the meridian, they should be made to approach its centre until the points of equilibrium are at about 80° east, 80° west and south, the observations being made with the needle at 80° east or 80° west. An objection to this method of adjusting this needle by means of magnets, and to which I have already re- ferred in a note, is that any change in their temperature will have a very sensible effect on the direction of the needle in this position ; and should such change take place dui'ing the ob- servations, corrections must be applied to the results before any accurate conclusions can be drawn from them. As, how- ever, an aurora is not generally of long continuance, any change in the temperature of the magnets during the observations is much more easily guarded against than where the observations have to be continued during successive days and at different seasons of the year. I have before remarked that this incon- venience will be, in a great measure, obviated by employing the torsion of a fine wire, or a very fine filament of glass, to retain the needle at about 80° from the meridian. In this case, the ratio of the force of torsion to the terrestrial force acting upon the needle having been determined, a measure will be obtained of the changes which take place in the intensity of the terres- trial force during the occurrence of an aurora. It is very de- sirable that it should be ascertained whether the effects on the needle are simultaneous with any particular class of phaenomena connected with the aurora ; whether these effects are dependent on the production of beams and corruscations, or on the forma- tion of luminous arches ; or whether any difference exists in the effects produced by these. In order to determine this, it is ne- cessary that the times of the occurrence of the different phae- nomena, and also of the changes in the directions of the needles, should be accurately noted ; and for such observations, three observers appear to be indispensable. REPORT ON THE MAGNETISM OF THE EARTH. 127 Whether the direction of the needle may be influenced by the electrical state of the clouds, is much more doubtful than the influence of the aurora. I am not aware of any extended series of observations made with a view to determine this point. Having adjusted, in a particular manner, a needle between two magnets, so that the directive force was considerably diminished, I found that the changes in the positions of electric clouds was accompanied by changes in the direction of the needle ; but, although the observations indicate that the needle was thus affected, they are of too limited a nature to draw any general conclusion from*. Some observations of Captain Sir Kverard Home, however, indicate the same kind of influence. In a con- versation which I had with him last year, having referred to the effect I had observed to be produced by the sun's rays, of bring- ing a vibrating needle to rest, it brought to his mind a similar effect which he observed during a thunder-storm. He has fa- voured me with his observations, and from these it appears that, in two instances, a needle came sooner to rest during a thunder-storm than it had previous or subsequent to it. I'he arc at which the vibrations ceased to be counted is not recorded, but the number of vibrations was reduced in one case from 100 to 40, and in another from 200 to 120. I have, in consequence of these observations, requested Lieutenant Barnett of the Royal Navy, who is engaged in the survey of the southern coast of the Gulf of Mexico, to make similar observations, should he have an opportunity; and as thundei'-storms are so frequent, and of svich intensity on that coast, I think he may obtain some im- portant results as connected with the influence of the electric state of the atmosphere upon the vibrations and direction of the needle. Upon a review of all the phasnomena of terrestrial magnetism, and considering the intimate relation which has been established between magnetism and electricity, by which it appears that, if not identical, they are only different modifications of the same principle, there can, I think, be little doubt that they are due to electric curx*ents circulating round the earth. How these currents are excited, whether by heat, by the action of another body, or in consequence of rotation, we are not at present able to determine ; but however excited, they must, though not wholly dependent upon them, be greatly modified by the phy- sical constitution of the earth's surface. We are, therefore, not to expect that symmetry in their course which would be the • Philosophical Transaction!!, 1823, p. 364. The arrangements which I have just described for determining the influence of the aurora borealis arc well adapted for deciding this point. 128 THIRD REPORT — 1833. consequence of a symmetrical constitution of that surface. But even if such symmetry did exist, the action of all the currents at different stations on the surface could scarcely be referred to the same two points as centres of force ; and without this symmetry, it would be absurd to expect it. The hypothesis, therefore, of only two poles, as explanatory of the phaenomena, must be rejected ; and if we are to refer these phasnomena to centres of action, we must, besides two principal ones, admit the existence of others depending, upon local causes. It has been said that if we refer the magnetism of the earth to another body, we only remove the difficulty, and gain little by the supposition*. It, however, appears to me, that if we could show that the magnetism of the earth is due to the action of the sun, independent of its heat, — which, however, I think the more probable cause, — the problem would be reduced to the same class as that of accounting for the light of the sun, the heating and chemical properties of its rays : we only know the fects, and are not likely to know more. If difficulties meet us at every step when we attempt to ex- plain the general phaenomena of terrestrial magnetism, these difficulties become absolutely insurmountable when we come to the cause of their progressive changes. Here, at least, we must for the present be satisfied with endeavouring to discover whether these changes are governed by any general laws : should they be so, their cause may possibly be discovered. Diligent and careful observation is the only means by which we can hope to attain this end, and indeed is that on which we must principally rely for gaining a more correct knowledge of all the phaenomena, and of their causes ; and, consequently, im- provements in the methods of observation, and in the instrvi- ments to be employed, become of the highest importance. This Report has already so far exceeded the limits within which I wished to have confined it, that I must restrict the re- marks on this part of the subject to a few points. In the observations of Humboldt, in those of M. Rossel, of Captain Sabine, and of Captain Foster, the terresti'ial magnetic intensity had been determined by the vibrations of a dipping needle in the plane of the magnetic meridian ; but as there is by this means, in consequence of the friction upon the axis, a difficulty in obtaining a sufficient number of vibrations to ensure accuracy, and a dipping instrument is besides ill adapted for carriage. Professor Hanstepn proposed to determine the same by means of a small needle suspended horizontally by a few • Hansteen's Inquiries concerning the Magnetism of the Earth. REPORT ON THE MAGNETISM OP THE EARTH. 129 untwisted fibres of silk. The advantages, however, attending this method of Professor Hansteen, I consider to be more ap- parent than real ; for without determining the dip, the hori- zontal force, deduced from the vibrations of the horizontal needle, cannot be reduced to the force in the direction of the dip ; and if the dip is determined, two instruments become ne- cessary where, before, only one was requisite. In order to obviate the inconveniences attending each of these methods, I have proposed a construction for a dipping needle, by means of which the observations which determine the di- rection of the terrestrial force will also give a measure of its intensity. The general principle of the construction is simply, that the centre of gravity of the needle should not be in its centre of figure, but in a line drawn from that centre at right angles, both to its axis of motion and to its magnetic axis ; so that, by two observations, one with the centre of gravity up- wards, and the other with it downwards, the dip, and likewise the relation which the static momentum of its weight bears to that of the terrestrial magnetic force acting upon the magnetism of the needle, may be determined. The principles on which these determinations depend, and the advantages which I pro- pose from the adoption of this construction, are fully described in a paper read before the Royal Society, and which will appear in the Philosophical Transactions of this year. Professor Gauss has proposed a method of determining the intensity and the changes it undergoes, by v/hich he hopes to reduce magnetical observations to the accuracy of astronomical ones. By the vibrations of a magnetized bar he determines the product of the terrestrial magnetic intensity by the static mo- mentum of its free magnetism. By introducing a second bar, and by observing at different distances the joint effects of the first, and of the terrestrial magnetism on this, he determines the ratio of the terrestrial intensity to the static momentum of the free magnetism of the first. Eliminating this last from the two equations, he obtains an absolute measure of the terres- trial magnetic intensity, independent of the magnetism of the bar. This is a most important result, for we shall thus be en- abled to determine the changes which the terrestrial intensity undergoes in long intervals of time. It is, however, to be ob- served, that it is only the horizontal intensity which is thus determined, and that, in order to determine the intensity of the whole force, another element, namely, the dip, must also be ob- served ; and I fear much that the introduction of this element will, in a great measure, counteract that accuracy of which the methods proposed for determining the times of vibration appear 1833. K 130 THIRD REPORT — 1833. capable. This must be an objection, even where the observa- tions are made in a fixed observatory ; but where an apparatus has to be moved from one station to another, I think the method could scarcely be applied successfully, principally on account of the delicacy of the preliminary observations, and of the time requisite for making them, in addition to that required for the observations by which the terrestrial intensity and its variations are to be determined. However greatly I may admire the saga- city which Professor Gauss has shown in devising means for the determination of an absolute measure of the horizontal in- tensity, I cannot avoid seeing the difficulties which may occur in its practical application. The method which Professor Gauss proposes, and has prac- tised, of observing the course of the daily variation, and of de- termining the time of vibration, by means of a plane mirror fixed on the end of the needle, perpendicularly to its axis, and observing the reflected image of the divisions of a scale by means of a theodolite fixed at a distance, appears to admit of the greatest possible precision, and will probably supersede other methods of observing the daily variation. I have adverted to the necessity of careful and dihgent ob- servation of all the phaenomena of terrestrial magnetism, as the surest means of arriving at a knowledge of their causes : it is with i-eluctance I state it, but I believe it to be a fact, that this is the only country in Europe in which such observations are not regularly carried on in a national observatory. Such an omission is the more to be regretted, seeing that no one has, I believe, cai'ried on a regular series of observations on the diurnal va- riation, since the valuable ones by Colonel Beaufoy were inter- rupted by his death, this interruption happening at a time when it was peculiarly desirable that the series should be unbroken. At this time the needle near London had begun to show a re- turn towards the true meridian ; but whether this was one of those oscillations which have occasionally been observed, or that, having really attained its maximum of westerly deviation, it was returning in the contrary direction, is, I believe, undecided at the present moment. Of all the data requisite for deter- mining the laws which govern the phaenomenon of the variation, the time of the maxima and their magnitude are the most im- portant. I trust that ere long the important desideratum will be supplied of a regular series of magnetical observations in the national Observatory of Great Britain. Royal Military Academy, 22nd June, 1833. [ liJl ] Report on the present State of the Analytical Theory of Hydro- statics and Hydrodynamics. By the Rev. J. Challis, late Fellow of Trinity College Cambridge. The problems relating to fluids, which have engaged the atten- tion of mathematicians, may be classed under two heads, — those which involve the consideration of the attractions of the con- stituent molecules, and the repulsion of their caloric ; and those in which these forces are not explicitly taken account of. In the latter class the reasoning is made to depend on some pro- perty derived from observation. For instance, water is observed to be very difficult of compression ; and this has led to the assumption of iabsolute incompressibilityj as the basis of the mathematical reasoning : air at rest, and under a given state of temperature, is observed to maintain a certain relation between the pressure and the density ; hence the fundamental property of the fluid which is the subject of calculation is assumed to be the constancy of this relation, to the exclusion of all the circum- stances which may cause it to vary. The fluids treated of in this kind of problems are rather hypothetical than real, yet not so different from real fluids but that the mathematical deduc- tions obtained respecting them admit of having the test of ex- periment applied. I propose in this Report to confine myself entirely to problems of the second class, — those in the common theory of fluids. The reasons for making this limitation ai'e, that both kinds together would afford too ample matter for one Report, and that those which I have selected are distinguished from the others by the different purpose in regard to science which correct solutions of them would answer : for the treat- ment of any hydrostatical or hydrodynamical questions which involve the consideration of molecular attraction and the repul^ sion of heat, must proceed upon certain hypotheses respecting the mode of action of these forses, and the intei-ior constitution of the fluid, as these are circumstances which from their nature cannot be data of observation ; and hence, assuming the ma- thematical reasoning founded on the hypotheses to be correct, a satisfactory comparison of the theoretical deductions with facts must serve principally to establish the truth of the hypo- theses, and so to let us into secrets of natui'e which probably could never be known by any other process. But when the k2 132 THIRD REPORT— 1833. basis of calculation, as in the questions that will come before us, is some observed and acknowledged fact, solutions which satisfy experiments will first of all serve to confirm the truth of the mathematical reasoning, and then give us confidence in the theoretical results, which, as often happens, cannot readily re- ceive the test of experiment. Calculations of this kind do not add much to our conviction that the facts applied as the test of the theory are really consequences of those which are the basis of it. For instance, we feel satisfied, independently of any ma- thematical reasoning, that the motions of waves on the surface of water are consequences of the incompressibility of the fluid, and the law of equal pressure. But the purpose which these calculations answer of confirming methods of applying analysis is very important, particularly in regard to the higher class of physical questions, which M. Poisson has proposed to refer to a distinct department of science, under the title of Math^ma- tique Physique, viz. those that require in their theoretical treat- ment some hypotheses respecting the interior constitution of bodies, and the laws of corpuscular action : for in questions of this nature, as well as in problems in the common theory of fluids, the mathematical reasoning conducts to partial differen- tial equations ; and if the method of treating these, and of drawing inferences from their integrals, be established in one kind, it may be a guide to the method to be adopted in the other. It is plainly, then, desirable that the mathematical pro- cesses be first confirmed in the cases in which the basis of rea- soning is an observed fact, that the reasoning may proceed with certainty in those cases where it is based on an hypothesis, the truth of which it proposes to ascertain. The subjects of this Report may now be stated to be, the leading hydrostatical and hydrodynamical problems recently discussed, which proceed upon the supposition of an incom- pressible fluid, or of a fluid in which the quotient of the pres- sure divided by the density is a constant ; and the end it has in view is, to ascertain to what extent, and with what success, analysis has been employed as an instrument of inquiry in these problems. I am desirous it should be understood that I have not attempted to make a complete enumeration either of the questions that have been discussed in this department of science, or of the labours of mathematicians in those which have come under notice. It has rather been my endeavour to give some idea of the most approved methods of treating the leading problems, and the possible sources of error or defect in the solutions. In taking this course I hope I may be considered to .have acted sufficiently in accordance with the recommendation REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 133 of the Committee for Mathematics, which was the occasion of my receiving the honour of a request to take this Report in hand. With the hmitation above stated as to the subjects our Re- port is to embrace, we shall have scarcely anything to say on the analytical theory of hydrostatics. The problems of interest in this department were early'solved, and present no difficulty in principle, and little in the detail of calculation. The deter- mination of the height of mountains by the barometer is a hydrostatical question, the difficulty of which does not consist in the analytical calculation, but only in ascertaining the law of the distribution of the atmospheric temperature. We shall not have to speak of the theories that have been invented to over- come this difficulty. Neither does it fall within the scope of this Report to notice the very valuable memoir of M. Poisson on the equilibrium of fluids *, which has for its object the deriva- tion of the general equations of equilibrium from a consideration of molecular attraction and the repulsion of caloric, and seems to have been composed in immediate reference to the theory of capillai'y attraction, which the author svibsequently pub- lished. With regard to the problem of capillary attraction, we may remark, that it is not possible by any supposition respect- ing the forces which sustain or depress the fluid in the tube, to solve it as a question in the common theory of hydrostatics. M. Poisson has shown the insufficiency of Laplace's theory, and by taking into account the molecular forces and the eiFect of heat, has proved that the explanation of the phasnomenon is essentially dependent on a modification of the property which is the basis of the common theory, viz, the incompressibility of the fluid. It does not fall within our province to say more on the celebrated theory of M. Poisson. One improvement I consider to have been recently made in the common theory of fluids. It has been usual to take the law of equal pressure as a datum of observation. Professor Airy, in his Lectures in the University of Cambridge, has shown that this property may be derived, by reasoning according to esta- blished mechanical principles, from another of a simpler kind, the notion of which may be gathered from observation, viz. that the division of a perfect fluid may be effected without the application of sensible force ; from which it immediately follows that the state of equilibrium or motion of a fluid mass is not altered by mere separation of its parts by an indefinitely thin partition. A definition of fluids founded on this principle, and • Memoircs de I' Academic des Sciences, Paris, torn. ix. 1830. 134 THIRD REPORT — 1833. a proof of the law of equal pressure, are given at the beginning of the Elements of Hydrostatics and Hydrodynamics of Pro- fessor Miller *. Several advantages attend this mode of com- mencing the mathematical treatment of fluids. The principle is one which perfectly characterizes fluids, as distinguished in the internal arrangement of their particles from solids. It may be rendered familiar to the senses. It is, I think, necessary for the solutions of some hydrostatical and hydrodynamical pro- blems, particularly those of reflection f. Lastly, in reference to the department of science proposed to be called Physical Mathematics, the propositions of the common theory ought to be placed on the simplest possible basis, because the questions of most interest in that department are those which have in view the explanation of the phasnomena that are the founda- tions of the reasoning in the other kind. The solution of one such question is a great step in scientific generalization. It is plainly, therefore, of importance that the fact proposed for ex- planation should be the simplest that direct observation can come at. The analytical theory of hydrodynamics is of a much more difficult nature than that of hydrostatics. The assumptions it is necessary to make to obtain even approximate solutions of the simplest problems of fluid motion betray the difficulty and im- perfection of this part of science. There are cases, however, of steady motion, that is, of motion which has arrived at a pei'ma- nent state, so that the velocity is constantly the same in quantity and direction at the same point, which require a much more simple analysis than those which do not satisfy this condition. It does not appear that the equations applicable to this kind of mo- tion were obtained in any general manner till they were given in an Elementary Treatise on Hydrostatics and Hydrodynaynics by Mr. Moseley %, who has derived them from a principle of so simple a nature, that, as it can be stated in a few words, it may be mentioned here. When the motion is steady, each particle in passing from one point to another, passes successively through the states of motion of all the particles which at any instant lie on its path. This principle is valuable for its generality : it is equally applicable to all kinds of fluids, and will be true, whe- ther or not the efl^ect of heat be taken into accoimt, if only the condition of steadiness remains. The equations of motion are readily derived from it, because it enables us to consider the .* Cambridge 183 J. t Dr. Young employed an equivalent principle to determine the manner of the reflection of waves of water. See his Natural Philosophy, vol. ii. p. 64. t Cambridge 1830. REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 135 motion of a single particle, in the place of the motion of an aggregate of particles. Though this mode of deriving them is the best possible on account of its simplicity, it was yet de- sirable to know how they may be obtained from the general equations of fluid motion. In a paper contained in the Trans- actions of the Philosophical Society of Cambridge *, the author of this Report has given a method of doing this, both for incom- pressible and elastic fluids, and has shown that a term in the general formulae which gives rise to the complexity common to most hydrodynamical questions, disappears for this kind of motion. Euler had already done the same for incompressible fluids f . The instances in nature of fluid motion of the steady kind are far from uncommon ; and it is probable that when the equations applicable to them are better known, and studied longer, they may be employed in very interesting researches. The motion of the atmosphere, as affected by the rotation of the earth, and a given distribution of the temperature due to solar heat, seems to be an instance of this kind. We will now proceed to consider in order the principal hydro- dynamical problems that have recently engaged the attention of mathematicians. For convenience we shall class them as follows : — I. Motion in pipes and vessels. II. The velocity of propagation in elastic fluids. III. Musical vibrations in tubes. IV. Waves at the surface of water. V. The resistance to the motion of a ball-pendulum. I. The motion of fluids in pipes and vessels has not been treated with any success, except in the cases in which the con- dition of steadiness is fulfilled. The paper above alluded to, in the Transactions of the Philosophical Society of Catnbridge, contains some applications of the equation of steady motion for incompressible fluids, to determine the velocity of water issuing from different kinds of adjutages in vessels of any shape : also a theoretical explanation of a phsenomenon which a short while ago excited some attention, — that of the attraction of a disc to an orifice through which a steady current either of water or air is issuing. In the Memoirs of the Paris Academy of Sciences X there is an Essay by M. Navier on the motion of elastic fluids in ves- sels, and through different kinds of adjutages into the sur- rounding air, or from one vessel into another. For the sake of simplicity the author considers the fluid to be subject to a con- stant pressure, and consequently the motion to have arrived at a state of permanence. His calculations are founded upon the • Vol. iii. Part III. f Memoires de I'/lcadimie de Berlin, 1755, p. 344. X Tom. ix. 1830. 136 THIRD REPORT— 1883. hypothesis of parallel slices, which assumes the velocity to be the same, and in the same direction, and the density to be the same at all points of any section transverse to the axis of the vessel or pipe. This hypothesis is one of those that the theory of hydrodynamics has borrowed from experience to supply its defects. Lagrange has, however, shown theoretically* that it always furnishes a first approximation, the breadth of the ves- sel being considered a quantity of the first order, and the effect of the adhesion of the fluid to the sides of the vessel being neg- lected. It is right to observe, that in the problems M. Navier has considered, this hypothesis might have been in a great measure dispensed with : the expression he has given, — more correct than that commonly adopted for the velocity of issuing through a small aperture by which airs of different densities communicate, — ^might have been obtained by employing the equa- tion above mentioned of steady motion, as, in fact, Mr. Moseley has done f. This would be a preferable mode of treating such questions, because in every instance in which these auxiliary hypotheses are got rid of, something is gained on the side of theory. This memoir contains another hypothesis, which can- not be so readily dispensed with. Theory is at present quite inadequate to determine the retardation in the flow of fluids occasioned by sudden contractions or widenings in the bore of the pipe. It is found by experiments with water, that the re- tardation is sufficiently represented by taking account of the loss of vis viva which, on the hypothesis of parallel slices, will result from the sudden changes of velocity which must be sup- posed to take place at the abrupt changes in the bore of the pipe. M. Navier extends these considerations to elastic fluids. The theory manifests a sufficient agreement with the experi- ments it is compared with, and is valuable on account of the applications it may receive. II. The most interesting class of problems in hydrodynamics are perhaps those which relate to small oscillations. Newton was the first to submit the vibrations of the air to mathema- tical calculation. The propositions in the second book of the Principia, devoted to this subject, and to the determination of the velocity of sound, may be ranked among the highest pro- ductions of his genius. He has assumed that the vibratory motion of the particles follows the law of the motion of an oscil- lating pendulum. It was soon discovered that many other assumed laws of vibration would, by the same mode of reasoning, * Mecanique Analytique, P&tt II. § xi. art. 34. t Elementary Treatise, p. 204. REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 137 conduct to the same velocity of propagation. This, which was thought to be an objection to the reasoning, is an evidence of its correctness : for the plain consequence is, that the velocity of propagation is independent of the kind of vibration which we may arbitrarily impress on the fluid ; — and so experience finds it to be. When the partial differential equation, which applies equally to the vibrations of the air and those of an elastic chord, had been formed and integrated, a celebrated discussion arose between Euler and D'Alembert as to the extent to which the integral could be applied ; whether only to cases in which the motion was defined by a continuous curve, or also to motion defined by a broken and discontinuous line. It is well known that the question was set at rest by Lagrange, in two Disser- tations published in vol. i. and vol. ii. of the Miscellanea Tau- rinensia. The difficulty that arrested the attention of these eminent mathematicians was one of a novel kind, and peculiar to physical questions that require for their solution the integrals of partial differential equations. The difficulty of integration, which is the obstacle in most instances, had been overcome by D'Alembert. It remained to draw inferences from the inte- gral, — to interpret the language of analysis. When an aggre- gate of points, as a mass of fluid or an elastic chord, receives an arbitrary and irregular impulse, any point not immediately acted upon may have a correspondent irregular movement after the initial disturbance has ceased. This is a matter of experi- ence. Was it possible, then, that these irregular impulses, and the consequent motions, were embraced by the analytical calcu- lation ? From Lagrange's researches it follows that the func- tions introduced by integration are arbitrary to the same degree that the motion is so practically, and that they will therefore apply to discontinuous motions. (Of course we must except the practical disturbances which the limitations of the calcula- tion exclude, — those which are very abrupt, or very large.) This has been a great advance made in the application of ana- lysis to physical questions. Had a diff'erent conclusion been arrived at, many facts of nature could never have come under the power of calculation. The Researches of Lagrange, which will ever form an epoch in the science of applied mathematics, estabUsh two points principally : First, That the arbitrary func- tions, as we have been just saying, are not necessarily conti- nuous : Secondly, That (in the instance he considered) they are equivalent to an infinite series of terms having arbitrary con- stants for coefficients, and proceeding according to the sines of multiple arcs. This latter result, which appears to be true for 138 THIRD REPORT — 1833. all linear partial differential equations of the second order, with constant coefficients, is valuable as presenting an analogy be- tween arbitrary constants and arbitrary functions. But the way in which Lagrange, after establishing these two points, proceeds to find the velocity of propagation, does not appear to me equally satisfactory with the rest of his reasoning. His method seems to be a departure from the principle which may be gathered from that of Newton. For, as was mentioned above, the reasoning of the Principia shows that the velocity of propagation is independent of all that is arbitrary. It seems important to the truth of the analytic reasoning, that it should not only obtain a constant velocity of propagation, but arrive at it by a process which is independent of the arbitrary nature of the functions ; whereas the method which the name of La- grange has sanctioned, is essentially dependent on the discon- tinuity of the functions, that is, on their being arbitrary. With a view of calling attention to this difficulty, and as far as possi- ble removing it, the author of this Report read a paper before the Philosophical Society of Cambridge, Avhich is published in Vol. iii. Part I. of their Transactions. I am far from assert- ing that that Essay has been successful; but some service, I think, will be done to science if it should lead mathematicians to a reconsideration of the mode of mathematical reasoning to be employed in regard to the ajjplications of arbitrary ftinctions. If the determination of the velocity of propagation in elastic fluids were the only problem affected by this treatment of arbi- trary functions, it would not be worth while to raise a question respecting the principle of the received method, as no doubt attaches to the result obtained by it ; but there are other pro- blems, (one we shall have to consider,) the correct solutions of which mainly depend on the construction to be put upon these functions. The difficulty I am speaking of, which is one of a delicate and abstx*act nature, will perhaps be best understood by the following queries, which seem calculated to bring the point to an issue : — Can the arbitrary functions be immediately applied to any but the parts of the fluid immediately acted upon by the arbitrary disturbance, and to parts indefinitely near to these ? To apply them to parts more remote, is it not necessary first to obtain the law of propagation ? And do not the arbi- trary functions themselves, by the quantities they involve, fur- nish us with means of ascertaining the law of propagation, independently of any consideration of discontinuity ? Euler and Lagrange determined the velocity of propagation in having regard to the three dimensions of the fluid, on the li- mited supposition that the initial disturbance is the same as to REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 139 density and velocity, at the same distance in every direction from a fixed point, which is the centre of it. Laplace first dis- pensed with this limitation in the case in which two dimensions only of the fluid are taken account of*. The principal cha- racter of his analysis is a new method of employing definite integrals. Finally, M. Poisson solved the same problem for three dimensions of the fluid f. This memoir deserves to be particularly mentioned for the interesting matter it contains. The object of the author is to demonstrate, in a more general manner than had been before done, some circumstances of the motions of elastic fluids which are independent of the particular motions of the fluid particles, such as propagation and reflection. The general problem of propagation just mentioned he solves by developing the integral of the partial differential equation of the second order in x, y, z, and t, applicable to this case, in a series proceeding according to decreasing powers of the di- stance from the centre of disturbance, as it cannot be obtained in finite terms, and then transforming the series into a definite integral, — a method which has of late been extensively em- ployed. The crossing of waves simultaneously produced by disturbances at several centres, is next considered, and this leads to a general solution of the problem of reflection at a plane surface. For the case in which the motions of the aerial parti- cles are not supposed small, the velocity of propagation along a line of air is shown to be the same as when they are small. This result is an inference drawn from the arbitrary disconti- nuity of the motion, on which it does not seem to depend. In a paper before alluded to J, the same result is obtained without reference to the principle of discontinuity. M. Poisson treats also of propagation in a mass of air of variable density, such as the earth's atmosphere. His analysis is competent to prove, in accordance with experience, that the velocity of sound is the same as in a mass of uniform density, and that its intensity at any place depends, in addition to the distance from the point of agitation, only on the density of the air where the disturbance is made. So that a bell rung in the upper regions of the air will not sound so loud as when rung by the same effort below, but will sound equally loud at all equal distances from the place where it is rung. In seeking for the general equations of the motion of fluids, (first obtained by Euler,) a quantity § is met with which, if it be * Memoires de V Academic, An 1779. , t " Memoire sur la Theorie du Son," Journal de I'Ecole Polytechnique, torn. vii. cah. xiv. J Transactions of the Philosophical Society of Cambridge, vol. iii. Part III. § In M. Poisson's writings this quantity is udx + v dy + ir dz. 140 THIRD REPORT 1833. an exact differential of a function of three variables, i-enders the subsequent analytical reasoning much simpler than it would be in the contrary case. This simplification has been proved by Lagrange to obtain in most of the problems of interest that are proposed for our solution *. Euler showed that the differential is inexact when the mass of fluid revolves round an axis so that the velocity is some function of the distance from the axisj-. But no general method exists of distinguishing the ipstances in which the quantity in question is a complete differential, and when it is not. Nor is it known to what physical circumstance this peculiarity of the analysis refers. To clear up this point is a desideratum in the theory of hydrodynamics. M. Poisson has left nothing to be desired in the generality with which he has solved the problem of propagation of motion in elastic fluids; for in the Memoirs of the Academy of Paris \ he has given a solution of the question, without supposing the initial disturb- ance to be such as to make the above-mentioned quantity an exact differential. His conclusions are, that the velocity of propagation is the same as when this supposition is made ; that the part of the motion which depends on the initial condensa- tions or dilations follows the same laws as in that case, but the part depending on the initial velocity does not return com- pletely to a state of repose after a determinate interval of time ; that at great distances from the place of agitation there is no essential difference between the motion in the two cases. HI. We turn now to the theory of musical vibrations of the air in cylindrical tubes of finite length. Little has been effected by analysis in regard to this interesting subject. The principal difficulty consists in determining the manner in which the mo- tion is affected by the extremities of the tube, whether open or closed, but particularly the open end. Those who first handled the question reasoned on the hypotheses, that at the open end the air is always of the same density as the external air to maintain an equilibrium with it, and at the closed end always stationary by reason of the stop. The latter supposition will be true only when the stop is perfectly rigid. It does not ma- terially affect the truth of the reasoning ; but if the other sup- position were strictly true, the sound from the vibrating column of air in the tube would not cease so suddenly as experience shows it does, when the disturbing cause is removed ; neither on this hypothesis could the external air be acted on so as to receive alternate condensations and rarefactions, and transmit * Mecanique Analytique, Part II. § xi. art. 16. t Memoires de I' Academic de Berlin, 1 755, p. 292. % torn. X. 1831. REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 141 sonorous waves. These objections to the old theory have been stated by M. Poisson, who proposes a new mode of considering the problem*. He reasons on an hypothesis which embraces both the case of an open and a closed end, viz. that the velo- city at each is in a constant ratio to the condensation. This ratio will be very large for the open end, and a very small frac- tion for the closed end. Its exact value in the latter case de- pends on the elasticity of the stop, and in the other on the mode of action of the vibrations on the external air, — to determine which is a problem of great difficulty, which M. Poisson has forborne to meddle with. His theory is not competent to assign a priori either the series of tones or the gravest that can be sounded by a tube of given length, but is more successful in determining the number of nodes and loops, and the intervals between them, when a given tone is sounded. To find the di- stances of the nodes and loops from the extremities of the tubes, he has recourse to the hypotheses of the old theory, which make the closed end the position of a node, and the open end the position of a loop. This, he says, will not be sensibly dif- ferent from the truth, if, in the one case, the stop be very un- yielding, and, in the other, the diameter of the tube be small. Recent researches on this subject, which we shall presently speak of, show that when the diameter is not very small the position of the loop is perceptibly distant from the open end. The latter part of M. Poisson's memoir contains an applica- tion of the principles of the foregoing part to the vibrations of air in a tube composed of two or more cylinders of different dia- meters, and to the motion of two different fluids superimposed in the same tube. In the course of this latter inquiry, the au- thor determines the reflection which sound experiences at the junction of two fluids ; and by an extension of like considerations to luminous undulations, obtains the same expressions for the relative intensities of light perpendicularly incident, and re- flected at a plane surface, as those given by Dr. Young in the Article Chromatics of the Supplement to the Encyclopcedia Britannica. This subject was afterwards resumed by M. Pois- son at greater length in a very elaborate memoir " On the Mo- tion of two Elastic Fluids superimposed f," which is chiefly remarkable for the bearing which the results have upon the theory of light. At the last meeting, in May this year, of the Philosophical So- ciety of Cambridge, a paper was read by Mr. Hopkins, in which, * Memoires de V Acad^mie des Sciences, Paris, An 1817. t Ibid. torn. X. p. 317. 142 THIUD REPORT — 1833. by combining analysis witb a delicate set of experiments, re- sults are obtained which are a valuable addition to this part of the theory of fluid motion. His experiments were made on a tube open at both ends, and the column of air within it was put in motion by the vibrations of a plate of glass applied close to one end. The following are the principal results. The nodes are not points of quiescence, but of minimum vibration ; — the extremity of the tube most remote from the disturbance is not a place of maximum vibration, but the whole system of places of maximum and minimum vibration is shifted in a very sensible degree towards it ; — the distances of the places of maximum and minimum vibration from each other, and from that extre- mity, remain the same for the same disturbance, whatever be the length of the tube. This last fact Mr. Hopkins proves by his analysis must obtain. The shifting of the places of maximum and minimum vibration is not accounted for by the theory : nor is it probable that it can be, unless the consideration of the mode of action of the vibrations on the external air be entered upon, — an important inquiry, but, as I said before, one of great difficulty. I think also that the effect of the vibrations of the tube itself on the contained air ought to be taken into account. IV. The problem of waves at the surface of water is princi- pally interesting as furnishing an exercise of analysis. The general difterential equations of fluid motion assume a very sim- ple form for the case of oscillations of small velocity and extent, and seem to oiFer a favourable opportunity for the application of analytical reasoning. Yet mathematicians have not succeed- ed in giving a solution of the problem in any degree satisfactory, which does not involve calculations of a complex nature. We need not stay to inquire in what way Newton found the velo- city of the propagation of waves to vary as the square root of their breadths : he was himself aware of the imperfection of his theory. The question cannot be well entered upon without partial differential equations. Laplace was the first to apply to it a regular analysis. His essay is inserted at the end of a memoir on the oscillations of the sea and the atmosphere, in the volume of the Paris Academy of Sciences for the year 1776. The differential equations of the motion are there formed on the supposition that the velocities and oscillations are always so small that their products, and the powers superior to the first, may be neglected. The problem without this limitation be- comes so complicated that no one has dared to attempt it. La- place's reasoning conducts to a linear partial differential equa- tion of the second order, consisting of two terms, which is readily integrated; but on account of the difficulty of obtaining a REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 143 general solution from this integi'al, he makes a particular sup- position, which is equivalent to considering the fluid to be de- ranged from its state of equilibrium by causing the surface in its whole extent to take the form of a trochoid, i. e. a serpentine curve, of which the vertical ordinate varies as the cosine of the horizontal abscissa. The solution in question is of so limited a nature, that we may dispense with stating the results arrived at. In the volume of the Memoirs of the Academy of Berlin for the year 1786, Lagrange has given* a very simple way of proving, in the Newtonian method of reasoning, that the ve- locity of propagation of waves along a canal of small and con- stant depth and uniform width, is that acquired by a heavy body falling through half the depth. In the Mecanique Ana' li/tiquef the same result is obtained analytically. The princi- pal feature of the analysis in this solution is, that the linear partial differential equation of the second order and of four va- riables, to which the reasoning conducts, is integrated approxi- mately in a series. Lagrange is of opinion, that on account of the tenacity and mutual adherence of the parts of the fluid, the motion extends only to a small distance vertically below the surface agitated by the waves, of whatever depth the fluid may be ; and that his solution will consequently apply to a mass of fluid of any depth, and will serve to determine, from the ob- served velocity of propagation, the distance to which the motion extends downwards. The problem of waves was proposed by the French Institute for the prize subject of 1816. M. Poisson, whose labours are preeminent in every important question of Hydrodynamics, had already given this his attention. His essay, which was the first deposited in the bureau of the Institute, was read Oct. 2, 1815, just at the expiration of the period allowed for competition. It forms the first part of the memoir " On the Theory of Waves," published in the volume of the Academy for the year 1816, and contains the general formulas required for the complete solution of the problem, and the theory, derived from these formulae, of waves propagated with a nniformly accelerated motion. In the month of December following, an additional paper was read by M. Poisson on the same subject, which forms the second part of the memoir just mentioned, and contains the theory of waves propagated with a constant velocity. These are much more sensible than the waves propagated with an accelerated motion, and are in fact those which are commonly seen to spread in • p. 192. \ Part II. sect. xi. ait. 3G. 144 THIRD REPORT — 1833. circles round any disturbance made at the surface of water. No theory of waves which does not embrace these can be con- sidered complete. In the essay of M. Cauchy, which obtained the prize, and is printed in the Mdmoires des Savans*, the theory of only the first kind of waves is given. This essay, however, claims to be more complete than the first part of M. Poisson's memoir, because it leaves the function relative to the initial form of the fluid surface entirely arbitrary, and conse- quently allows of applying the analysis to any form of the body immersed to produce the initial disturbance. M. Poisson re- stricts his reasoning to a body, of the form of an elliptic para- boloid, immersed a little in the fluid, with its vertex downward and axis vertical ; and as this form may have a contact of the second order, with any continuous surface, the reasoning may be legitimately extended to any bodies of a continuous form, but not to such as have summits or edges, like the cone, cy- linder and prism. This restriction having been objected to as a defect in the theoryf , M. Poisson answers J that his analysis is not at fault, but that one of the differential equations of the problem, which expresses the condition that the same particles of water remain at the surface during the whole time of motion, very much restricts the form which the immersed body may be supposed to have. When the initial motion is produced by the immersion of a body whose surface presents summits or edges, it is not possible, he thinks, to represent the velocities of the fluid particles by analytical formulae, especially at the first in- stants of the agitation, when the motion must be very complicated, and the same points will not remain constantly at the surface. With the exception of the particular we have been mention- ing, the two essays do not present mathematical processes es- sentially different in principle. Attached to that of M. Cauchy, which was published subsequently to M. Poisson's memoir, are Valuable and copious additions, serving to clear up several points of analysis that occur in the course of the work, and re- ferring chiefly to integration by series and definite integrals, and to the treatment of arbitrary functions. Among these is a lengthened discussion of the theory of the waves uniformly propagated, the existence of which, as indicated by the analysis, had escaped the notice of both mathematicians in their first re- searches. In this discussion the velocities of propagation are determined of the two foremost wa,ves produced by the immer- ♦ vol. lii. f Bulletin de la Societe Philomatiqiie, Septembre 1818, p. 129. X "Note sur le Probleme des Ondes," torn. viii. of Mimoires de I'Acadhnie des Sciences, p. 371. REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 145 sion and sudden elevation of bodies of the forms of a parabo- loid, a cylinder, a cone, and a solid, generated by the revolution of a parabola about a tangent at its vertex. To bodies of the last three forms, M. Poisson objects to extending the reasoning; and in the " Note" above referred to, attempts to show that such an extension leads to results inconsistent with the principle of the coexistence of small vibrations. If we are not permitted to receive the analysis of M. Cauchy in all the generality it lays claim to, we must at least assent to the reasonableness of the following conclusion it pretends to arrive at, viz. that "the heights and velocities of the different waves produced by the immersion of a cylindrical or prismatic body depend not only on the width and height of the part immersed, but also on the form of the surface which bounds this part." There is also much appearance of probability in a remark made by the same mathematician, that the number of the waves produced may depend on the form of the immersed body and the depth of immersion. We proceed to say a few words on the contents of M. Pois- son's memoir. He commences by showing, as well by a priori reasoning as by an appeal to facts, that Lagrange's solution cannot be extended to fluid of any depth. In his own solution he supposes the fluid to be of any uniform depth, but princi- pally has regard to the case which most commonly occurs of a very great depth : he neglects the square of the velocity of the oscillating particles, as all have done who have attempted this problem, and assumes, that a fluid particle which at any instant is at the surface, remains there during the whole time of the motion. This latter supposition seems necessary for the con- dition of the continuity of the fluid. With regard to the neg- lect of the square of the velocity, it does not seem that we can tell to what extent it may affect the calculations so well as in the case of the vibrations of elastic fluids, where the velocity of the vibrating particle is neglected in comparison of a known and constant velocity, that of propagation. M. Poisson treats first the case in which the motion takes place in a canal of uniform width, and, consequently, abstraction is made of one horizontal dimension of the fluid ; and afterwards the case in which the fluid is considered in its three dimensions. The former requires for its solution the integration of the same differential equation of two terms * as that occurring in Laplace's theory. No use is made of the common integral of this equation, as, on account of the impossible quantities it involves, it would be difficult * In M. Poissoii's works this equation is '-1? -f — = 0, 1833. T. 146 THIRD REPORT — 1833. to make it serve to determine the laws of propagation. It is remarkable that this integral is not necessary for solving the problem, although, as M. Poisson has shown in his first me- moir, " On the Distribution of Heat in Solid Bodies," and M. Cauchy in the Notes added to his " Theory of Waves," a solu- tion may be derived from it equivalent to that which they have given without its aid. We may be permitted to doubt whether its meaning is yet fully understood, and to hope that, by over- coming some difficulty in the interpretation of this integral, the problem of waves may receive a simpler solution than has hi- therto been given. Be this as it may, the process of integration adopted by M. Poisson leaves nothing to be wished for in regard to generality. It is easy to obtain an unlimited number of pai-- ticular equations not containing arbitrary functions, which will satisfy the differential equation in question, and to combine them all in an expression for the principal variable (ip), deve- loped in series of real or imaginary exponentials. This will be the most general integral the equation admits of, and (to use the words of M. Poisson,) " there exist theorems, by means of which we may introduce into expressions of this nature, arbi- trary functions, which represent the initial state of the fluid : the difficulty of the question consists then in discussing the re- sulting formulas, and discovering from them all the laws of the phasnomenon. The theory of waves furnishes at present the most complete example of a discussion of this sort." In a Report like the present, it is not possible to give any very precise idea of the analysis which has been employed for solving the problem of waves. I have thought it proper to call attention to a process of reasoning which has been very exten- sively employed by the French mathematicians of the present day, and indeed may be considered to be the principal feature of their calculations in the more recent applications of mathe- matics to physical and mechanical questions. To understand fully the nature and power of the method, the works of Fourier, particularly The Analytical Theory of Heat ^ the Notes, before spoken of, to M. Cauchy's " Theory of Waves," and the two memoirs of M. Poisson " On the Distribution of Heat in Solid Bodies," must be studied. I will just refer to some parts of the writings of the last-mentioned geometer, where he has been careful to state in a concise manner the principle of the method in question. There are some remarks on the generality of a main step in the process in the Bulletin de la Societe Philoi7ia- tique*. The note before spoken of in the eighth volume of the * An 1817, p. ISO. REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 147 Memoirs of the Academy concludes with a brief account of the history and principle of this way of expressing the complete integral by a series of particular integrals, and introducing the arbitrary function. But 1 would chiefly recommend the peru- sal of the remarks at the end of a memoir by this author " On the Integration of some linear partial Differential Equations ; and particularly the general Equation of the Motion of Elastic Fluids." To the memoir itself I beg to refer, by the way, as presenting a demonstration of the constancy of the velocity of propagation from an irregular disturbance in an elastic fluid, more simple and direct than that in the Journal de VEcole Po- lytechniqiie. It contains also a general integral of the linear partial differential equation of three terms, which occurs in the problem of waves for the case in which the three dimensions of the fluid are taken account of; but the author does not consider this integral of much utility, because of the impossible quantities involved in it, and rather recommends the method of express- ing the principal variable by infinite series of exponentials. In fact, in the " Theory of AVaves " this case is treated in a manner exactly analogous to that in which abstraction is made of one dimension of the fluid. It may be useful to state some of the principal results ob- tained by theory respecting the nature of waves, to give an idea of what the independent power of analysis has been able to ef- fect. With respect, first, to the canal of uniform width, the law of the velocity of propagation found by Lagrange is confirmed by M. Poisson's theory when the depth is small, but not other- wise. When the canal is of unlimited depth, the following are the chief results : (1.) An impulse given to any point of the surface affects in- stantaneously the whole extent of the fluid mass. The theory determines the magnitude and direction of the initial velocity of each particle resulting from a given impulse. (2.) " The summit of each wave moves with a uniformly acce- lerated motion." This must be understood to refer to a series of very small waves, called by M. Poisson dents, which perform their move- ments as it were on the surface of the larger waves, which he calls " les ondes denteUes." Each wave of the series is found to have its proper velocity, independent of the primitive im- pulse. Waves of this kind have been actually observed : they are small from the first, and quickly disappear. (3.) At considerable distances from the place of disturbance, l2 148 THIRD REPORT — 1833. there are waves of much more sensible magnitude than the pre- ceding. Their summits are propagated with a uniform velocity, which varies as the square root of the breadth a fleur deau of the fluid originally disturbed. Yet the different waves which are formed in succession are propagated with different veloci- ties : the foremost travels swiftest. The amplitude of oscilla- tions of equal duration are reciprocally proportional to the square root of the distances from the point of disturbance. (4.) The vertical excursions of the particles situated directly below the primitive impulse, vary according to the inverse ratio of the depth below the surface. This law of decrease is not so I'apid but that the motion will be very sensible at very consider- able depths : it will not be the true law, as the theory proves, when the original disturbance extends over the whole surface of the water, for the decrease of motion in this case will be much more rapid. The results of the theory, when the three dimensions of the fluid are considered, are analogous to the preceding, (1), (2), (3), (4), and may be stated in the same terms, excepting that the am- plitudes of the oscillations are inversely as the distances from the origin of disturbance, and the vertical excursions of the par- ticles situated directly below the disturbance vary inversely as the square of the depth. There is a good analysis of M. Poisson's theory, and a com- parison of many of the results with experiments, in a Treatise by M. Weber, entitled Wellenlehre aiif Experimente gegrim- det*. The experiments of M. Weber were made in a manner not sufficiently agreeing with the conditions supposed in the theoi'y to be a correct test of it. They, however, manifest a general accordance with it, and confii'm the existence of the small accelerated waves near the place of distui'bance, and of a sensible motion of the fluid particles at considerable depths below the surface. In one particular, in which the theory ad- mits of easy comparison with experiment, it is not found to agree. When the body employed to cause the initial agitation of the water is an elliptic paraboloid, with its vertex downwards and axis vertical, and consequently the section in the plane of the surface of the water an ellipse, theory determines the velo- city of propagation to be greater in the direction of the major axis than in that of the minor in the proportion of the square root of the one to the square root of the other. This result, which it must be confessed has not an appearance of probabi' lity, is not borne out by experience. * Leipzig, 182,5. REPOUT ON HYDROSTATICS AND HYDRODYNAMICS. 149 The theory has been also put to the test of expermient by M. Bidone, who succeeded in overcoming in great measure an obstacle in the way of making the experiments according to the conditions supposed in the theory, arising from the adhesion of the water to the immersed body*. His observations confirm the existence and laws of motion of the accelerated waves. V. Scarcely anything worth mentioning has been effected by theory in regard to the resistance of fluids to bodies moving in them. The defect of every attempt hitherto made has arisen from its proceeding upon some hypothesis respecting the law of the resistance ; for instance, that it varies as the ve- locity, or as the square of the velocity: whereas the law, which cannot be known a priori, ought to be a result of the calcula- tion, which should embrace not only the motion of the body, but that of every particle of the fluid which moves simulta- neously with it. The only problem that has been attempted to be solved on this principle, is one of very considerable in- terest, relating to the correction to be applied to the pendulum to effect the reduction to a vacuum. The memoir of M. Pois- son, " On the Simultaneous Motions of a Pendulum and of the surrounding Air," was read before the Royal Academy of Paris in August 1831, and is inserted in vol. xi. of iheiv 3femoires. He takes the case of a spherical ball suspended by a very slen- der thread, the effect of which is neglected in the calculations ; the ball is supposed to perform oscillations of very small ampli- tude, so that the air in contact with its surface is sensibly the same during the motion. A simpler problem of resistance can- not be conceived. M. Poisson considers the effect which the friction of the particles of air against the surface of the ball may have on its motion, and comes to the conclusion that the time of the oscillations is not affected by it, but only their ex- tent. The most important result of the theoretical calculation is, that the correction which has been usually applied for the reduction to a vacuum, and calculated without considering the motion of the air, must be increased by one half. This he finds to agree sufficiently with some experiments of Captain Sabine. He also adduces forty-four experiments of Dubuat, made fifty years ago, upon oscillations in water, and three upon oscilla- tions in air. These give nearly the same numerical result, and agreeing nearly with the value 1^. The experiments, however, of M. Bessel give results which coincide with Dubuat's for os- cillations in water, but determine the correction in air for re- duction to a vacuum to be very nearly double that hitherto • See vol. XXV. of the Memoirs of the Royal Academy of Turin. 150 THIRD REPORT — 1833. applied, instead of once and a half. M. Poisson thinks that the calculations of M. Bessel leave some room for doubt, and objects to the discordance of the values obtained for air and watei', which, according- to his own theory, ought to agree. More recent experiments of Mr. Baily *, which, from their num- ber and variety, and the care taken in performing them, are entitled to the utmost confidence, give the value 1*864 for spheres of different materials one inch and a half in diameter, and 1 '748 for spheres two inches in diameter, the latter being nearly the size of those for which M. Bessel obtained 1*946. The theory of M. Poisson does not recognise any difference in the value of the coefficient for spheres of different diameters. The discrepancies that thus appear between theory and expe- riment, and between the experiments themselves, show that there is much that requires clearing up in this important sub- ject. As far as theory is concerned, it is easily conceivable that much must depend upon the way in which the law of trans- mission of the motion from the parts of the fluid immediately acted on by the sphere to the parts more remote is to be deter- mined : and, as it is the province of this Report to point out any possible source of error in theory, I will venture again to express my doubts of the correctness of the principle em- ployed in the solution of this problem, of making the deter- mination of the law of transmission depend on the arbitrary discontinuity of the functions introduced by integration, the law itself not being arbitrary f. A singular fact, relating to the resistance to the motion of bodies partly immersed in water, has been recently established by experiments on canal navigation, by which it appears that a boat, drawn with a velocity of more than four or five miles an hour, rises perceptibly out of the water, so that the water-line is not so distant from the keel as in a state of rest, and the re- sistance is less than it would be if no such effect took place. Theory, although it has never predicted anything of this na- ture, now that the fact is proposed for explanation, will proba- bly soon be able to account for it on known mechanical prin- ciples. The foregoing review of the theory of fluid motion, incom- * PhHosophlcal Transactions for 1832, p. 399. t In an attempt at this problem made by myself, and published subsequently to the Meeting of the Association, the value of the coefficient is found to be 2, without accounting for any difference for spheres of different diameters. See the London and Edinburgh Philosophical Magazine and Journal for Septem- ber 1833. REPORT ON HYDROSTATICS AND HYDRODYNAMICS. 151 plete as it is, may suffice to show that this department of science is in an extremely imperfect state. Possibly it may on that ac- count be the more likely to receive improvements ; and I am disposed to think that such will be the case. But these im- provements, I expect, will be available not so much in practical applications, as in reference to the great physical questions of light, heat and electricity, which have been so long the subjects of experiment, and the theories of which require to be perfected. For this purpose a more complete knowledge of the analytical calculation proper for the treatment of fluids in motion may be of great utility. I [ 153 ] Report on the Progress and Present Stale of our Knowledge of Hydraulics as a Branch of Engifieering. By George Rennie, Esq., F.R.S., §-c. ^-c. Part I. The paper now communicated to the British Association for the Advancement of Science comprises a Report on the pro- gress and present state of our knowledge of Hydrauhcs as a branch of Engineering, with reference to the principles already established on that subject. Technically speaking, the term hydraulics signifies that branch of the science of hydrodynamics which treats of the motion of fluids issuing from orifices and tubes in reservoirs, or moving in pipes, canals or rivers, oscillating in waves, or opposing a resistance to the progress of solid bodies at rest. We can readily imagine that if a hole of given dimensions be pierced in the sides or bottom of a vessel kept constantly full, the expenditure ought to be measured by the amplitude of the opening, and the height of the liquid column. If we isolate the column above the orifice by a tube, it ap- pears evident that the fluid will fall freely, and follow the laws of gravity. But experiment proves that this is not exactly the case, on account of the resistances and forces which act in a contrary direction, and destroy part of, or the whole, effect. The development of these forces is so extremely complicated that it becomes necessary to adopt some auxiliary hypothesis or abbreviation in order to obtain approximate results. Hence the science of hydrodynamics is entirely indebted to experi- ment. The fundamental problem of it is to determine the efflux of a vein of water or any other fluid issuing from an aperture made in the sides or bottom of a vessel kept constantly full, or allowed to empty itself. Torricelli had demonstrated that, abstracting the resistances, the velocities of fluids issuing from very small orifices followed the subduplicate ratio of the pres- sures. This law had been, in a measure, confused by sub- sequent writers, in consequence of the discrepancies which appeared to exist between the theory and experiment ; until Varignon remarked, that when water escaped from a small opening made in the bottom of a cylindrical vessel, there ap- peared to be very little, or scarcely any, sensible motion in the 154 THIRD REPORT — 1833. particles of the water ; from which he concluded that the law of acceleration existed, and that the particles which escaped at every instant of time received their motion simply from the pressure produced by the weight of the fluid column above the orifice, and that the weight of this column of fluid ought to represent the pressure on the particles which continually escape from the orifice ; and that the quantity of motion or expenditure is in the ratio of the breadth of the orifice, multiplied by the square of the velocity, or, in other words, that the height of the water in the vessel is proportional to the square of the ve- locity with which it escapes ; which is precisely the theorem of Torricelli. This mode of reasoning is in some degree vague, because it supposes that the small mass which escapes from the vessel at each instant of time acquires its velocity from the pressure of the column immediately above the orifice. But supposing, as is natural, that the weight of the cohunn acts on the particle during the time it takes to issue from the vessel, it is clear that this particle will receive an accelerated motion, whose quantity in a given time will be proportional to the pressure multiplied by the time : hence the product of the weight of the column by the time of its issuing from the orifice, will be equal to the product of the mass of this particle by the velocity it will have acquired ; and as the mass is the product of the opening of the orifice, by the small space which the particle describes in issuing from the orifice, it follows that the height of the column will be as the square of the velocity ac- quired. This theory is the more correct the more the fluid approaches to a perfect state of repose, and the more the dimensions of the vessel exceed the dimensions of the orifice. By a contrary mode of reasoning this theory became insufficient to determine the motions of fluids through pipes of small dia- meters. It is necessary, therefore, to consider all the motions of the particles of fluids, and examine how they are changed and altered by the figure of the conduit. But experiment teaches us that when a pipe has a different direction from the vertical one, the different horizontal sections of the fluid preserve their parallelism, the sections following taking the place of the pre- ceding ones, and so on ; from which it follows (on account of the incompressibility of the fluid) that the velocity of each horizontal section or plate, taken vertically, ought to be in the inverse ratio of the diameter of the section. It suffices, therefore, to determine the motion of a single section, and the problem then becomes analogous to the vibration of a com- pound pendulum, by which, according to the theory of James Bernoulli, the motions acquired and lost at each instant of time ON HYDRAULICS AS A BRANCH OP ENGINEERING. 155 form an equilibrium, as may be supposed to take place with the different sections of a fluid in a pipe, each section being animated with velocities acquired and lost at every instant of time. The theory of Bernoulli had not been proposed by him until long after the discovery of the indirect principle of vis viva by Huygens. The same was the case with the problem of the mo- tions of fluids issuing from vessels, and it is surprising that no advantage had been taken of it earlier. Michelotti, in his experi- mental researches de Separatione Fluidorum in Corpore Ani- mali, in rejecting the theory of the Newtonian cataract, (which had been advanced in Newton's Mathematical Principles, in the year 1687, but afterwards corrected in the year 1714,) sup- poses the water to escape from an orifice in the bottom of a vessel kept constantly full, with a velocity produced by the height of the superior surface ; and that if, immediately above the lowest plate of water escaping from the orifice, the column of water be frozen, the weight of the column will have no effect on the velocity of the water issuing from the orifice ; and that if this solid column be at once changed to its liquid state, the effect will remain the same. The Marquis Poleni, in his work De Castellis j)er quce derivantur Fliiviornm Aquce, published at Padua in the year 1718, shows, from many experiments, that if A be the orifice, and H the height of the column above it, the quantity of water which issues in a given time is represented by 2 A H X fTTTjT^j whereas if it spouted out from the orifice with a velocity acquired by falling from the height H, it ought to be exactly 2 A H, so that experiment only gives a little more than half the quantity promised by the theory ; hence, if we were to calculate from these experiments the velocity that the water ought to have to furnish the necessary quantity, we should find that it would hardly make it reascend ^rd of its height. These experiments would have been quite contrary to expectation, had not Sir Isaac Newton observed that water issuing from an orifice f ths of an inch in diameter, was contracted Ijths of the diameter of the orifice, so that the cylinder of water which actually issued was less than it ought to have been, according to the theory, in the ratio of 441 to Q9,^ ; and aug- menting it in this proportion, the opening should have been ~ A H jTqqq, or l^ths of the quantity which ought to have issued on the supposition that the velocity was in the ratio of the square root of the height ; from which it was inferred that the theory was correct, but that the discrepancy was owing to cer- 156 THIRD REPORT — 1833. tain resistances, which experiment could alone determine. The accuracy of the general conclusion was affected by several assumptions, namely, the perfect fluidity and sensibility of the mass, which was neither affected by friction nor cohesion, and an infinitely small thickness in the edge of the aperture. Daniel Bernoulli, in his great work, Hydrody namica , seu de Viribns et Motibus Fluidorum Commentaria, published at Stras- burgh in the year 1738, in considering the efflux of water from an orifice in the bottom of a vessel, conceives the fluid to be divided into an infinite number of horizontal strata, on the fol- lowing suppositions, namely, that the upper surface of the fluid always preserves its horizontality ; that the fluid forms a con- tinuous mass ; that the velocities vary by insensible gradations, like those of heavy bodies ; and that every point of the same stratum descends vertically with the same velocity, which is inversely proportional to the area of the base of the stratum ; that all sections thus retaining their parallelism are contiguous, and change their velocities imperceptibly ; and that there is always an equality between the vertical descent and ascent, or vis viva : hence he arrives, by a very simple and elegant pro- cess, to the equations of the problem, and applies its general formulae to several cases of practical utility. When the figure of the vessel is not subject to the law of continuity, or when sudden and finite changes take place in the velocities of the sections, there is a loss of vis viva, and the equations require to be modified. John Bernoulli and Maclaurin arrived at the same conclusions by different steps, somewhat analogous to the cataract of Newton. The investigations of D'Alembert had been directed principally to the dynamics of solid bodies, until it occurred to him to apply them to fluids ; but in following the steps of Bernoulli he discovered a formula applicable to the motions of fluid, and reducible to the ordinary laws of hydro- statics. The application of his theory to elastic and non-elastic bodies, and the determination of the motions of fluids in flexible pipes, together with his investigations relative to the resistance of pipes, place him high in the ranks of those who have contri- buted to the perfection of the science. The celebrated Euler, to whom every branch of science owes such deep obligations, seems to have paid particular attention to the subject of hydrodynamics ; and in attempting to reduce the whole of it to uniform and general formulae, he exhibited a beautiful example of the application of analytical investigation to the solution of a great variety of problems for which he was so famous. The Memoirs of the Academy of Berlin, from the year 1768 to 1771, contain numerous papers relative to fluids ON HYDRAULICS AS A BRANCH OF ENGINEERING. 157 flowing from orifices in vessels, and through pipes of constant or variable diameters. " But it is greatly to be regretted," says M. Prony, " that Euler had not treated of friction and cohesion, as his theory of the linear motion of air would have applied to the motions of fluids through pipes and conduits, had he not always reasoned on the hypotheses of mathematical fluidity, independently of the resistances which modify it." In the year 1765 a very complete work was published at Milan by PauJ Lecchi, a celebrated Milanese engineer, entitled Idrostatica esaminata ne suoi Principi e Stabilite nelle suoi Regole delta Mensura delta Aeque correnti, containing a com- plete examination of all the different theories which had been proposed to explain the phgenomena of effluent water, and the doctrine of the resistance of fluids. The author treats of the velocity and quantity of water, whether absolutely or relatively, which issues from orifices in vessels and reservoirs, according to their different altitudes, and inquires how far the law applies to masses of water flowing in canals and rivers, the velocities and quantities of which he gives the methods of measuring. The extensive and successful practice of Lecchi as an engineer added much to the reputation of his work*. In the year 1764 Professor Michelotti of Turin undertook, at the expense of the King of Sardinia, a very extensive series of experiments on running water issuing through orifices and additional tubes placed at different heights in a tower of the finest masonry, twenty feet in height and three feet square inside. The water was supplied by a channel two feet in width, and under pressures of from five to twenty-two feet. The effluent waters were conveyed into a reservoir of ample area, by canals of brick-work lined with stucco, and having various forms and declivities ; and the experiments, particularly on the efilux of water through differently shaped orifices, and addi- tional tubes of different lengths, were most numerous and accurate, and Michelotti was the first who gave representations of the changes which take place in the figure of the fluid vein, after it has issued from the orifice. His expei'iments on the velocities of rivers, by means of the bent tube of Pitot, and by an instrument resembling a water-wheel, called the stadera idraulica, are numerous and interesting ; but, unfortunately, their reduction is complicated with such various circumstances that it is difficult to deinve from them any satisfactoi'y conclu- sions. But Michelotti is justly entitled to the merit of having made the greatest revolution in the science by experimental * Sec also Memorie Idrostatico-storiclie, 1773. 158 THIRD REPORT 1833. investigation *. The example of Michelotti gave a fresh sti- mulus to the exertions of the French philosophers, to whom, after the ItaUans, the science owes the greatest obligations. Accordingly, the Abb^ Bossut, a most zealous and enhght- ened cultivator of hydrodynamics, undertook, at the expense of the French Government, a most extensive and accurate se- ries of experiments, which he published in the year 1771, and a more enlarged edition, in two volumes, in the year 1786, entitled Traits Theorique et Experimental (THijdro- namique. The first volume treats of the general principles of hydrostatics and hydraulics, including the pressure and equili- brium of non-elastic and elastic fluids against inflexible and flexible vessels ; the thickness of pipes to resist the pressure of stagnant fluids ; the rise of water in barometers and pumps, and the pressure and equilibrivun of floating bodies ; the ge- neral principles of the motions of fluids through orifices of dif- ferent shapes, and their friction and resistance against the orifices ; the oscillations of water in siphons ; the percussion and resistance of fluids against solids ; and machines moved by the action and reaction of water. The second volume gives a great variety of experiments on the motions of water through orifices and pipes and fountains ; their resistances in rectan- gular or curvilinear channels, and against solids moving through them ; and lastly, of the fire- or steam-engine. In the course of these experiments he found that when the water flowed through an orifice in a thin plate, the contraction of the fluid vein diminished the discharge in the ratio of 16 to 10; and when the fluid was discharged through an additional tube, two or three inches in length, the theoretical discharge was diminished only in the ratio of 16 to 13. In examining the effects of fric- tion, Bossut found that small orifices discharged less water in proportion than large ones, on account of friction, and that, as the height of the reservoir augmented, the fluid vein contracted likewise ; and by combining these two circumstances together, he has furnished the means of measuring with precision the quantity of water discharged either from simple orifices or additional tubes, whether the vessels be constantly full, or be allowed to empty themselves. He endeavoured to point out the law by which the diminution of expenditure takes place, according to the increase in the length of the pipe or the num- ber of its bends ; he examined the effect of friction in dimi- nishing the velocity of a stream in rectangular and curvilinear channels ; and showed that in an open canal, with the same * Sperimenti Idraulici, 1767 and 1771. ON HYDRAULICS AS A BRANCH OF EGINEERING. 159 height of reservoh", the same quantity of water is always dis- charged, whatever be the decUvity and length ; that the ve- locities of the waters in the canal are not as the square roots of the declivities, and that in equal declivities and depth of the canal the velocities are not exactly as the quantities of water discharged ; and he considers the variations which take place in the velocity and level of the waters when two rivers unite, and the manner in which they establish their beds. His experiments, in conjunction with D'Alembert and Con- dorcet, on the resistance of fluids, in the year 1777, and his subsequent application of them to all kinds of surfaces, in- cluding the shock and resistance of water-wheels, have justly entitled him to the gratitude of posterity. The Abbe Bossut had opened out a new career of experiments ; but the most dif- ficult and important problem remaining to be solved related to I'ivers. It was easy to perform experiments with water running through pipes and conduits on a small scale, under given and determined circumstances : but when the mass of fluid rolled in channels of unequal capacities, and which were composed of every kind of material, from the rocks amongst which it accu- mulated to the gravel and sand through which it forced a pass- age, — at first a rapid and impetuous torrent, but latterly hold- ing a calm and majestic course, — sometimes forming sand-banks and islands, at other times destroying them, at all times capri- cious, and subject to variation in its force and direction by the slightest obstacles, — it appeared impossible to submit them to any general law. Unappalled, however, by these difficulties, the Chevalier Buat, after perusing attentively M. Bossut's work, undertook to solve them by means of a theorem which appeared to him to be the key of the whole science of hydraulics. He consi- dered that if water was in a perfect state of fluidity, and ran in a bed from which it experienced no resistance whatever, its motion would be constantly accelerated, like the motion of a heavy body descending an inclined plane ; but as the velocity of a river is not accelerated ad infinitum, but arrives at a state of uniformity, it follows that there exists some obstacle which destroys the accelerating force, and prevents it from impressing upon the water a new degree of velocity. This obstacle must therefore be owing either to the viscidity of the water, or to the resistance it experiences against the bed of the river ; from which Dubuat derives the following principle : — That when water runs uniformly in any channel, the accelerating force which obliges it to run is equal to the sum of all the resistances which it experiences, whether arising from the viscidity of the water or the friction of its bed. Encouraged by this discovery, 160 THIRD REPORT — 1833. and by the application of its principles to the solution of a gi'eat many cases in practice, Dubuat' was convinced that the motion of water in a conduit pipe was analogous to the uniform motion of a river, since in both cases gravity was the cause of motion, and the resistance of the channel or perimeter of the pipes the modifiers. He then availed himself of the experiments of Bossut on conduit pipes and artificial channels to explain his theory : the results of which investigations were published in the year 1779. M. Dubuat was, however, sensible that a theory of so much novelty, and at variance with the then received theory, required to be supported by experiments more numerous and direct than those formerly undertaken, as he was constrained to suppose that the friction of the water did not depend upon the pressure, but on the surface and square of the velocity. Accordingly, he devoted three years to making fresh experi- ments, and, with ample funds and assistance provided by the French Government, was enabled to publish his great work, entitled Principes d' Hydraidique verifies par un grand nombre d' Experiences, faittis par Ordre du Gouvernement, 2 vols. 1786, (a third volume, entitled Principes dHydrauUque et Hydro^ namique, appeared in 1816); — in the first instance, by repeating and enlarging the scale of Bossut's experiments on pipes (with water running in them) of difFei'ent inclinations or angles, of from 90° to iqMj*^ P^'^* ^^ ^ right angle, and in channels of from 1 \ line in diameter to 7 and 8 square toises of surface, and sub- sequently to water running in open channels, in which he ex- perienced great difficulties in rendering the motion uniform : but he was amply recompensed by the results he obtained on the diminution of the velocity of the different parts of a uniform current, and of the relation of the velocities at the surface and bottom, by which the water works its own channel, and by the knowledge of the resistances which different kinds of beds pro- duce, such as clay, sand and gravel; and varying the experiments on the effect of sluices, and the piers of bridges, &c., he was ena- bled to obtain a formula applicable to most cases in practice*. Thus, let V = mean velocity per second, in inches. d = hydraulic mean depth, or quotient which arises from dividing the area or section of the canal, in square inches, by the perimeter of the part in contact with the water, in linear inches. s = the slope or declivity of the pipe, or the sur- face of the water. g = 16'087, the velocity in inches which a body acquires in fidling one second of time. * Edinburgh Encyclopedia, Art. Hydrodynamics, by Brewster. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 161 n = an abstract number, which was found by ex- periment to be equal to 243*7. then . = _^Il{^^I^:}L_ _ 0-3 (^^ _ o-i). ys — log. V s + I'o Such are some of the objects of M. Dubuat's work. But his hypotheses are unfortunately founded upon assumptions which render the appUcations of his theory of little use. It is evident that the supposition of a constant and uniform velocity in rivers cannot hold : nevertheless he has rendered great services to the science by the solution of many important questions relating to it ; and although he has left on some points a vast field open to research, he is justly entitled to the merit of originality and accuracy. Contemporary with Dubuat was M. Chezy, one of the most skilful engineers of his time : he was director of the Ecole des Fonts et Chauss^es, and reported, conjointly with M. Perronet, on the Canal Yvette. He endeavoured to assign, by experiment, the relation existing between the inclination, length, trans- versal section, and velocity of a canal. In the course of this investigation he obtained a very simple expression of the velo- city, involving three different variable quantities, and capable, by means of a single experiment, of being applied to all cur- rents whatever. He assimilates the resistance of the sides and bottom of the canal to known resistances, which follow the law of the square of the velocity, and he obtains the following sim- ple formula: V = S^-, where g" is = 16"087 feet, the velocity acquired z s by a heavy body after falling one second. d = hydraulic mean depth, equal to the area of the section divided by the perimeter of the part of the canal in contact with the water. s = the slope or declivity of the pipe. « = an abstract number, to be determined by experiment. In the year 1784, M. Lespinasse published in the Memoirs of the Academy of Sciences at Toulouse two papers, contain- ing some interesting observations on the expenditure of water through large orifices, and on the junction and separation of rivers. The author had performed the experiments contained in his last paper on the rivers Fresquel and Aude, and on that part of the canal of Languedoc below the Fresquel lock, towards its junction with that river. As we before stated, M. Dubuat had classified with much 1833. M 162 THIRD REPORT — 1833. sagacity his observations on the different kinds of resistance experienced in the motion of fluids, and which might have led him to express the sum of the resistances by a rational function of the velocity composed of two or three terms only. Yet the merit of this determination was reserved to M. Coulomb, who, in a beautiful paper, entitled " Experiences destinees a deter- miner la Coherence des Fluides et les Lois de leurs Resistances dans les Mouvemens tres lents," proves, by reasoning and facts, 1st, That in extremely slow motions the part of the resist- ance is proportional to the square of the velocity. 2ndly, That the resistance is not sensibly increased by in- creasing the height of the fluid above the resisting body. 3rdly, That the resistance arises solely from the mutual co- hesion of the fluid particles, and not from their adhesion to the body upon which they act. 4thly, That the resistance in clarified oil, at the temperature of 69° Fahrenheit, is to that of water as 17*5 : 1 ; a proportion which expresses the ratio of the mutual cohesion of the par- ticles of oil to the mutual cohesion of the particles of water. M. Coulomb concludes his experiments by ascertaining the resistance experienced by cylinders that move very slowly and perpendicularly to their axes, &c. This eminent philosopher, who had applied the doctrine of torsion with such distinguished success in investigating the phasnomena of electricity and magnetism, entertained the idea of examining in a similar manner the resistance of fluids, con- trary to the doctrines of resistance previously laid down. M. Coulomb proved, that in the resistance of fluids against solids, there was no constant quantity of sufficient magnitude to be detected ; and that the pressure sustained by a moving body is represented by two terms, one which varies as the simple velocity, and the other with its square. The apparatus with which these results were obtained con- sisted of discs of various sizes, which were fixed to the lower extremity of a brass wire, and were made to oscillate under a fluid by the force of torsion of the wire. By observing the successive diminution of the oscillations, the law of resistance was easily found. The oscillations which were best suited to these experiments continued for twenty or thirty seconds, and the amplitude of the oscillation (that gave the most regular re- sults) was between 480 the entire division of the disc, and 8 or 10 divisions from zero. The first who had the happy idea of applying the law of Coulomb to the case of the velocities of water running in na- tural or artificial channels was M. Girard, Ingenieur en chef ON HYDRAULICS AS A BRANCH OF ENGINEERING. 163 des Fonts et Chaussees, and Director of the Works of the Canal I'Ourcq at Paris *. He is the author of several papers on the theory of running waters, and of a valuable series of experiments on the motions of fluids in capillary tubes. M. Coulomb had given a common coefficient to the two terms of his formula representing the resistance of a fluid, — one pro- portional to the simple velocity, the other to the square of the velocity. M. Girard found that this identity of the coefficients was applicable only to particular fluids under certain circum- stances ; and his conclusions were confirmed by the researches of M. Prony, derived from a great many experiments, which make the coefficients not only different, but very inferior to the value of the motion of the filaments of the water contiguous to the side of the pipe. The object of M. Girard's experiments was to determine this velocity ; and this he has effected in a very satisfactory manner, by means of twelve hundred experiments, performed with a series of copper tubes, from 1*88 to 2'96 millimetres in diameter, and from 20 to 222 centimetres in length ; from which it appeared, that when the velocity was expressed by 10, and the temperature was 0, centigrade, the velocity was increased four times when the temperature amounted to 85°. When the length of the capillary tube was below that limit, a variation of temperature exercised very little influence upon the velocity of the issuing fluid, &c. It was in this state of the science that M. Prony (then having under his direction different projects for canals,) undertook to reduce the solutions of many important problems on running water to the most strict and rigorous principles, at the same time capable of being applied with facility to practice. For this purpose he selected fifty-one experiments which corresponded best on conduit pipes, and thirty-one on open conduits. Proceeding, therefore, on M. Giraird's theory of the analogy between fluids and a system of corpuscular solids or material bodies, gravitating in a curvilinear channel of indefinite length, and occupying and abandoning successively the dif- ferent parts of the length of channel, he was enabled to express the velocity of the water, whether it flows in pipes or in open conduits, by a simple formula, free of logarithms, and requiring merely the extraction of the square root-f. • Essai stir le Mouvement des Eaux courantes : Paris 1804. Recherches sitr les Eaux publiqiies, ^-c. Devis general du Canal I'Ourcq, ^-c. + Memoires dfs Savans Ef rangers, 8fc. 1815. M 2 164 THIRD REPORT — 1833. Thus v= - 0-0469734 + a/ 0-00^2065 + 3041-47 x G, which gives the velocity in metres : or, in English feet, ?;= - 0-1541131 + \/~0W3751 + 32806-6 x G. When this formula is apphed to pipes, we must take G = :^DK, rj I 7 rr which is deduced from the equation K = j . When it is applied to canals^, we must take G = R I, which is deduced from the equation I = :p, R being equal to the mean radius of Dubuat on the hydraulic mean depth, and I equal to the sine of inclination in the pipe or canal. M. Prony has drawn up ex- tensive Tables, in which he has compared the observed velo- cities with those which are calculated from the preceding for- mulae, and from those of Dubuat and Girard. In both cases the coincidence of the observed results with the formulae are very i-emarkable, but particularly with the formulae of M. Prony. But the great work of M. Prony is his Noiwelle Architecture Hydraulique, published in the year 1790. This able produc- tion is divided into five sections, viz. Statics, Dynamics, Hydro- statics, Hydrodynamics, and on the physical circumstances that influence the motions of Machines. The chapter on hydro- dynamics is particularly copious and explanatory of the motions of compressible and incompressible fluids in pipes and vessels, on the principle of the parallelism of the fluid filaments, and the efflux of water through different kinds of orifices made in vessels kept constantly full, or permitted to empty themselves ; he details the theory of the clepsydra, and the curves described by spouting fluids ; and having noticed the different phaenomena of the contraction of the fluid vein, and given an account of the ex- periments of Bossut, M. Prony deduces formulae by which the re- sults may be expressed with all the accuracy required in practice. In treating of the impulse and resistance of fluids, M. Prony explains the theory of Don George Juan, which he finds con- formable to the experiments of Smeaton, but to differ very ma- terially from the previously received law of the product of the surfaces by the squares of the velocities, as established by the joint experiments of D'Alembert, Condorcet and Bossut, in the year 1775. The concluding part of the fourth section is de- voted to an examination of the theory of the equilibrium and motion of fluids according to Euler and D'Alembert ; and by a rigorous investigation of the nature of the questions to be de- termined, the whole theory is reduced to two equations only, in narrow pipes, according to the theory of Euler, showing its approximation to the hypothesis of the parallelism of filaments. I ON HYDRAULICS AS A BRANCH OF ENGINEERING. 165 The fifth and last section investigates the different circum- stances (such as friction, adhesion and rigidity,) which influence the motions of machines. A second volume, published in the year 1796, is devoted to the theory and practice of the steam-engine. Previously to the memoir of M. Prony, Sur le Jaugeage des Eaux courantes, in the year 1802, no attempt had been made to establish vi^ith cer- tainty the correction to be applied to the theoretical expendi- tures of fluids through orifices and additional tubes. The phag- nomenon had been long noticed by Sir Isaac Newton, and illus- trated by Michelotti by a magnificent series of experiments, which, although involving some intricacies, have certainly formed the groundwork of all the subsequent experiments upon this particular subject. By the method of interpolation, M. Prony has succeeded in discovering a series of formulas applicable to the expenditures of currents out of vertical and horizontal orifices, and to the con- traction of the fluid vein ; and in a subsequent work, entitled Recherches sur le Mouvemens des Eaux courantes, he establishes the following formulae for the mean velocities of rivers. When V = velocity at the surface, and U = mean velocity, U = 0-816458 V, which is about f V. These velocities are determined by two methods. 1st, By a small water-wheel for the velocity at the surface, and the im- proved tube of Pitot for the velocities at different depths below the surface. If h = the height of the water in the vertical tube above the level of the current, the velocity due to this height will be deter- / metres mined by the formula V= V2gh=\/ 19-606 h = 4-429 -//*. When water runs in channels, the inclination usually given amounts to between ^^^q*^ ^^^ ^o o*^ P^'^' ^^ *^^ length, which will give a velocity of nearly 1^ mile per hour, sufficient to allow the water to run freely in earth. We have seen the incli- nation very conveniently applied in cases of drainage at xzW*-^ and y^o*^' ^^^ some rivers are said to have ^^%o*^ only. M. Prony gives the following formulas, from a great number of observations : If U = mean velocity of the water in the canal, I = the inclination of the canal per metre, R = the relation of the area to the profile of its perimeter, we shall have U = - 0-07 + V0'005 + 3233. R71; 166 THIRD REPORT 1833. and for conduit pipes, calling U = the mean velocity, Z = the head of water in the inferior orifice of the pipe, L = the length of the pipe in metres, D = the diameter of the pipe, we shall have U = - 0-0248829 + \/0-000619159 + 717-857 DZ or, where the velocity is small, U = 26-79 ^DZ; L that is, the mean velocities approximate to a direct ratio com- pounded of the squares of the diameters and heads of water, and inversely as the square root of the length of the pipes : and by experiments made with great care, M. Prony has found that the formula \J = - 0-0248829 + -/ 0-000619159 + 717-857 DZ L scarcely differs more or less from experiments than ^^^ or gL. The preceding formulae suppose that the horizontal sections, both of the reservoir and the recipient, are great in relation to the transverse section of the pipe, and that the pipe is kept constantly full *. In comparing the formulae given for open and close canals, M. Prony has remarked that these formulae are not only similar, but the constants which enter into their composition are nearly the same ; so that either of them may represent the two series of phaenomena with sufficient exactness. The following formula applies equally to open or close canals : U = - 0-0469734 + ^Z (^0-0022065 + 3041-47 ^V But the most useful of the numerous formulae given by M. Prony for open canals is the following : * According to Mr. Jardine's experiments on the quantity of water delivered by the Coniston Main from Coniston to Edinburgh, the following is a compa- rison : Scots Pints. Actual delivery of Coniston Main 189-4 Ditto by Eytelwein's formula 189'77 Ditto by Girard's formula 188-26 Ditto by Dubuat's formula 188-13 Ditto by Prony "s simple formula 192-32 Ditto by Prony 's tables 180-7 ON HYDRAULICS AS A BRANCH OF ENGINEERING. 167 Let g = the velocity of a body falling in one second, w = the area of the transverse section, p — the perimeter of that section, I = the inclination of the canal, Q = the constant volume of water through the section, U = the mean velocity of the water, R = the relation of the area to the perimeter of the section; then 1st, 0-000436 U + 0-003034 V^- = glR = gl-; 2ndly, U = ^; 3rdly, R w' - 0-0000444499 .wj- 0-000309314 y = 0. This last equation, containing the quantities w QIw and R = — , p shows how to determine one of them, and, knowing the three others, we shall have the following equations : 4thly, p = 0.000436 Qw + 0-003034 Q^' rfui T P (0-0000444499 Qw + 0-000 309314 Q^) 6thly, . = 0-000436 ± ^[(0-000^ + 4 |-003034) g RI] Q^ These formulae are, however, modified in rivers by circum- stances, such as weeds, vessels and other obstacles in the rivers ; in which case M. Girard has conceived it necessary to introduce into the formulas the coefficient of correction = 1-7 as a multiplier of the perimeter, by which the equations will be, • p-1'7 (0-000436 U + 0-003034 W) = gl w. The preceding are among the principal researches of this distinguished philosopher *. In the year 1798, Professor Venturi of Modena published a very interesting memoir, entitled Sur la Communication lat^- rale du Mouvement des Fluides. Sir Isaac Newton was well acquainted with this communication, having deduced from it the propagation of rotary motion from the interior to the exte- rior of a whirlpool ; and had affirmed that when motion is pro- pagated in a fluid, and has passed beyond the aperture, the • Recherchcs Physico-mathematiques sur la Thcorie des Eaux courantes, par M. Prony. 168 THIRD REPORT — 1833. motion diverges from that opening, as from a centre, and is propagated in right Unes towards the lateral parts. The sim- ple and immediate application of this theorem cannot be made to a jet or aperture at the surface of still water. Circumstances enter into this case which transform the results of the principle into particular motions. It is nevertheless true that the jet communicates its motion to the lateral parts without the orifice, but does not repel it in a radial divergency. M. Venturi illus- trates his theory by experiments on the form and expenditure of fluid veins issuing from orifices, and shows how the velocity and expenditure are increased by the application of additional tubes; and that in descending cylindrical tubes, the upper ends of which possess the form of the contracted vein, the expense is such as corresponds with the height of the fluid above the inferior extremity of the tube. The ancients remarked that a descending tube applied to a reservoir increased the expendi- ture*. D'Alembert, Euler and Bernoulli attributed it to the pressure of the atmosphere. Gravesend, Guglielmini and others sought for the cause of this augmentation in the weight of the atmosphere, and determined the velocity at the bottom of the tube to be the same as would ai'ise from the whole height of the column, including the height of the reservoir. Guglielmini supposed that the pressure at the orifice below is the same for a state of motion as for that of rest, which is not true. In the experiments he made for that purpose, he paid no regard either to the diminution of expenditure produced by the irregularity of the inner surface of the tubes, or the augmentation occa- sioned by the form of the tubes themselves. But Venturi esta- blished the proposition upon the principle of vertical ascension combined with the pressure of the atmosphere, as follows : 1st, That in additional conical tubes the pressure of the at- mosphere increases the expenditure in the proportion of the exterior section of the tube to the section of the contracted vein, whatever be the position of the tube. Sndly, That in cylindrical pipes the expenditure is less than through conical pipes, which diverge from the contracted vein, and have the same exterior diameter. This is illustrated by experiments with differently formed tubes, as compared with a plate orifice and a cylindrical tube, by which the ratios in point of time were found to be 41", 31" and 27", showing the advan- tage of the conical tube. 3rdly, That the expenditure may be still fiirther increased, * " Calix devaxus amplius rapit." — Frontinus de Aqueductibus. See also Pneumatics of Hero. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 169 in the ratio of 24 to 10, by a certain form of tube, — a circum- stance of which he supposes the Romans were well aware, as appears from their restricting the length of the pipes of con- veyance from the pubHc reservoirs to fifty feet ; but it was not perceived that the law might be equally evaded by applying a conical frustrum to the extremity of the tube. M. Venturi then examines the causes of eddies in rivers; whence he deduces from his experiments on tubes with en- larged parts, that every eddy destroys part of the moving force of the current of the river, of which the course is permanent and the sections of the bed unequal, the water continues more elevated than it would have done if the whole river had been equally contracted to the dimensions of its smallest section, — a consequence extremely important in the theory of rivers, as the retardation experienced by the water in rivers is not only due to the friction over the beds, but to eddies produced from the irregularities in the bed, and the flexures or windings of its course : a part of the current is thus employed to restore an equilibrium of motion, which the current itself continually de- ranges. As respects the contracted vein, it had been pretended by the Marquis de Lorgna* that the contracted vein was no- thing else but a continuation of the Newtonian cataract ; and that the celerity of the fluid issuing from an orifice in a thin plate is much less than that of a body which falls from the height of the charge. But Venturi proved that the contraction of the vein is incomparably greater than can be produced by the acceleration of gravity, even in descending streams, the contraction of the stream being 0*64, and the velocity nearly the same as that of a heavy body which may have fallen through the height of the charge. These experimental principles, which are in accordance with the results of Bossut, Michelotti and Poleni, are strictly true in all cases where the orifice is small in proportion to the section of the reservoir, and when that orifice is made in a thin plate, and the internal afflux of the filaments is made in an uniform manner round the orifice itself. Venturi then shows the form and contraction of the fluid vein by in- creased charges. His experiments with the cone are curious ; and it would have been greatly to be regretted that he had stopped short in his investigations, but for the more extensive researches of Bidone and Lesbros. M. Hachette, in opposition to the theory of Venturi, assigns, as a cause of the increase by additional tubes, the adhesion of the fluid to the sides of the . tubes arising from capillary attraction. * Memorie della Societa Ifaliana, vol. iv. 170 THIRD REPORT — 183.3. In the year 1801, M. Eytelwein, a gentleman well known to the pubUc by his ti'anslation of M. Dubuat's work into German, (with important additions of his own,) published a valuable compendium of hydraulics, entitled Handbuch der Mechanik und der Hydraidik, in which he lays down the following rules. 1 . That when water flows from a notch made in the side of a dam, its velocity is as the square of the height of the head of the water ; that is, that the pressure and consequent height are as the square of the velocity, the proportional velocities being nearly the same as those of Bossut. 3. That the contraction of the fluid vein from a simple orifice in a thin plate is reduced to 0*64. S. For additional pipes the coefficient is 0*65. 4. For a conical tube similar to the curve of contraction 0*98. 5. For the whole velocity due to the height, the coefficient by its square must be multiplied by 8*0458. 6. For an orifice the coefficient must be multiplied by 7*8. 7. For wide openings in bridges, sluices, &c., by 6'9. 8. For short pipes 6"6. 9. For openings in sluices without side walls 5"1. Of the twenty-four chapters into which M. Eytelwein's * work is divided, the seventh is the most important. The late Dr. Thomas Young, in commenting upon this chapter, says : . " The simple theorem by which the velocity of a river is de- termined, appears to be the most valuable of M. Eytelwein's improvements, although the reasoning from which it is deduced is somewhat exceptionable. The friction is nearly as the square of the velocity, not because a number of particles proportional to the velocity is torn asunder in a time proportionally short, — for, according to the analogy of soUd bodies, no more force is destroyed by friction when the motion is rapid than when slow, — but because when a body is moving in lines of a given curva- ture, the deflecting forces are as the squares of the velocities ; and the particles of water in contact with the sides and bottom must be deflected, in consequence of the minute irregularities of the surfaces on which they slide, nearly in the same curvi- linear path, whatever their velocity may be. At any rate (he continues) we may safely set out with this hypothesis, that the principal part of the friction is as the square of the velocity, and the friction is nearly the same at all depths f; for Professor . Robison found that the time of oscillation of the fluid in a bent * See Nicholson's translation of Eytelwein's work. t See my "Experiments on the Friction and Resistance of Fluids," Philo- sopkical Transactions for 1831. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 171 tube was not increased by increasing the pressure against the sides, being nearly the same when the principal part was si- tuated horizontally, as when vertically. The friction will, how- ever, vary, according to the surface of the fluid which is in contact with the solid, in proportion to the whole quantity of fluid ; that is, the friction for any given quantity of water will be as the surface of the bottom and sides of a river directly, and as the whole quantity in the river inversely ; or, supposing the whole quantity of water to be spread on a horizontal sur- face equal to the bottom and sides, the friction is inversely as the height at which the river would then stand, which is called the hydraulic mean depth*." It is, therefore, calculated that the velocities will be a mean proportional between the hydraulic mean depth and the fall, or -j-^tlis of the velocity per second. Professor Robison informs us, that by the experiments of Mr. Watt on a canal eighteen feet wide at the top, seven feet at the bottom, and four feet deep, having a fall of four inches per mile, the velocities were seventeen inches per second at the surface, fourteen inches per second in the middle, and ten inches per second at the bottom, making a mean velocity of fourteen inches per second ; then finding the hydraulic mean depth, and dividing the area of the section by the perimeter, we have „„ „ , or 29*13 inches ; and the fall in two miles being eight inches, we have v'(8 x 29*13) = 15*26 for the mean proportional of ■j-^ths, or 13*9 inches, which agrees very nearly with Mr. Watt's velocity. The Professor has, however, deduced from Dubuat's elabo- rate theories 12*568 inches. But this simple theorem applies only to the straight and equable channels of a river. In a curved channel the theorem becomes more complicated ; and, from observations made in the Po, Arno, Rhine, and other rivers, there appears to be no general rule for the decrease of velocity going downwards. M. Eytelwein directs us to deduct from the superficial velocity ^-^^ for every foot of the whole depth. Dr. Young thinks y^g*^^ ^^ *^^ superficial velocity suf- ficient. According to Major Rennell, the windings of the river Ganges in a length of sixty miles are so numerous as to reduce the declivity of the bed to four inches per mile, the medium rate of motion being about three miles per hour, so that a mean hydraulic depth of thirty feet, as stated to be f rds of the velocity per second, will be 4*47 feet, or three miles per hour. Again, the river when full has thrice the volume of water in it, and its motion is also accelerated in the proportion of 5 to 3 ; * See Nicholsons Journal for 1802, vol. iii. p. 31. 172 THIRD REPORT — 1833. ' and, assuming the hydraulicmean depth to be doubled at the time of the inundation, the velocity will be increased in the ratio of 7 to 5 ; but the inclination of the surface is probably increased also, and consequently produces a further velocity of from 1*4 to 1*7. M. Eytelwein agrees with Gennete*, that a river may absorb the whole of the water of another river equal in magnitude to itself, without producing any sensible elevation in its surface. This apparent paradox Gennete pretends to prove by experiments, from observing that the Danube absorbs the Inn, and the Rhine the Mayne I'ivers ; but the author evi- dently has not attended to the fact, as may be witnessed in the junction of rivers in marshes and fenny countries, — the various rivers which run through the Pontine and other marshes in Italy, and in Cambridgeshire and Lincolnshire in this country : hence the familiar expression of the waters being overridden is founded in facts continually observed in these districts. We have also the experiments of Brunings in the Architecture Hy- draulique Generate de Wiebeking, Wattmann's M^moires sur I'Art de construire les Canaux, and Funk Sur F Architecture Hydraidique generate, which are sufficient to determine the coefficients under different circumstances, from velocities of fths to 7| feet, and of transverse sections from J- to 19135 square feet. The experiments of Dubuat were made on the canal of Jard and the river Hayne ; those of Brunings in the Rhine, the Waal and Ifrel; and those of Wattmann in the drains near Cuxhaven. M. Eytelwein's paper contains formulae for the contraction of fluid veins through orifices f, and the resistances of fluids passing through pipes and beds of canals and rivers, according to the experiments of Couplet, Michelotti, Bossut, Venturi, Dubuat, Wattmann, Brunings, Funk and Bidone. In the ninth chapter of the Handbuch, the author has en- deavoured to simplify, nearly in the same manner as the motion of rivers, the theory of the motion of water in pipes, observing that the head of water may be divided into two parts, one to produce velocity, the other to overcome the friction ; and that the height must be as the length and circumference of the sec- tion of the pipe directly, or as the diameter, — and inversely as the area of the section, or as the square of the diameter. * Experiences sur le Cours des Fleuves, ou Lettre a un Magistral Hollandais, par M. Gennete. Paris 1760. + " Recherches sur le Mouvement de I'Eau, en ayant egard a la Contraction qui a lieu au Passage par divers Orifices, et a la Resistance qui retard le Mouve- ment, le long des Parois des Vases ; par M. Eytelwein," — Memoires de I'Aca- dimie de Berlin, 1814 and 1815. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 173 In the allowance for flexure, the product of its square, multi- plied by the sum of the sines of the several angles of inflection, and then by "0038, will give the degree of pressure employed in overcoming the resistance occasioned by the angles ; and de- ducting this height from the height corresponding to the velo- city, will give the corrected velocity*. M. Eytelwein investigates, both theoretically and experi- mentally, the discharge of water by compound pipes, — the mo- tions of jets, and their impulses against plane and oblique sur- faces, as in watei'-wheels, in which it is shown that the hydraulic pressure must be twice the weight of the generating column, as deduced from the experiments of Bossut and Langsdorft ; and in the case of oblique surfaces, the effect is stated to vary as the square of the sine of the angle of incidence ; but for motions in open water about fths of the difference of the sine from the radius must be added to this square. The author is evidently wrong in calculating upon impulse as forming part of the motion of overshot wheels; but his theory, that the perimeter of a water-wheel should move with half the velocity of a given stream to produce a maximum effect, agrees perfectly with the experiments of Smeaton and others if. The author concludes his highly interesting work by exa- mining the effects of air as far as they relate to hydraulic ma- chines, including its impulse against plane surfaces on siphons • Hence, if / denote the height due to the friction, d = the diameter of the pipe, a = a constant quantity, we shall have, / = V^ ^^ and V^ = •^—,. •' d al But the height employed in overcoming the friction corresponds to the differ- ence between the actual velocity and the actual height, that is, /= A 5-, where b is the coefficient for finding the velocity from the height. Hence we have, V^ = =^-= and V = V — — -. ab' I ab^l -\- d Now Dubuat found h to be 6-6, and aV^ was found to be 0-0211, particularly when the velocity is between six and twenty-four inches per second. Hence or more accurately, V =: 50 v ( -, ^, |. •' \/+ 50 d/ + The author of this paper has made a great many experiments on the max- imum effect of water-wheels ; but the recent experiments of the Franklin Insti- tution, made on a more magnificent scale, and now in the course of trial, eclipse everything that has yet been effected on this subject. See also Poncelet, Me- moire sur les Roues Hydrauliques, and Aubes Courbes par dessovs, ^fc. 1 827. 174 THIRD REPORT — 1833. and pumps of different descriptions, horizontal and inclined helices, bucket-wheels, throwing-wheels, and lastly, on instru- ments for measuring the velocity of streams of water. A very detailed account of the work was given in the Journal of the Royal Institutioti, by the late Dr. Young. But it is due to MM. Dubuat and Prony to state, that M. Eytelwein has exactly followed the steps of these gentlemen in his Theory of the Motion of Water in open Channels. In the year 1809 a valuable series of experiments upon the motions of waters through pipes, was made by MM. Mallet and Vici at Rome, and afterwards by M. Prony*. It had been proved, by experiments made with great care, that the diminution of velocity, and consequent expenditure in pipes, was not in the ratio of the capacity of the pipes, as Fron- tinus had supposed in his valuation of the product of the an- cient module or calice ; and as it was desirable to ascertain the actual product of the three fountains now used at Rome, a se- ries of experiments was undertaken by these gentlemen ; the principal result of which was, that a pipe, of which the gauge was five oncesf, furnished fth more water than five pipes of one once, on account of the diminution of the velocity by friction in the ratio of the perimeters of the orifices as com- pared with their areas. M. Mallet also made a great many researches relative to the distribution of water in the different cities and towns of En- gland and France, with a view to their application at Paris ; of all of which he has published an account. The researches that had been made hitherto on the expendi- ture of water through orifices, had for their object the deter- mination of the velocity and magnitude of the section, by which it is necessary to multiply the velocity to obtain the expense. But although these be the first elements for consideration, they are not sufficient ; for the fluid vein presents other phaenomena equally important, both in the theory and its application, namely, the form and direction of the vein after it has issued from the orifice. The former phaenomena, as we before stated, had been long noticed by Michelotti and others, but nothing precise had been established on the forms and remarkable phag- nomena of the fluid vein itself. Venturi had given three ex- amples. M. Hachette, in two memoirs presented to the Academic Royale des Sciences in 1815 and 1816, also considered the * Notices Historiques, par M. Mallet. Paris 1830. t French measure, or 0-03059 French kilolitres. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 175 form of veins ; and in his Traits des Machines, he states that he had ah'eady given a description of veins issuing from circu- lar, elliptical, triangular and square orifices, without having entered into any detail respecting them, so that that part of the subject was in a great measure involved in doubt. In 1829 a paper, entitled "Experiences sur la Forme et sur la Direction des Veines et des Courans d'Eau, lances par diverses Ouvertures," was read to the Academy of Sciences at Turin by M. Bidone, giving an account of a series of experiments made in the years 1826 and 1827, in the Hydraulic Establishment of the Royal University. The results of these experiments are divided into five articles. The first gives a description of the apparatus and mode of proceeding, and the figures obtained from veins expended from rectilinear and curvilinear orifices, with salient angles pierced in vertical plates, and whose perimeters are formed by straight and curved lines, varying upwards of fifty different ways, with vai-iable and invariable changes, from zero to twenty-two French feet : the area of water was equal to one square inch. The sections of the veins were taken at different distances from the aperture. The results are extremely curi- ous, as illustrating the influence of pressure and divergence on part of a fluid mass not i?i equilibrio, and may be assimilated to the phsenomena presented by the undulation of streams of light. The author contents himself with stating the results, which are further illustrated by diagrams. In a second paper, read to the Accademia delle Scienze in April following of the same year (1829), M. Bidone enters into a theoretical consideration of his experiments, in which he re- presents the greatest contraction of the fluid vein to take place at a distance not exceeding the greatest diameter of the orifice, whatever be the shape ; from which it results that the expres- sion for the expense of the orifice is equal to the sum of the product of each superficial element multiplied by the velocity of the fluid vein ; and as it was determined by experiment that the area of the vena contracta was from 0"60 to 0*62 of the area of the orifice, it follows that this coefficient of con- traction, multiplied by the velocity due to the charge, repre- sents the expenditure. M. Bidone considers the case of a fluid vein reduced to a state of permanence, and expended from a very small orifice, as compared with the sections of the containing vessel, accord- ing to the theory of Euler; and finds that the magnitude of the section of the contracted vein does not depend upon the velocity of the component filaments, but solely on their direction, a re- sult conformable to experiment. 176 THIRD REPORT — 1833. He then determines, from the results of M. Venturoli*, the absolute magnitude of the contracted section of the vein (issuing from a circular orifice) to be exactly f rds of the orifice, the correction due to the contraction depending upon the ad- hesion and friction of the fluid against the perimeter of the ori- fice, and the ratio of the area of the vein to the area of the orifice : the same for all orifices. Hitherto the magnitude of fluid veins, as determined by direct measurements, had given greater coefficients than the effective expenditure allowed. Michelotti, with a pressure of twenty feet, with orifices of one and two inches in diameter, found the coefficient 0*649 Bossut 0-660 Borda 0-646 Venturi 0-640 Eytelwein 0-640 Hachette 0-690 Newton 0-707 Helsham 0-705 Brindley and Smeaton 0-631 Banks 0-750 Rennief 0-621 In several experiments the ratio rarely exceeded 0-620 ; so that the discrepancy must have arisen from inaccuracies in the measurement of the fluid vein and orifice. In the year 1827, it having been considered desirable to re- peat the experiments of Bossut and Dubuat, application was made to the French Government by General Sabatier, Com- mander-in-chief of the Mihtary School at Metz, for permission to undertake a series of experiments on a scale of magnitude sufficient to estabhsh the principles laid down by those authors, and serve as valuable practical rules for future calculations. The apparatus consisted, 1st, of an immense basin, having an area of 25,000 square metres ; 2nd, of a smaller reservoir, having a superficial area of 1 500 square metres, and a depth of 3-70 metres, so contrived, by means of sluices, as to have a complete command of the level of the water during the experi- ment ; 3rd, of a basin directly communicating with the second basin, 3-68 metres in length, and 3 metres in width, to receive the product of the orifices ; 4th, a basin or gauge capable of containing 24,000 litres. * Elementi di Meccanica e d'Idraulka: Milano 1818. Recherche Geome- triche fatte nella Scuola degli Ingegneri pontifici d'Acque e Strode, Fanno 1821. Milano. t "On the Friction and Resistance of Fluids," Philosophical Transactions of 1831. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 177 The time was constantly noticed by an excellent stop-watch, made by Breguet; and the opening of the orifices, the charges of the fluid in the reservoir, as well as the level of the water m the gauge basin relative to each expense of fluid, were always measured to the tenth of a millimetre, so that, even under the most unfavourable circumstances, the approximation was at least to g^odth part of the total result. The total disposable fall or height, counting from the ordinary surface of the Moselle river, was four metres, from which two metres were deducted for the gauge basin, leaving only a fall of two metres under the most favourable circumstances ; and in the subsequent experiments of 1828 the height never exceeded 1-60 metre, sufficiently high for all practical purposes. An apparatus was provided for regulating the height of the orifice and the surface of the water in the reservoirs, and for tracing with the greatest accu- racy the forms and sections of the fluid veins before and after issuing from the orifices, and the depressions experienced by the surface of the water previously to its issuing from an open- ing of twenty centimetres square, the upper side of which was on a level with the surface of the water in the reservoir. These depressions are recorded in the Tables, 1st, On the expenditure of water through rectangular verti- cal orifices, twenty centimetres square, and varying in height from one to twenty centimetres, under charges of from -0174 of a metre to 1"6901 metre: 2ndly, On the expenditures of water from the similar-sized orifices, open at the top, but under charges of from two to twenty-two centimetres. The whole is comprised in eleven Tables of 241 experiments, to which is added a twelfth Table, showing the value of the co- efficients of contraction for complete orifices, from twenty cen- timetres square to one centimetre, calculated according to the following formula: D for the height of the orifices, where* D = lo\/¥gli = l{Ji-¥) ^/2g ^^—^ being the theo- retical expense relative to the velocity ; or the theoretical expense, having regard to the influence of the orifice. • That is, where/ = 020 metre, being thehorizontal breadth of all the orifices; h = the charge of the fluid on the lower part of the orifice ; A'= the charge in the upper or variable side of the orifice ; oz= k — h' the thickness of the vein of water. 18.^3. N 178 THIRD REPORT 1833. -' The conclusions to be derived from these Tables are, 1st, That for complete orifices of twenty centimetres square and high charges, the coefficient is 0*000 ; with the charge equal to four or five times the opening of the orifice, the co- efficient augments to 0"605 ; but beyond that charge the co- efficient diminishes to 0"593. 2ndly, That the same law maintains for orifices of ten and five centimetres in height, the coefficients being for ten centi- metres 0'61], 0-618, 0"611 respectively, and for five centi- metres in height 0-618, 0-631, 0-623. Lastly, That with orifices of three, two and one centimetres in height, the law changes very rapidly, and the coefficients increase as the opening of the orifice becomes less, being for one centimetre 0-698, the smallest height of the orifice, to 0-640 for three centimetres. These remarkable discrepancies from the results of Bidone and others are attributed by MM. Lesbroa and Poncelet to diflferences in the construction of the apparatus or in the mode of measurement adopted by the latter gentlemen ; but in gene- ral the coincidences are sufficiently satisfactory, and they are the more accurately confirmed by the subsequent investigations of MM. D'Aubuisson and Castel at Toulouse *. As respects water issuing from the openings or notches made in the sides of dams, or what we should term incomplete orifices, it appears that the coefficient obtained by the ordinary formula of Dubuat, or I h V 2gh, augments from the total charge of twenty-two cen- timetres when it is from 0-389 to two centimetres when it be- comes 0-415 ; hence we may safely adopt M. Bidone's coefficient of 0-405, or, according to MM. Poncelet and Lesbros' theory 0-400, for calculating expenditures through notches in dams. From these and other experiments the authors are led to con- clude, that the law of continuity maintains for indefinite heights both with complete and incomplete orifices, and that the same coefficient can be obtained by adopting in both cases the same formula. The avithors observe that the area of the section of the greatest contraction of the vein, considered as a true square, is exactly two thirds of the area of the orifice ; a fact which goes to prove that there is no certain comparison be- tween the mean theoretical or calculated velocities, by means of the formula now used, and the mean effective velocities de- rived from the expenditure. The authors conclude their memoir by recommending their experiments for adoption in all cases of plate orifices situated * 4nnales de Chimie et de Physique for 1830, torn. xliv. p. 225. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 17i) at a distance from the sides and bottom of the reservoir, pro- mising to investigate with similar accuracy in a future memoir the cases which may occur to the contrary. A note is appended to the memoir by M. Lesbros, contain- ing formulae for calculating the effective expenditure of com- plete orifices ; and also a Table of constants, which gives the effective expenditure of each orifice as compared with experi- ment. We have been thus particular in detaihng the results of MM. Lesbros and Poncelet's work, because they have com- prehended all the cases upon which there remained any doubts, and with very few exceptions are in accordance with the expe- riments of Brunacci, Navier, Christian, Gueymard, D'Aubuis- son, and by the author of this paper *. So that in point of accuracy and laborious investigation, the authors of these va- luable accessions to our knowledge, not only merit our grati- tude, but have very amply i-ephed to the liberality of the French Government. Having thus endeavoured to elucidate the labours of the foreign philosophers who have contributed so greatly to the progress of hydraulics, it only remains for us to notice the scanty contributions of our countrymen to the science. While France and Germany were rapidly advancing upon the traces of Italy, England remained an inactive spectator of their pro- gress, contented with the splendour of her own Newton, to receive from foreigners whatever was original or valuable in the science. The Philosophical Transactions, rich as they are in other respects, scarcely contain a single paper on this subject founded on any experimental investigations. Some erroneous and inconclusive inferences from Newton, by Dr. Jurin ; a paper on the Measure of Force, by Mr. Eames ; a paper on Wiers, by Mr. Roberts ; another on the Motion and Resistance of Fluids, by Dr. Vince ; and a summary of Bossut and Dubuat's Experiments on the Motions of Fluids through Tubes, by Dr. Thomas Young, comprise nearly the whole of the papers on hydraulics in the Philosophical Transactions. The various treatises on the subject published by Maclaurin, Emerson, Dr. Matthew Young, Desaguliers, Clare and Switzer, with the exception of the theoretical investigations, are compiled principally from the works of foreigners ; and it was not until the subject was taken up by Brindley, Smeaton, Robison, Banks and Dr. Thomas Young, that we were at all aware of our defi- ciency. Practical men were either necessitated to follow the un- certain rules derived from their predecessors, or their own expe- rience and sagacity, for the little knowledge they possessed. * Philosophical Transactions for 18SI, N 2 180 THIRD RKPORT — 1833. On the suliject of lij'drometry ^ve were equally ignorant ; and although the Italian collection had been published several years previously, and was well known on the Continent, it was not until Mr. Mann published an abstract of that collection that we were at all aware of the state of the science abroad. Under these circumstances the author of this paper was in- duced, in the year 1830, to undertake a series of experiments to ascertain, 1st, The friction of water against the surface of a cylinder, and discs revolving in it, at different depths and ve- locities : from which it appeared, that with slow velocities the friction approximated the ratio of the surfaces, but that an in- crease of surface did not materially affect it with increased velo- cities ; and that with equal surfaces the resistances approxi- mated to the squares of the velocities. 2ndly, To ascertain the direct resistances against globes and discs revolving in air and water alternately : from which it resulted, that the resistances in both cases were as the squares of the velocities; and that the mean resistances of circular discs, square plates, and globes of equal area, in atmospherical air, were as under : Circular discs . . 25-180 MS Square plates . . 22'010 in air, . . 1 '36 in water. Round globes . . 10-627 0-75 3rdly, That with circular orifices made in brass plates of gJg-th of an inch in thickness, and having apertures of i, ^, f , y of an inch respectively, under pressures varying from one to four feet, the average coefficients of contraction were, for altitudes of 1 foot 0-619 4 feet 0-621 For additional tubes of glass the coefficient was,. for 1 foot 0-817 4 feet 0-806 4thly, That the expenditures through orifices, additional tubes, and pipes of different lengths, of equal areas and under the same altitude as compared with the expenditui-e through a pipe of 30 feet in length, are as 1 : 3 for orifices, 1 : 4 for additional tubes, 1 : 3-7 for a pipe 1 foot in length, 1 : 2-6 8 feet , 1 : 2-0 4 , 1 : 1-4 2 . 5thly, That with bent rectangular pipes ^ an inch in diameter, and 15 feet in length, the expenditures were diminished with fourteen bends two thirds, as compared with a straight pipe. ON HYDRAULICS AS A BRANCH OF ENGINEERING. 181 and with twenty-four right angles, one third ; but did not seem to observe any decided law. In several experiments tried on a great scale, the results gave from one fifth to one sixth of the altitude for the fric- tion. In the case of the Coniston main, which conducts the water from the reservoir at Coniston to the castle of Edin- burgh, the diameter of which is 41 inches, the length 14,930 feet, and the altitude 51 feet, it was proved by Mr. Jardine that the formulae of Dubuat and Eytelwein approximated to the real results very nearly ; and in some experiments made on a. great scale by the author of this paper, these formulae were found equally applicable. In several experiments made in the year 1828, on the water-works at Grenoble, by M. Gueymard, it was found that pipes of six and eight French inches in dia- meter furnished only two thirds of the water indicated by the formulee of M. Prony ; but when of nine inches diameter, the formula approximated very nearly. In M. Gueymard's expe- riment the altitude of the reservoir above the point of delivery was 8*453 metres, or 27'73 English feet. The height to which the water was required to be elevated was 5'514 metres, or 18 feet ; the volume of water required was 954 litres, or 33*6 cubic feet; the length of the pipe was 3200 metres, or 10498 feet. There were eight gentle curves in the system, but en- larged beyond the average diameter of the parts of the pipe ; from which it resulted that the height to which the water was delivered was only two thirds of the height of the reservoir*. In the preceding short but imperfect history of the science of hydraulics we have confined our attention to the experi- mental researches that have been made on spouting fluids only. In a future communication I hope to examine the state of our knowledge of the natural phaenomena of rivers, and the causes by which they are influenced ; at present it is extremely limited, and although we have many works upon the subject, very little seems to be known either of their properties or of the laws by which they are governed. • According to M. Prony 's theory, the height raised would only have been ■5-514 metres instead of 5-671 metres. The difficulty, however, of making ex- periments on a great scale will always prove an obstacle to the right solution of the question, in as much as it exacts that the pipe be of the same dia- meter throughout, that is, perfectly straight, and free from bends, and the charge of water invariable. For this purpose M. Prony has calculated Tables showing the relation subsisting between the expenditure, diameter, length, the total inclination of the pipes, and the difference of pressure at its extremities. 182 THIRD REPORT — lS'6o. Appendix. Since the foregoing Report was read to the British Associa- tion a paper, entitled " Memoire sur la Constitution des Veines Liquides lancees par des Orifices Circulaires en mince paroi," has been communicated to the Academy of Sciences at Paris, by M. FeUx Savart, 26 Aout 1833. The author, after detailing very minutely the different phsenomena presented by liquid veins issuing from circular orifices perforated in thin plates, attached to the bottom and sides of vessels, illustrates his po- sitions by a series of curious experiments on the vibrations and sounds of the drops which issue from the annular rings or pipes formed by the troubled part of the liquid. The results of these experiments are best given in his own words. " 1°. Toute veine liquide lancee verticalement de haut en bas par un orifice circulaire pratique dans une parol plane et hori- zontale est toujours composee de deux parties bien distinctes par I'aspect et la constitution. La partie qui touche a I'orifice est un solide de revolution dont toutes les sections horizontales vont en decroissant graduellement de diametre. Cette premiere partie de la veine est calme et transparente, et ressemble a un tige de cristal. La seconde partie, au contraire, est toujours agitde, et paralt denuee de transparence, quoiqu'elle soit ce- pendant d'une forme assez reguliere pour qu'on puisse facile- ment voir quelle est divisee en un certahi nombre de ren- flemens allonges dont le diametre maximum est toujours plus grand que celui de I'orifice. " 2°. Cette seconde partie de la veine est composee de gouttes bien distinctes les unes des autres, qui subissent pendant leur chute, des changemens periodiqvies de forme, auxquels sont dues les apparences de ventres ou renflemens reguherement espaces que I'inspection directe fait reconnaitre dans cette partie de la veine, dont la continuite apparente depend de ce que les gouttes se succedent a des intervalles de temps qvii sont moindres que la duree de la sensation produite sur la retine par chaque goutte en particulier. " 3°. Les gouttes qui forment la partie trouble de la veine sont produites par des renflemens annulaires qui prennent naissance tres pres de I'orifice, et qui se propagent a des inter- valles de temps ^gaux, le long de la partie limpide de la veine, en augmentant de volume a mesure qu'ils descendent, et qui enfin se separent de I'extremite inferieure de la partie limpide et continue a des intervalles de temps egaux a ceux de leur production et de leur propagation. ON HYDRAULICS AS A BRANCH OF ENGINEERING, 183 "4°. Ces renflemens annulaires sent engendr^s par une suc- cession periodique de pulsations qui ont lieu a rorifice meme ; de sorte que la vitesse de lecouleuient, au lieu d'etre uniforme, est periodiquement variable. "5". Le nombre de ces pulsations, m^me pour des charges foibles, est toujours assez grand, dans un temps donne, pour qu'elles soient de I'ordre de celles qui, par la frequence de leur retour, peuvent donner lieu a des sons perceptibles et compa- rables. Ce nombre ne depend que de la vitesse de I'ecoule- ment, a laquelle il est directement proportionnel, et du diametre des orifices, auquel il est inversement proportionnel. II ne pa- rait alter e ni par la nature du liquide, ni par la temperature. "6°. L'amplitude de ces pulsations pent etre considerable- ment augmentee par des vibrations de m^me periode commu- niquees a la masse entiere du liquide et aux parois du reservoir qui le contient. Sous cette influence etrangere, les dimensions et I'etat de la veine peuvent subir des changemens remarqua- bles : la longueur de la partie limpide et continue pent se reduire presqu'a rien, tandis que les ventres de la partie trouble acquierent vine regularite de forme et une transparence qu'ils ne possedent pas ordinairement. Lorsque le nombre des vibra- tions communiquees est diiferent de celui des pulsations qui ont lieu a I'orifice, leur influence peut meme aller jusqu'a changer le nombre de ces pulsations, mais seulement entre de certaines limites. " 7°. La depense ne paralt pas alteree par l'amplitude des pulsations, ni meme par leur nombre. " 8°. La resistance de I'air n'influe pas sensiblement sur la forme et les dimensions des veines, non plus que sur le nombre des pulsations. " 9°. La constitution des veines lancees horizontalement ou m&me obliquement de bas en haut ne difFere pas essentiellement de celle des veines lancees verticalement de haut en bas ; seule- ment le nombre des pulsations a I'orifice parait devenir d'autant moindre que le jet approche plus d'etre lance verticalement de bas en haut. " 10°. Quelle que soit la direction de la veine, son diametre decroit toujours tres rapidement jusqu'a une petite distance de I'orifice ; mais quand la veine tombe verticalement, le decroisse- ment continue jusqu'a ce que la partie limpide se perde dans la partie trouble : il en est encore de m^me quand la veine est lancee horizontalement, quoiqu'alors le decroissement suive une loi moins rapide. Lorsque le jet est lance obliquement de bas en haut, et qu'il forme avec I'horizon un angle de 25° a 45°, toutes les sections normales a la courbe qu'il decrit deviennent 184 THIRD REPORT — 1833. sensiblement %ales entre elles, a partir de la partie contractee que touche a rorifice. Enfin, pour des angles plus grands que 45°, le diametre de la veine va en augmentant depuis la partie contractee jusqu'a la naissance de la portion trouble ; de sorte que c'est seulement alors qu'il existe une section qu'on peut k juste titre appeler section contractee." 185 Report on the Recent Progress and Present State of certain Branches of Analysis. By George Peacock, M.A., F.R.S., F.G.S., F.Z.S., F.R.A.S., F.C.P.S., Fellow and Tutor of Trinity College, Cambridge. The present Report was intended in the first instance to have comprehended some notice of the recent progress and present state of analytical science in general, including algebra, the application of algebra to geometry, the differential and integral calculus, and the theory of series : a very little progress, how- ever, in the inquiries which were required for the execution of this tmdertaking convinced me of the necessity of confining them within much narrower limits, unless I should have ven- tured to occupy a much larger space in the annual publication of the Proceedings and Reports of the British Association than could be properly or conveniently allotted to one department of science, when so many others were required to be noticed. It is for these reasons that I shall restrict my observations, in the following Report, to Algebra, Trigonometry, and the Arithmetic of Sines ; at the same time I venture to indulge a hope of being allowed, upon some future occasion, to bring before the Members of the Association some notice of those higher branches of analysis which at present I feel myself compelled, though reluctantly, to omit. Algebra. — The science of algebra may be considered under two points of view, the one having reference to its principles, and the other to its applications : the first regards its complete- ness as an independent science ; the second its usefulness and power as an instrument of investigation and discovery, whether as respects the merely symbolical results which are deducible from the systematic developement of its principles, or the ap- plication of those results, by interpretation, to the physical sciences. Algebra, considered with reference to its principles, has re- ceived very little attention, and consequently very little im- provement, during the last century ; whilst its applications, using that term in its largest sense, have been in a state of continued advancement. Many causes have contributed to this comparative neglect of the accurat^^ and logical examination of the first principles of algebra : in the first place, the proper 186 THIRD REPORT — 18o3. assumption and establishment of those principles involve meta- physical difficulties of a very serious kind, which present them- selves to a learner at a period of his studies when his mind has not been subjected to such a system of mathematical discipline as may enable it to cope with them : in tlie second place, we are commonly taught to approach those difficulties under the cover of a much more simple and much less general science, by steps which are studiously smoothed down, in order to render the transition from one science to the other as gentle and as little startling as possible ; and lastly, from the peculiar relation which the first principles of algebra, in common with those of other sciences of strict demonstration, bear to the great mass of facts and reasonings of which those sciences are composed. It is this last circumstance which constitutes a marked distinc- tion between those sciences which, like algebra and geometry, are founded upon assumed principles and definitions, and the physical sciences : in one case we consider those principles and definitions as ultimate facts, from which our investigations pro- ceed in one direction only, giving rise to a series of conclusions which have reference to those facts alone, and whose correct- ness or truth involves no other condition than the existence of a necessary connexion between them, in whatever manner the evidence of that existence may be made manifest ; whilst in the physical sciences there are no such ultimate facts which can be considered as the natural or the assumable limits of our inves- tigations. It is true, indeed, that in the application of algebra or geometry to such sciences, we assume certain facts or prin- ciples as possessing a necessary existence or truth, investing them, as it were, with a strictly mathematical character, and making them the foundation of a system of propositions, whose connexion involves the same species of evidence with that of the succession of propositions in the abstract sciences ; but in as- signing to such propositions their proper interpretation in the physical world, our conclusions are only true to an extent which is commensurate with the truth and universality of application of our fundamental assumptions, and of the various conditions by which the investigation of those propositions has been sup- posed to be limited ; in other words, such conclusions can be considered as approximations only to physical truth ; for such assumed first principles, however vast may be the superstruc- ture which is raised upon them, form only one or more links in the great chain of propositions, the tennination and foundation of which must be for ever veiled in the mystery of the first cause. It is not my intention to enter upon the examination of the REPORT ON CERTAIN BRANCHES OF ANALYSIS, 187 general relations which exist between the speculative and physi- cal sciences, but merely to point out the distinction between the ultimate objects of our reasonings in the one class and in the other : in the first, we merely regard the results of the science itself, and the logical accuracy of the reasoning by which they are deduced from assumed first principles; and all our conclu- sions possess a necessary existence, without seeking either for their strict or for their approximate interpretation in the nature of things : in the second, we found our reasonings equally upon assumed first principles, and we equally seek for logical accu- racy in the deduction of our conclusions from them ; but both in the principles themselves and in the conclusions from them, we look to the external world as furnishing by interpretation corresponding principles and corresponding conclusions ; and the physical sciences become more or less adapted to the ap- plication of mathematics, in proportion to the extent to which our assumed first principles can be made to approach to the most simple and general facts or principles which are discover- able in those sciences by observation or experiment, when di- vested of all incidental and foreign causes of variation ; and still more so, when the causes of such variation can be di- stinctly pointed out, and when their extent and influence are reducible to approximate at least, if not to accurate estimation. The first principles, therefore, which form the foundation of our mathematical reasonings in the physical sciences being neither arbitrary assumptions nor necessary truths, but really forming part of the series of propositions of Avhich those sci- ences are composed, can never cease to be more or less the subject of examination and inquiry at any point of our re- searches : they form the basis of those interpretations which are perpetually required to connect our mathematical with the corresponding physical conclusions ; and even supposing the immediate appeal to them to be superseded, as will frequently be the case, by other propositions which are deducible from them, they still continue to claim our attention as the proposi- tions which terminate those physical and logical inquiries at which our mathematical reasonings begin. But in the abstract sciences of geometry and algebra, those principles which are the foundation of those sciences are also the proper limits of our inquiries ; for if they are in any way connected with the phy- sical sciences, the connexion is arbitrary, and in no respect af- fects the truth of our conclusions, which respects the evidence of their connexion with the first principles only, and does not require, though it may allow, the aid of physical interpretation. It is true that there exists a connexion between physical and 188 THIRD REPORT — 1833. speculative geometry, as well as between physical and specula- tive mechanics ; and if in speculative geometry we regarded the actual construction and mensuration of the figures and solids in physical geometry alone, the transition from one science to the otlier being made by interpretation, then speculative geo- metry and speculative mechanics must be regarded as sciences which were similar in their character, though different in their objects : but we cultivate speculative geometry without any such exclusive reference to physical geometry, as an in- strument of investigation more or less applicable, by means of interpretation, to all sciences which are reducible to mea- sure, and whose abstract conclusions, in whatever manner suggested or derived, possess a great practical value altogether apart from their applications to practical geometry ; whilst the conclusions in speculative mechanics are valuable from their applications to physical mechanics only, and are not other- wise separable from the conclusions of those abstract sciences which are employed as instruments in their investigation. This separation of speculative and physical geometry was perfectly understood by the ancients, though their views of its application to the physical sciences were extremely limited; and it is to the complete abstraction of the principles of specu- lative geometry that we must in a great measure attribute the vast discoveries which were made by its aid in the hands of Newton and his predecessors, when a more enlarged and phi- losophical knowledge of the laws of nature supplied those phy- sical axioms or truths which were required as the medium of its applications ; and though it was destined to be superseded, at least in a great degree, by another abstract science of much greater extent and applicability, yet it was enabled to maintain its ground for a considerable time against its more powerful rival, in consequence of the superior precision of its prin- ciples and the superior evidence of its conclusions, when con- sidered with reference to the form under which the principles and conclusions of algebra were known or exhibited at that period. Algebra was denominated in the time of Newton specious or universal arithmetic, and the view of its principles which gave rise to this synonym (if such a term may be used) has more or less prevailed in almost every treatise upon this subject which has appeared since his time. In a similar sense, algebra has been said to be a science which arises from that generalization of the processes of arithmetic which results from the use of symbolical language : but though in the exposition of the prin- ciples of algebra, arithmetic has always been taken for its foun- REPORT ON CERTAIN BRANCHES OF ANALYSIS, 189 dation, and the names of the fundamental operations in one science have been transferred to the other without any imme- diate change of their meaning, yet it has generally been found necessary subsequently to enlarge this very narrow basis of so very general a science, though the reason of the necessity of doing so, and the precise point at which, or the extent to which, it was done, has usually been passed over without notice. The science which was thus formed was perfectly abstract, in what- ever manner we arrived at its fundamental conclusions ; and those conclusions were the same whatever view was taken of their origin, or in whatever manner they were deduced ; but a serious error was committed in considering it as a science which admitted of strict and rigorous demonstration, when it certainly possessed no adequate principles of its own, whether assumed or demonstrated, which could properly justify the character which was thus given to it. There are, in fact, two distinct sciences, arithmetical and symbolical algebra, which are closely connected with each other, though the existence of one does not necessarily deter- mine the existence of the other. The first of these sciences would be, properly speaking, universal arithmetic : its general symbols would represent numbers only ; its fundamental ope- rations, and the signs used to denote them, would have the same meaning as in common arithmetic ; it would reject the inde- pendent use of the signs + and — , though it would recognise the common rules for their incorporation, when they were preceded by other quantities or symbols : the operation of subtraction would be impossible when the subtrahend was greater than the quantity from which it was required to be taken, and there- fore the proper imjjossible quantities of such a scienee-would be the negative quantities of symbolical algebra ; it would re- ject also the consideration of the multiple values of simple roots, as well as of the negative and impossible roots of equa- tions of the second and higher degree : it is this species of al- gebra which alone can be legitimately founded upon arithmetic as its basis. Mr. Frend *, Baron Maseres, and others, about the latter end of the last century, attempted to introduce arithmetical * The Principles of Algebra, by William Frend, 1796; and The true The- ory of Equations, established on Mathematical Demonstration, 1799. The fol- lowing extracts from his prefaces to these works will explain the nature of his views : " The ideas of number are the clearest and most distinct of the human mind : the acts of the mind upon them are equally simple and clear. There cannot be confusion in them, unless numbers too great for the comprehension of the 190 THIRD REPORT— 1833. to the exclusion of sijmboUcal algebra, as the only form of it which was capable of strict demonstration, and which alone, therefore, was entitled to be considered as a science of strict and logical reasoning. The arguments which they made use of were unanswerable, M^hen advanced against the form under which the principles of algebra were exhibited in the elemen- tary and all other works of that period, and which they have continued to retain ever since, with very trifling and unimpor- tant alterations ; and the system of algebra which was formed by the first of these authors was perfectly logical and complete, the connexion of all its parts being capable of strict demon- stration; but there were a great multitude of algebraical re- sults and propositions, of unquestionable value and of unques- tionable consistency with each other, which were irreconcila- ble with such a system, or, at all events, not deducible from it ; and amongst them, the theory of the composition of equations, which Harriot had left in so complete a form, and which made it necessary to consider negative and even impossible quan- learner are employed, or some arts are used which are not justifiable. The first error in teaching the first principles of algebra is obvious on per\ising a few pages only of the first part of Maclaurin's Algebra. Numbers are there divided into two sorts, positive and negative : and an attempt is made to explain the nature of negative numbers, by allusions to book debts and other arts. Now when a person cannot explain the pi-inciples of a science, without reference to a metaphor, the probability is, that he has never thought accurately upon the subject. A number may be greater or less than another number : it may be added to, taken from, multiplied into, or divided by, another number ; but in other respects it is very intractable; though the whole world should be destroyed, one will be one, and three will be three, and no art whatever can change their nature. You may put a mark before one, wiiich it will obey ; it submits to be taken away from another number greater than itself, but to attempt to take it away from a number less than itself is ridiculous. Yet this is attempted by algebraists, who talk of a number less than nothing, of multiplying a negative number into a negative number, and thus producing a positive number, of a number being imaginary. Hence they talk of tvyro roots to every equation of the second order, and the learner is to try which will succeed in a given equa- tion : they talk of solving an equation which requii-es two impossible roots to make it soluble : they can find out some impossible numbers, which being multiplied together produce unity. This is all jargon, at which common sense recoils ; but from its having been once adopted, like many other figments, it finds the most strenuous supporters among those who love to take things upon trust and hate the colour of a serious thought." " From the age of Vieta, the father, to this of Maseres, the restorer of alge- bra, many men of the greatest abilities have employed themselves in the pursuit of an idle hypothesis, and have laid down rules not founded in truth, nor of any sort of use in a science admitting in every step of the plainest principles of reasoning. If the name of Sir Isaac Newton appears in this list, the number of advocates for errour must be considerable. It is, however, to be recollected, that for a much longer period, men scarcely inferiour to Newton in genius, and his equals, probably, in industry, maintained a variety of positions in philoso- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 191 titles as having a real existence in algebra, however vain might be the attempt to interpret their meaning. Both Mr. Frend and Baron Maseres were sensible of the con- sequences of admitting the truth of this theory of the compo- sition of equations as far as their system was concerned, and it must be allowed that they have struggled against it with con- siderable ingenuity: they admitted the possibility of multiple real, that is, positive roots, and which are all equally congruous to the problem whose solution was required through the medium of the equation, indicating an indetermination in the problem proposed : but it would be easy to propose problems leading to equations whose roots were real and positive, and yet not con- gnious to the problem proposed, whose existence must be ad- mitted upon their own principles ; and if so, why not admit the existence of other roots, whether negative or impossible, to which the algebraical solution of the problem might lead, though they might admit of no very direct interpretation, in conformity with the expressed conditions of the problem*? phy, which were overthrown by a more accurate investigation of nature ; and if the name Ptolemy can no longer support liis epicycles, nor that of Des Cartes his vortices, Newton's dereliction of the principles of reasoning cannot establish the fallacious notion, that every equation has as many roots as it has dimensions." " This notion of Newton and others is founded on precipitation. Instead of a patient examination of the subject, an hypothesis which accounts for many appearances is formed; where it fails, unintelligible terms are used; in thos6 terms indolence acquiesces ; much time is wasted on a jargon whicli has the appearance of science, and real knowledge is retarded. Thus volumes upon volumes have been written on the stupid dreams of Athanasius, and on the im- possible roots of an equation of w dimensions." This work of Mr. Frend, though containing many assertions which show great distrust of the results of algebraical science which were in existence at the time it was written, presents a very clear and logical view of the principles of arithmetical algebra. The voluminous labours of Baron Maseres are contained in his Scriptores Logarithmici, and in a thick volume of Tracts on the Resolution of Cubic and Biquadratic Equations. He seems generally to have forgotten that an}' change had taken place in the science of algebra between the age of Ferrari, Cardan, Des Cartes, and Harriot, and the end of the 1 8th century ; and by considering all algebraical formulae as essentially arithmetical, he is speedily overwhelmed by the same multiplicity of cases (which are all included in the same really al- gebraical formula) which embarrassed and confounded the first authors of the science. * Thus, in the solution of the following problem : " Sold a horse for 24?., and by so doing lost as much per cent, as the horse cost me : required the prime cost of the horse ?" we arrive at the equation 100 X —x" = 2400 ; if we subtract both sides of this equation from 2500, we get 2500 — 100 a; + x- = 100, or X- — 100 X -\- 2500 = 100, inastnuch as the quantities upon each side of the sign = are in both cases 192 THIRD REPORT — 1833. If the authors of this attempt at algebraical reform had been better acquainted with the more modern results of the science, they would have felt the total inadequacy of the very limited science of arithmetical algebra to replace it ; and they would probably have directed their attention to discover whether any principles were necessary to be assumed, which were not neces- sarily deducible as propositions from arithmetic or arithmetical algebra, though they might be suggested by them. As it was, however, these speculations did not receive the consideration which they really merited ; and it is very possible that the attempt which was made by one of their authors to connect the errors in reasoning, which he attacked, as forming part only of a much more extensive class to which the human mind is liable from the influence of prejudice or fashion, had a tendency to divert men of an enlarged acquaintance with the results of algebra from such a cautious and sustained examination of them as was required for their refutation, or rather for such a correc- tion of them as was really necessary to establish the science of algebra upon its proper basis. I know that it is the opinion of many persons, even amongst the masters * of algebraical science, that arithmetic does supply identical with each other : if we extract the square root on both sides, re- jecting the negative value of the square root, we get in the first case 50 — a; = 10, and in the second, a? — 50 = 10. The first of these simple equations gives us « = 40, and the second x = 60, both of which satisfy the conditions of the problem proposed : the two roots which are thus obtained, strictly by means of arithmetical algebra, show that the pro- blem proposed is to a certain extent indeterminate. Mr. Frend and Baron Maseres contended that multiple real roots, which are always the indication of a similar indetermination in the problems which lead to such equations, might be obtained by arithmetical algebra alone, and that all other roots were useless fictions, which could lead to no practical conclusions. But it is very easy to show, that incongruous and real, as well as negative and impossible roots, may equally indicate the impossibility of the problem proposed : thus, if it was proposed " to find a number the double of whose square exceeds three times the number itself by 5," we shall find 4 and — 1 for the roots of the resulting equation, both of which equally indicate the impossibility of the pro- blem proposed, if by number be meant a lohole positive number. * Cauchy, who has enriched analysis with many important discoveries, and who is justly celebrated for his almost unequalled command over its lan- guage, has made it the principal object of his admirable work, entitled Cours d' Analyse de I'Ecole Royale Polytechnique, to meet the difficulties which pre- sent themselves in the transition from arithmetical to symbolical algebra : and though he admits to the fullest extent the essential distinction between them, in the ultimate form which the latter science assumes, yet he considers the principles of one as deducible from those of the other, and presents the rules for the concurrence and incorporation of signs ; for the inverse relation of the operations called addition and subtraction, multiplication and division ; for REPORT ON CJERTAIN BRANCHES OF ANALYSIS. 198 a sufficient basis for symbolical algebra considered under its most general form ; that symbols, considered as representing numbers, may represent every kind of concrete magnitude ; the indifference of the order of succession of different algebraical operations, as so many theorems founded upon the ordinary principles and reasonings of arithmetic. In order to show, however, the extraordinary vagueness of the reasoning which is employed to establish these theorems, we will notice some of them in detail : On repvesente, says he, les grandeurs qui doivent servir d'ac- croissements, par des nomhres precedes du signs +. <?' l^s grandeurs qui doivent servir de diminutions par des nomhres precedes du signe — . Cela pose, les signes -\- et — places devant les nombres peuvent itre compares, suivani la remarque qui en a He faite^, a. des adjectifs place's aupres de leurs suhstantifs. It is unques- tionable, however, that in the most common cases of the interpretation of specific magnitudes affected with the signs + and — , there is no direct refer- ence either to increase or diminution, to addition or to subtraction. He sub- sequently gives those signs a conventional interpretation, as denoting quan- tities which are opposed to each other ; and assuming the existence of quan- tities affected by independent signs, and denoting + A by a, and — A by 6, he savs that 4-«=-|-A + h=z — K — a = — A — b = + A; and therefore, -f(+A) = + A +(_A) = -A - (-h A) = - A - (- A) = -h A ; which he considers as a sufficient proof of the rule of the concurrence of signs in whatever operations they may occur ; though it requires a very slight examination of this process of reasoning to show that it involves several ar- bitrary assumptions and interpretations which may or may not be consistent with each other. In the proofs which he has given of the other fundamental theorems which we have mentioned above, we shall find many other instances of similar confusion both in language and in reasoning : thus, " subtraction is the inverse of addition in arithmetic ; then therefore, also, subtraction is the inverse of addition in algebra, even when applied to quantities affected with the signs -\- and — , and whatever those quantities may be." But is this a conclusion or an assumption? or in what manner can we explain in words the process which the mind follows in effecting such a deduction ? " If a and h be whole numbers, it may be proved that a 6 is identical with h a : therefore, a b is identical with b a, whatever a and b may denote, and whatever may be the interpretation of the operation which connects them." But any attempt to establish this conclusion, without a previous definition of the meaning of the operation of multiplication when applied to such quanti- ties, will show it to be altogether impracticable. The system which he has fol- lowed, not merely in the establishment of the fundamental operations, but likewise in the interpretation of what he terms symbolical expressions and symbolical equations, requires the introduction of new conventions, which are not the less arbitrary because they are rendered necessary for the purpose of making the results of the science consistent with each other : some of those conventions I believe to be necessary, and others not ; but in almost every in- stance I should consider them introduced at the wrong place, and more or less inconsistently with the professed grounds upon which the science is founded. ' By Buee in the Philosophical Transactions, 1806. 1833. o 194 THIRD KEPOUT — 1833. that the operations of addition, subtraction, multipUcation and division are used in one science and in the other in no sense which the mind may not comprehend by a practicable, though it may not be by a very simple, process of generalization ; that we may be enabled by similar means to conceive both the use and the meaning of the signs + and — , when used independ- ently ; and that though we may be startled and somewhat em- barrassed by the occurrence of impossible quantities, yet that investigations in which they present themselves may generally be conducted by other means, and those difficulties may be evaded which it may not be very easy or very prudent to en- counter directly and openly. In reply, however, to such opinions, it ought to be remarked that arithmetic and algebra, under no view of their relation to each other, can be considered as one science, whatever may be the nature of their connexion with each other ; that there is nothing in the nature of the symbols of algebra which can es- sentially confine or limit their signification or value ; that it is an abuse of the term generalization* to apply it to designate the process of mind by which we pass from the meaning of a — b, when a is greater than b, to its meaning when a is less than b, or from that of the product a b, when a and b are abstract num- bers, to its meaning when a and b are concrete numbers of the same or of a different kind ; and similarly in every case where a result is either to be obtained or explained, where no pre- vious definition or explanation can be given of the operation upon which it depends : and even if we should grant the legiti- macy of such generalizations, we do necessarily arrive at a new science much more general than arithmetic, whose principles, however derived, may be considered as the immediate, though not the ultimate foundation of that system of combinations of symbols which constitutes the science of algebra. It is more natural and philosophical, therefore, to assume such principles as independent and ultimate, as far as the science itself is con- cerned, in whatever manner they may have been suggested, so that it may thus become essentially a science of sypibols and their combinations, constructed upon its own rules, which may * The operations in arrithmetical algebra can be previously defined, whilst those in symbolical algebra, though bearing the same name, cannot : their meaning, however, when the nature of the symbols is known, can be generall_y, but by no means necessarily, interpreted. The process, therefore, by which we pass from one science to the other is not an ascent from particulars to generals, which is properly called generalization, but one which is essentially arbitrary, though restricted with a specific view to its operations and their results admit- ting of such interpretations as may make its applications most generally useful. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 195 be applied to arithmetic and to all other sciences by intei'preta- tion : by this means, interpretation -w'tM follow, and woiprecede, the operations of algebra and their results ; an order of suc- cession which a very slight examination of their necessary changes of meaning, corresponding to the changes in the spe- cific values and applications of the symbols involved, will very speedily make manifest. But though the science of arithmetic, or of arithmetical al- gebra, does not furnish an adequate foundation for the science of symbolical algebra, it necessarily suggests its principles, or rather its laws of combination ; for in as much as symbolical al- gebra, though arbitrary in the authority of its principles, is not arbitrary in their application, being required to include arith- metical algebra as well as other sciences, it is evident that their rules must be identical with each other, as far as those sciences proceed together in common : the real distinction between them will arise from the supj^osition or assumption that the symbols in symbolical algebra are perfectly general and unlimited both in value and rejjresentation, and that the operations to which they are subject are equally general likewise. Let us now consider some of the consequences of such an assumption. A system of symbolical algebra will require the assumption of the independent use of the signs + and — . For the general rule in arithmetical algebra* informs us, that the result of the subtraction of 6 + c from a is denoted hy a — b — c, or that a — {b + c) = a — b — c, its application being limited by the necessity of supposing that 6 + e is less than a. The general hypothesis made in symbolical algebra, namely, that symbols are unlimited in value, and that operations are equally applicable in all cases, would necessarily lead us to the conclusion that a — {b + c) = a — b — c for all values of the symbols, and therefore, also, when 6 = a, in which case we have a — {a + c) = a — a — c=:— c. In a similar manner, also, we find a — {a — c) = a — a + c=^ -|-e = cf. We are thus necessarily led to the assumption of the exist- ence of such quantities as — c and + c, or of symbols preceded • Whatever general symbolical conclusions are true in arithmetical algebra must be true likewise in symbolical algebra, otherwise one science could not include the other. This is a most important principle, and will be the subject of particular notice hereafter. t For it appears from arithmetical algebra that a — a^O, and that a — a 196 THIRD REPORT 1833. by the independent signs * + and — , which no longer denote operations, though they may denote affections of quantity. It appears likewise that + c is identical with c, but that — c is a quantity of a different nature from c : the interpretation of its meaning must depend upon the joint consideration of the spe- cific nature of the magnitude denoted by a, and of the symbolical conditions which the sign — , thus used, is required to satisfyj-. In a similar manner, the result of the operation, or rather the operation itself, of extracting the square root of such a quantity as « — 6 is impossible, unless a is greater than b. To remove the limitation in such cases, (an essential condition in symbolical algebra,) we assume the existence of a sign such as -v/ — I ; so that if we should suppose b =■ a -\- c, we should get V {a — b) = '/{« — (« + (?)} = V [a — a—c) = V [ — c) = V' — \ cX- In a similar manner, in order to make the ope- ration universally applicable, when the ?«"' root of a — 6 is required, we assume the existence of a sign v^ — 1, for which, as will afterwards appear, equivalent symbolical forms can al- ways be found, involving a/ — 1 and numerical quantities. By assuming, therefore, the independent existence of the signs +, — , X/\, and v^ — 1, (1)", and ( — 1)''§, we shall obtain a symbolical result in all those operations, which we call addi- tion, subtraction, multiplication, division, extraction of roots, and raising of powers, though their meaning may or may ?iot be identical with that which they possess in arithmetic. Let us now inquire a little further into the assumptions which deter- mine the symbolical character and relation of these funda- mental operations. The operations called addition and subtraction are denoted by the signs + and — . They are the inverse of each other. * That is, not preceded by other symbols as in the expressions a — c and a + c. \ Amongst these conditions, the principal is, that if — c be subjected to the operation denoted by the sign — , it will become identical with + c: thus, a — ( — c) ■:= a-\- c. It does not follow, however, that the sign — thus used, must necessarily admit of interpretation. X The symbolical form, however, of this and of similar signs is not arbi- trary, but dependent upon the general laws of symbolical combination. § I do not assert the necessity of considering such signs as V — 1. (1)", ( — 1)», as forming essentially a part of the earliest and most fundamental as- sumptions of algebra : the necessity for their introduction will arise when those operations with which they are connected are first required to be con- sidered, and will in all cases be governed by the general principle above men- tioned. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 197 In the concurrence of the signs + and — , in whatever man- ner used, if two hke signs come together, whether + and + , or — and — , they are replaced by the single sign + ; and when two unlike signs come together, whether + and — , or — and + , they are replaced by the single sign — . When different operations are performed or indicated, it is indifferent in what order they succeed each other. The operations called multiplication and division are de- noted by the signs x and h-, or more frequently by a conven- tional position of the quantities or symbols with respect to each other : thus, the product of a and b is denoted by a x b, a . b, or a b ; the quotient of a divided by b is denoted by « -r- A, or by -V. •' b The operations of multiplication and division are the inverse of each other. In the concurrence of the signs + and — in multiplication or division, if two like signs come together, whether + and + , or — and — , they are replaced by the single sign + ; and if two un- like signs come together, whether + and — , or — and +, they are replaced by the single sign — . When different operations succeed each other, it is not indif- rent in what order they are taken. We arrive at all these rules, when the operations are defined and when the symbols are numbers, by deductions, not from each other, but from the definitions themselves : in other words, these conclusions are not dependent upon each other, but upon the definitions only. In the absence, therefore, of such defini- tions of the meaning of the operations which these signs or forms of notation indicate, they become assumptions, which are independent of each other, and which serve to define, or rather to interpret* the operations, when the specific nature of the symbols is known ; and which also identify the results of those operations \mth the corresponding restdts in arithmetical alge- bra, lohen the symbols are numbers and when the operations are arithmetical operations. The rules of symbolical combination which are thus assumed * To define, is to assign beforehand the meaning or conditions of a term or operation ; to interpret, is to determine the meaning of a term or operation conformably to definitions or to conditions previously given or assigned. It is for this reason, that we define operations in arithmetic and arithmetical alge- bra conformably to their popular meaning, and we interpret them in symboli- cal algebra conformably to the svmbolical conditions to which they are sub- ject. 198 THIRD REPORT — 1833. have been suggested only by the corresponding rules in arith- metical algebra. They cannot be said to he founded xx^on them, for they are not deducible from them ; for though the opera- tions of addition and subtraction, in their arithmetical sense, are applicable to all quantities of the same kind, yet they ne- cessarily require a different meaning when applied to quanti- ties which are different in their nature, whether that difference consists in the kind of quantity expressed by the unaffected symbols, or in the different signs of affection of symbols de- noting the same quantity ; neither does it necessarily follow that in such cases there exists any interpretation which can be given of the operations, which is competent to satisfy the re- quired symbolical conditions. It is for such reasons that the investigation of such interpretations, when they are discover- able, becomes one of the most important and most essential of the deductive processes which are required in algebra and its applications. Supposing that all the operations which are required to be performed in algebra are capable of being symbolically de- noted, the results of those operations will constitute what are called equivalent forms, the discovery and determination of which form the principal business of algebra. The greatest part of such equivalent forms result from the direct applica- tion of the rules for the fundamental operations of algebra, when these rules regard symbolical combinations only : but in other cases, the operations which produce them being nei- ther previously defined nor reduced to symbolical rules, unless for some specific values of the symbols, we are compelled to resort, as we have already done in the discovery and assump- tion of the fundamental rules of algebra themselves, to the re- sults obtained for such specific values, for the purpose of dis- covering the rules which determine the symbolical natiu-e of the operation for all values of the symbols. As this principle, which may be termed the principle of the permanence of equi- valent forms, constitutes the real foundation of all the rules of symbolical algebra, when viewed in connexion with arithmeti- cal algebra considered as a science of suggestion, it may be proper to express it in its most general form, so that its autho- rity may be distinctly appealed to, and some of the most im- portant of its consequences may be pointed out. Direct proposition : Whatever form is algebraically/ equivalent to another when expressed in general symbols, must contifiue to be equivalent, whatever those sijmbols denote. Converse proposition : REPORT ON CERTAIN BRANCHES OF ANALYSIS. 199 Whatever equivalent form is discoverable in arithmetical algebra considered as the science of suggestion, when the sym- bols are general in their form, though specif c in their value, will continue to be an equivalent form when the symbols are general in their nature as well as in their form *. The direct proposition must be true, since the laws of com- bination of symbols by which such equivalent forms are de- duced, have no reference to the specific values of the symbols. The converse proposition must be true, for the following reasons : If there be an equivalent form when the symbols are general in their nature as well as in their form, it must coincide with the form discovered and proved in arithmetical algebra, where the symbols are general in their form but specific in their na- ture ; for in passing from the first to the second, no change in its form can take place by the first proposition. Secondly, we may assume the existence of such an equivalent form in symbols which are general both in their form and in their nature, since it will satisfy the only condition to which all such forms are subject, which is, that of perfect coincidence with the results of arithmetical algebra, as far as such results are common to both sciences. Equivalent forms may be said to have a necessary existence when the operation which produces them admits of being de- fined, or the rules for performing it of being expressly laid down : in all other cases their existence may be said to be conventional or assumed. Such conventional results, however, are as much real results as those which have a necessary ex- istence, in as much as they satisfy the only condition of their existence, which the principle of the permanence of equivalent forms imposes upon them : thus, the series for (1 + ;r)" has a necessary existence whenever the nature of the operation upon \ + X which it indicates can be defined ; that is, when « is a whole or a fractional, a positive or negative, number f ; but for all other values of n, where no previous definition or interpre- tation of the nature of the operation which connects it with its equivalent series can be given, then its existence is conventional only, though, symbolically speaking, it is equally entitled to be considered as an equivalent form in one case as in the other. It is evident that a system of symbolical algebra might be • Peacock's Algebra, Art. 132. + The meaning of (1 + «)" cannot properly be said to be defined when n is a fractional number, whether positive or negative, or a negative whole num- ber, but to be ascertained by interpretation conformably to the principle of the permanence of equivalent forms. 200 THIRD REPORT 1833. formed, in which the symbols and the conventional operations to which they were required to be subjected would be perfectly general both in value and application. If, however, in the con- sti-uction of such a system, we looked to the assumption of such rules of operation or of combination only, as would be sufficient, and not more than sufficient, for deducing equivalent forms, without any reference to any subordinate science, we shovdd be altogether without any means of interpreting either our opera- tions or their results, and the science thus formed would be one of symbols only, admitting of no applications whatever. It is for this reason that we adopt a subordinate science as a sci- ence of suggestion, and we frame our assumptions so that our results shall be the same as those of that science, when the symbols and the operations upon them become identical like- wise : and in as much as arithmetic is the science of calculation, comprehending all sciences which are reducible to measure and to number ; and in as much as arithmetical algebra is the imme- diate form which arithmetic takes when its digits are replaced by symbols and when the fundamental operations of arithmetic are applied to them, those symbols being general in form, though specific in value, it is most convenient to assume it as the subordinate science, which our system of symbolical algebra must be required to comprehend in all its parts. The principle of the permanence of equivalent forms is the most general ex- pression of this law, in as much as its truth is absolutely neces- sary to the identity of the results of the two sciences, when the symbols in both denote the same things and are subject to the same conditions. It was with reference to this principle that the fundamental assumptions respecting the operations of ad- dition, subtraction, multiplication and division were said to be suggested by the ascertained rules of the operations bearing the same names in arithmetical algebra. The independent use of the signs -}- and — , and of other signs of affection, was an as- sumption requisite to satisfy the still more general principle of symbolical algebra, that its symbols should be unlimited in value and representation, and the operations to which they are sub- ject unlimited in their application. In arithmetical algebra, the definitions of the operations de- termine the rules ; in symbolical algebra^ the rules determine the meaning of the operations, or more properly speaking, they furnish the means of interpreting them : but the rules of the former science are invariably the same as those of the latter, in as much as the rules of the latter are assumed with this view, and merely differ from the former in the universality of their applications : and in order to secure this universality of their REPORT ON CERTAIN BRANCHES OF ANALYSIS. 201 applications, such additional signs* are assumed, and of such a symbolical foi-m, as those applications may render necessary. We call those rules, or their equivalent symbolical consequences, assumptions, in as much as they are not deducible as conclusions from any previous knowledge of those operations which have corresponding names: and we might call them arhitrary as- sumptions, in as much as they are arhitrar'ily imposed upon a science of symbols and their combinations, which might be adapted to any other assumed system of consistent rules. In the assumption, therefore, of a system of rules such as will make its symbolical conclusions necessarily coincident with those of arithmetical algebra, as far as they can exist in common, we in no respect derogate from the authority or completeness of sym- bolical algebra, considered with reference to its own conclu- sions and to their connexion with each other, at the same time that we give to them a meaning and an application which they would not otherwise possess. It follows from this view of the relation of arithmetical and symbolical algebra, that all the results of arithmetical algebra which are general in form are true likewise in symbolical algebra, whatever the symbols may denote. This conclusion may be said to be true in virtue of the principle of the perma- nence of equivalent forms, or rather it may be said to be the proper expression of that principle. Its consequences are most important, as far as the investigation of the fundamental pro- positions of the science are concerned, in as much as it enables us to investigate them in the most simple cases, when the operations which produce them are perfectly defined and un- derstood, and when the general symbols denote positive whole numbers. If the conclusions thus obtained do not involve in their expression any conditiofi which is essentially connected v-it/i the specific values of the symbols, they may be at once transferred to symbolical algebra, and considered as true for all values of the symbols whatsoever^. Thus, coefficients in arithmetical algebra, such as m in m a, which are general in form, lead to the interpretation of such * There is no necessary limit to the multiplication of such signs : the signs + . — I (1)" and (—1)" and their equivalents (for the symbolical form of such signs is not arbitrary), comprehend all those signs of affection which aie re- quired by those operations with which we are at present acquainted. + Some formula; are essentially arithmetical : of this kind is 1 . 2 . 3 . . . r, in which r must be a whole number. The formula "K"' — ^) • • • (w — r+ 1) 1.2 . . . r IS symbolical with respect to m, but arithmetical with respect to r. Such cases, and their extension to general values of/-, will be more particularly considered herealter. 202 THIRD REPORT 1833. expressions as m a in symbolical algebra, when m is a number whole or fractional, and a any symbol whatsoever. When m, n and a are whole numbers, it very readily appears that ma + na = (/« + n) a, and that ma — n a ^= {m — n) a : the same con- clusions are true likewise for all values of ?n, n and a. In arithmetical algebra we assume a^, aP, «"*, &c., to represent a a, a a a, aaaa, &c., and we readily arrive at the conclusion that «"* X «" = «'"■'" ", when m and n are whole numbers : the same conclusion must be true also when m and n are any quantities whatsoever. In a similar manner we pass from the result {cf) " = «"", when « is a whole number, to the same conclusion for all values of the symbols *. The preceding conclusions are extremely simple and element- ary, but they are not obtainable for all values of the symbols by the aid of any other principle than that of the permanence of equivalent forms : they are assumptions which are made in conformity with that principle, or rather for the purpose of rendering that principle universal ; and it will of course follow that all interpretations of those expressions where m and n are not whole numbers must be subordinate to such assumptions. Thus, Tr~''"0~('o+'2") a, = a, and therefore -^ must mean one half of a, whatever a may be ; a- x a = a^ = a = a, and therefore a must mean the square root of a, whatever a may be, whenever such an operation admits of interpretation. In a similar manner -^ must mean one third part, and a^ the cube root of a, whatever a may be, and simi- larly in other cases : it follows, therefore, that the interpreta- tion of the meaning of a , a^, &c., is determined by the general * The general theorems ma -\- na = (m -\- n) a and ma — na= {m — n) a, m —VI ar'Xa!' = 0'" + " and \ = a*"" ", (a"')" = o™" and (o*") " =a»' which a" are deduced by the principle of the permanence of equivalent forms, and which are supplementary to the fundamental rules of algebra, are of the most essen- tial importance in the simplification and abridgement of the results of those operations, though not necessary for the formation of the equivalent results themselves. It also appears from the four last of the above-mentioned theorems that the operations of multiplication and division, involution and evolution, are performed by the addition and subtraction, multiplication and division, of the indices, when adapted to the same symbol or base. If such indices or logarithms be calculated and registered with reference to a scale of their corresponding numbers, they will enable us to reduce the order of arithmetical operations by two unities, if their orders be regulated by the following scale ; addition (1), sub- traction (2), multiplication (3), division (4), involution (5), and evolution (6). REPORT ON CERTAIN BRANCHES OF ANALYSIS. 203 principle of indices, and also that we ought not to say that we assume a to denote ^/a, and a^ to denote ^a, as is commonly done *, in as much as such phrases would seem to indicate that such assumptions are independent, and not subject to the same common principle in all cases. In all cases of indices which involve or designate the inverse processes of evolution, we must have regard likewise to the other great principle of symbolical algebra, which authorizes the existence of signs of affection. The square root of a may be either affected with the sign + or with the sign — ; for + or X + «*, and — «^ X — « , will equally have for their result + a or a, by the general rule for the concurrence of similar signs and the general principle of indices : in a similar manner d^ may be affected with the multiple sign of affection (1)^, if there are any symbolical values of (1)^ different from + 1 (equi- valent to the sign +), which will satisfy the requisite symbo- lical conditions f . It is the possible existence of such signs of affection, which is consequent upon the universality of alge- braical operations, which makes it expedient to distinguish be- tween the resvdts which are not affected by such signs, and the same results when affected by them. The first class of results or values are such as are alone considered in arithmeti- cal algebra, and we shall therefore term them arithmetical va- lues, though the quantities themselves may not be arithmetical : the second class may be termed algebraical values, in as much as they are altogether, as far as they are different from the arithmetical values, the results of the generality of the opera- tions of symbolical algebra. This distinction may generally be most conveniently ex- pressed by considering such a sign as a factor, or a symbolical quantity multiplied according to the rule for that operation into the arithmetical value : in this sense + 1 and — I may be considered as factors which are equivalent to the signs + and — , that is, equivalent to affecting the quantities into which they are multiplied with the signs + and — , according to the • Wood's Algebra, Definitions. f That is, it' there is any symbolical expression different from 1, such as ^ '-, and ^ , the cubes of which are identical with 1. In a similar manner we may consider the existence of multiple values of 1" or ( — 1) , and, therefore, of multiple signs of affection corresponding to them, as consequent upon the general laws of combination of symbolical algebra, and as results to be determined from those laws, and whose existence, also, is de- pendent upon them. 204 THIRD REPORT — 1833. general rule for the concurrence of signs. In a similar manner we may consider (1)* (a)* as equivalent to (r/)* ; (1)^ (o)^ as equivalent to («)^; (1)" a" as equivalent to «" ; (—1)" («)" as equivalent to ( — a)", and similarly in other cases : in all such cases the algebraical quantity into which the equivalent sign or its equivalent factor is multiplied, is supposed to possess its arithmetical value only*. The series for (1 + x)", when n is a whole number, may be exhibited under a general form, which is independent of the specific value of the index ; for such a series may be continued indefinitely in form, though all its terms after the (« + l)th must become equal to zero. Thus, the series (1 + £)• = (1)" (l + » 0,- + -f-^' ^ + + "i'ra'\"/".T" -'' + ^°-) indefinitely continued, in which n is particular in value (a whole number) though general in form, must be true also, in virtue of the principle of the permanence of equivalent forms, when n is general in value as well as in formf . This theorem, which, singly considered, is, of all others, the most important in analysis, has been the subject of an almost unlimited variety of demonstrations. Like all other theorems whose consequences present themselves very extensively in algebraical results, it is more or less easy to pass from some one of those consequences to the theorem itself: but all the demonstrations which have been given of it, with the excep- tion of the principle of one given by Euler^, have been con- fined to such values of the index, namely, whole or fractional numbers, whether positive or negative, as made not only the development depend upon definable operations, but like- wise assumed the existence of the series itself, leaving the form of its coefiicients alone undetermined. It is evident, however, that if there existed a general form of this series, its form could * This separation of the symbolical sign of affection from its arithmetical subject, or rather the expression of the signs of affection explicitly, and not im- plicitly, is frequently important, and affords the only means of explaining many paradoxes (such as the question of the existence of real logarithms of negative numbers), by which the greatest analysts have been more or less embarrassed. f If such a series should, for any assigned value of n, have more symbolical values than one, one of them will be the arithmetical value, inasmuch as one symbolical value of l" is always 1. X In the Nov. Comm. Petropol. for 1771. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 205 be detected for any value of the index whatever, which was general in form, and therefore, also, when that index was a whole number ; a case in which the interpretation of the opera- tion designated by the index, as well as the performance of the operation itself, was the most simple and immediate. That such a series, likewise, would satisfy the only sym- bolical conditions which the general principles of indices ini- poses upon the binomial, might be very easily shown; for if m and n be whole numbers, then if the two series {\ \ xf -V^\\ ^■ mx ■\- -\ ^ x^ -j- &c. J (1 + a-)« = 1« (l + n X + '--^^^-^^^2 +- &c.) be multiphed together, according to the rule for that purpose, we must obtain (1 4- xy^''^ l™-*-" ( 1 + (m+ n)x + ^ \2 ^^7 a series in which m + n has replaced m or n in its component factors : and in as much as we must obtain the same symbolical result of tliis multiplication, whatever be the specific values of m and n, it follows, that if the same form of these series repre- sents the development of (1 -|- or)" and (1 + x^, whatever m and n may be, then, likewise, the series for the product of (1 + a:)"' and (I + xY, or (1 + x)'"+", would be that which arose from putting wi + w in the place of m or n in each of the component factors. If, therefore, we assumed S {uri) and S (ra) to represent the series for (1 + xj" and (1 + x^, when m and n are any quantities whatsoever, then (1 + a-)" x (1 + .r)« = (1 + ^)'« + " = S (w -f- w) = S (;h) X S (w) ; or, in other words, the series will possess precisely the same symbolical properties with the binomial to which they are required to be equivalent. It is the equation a"" x «" = a*""*"", for all values of ra and n, which determines the interpretation of «"' or a", when such an interpretation is possible ; in other words, such quantities pos- sess no properties which are independent of that equation. The same remark of course extends to (1 + xj", for all values of w, and similarly, likewise, to those series which are equivalent to it. That all such series must possess the same form would be evident from considering that the symbolical properties of (1-1- J")" undergo no change for a change in the value of n, and that no series could be permanently equivalent to it whose form 206 THIRD REPORT 1833. was not equally permanent likewise. In assuming, therefore, the existence of such a permanent series, our symbolical conclu- sions are necessarily consistent with each other, and it is the interpretation of the operations which produce them, which must be made in conformity with them. It is true that we can extract the square or the cube root of 1 + x, and we can also determine the corresponding series by the processes of arith- metical algebra ; and we likewise interpret (1 + xf and (1 + xf to mean the square and the cube root of 1 + x, in confoi-mity with the general principle of indices. The coincidence of the series for (1 + xf and (1 + xf, whether produced by the processes of arithmetical algebra, or deduced by the principle of the permanence of equivalent forms from the series for (1 + x)", would be a proof of the correctness of our interpreta- tion, not a condition of the truth of the general principle itself. In order to distinguish more accurately the precise limits of hypothesis and of proof in the establishment of the fundamental propositions of symbolical algebra, it may be expedient to re- state, at this point in the progress of our inquiry, the order in which the hypotheses and the demonstrations succeed each other. We are supposed to be in possession of a science of arith- metical algebra whose symbols denote numbers or arithmetical quantities only, and whose laws of combination are capable of strict demonstration, without the aid of any principle which is not furnished by our knowledge of common arithmetic. The symbols in arithmetical algebra, though general in form, are not general in value, being subject to limitations, which are necessary in many cases, in order to secure the practicability or possibility of the operations to be performed. In order to effect the transition from arithmetical to symbolical algebra, we now make the following hypotheses : (1.) The symbols are unlimited, both in value and in repre- sentation. (2.) The operations upon them, whatever they may be, are possible in all cases. (3.) The laws of combination of the symbols are of such a kind as to coincide universally with those in arithmetical algebra when the symbols are arithmetical quantities, and when the operations to which they are subject are called by the same names as in arithmetical algebra. The most general expression of this last condition, and of its connexion with the first hypothesis, is the law of the perma- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 207 nence of equivalent forms, which is our proper guide in the establishment of the fundamental propositions of symbolical algebra, in the invention of the requisite signs, and in the de- termination of their symbolical form : but in the absence of the complete enunciation of that law, we may proceed with the in- vestigation of the fundamental rules for addition, subtraction, multiplication and division, and of the theorems for the collec- tion of multiples, and for the multiplication and involution of powers of the same symbol, which will, in fact, form a series of assumptions which are not arbitrary, but subordinate to the conditions which are imposed by our hypotheses : but if we suppose those conditions to be incorporated into one general law, whose truth and universality are admitted, then those as- sumptions become necessary consequences of this law, and must be considered in the same light with other propositions which follow, directly or indirectly, from the first principles of a demonstrative science. In the same manner, if we assume the existence of such signs as are requisite to secure the universality of the operations, the symbolical form of those signs, and the laws which regulate their use, will be determined by the same principles upon which the ordinary results of symbolical al- gebra are founded. The natural and necessary dependence of these two methods of proceeding upon each other being once established, we may adopt either one or the other, as may best suit the form of the investigation which is under consideration : the great and im- portant conclusion to which we arrive in both cases being, the transfer of all the conclusions of arithmetical algebra which are general in form (that is, which do not involve in their expres- sion some restriction which limits the symbols to discontinuous values,) to symbolical algebra, accompanied by the invention or use of such signs (with determinate symbolical forms) as may be necessary to satisfy so general an hypothesis. There are many expressions which involve symbols which are necessarily discontinuous in their value, either from the form in which they present themselves in such expressions or from some very obvious conventions in their use : thus, when we say that cos x = COS {2r i: -\- x), and — COS x = cos { (2 r -|- 1) tt 4 x} propositions which are only true when r is a whole number, the limitation is conveyed (though imperfectly) by the con- ventional use of 2 r and 2 r + 1 to express even and odd num- bers ; for otherwise there would be no sufficient reason for not 208 THIRD REPORT— 1833. using the simple symbol r both in one case and the other. In a similar manner, in the expression of Demoivre's theorem (cos 9 + \/^^l sin fl)" = cos (2 ?• w TT + ?i fl) + -v/ — 1 sin {2 r mr + n 6), we may suppose w to be any quantity whatsoever *, but r is ne- cessarily a whole number. In some cases, however, the construction of the formula it- self will sufficiently express the necessary restriction of the values of one or more of its symbols, without the necessity of resorting to any convention connected with their introduction : thvis, the formula 1 x 2 x 3 r, commencing from 1 , is essentially arithmetical, and limited by its form ro whole and positive values of r. The same is the case with the formula r (r — 1) . . . . 3 . 2 . 1, where some of the successive and strictly arithmetical values of the terms of the series r, r — 1, &c., are put down ; but the formula r (r — 1) ('' — 2) .... is subject to no such restriction, in as much as any number of such factors may be formed and multiplied together, whatever be the value of r. In a similar manner, the formula n {n — I) ... {n — r + I), r72 77. r which is so extensively used in analysis, is unlimited with re- spect to the symbol w, and essentially limited with respect to the symbol r : it is under such circumstances that it presents itself in the development of (1 + a;)". In the differential calculus we readily find '^ = n{n-l). ..{n-r + l)x"-', and in a similar manner also •^(.r« + C, x'^-' + C, x'-^ + . . C„) = w (w-1) . . (H-r+ 1>«-'-: dx^ 1 in both these cases the value of n is unlimited, whilst the value of r is essentially a positive whole number ; in other words, * The investigation of this formula (like the equivalent series for (1 -|- x) when n is a general symbol,) requires the aid of the principle of the perma- nence of equivalent forms, in common with all other theorems connected with the general theory of indices. The formula above given involves also impli- citly any sign of affection which the general value of n may introduce : for (cos 6 + V"^ sin tf)" = (1)" (cos n6+ V^-i sin n 6) = (cos 2 /• M TT + -\/— 1 sin 2rn-!r) (cos n6 + -v/ — 1 sin n 6) = cos {2 rn TT -{■ n d) -\- V' — 1 sin (2 r n rr -\- n 6) REPORT ON CERTAIN BRANCHES OF ANALYSIS. 209 the principle of equivalent forms might be extended to this for- mula (supposing it to be investigated for integral values of n and r,) as far as the symbol n is concerned only. This arithme- tical coefficient of differentiation (if such a term may be ap- pHed to it,) will present itself in the expression of the rth dif- ferential coefficient (r being a whole positive number,) of all al- gebi-aical functions *; and it is for this reason that we aveojjpa- rently debarred from considering fractional or general indices of differentiation when applied to such functions, and that we are consequently prevented from treating the differential and integral calculus as the same branch of analysis whose general laws of derivation are expressed by common formulae. But is it not possible to exhibit the coefficient of differentia- tion under some equivalent form which may include general values of the index of differentiation ? It is well known that the definite integral / d X \ log — ) (adopting Fourier's notation,) is equal to 1 . 2 . . . . 7z, when n is a whole number ; and that consequently^ under the same cir- cumstances, the coefficient of differentiation or / d X \ log — ) n (rt — 1) . . . (w — r + 1) = ^'^.(logiy ' and in as much as \hQ form of this equivalent expression is not restricted to integral and positive values of r, we may assume • IT, -f 1 ^ f^' « (-l)'-.2.3...(r+l)a;' - * Inus if M = -; — ; — r, we have -. — 5 =: 7^ — \ — srrxi X (1 - ^ ('^ - ^) , r(r-l)(r-2) (r - 3) _ ^^ ] . \ 2.3a;2+ 2.3.4.5 a;'' '/* I , d'u 1.2 r and if M = -TT. — I — 5;, we have V (1 + a;7 dxr — (1 — a"- ) »•+ 4 X ( 1+ JL . '• (>• - ^) , 1 ^(,.-l)(r-2) (r-3) , & 1 12= x" "^ 2= .4= a-4 J + This definite integral, the second of that class of transcendents to which Legendre has given the name of Intecjrales Euleriennes, was first considered by Euler in the fifth volume of the Commentarii PetropoUtani, in a memoir on the interpolation of the terms of the series 1 + 1x2 + 1x2x3 + 1x2X3X4 + &c., which is full of remarkable views upon the generalization of formula; and their interpretation. The same memoir contains the first solution of a pro- blem involving fractional indices of differentiation, 1833. P 210 THIRD REPORT — 1833. ■it to be permanent, so long as we do not at the same time as- sume the necessary existence and interpretation of equivalent results. If, however, such results can be found, either gene- rally, or for particular values (not integral and positive) of r, apart from the sign of integration, the consideration of the values of the corresponding differential coefficients will involve no other theoretical difficulty than that which attends the transition from integral to fractional and other values of common indices. Euler, in his Differential Calculus *, has given the name of inexplicable functions to those functions which are apparently resti'icted by their form to integral and positive values of one or more of the general symbols which they involve : of this kind are the functions Ix2x3x X, I '^ 2 '^ 3 '^ x' 111 1 1 H i. — 1" 2" ' -^" 1 4- " ~ ^ I ^ ~ ^^ a — (x — 1) b '^ a + b '^ a + 3b '^ a + {x + l)b' and innumerable others which present themselves in the theory of series. The attempts which he has made to interpolate the series of which such functions form the general terms, are properly founded upon the hypothesis of the existence of per- manent equivalent forms, though it may not be possible to ex- hibit the explicit forms themselves by means of the existing signs and symbols of algebra. In the cases which we have hitherto considered, the forms which were assumed to be per- manent had a real previous existence, which necessarily re- sulted from operations which were capable of being defined. In the case of inexplicable functions, the corresponding perma- nent forms which hypothetically include them, may be consi- dered as having an hypothetical existence only, whose form degenerates into that of the inexplicable function in the case of integral and positive values of the independent variable or va- riables. It is for the expression of such cases that definite integrals find their most indispensable usage. * Instiiutiones Calculi Differentialis, Capp. xvi. et xvii. See also an admira- ble posthumous memoir of the same author amongst the additions to the Edition of that work printed at Pavia in 1787- He had been preceded in such researches by Stirling, an author of great genius and originality, whose la- bours upon the interpolation of series and other subjects have not received the attention to which they are justly entitled. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 21 1 It is easy to construct formulae which may exhibit the possi- bility of their thus degenerating into others of a much more simple form, when one or more of the independent variables be- come whole numbers : of this kind is the formula « + <S sin (2 r TT + ^) + y sin (g rir + flp + &c. ,. a + /3sin9 + ysinfl' +,&c. ^ fv)> which is, or is not, identical with 4) (r), according as r is a whole or a fractional number : such functions are termed nticlulating functions by Legendre *. We can conceive also the possible existence of many other transcendents amongst the unknown and undiscovered results of algebra, which may possess a simi- lar property. The transcendent X '^ ^ O^s D' mentioned above, possesses many properties which give it an uncommon importance in analysis, and most of all from its fur- nishing the connecting link in the transition from integral and positive to general indices of differentiation in algebraical functions. If we designate, as Legendre has done, we shall readily derive the fundamental equation r(l +r) = rr(r) t (1) which is in a form which admits of all values of r. It appears * Traits des Fonctions Elliptiques, torn. xi. p. 476. t In as much as t^"-" ,„ _ r (1 + n) -x^ = X'' = A a;^ d j;"-' r (1 + r) and dn-r+l r (1 + W) (lx«-r + 1 * — r (r) '^ — D X —TAX , it follows that r A = B, and therefore also that "which is the equation (l) : and it is obvious that the transition from dn-r d»-r+ 1 .x" to (which is equivalent to the simple differentiation of A .V, when A is a constant coefficient), will lead to the same relation between T (1 + r) and r(r), ivhateiJe)-he the value of »•, whether positive or negative, whole or frac- tional. Legendre has apparently limited this equation to positive values of r. p2 212 THIRD REPORT — 183J}. also from this equation that if the values of the transcendent r (r) can be determined for all values of r which are included (Fonctions Elliptiques, torn. ii. p. 415,) a restriction which is obvious!)' unne- cessary. There are two cases in which the coefficient of .x"-'' in the equation d'- x" r (1 + n) dxr ~ r(l + n — r) requires to be particularly considered : the first is that in which this coeffi- cient becomes infinite ; the second, that in which it becomes equal to zero. The numerator F (1 -|- w) will be infinite when n is any negative whole number ; the denominator T (l +n — r) will become infinite when n — r is any negative whole number, and in no other case : if n be a negative whole number, and if r be a whole number, either positive or negative, such that n — r r(l -l-n) ^ . . is negative, then the coefficient fT/T^; — ;— Tn becomes finite, in as much as r (— i) (if t be a whole number) = j— ^ 1( — IV ' ^^^ ^ ^^^ disap- d~r 1 . „ . pears, therefore, by division : thus all the coefficients of , _^ • — are infinite, unless r be a negative whole number, such as — m, in which case it becomes 1 . 2 . . TO . ( — l)*", a result which is easily verified. In a similcir manner it d'' 1 would appear that the coefficients of , _y • — are infinite, when w is a posi- tive number, unless r be a negative whole number equal to, or greater than, n. The coefficient „ , , — — ^ will become equal to zero, when I + n — r is, and when 1 -|- n is not, equal to zero or to any negative whole number ; for, under such circumstances, the denominator is infinite and the numerator is finite. As the most important consequences will be found to result from these critical values of the coefficient of differentiation, we shall proceed to examine them somewhat in detail. (1.) The simple differentials or differential coefficients of constant quantities are equal to zero, whilst the differentials or differential coefficients to general indices (positive whole numbers being excepted,) are variable. Thus da d . a x' a T (l) . , a d~^a dx^ dx^ I'Ci) " V-^x' ^^-i a (r 1) , , 2 a/j: d-ifl r(l) = "rnr • "- * = ^v ■■ d^i = F(2j • «*"■"' = «^' and similarly in other cases. (2.) The differentials of zero to general indices (positive whole numbers being excepted) are not necessarily equal to zero. Thus, if we suppose C „ dro C r (1 - «) ,._„_^ . a = = ^prTTvi x-", we get r (0) ' ^ - dx^ — r (0) T(l — n — r) if n be a positive whole number, F (1 — ?^) = oo , and this expression is finite un- less r (1 — n — »■) = oo, in which case it is zero : if r be also a positive whole = Ca; + C REPORT ON CERTAIN BRANCHES OF ANALYSIS. S13 between any two successive whole numbers, they can be der termined for all other values of r. Euler * first assigned the number, it is always zero : if r = — 1, it is finite when m = 1 : if r := — 2, it is finite when >j = 1 or « = 2 : if r = — 3, it is finite when « = 1, or w = 2, or M = 3 : and generally if r be any negative whole number, there will be finite values corresponding to every value of n from 1 to — r ; wo thus get rf-'O _ n dx-i dx-^ 3 5 = r-S + Ci a; + C2 d x-^ 1 . 2 d-« _ C A-"-i , Ci . ^"-2 d^»- 1.2. .(».-]) + 1.2..(«-2) +U-2.* + U-i. This is the true theory of the introduction of complementary arbitrary func- tions in the ordinary processes of integration. More generally, if r be not a whole number, d^O _ _C_ Y{\-n ) ^_„_^ dx''- r (0) ■ r(i — M — ?•) which will be finite when n is a positive whole number and when 1 — m — r is not a negative whole number : thus if w be any number in the series 1, 2, 3 , . ., and iir=k, then d^O dx-~ C . X T r (-i) or c ari or r c (-4) X' ,1 and so on for ever : consequently. tFO _ ^ + ^ + C2 ■7 -f &c. in hi) ^?iii ttiti 2. d x^ x^ x^ a; - In a similar manner, we shall find .fi r\ 3 1 . C2 . C3 .... ? 2 = C a;^ + Ci a^ H r "I T + &c. tw tvjimtum, a a; - The knowledge of these complementary arbitrary functions will be found of great importance for the purpose of explaining some results of the general differentiation of the same function under different forms which would other- wise be irreconcileable with each other. (3.) The differential coefficient will be zero, when n is 7iot, and whenn — r is, a negative whole number. Thus, d^x „ d3«2_ d^ x^ _ dJ x~^ _^ d~^ ^~ ^-(\. = 0, — — 0, — 0, u, — _ u . d x^ dx^ d x^ d x^ dx _ 5 7 and similarly in other cases. (4.) The differentials of 00 are not necessarily equal to », but may he finite. If we represent 00 by C F (0), we shall find Commetitarii Petrop., vol. v. 1731. 214 THIRD REPORT — 1833. ' value of r ( "H" ) = V'"', by the aid of the very remarkable ex- pression for TT, which Wallis derived from his theory of inter- ^'(^)^° = Cr(0) . r(l).iZ!::=(_l)r_l.l.2..(r-l).^-r, d X'' r (1 — r) whenever / is a positive whole number. Conversely also, rf-i . a;-i r (0) T, r (0) = — -^ . x", where — ■^—' = oo . dx-^ T{1) r(l) d-^ . x -i _ r(o) d a;-2 r (2) d-^ . a;-i _ r (0) d a;-3 r (3) d-T . a-' _ r(0) x\ d x-r V {r) the arbitrary complementary functions being omitted. (5.) The occurrence of infinite values of the coefHcient of differentiation will generalljf be the indication of some essential change of form in the transition from the primitive function to its corresponding differential coefficients. Thus, <f-i 1 r (0) „ , , ^ . - = — ^-1 . aO = log X + C ; dx-i X r(l) o -r . this last result or value of -~— :■ . a:" being obtained by the ordinary process of integration : and generally, ^'_^^-^ = LM xr-i + c^'-'' + Ci ^'-' + &c dx-r r (r) re*) r(7-— 1) ' the first term of which is infinite, in all cases in which r is not a negative whole number, in which case it becomes equal to ( — 1)"'' 1.2... ( — r) x''-^, the complementary arbitrary functions also disappearing. If we suppose, however, r to be a positive whole number, and if we replace frjvs • "^ ^Y its transcendental value already determined, we shall get - , = Hog « + C !■ + „ + &c., dx-r r (r) '■ " ^ -I r (;• — 1) which may be replaced by ^' =m {'»«'+ (-')'rW.ra-0 + c} +-£l^,',+ . . .. which is in a form which is true for all values of r whatsoever, and which coincides, for integral values of r, with the form determined by the ordinary process of integration. More generally, d-r . x-» _ r (1 — r) ^~„+r ; C ■ A"-" , Ci ..r'-"- ' .dx-r r(l—n + r)'^ r(r — n + l)T{r — n) '^ ' ' " which is finite, whilst r is less than n ; and when r and n ai'e whole numbers, p ffi f\ becomes = (— 1)'' . — i ^ '' a''-", omitting complementary functions. 1 ()i) REPORT ON CERTAIN BRANCHES OF ANALYSIS. 215 polations ; and subsequently, by a much more direct process, which lead to the equation, r (r) r (1 r) = -r^ — (when r > < 1) : ^ < ^ sin r ir ^ If, under the same circumstances, r be greater than n, the coefficient of dif- ferentiation becomes infinite, and its value, determined as above, becomes x'—» ,, , Ci a*"-"-' = roOVir-n + l) {'°S*' + ^} + r (r - u) + ^'=- ^iWrcT^TTTT) ^^"° " + ^^ '^' "" ^'•-«+i) r («-r) + c} (r — n) which is in a form adapted to all values of r. The cases which we have considered above are the only ones in which the coefficient of differentiation will become ivfinite, in consequence of the intro- duction of log « in the expression of its value. We shall have occasion here- after to notice more particularly the meaning of infinite values of coefficients as indications of a change in the constitution of the function into which they are multiplied. (6.) Uti = (a X -I- i)«, then dru r(l-J-w)«'' . .V Ci"--! mrr = Y\\--\-i^-V) • («* + ^y-" + rT=7) + • • • • For if V = a X -j- b, then -j— = a and -7— j = ; and therefore d^ u T (\ ->t- 7i) /dv\- , C«'-> , „ Thus if M = (a; -|- \y, we get rf«^ ^ ^^' V (- i) x" r (- 4) a;^ 8 .„, ,^4 , C , C, , C2 (x+ir + ^ + ^ + ^ -I-&C. *^ X- x' X- If we replace (a; -f 1)2 by a'2 -)- 2 -|-a; 1, we shall get ^ _ L(^) ^.^ , 2 £11) J 4. Eil) _L d„4 - r a) '^ + 2 J, ^^^ * + r (i) • x4 C Ci r (— \) x^ r (— 4-) «* It thus appears that the two results may be made to coincide with each other, when (ar -)- 1)t jn the fi.rst of them is developed, by the aid of the proper arbitrary functions. The necessity of this introduction of arbitrary functions to restore the re- quired identity of the expressions deduced for the same differential coefficients, presents itself also in the ordinary processes of the integral calculus : thus, if u = (a; -I- 1)2, we find 216 THIRD REPORT — 1833. Legendre, following closely in the footsteps of this illustrious analyst, has succeeded in the investigation of methods by which the values of this transcendent F (r) may be calculated to any required degree of accuracy for all positive values of r, and has a"* x^ X" a-' 1 „ „ If we replace (« + 1)2 by a;2 + 2 a; + 1, we find d-^u X* '-^^ «:- r. . r. I^=r2 + T + T+C^+Ci. d-^ u It is obvious that these two values of -, 5 cannot be made identical, without the aid of the proper arbitrary functions. dr u (7.) Let u = f" where v =.f{x) : and let it be required to find TTr' d'' u llie general expression for -j-^ , when r is a whole number, is generally extremely complicated, though the law of formation of its terms can always be assigned. If the inexplicable expressions in the resulting series be re- placed by their proper transcendents, the expression may be generalized for any value of r. d V , .^d^ V , dr u li -J— =:p and if -j—^ = c, a constant quantity, then ~r^'=- n (n — 1) .... {n — r -\- I) v"-'' f c r (r — I) £_^ _i_ r (r— 1) (r — 2) (?• — 3) c^ v^ 7 V + 1 (ra — r + 1) 'y^l.2.Qi — r+l) {n— r + 2)"^'^ ^H r (1 +w) „ r r (r — 1) cc , „ ) + T(^ - --' + r(-;-i) --^ + ^- which is in a form adapted to any value of r. Tf _ ' 1 ■ dx^ + X- x^ Rational functions of x may be resolved into a series of fractions, whose denominators are of the form {x -f- a)", and whose numerators are constant quantities, whose rth differential coefficients may be found by the methods given above. Irrational functions must be treated by general methods similar to that followed in the example just given, which will be more or less com- plicated according to the greater or less number of successive simple differ- entials of the function beneath the radical sign, which are not equal to zero. X - (^- /o 3 . 5 2"'a-4 3.5.7 !) l 2* a;2 2^* x^ + ^ + ^.; + &c. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 217 given tables of its logarithmic values to twelve places of decimals, with colmiins of three orders of differences for 1000 equal in- tervals between 1 and 2 * ; and similar tables have been given by Bessel and by others. We may therefore consider ourselves to be in possession of its numerical values under all circum- stances, though we should not be justified in concluding from thence that their explicit general symbolical forms are either discoverable or that they are of such a nature as to be ex- pressible by the existing language and signs of algebra. The equation r (/•) = (r - 1) (r - 2) .... (r - 711) F (r - m), or r {r — m) = -, ^r-, \^ -, ., ^ ' (r — 1) (r — 2) . . (r — m) where m is a whole number, will explain the mode of passing- from the fundamental transcendents, when included between r = and I, or between r = 1 and 2, to all the other derived transcendents of their respective classes f. The most simple of such classes of transcendents, are those which correspond to (^) = s/-, which alone require for their determination the aid of no higher transcendents than circular arcs and logarithms. In all cases, also, if we consider F (r) as expressing the arithmetical yaiwe of the corresponding transcendent, its general form would require the introduction of the factor V, considered as the recipient of the multiple signs of aftection which are proper for each dif- ferential coefficient, if we use that term in its most general sense. In the note, p. 211, we have noticed the principal properties of these fractional and general differential coefficients, partly for the purpose of establishing upon general principles the basis of a new and very interesting branch of analysis %, and * Fonctions Elliptiques, torn. ii. p. 490. tTh„s,r(-L) = V». r(i-) = i. V^,r(±) = IJ Vx. ^/ Tc, &c. X The consideration of fractional and general indices of differentiation was first suggested by Leibnitz, in many passages of his Commercium EpistoKcum with John Bernouilli, and elsewhere ; but the first definite notice of their theory was given by Euler in the Petersburcjh Commentaries for 1731 : they have also been considered by Laplace and other writers, and particularly by Fourier, ill his great work. La Tkcorie de la Propagation de la Cliakur. The last of these illustrious authors has considered the general dillcrential coeffi- 218 THIRD REPORT 1833. partly for the purpose of illustrating the principle of the per- manence of equivalent forms in one of the most remarkable examples of its application. The investigations which we have given have been confined to the case of algebraical functions, cients of algebraical functions, through the medium of their conversion into transcendental functions by means of the very remarkable formula, 2 /*+ 00 /»+ CO <?•«= — ^ <p(«)d« / (p{a.)dqcosq{x — oi.), - 03 - CO which immediately gives us, 6r (px _ 2 /*+ CO />+ CD (?»• ~d^-lrj <^{«-)dciJ <P{cc)dq-^^r COS q(x-o,); - CD - 03 which can be determined, therefore, if ^ cos q (,x — a.) can be determined, and the requisite definite integrations effected. If, indeed, we grant the prac- ticability of such a conversion of (p («) in all cases, and if we suppose the difficulties attending the consideration of the resulting series, which arise from the peculiar signs, whether of discontinuity or otherwise, which they may implicitly involve, to be removed, then we shall experience no embarrass- ment or difficulty whatever in the transition from integral to general indices of differentiation. In the thirteenth volume of the Journal de I'Ecole Polytechnique for 1832, there are three memoirs by M. Joseph Liouville, all relating to general in- dices of differentiation, and one of them expressly devoted to the discussion of their algebraical theory. The author defines the differential coefficient of the order f<, of the exponential funetion e"""^ to be mf^ e'"'^, and consequently the ^th differential coefficient of a series of such functions denoted by 2 A^ e*" * must be represented by 2 A^ rtT e""'. If it be granted that we can properly define a general differential coefficient, antecedently to the exposition of any general principles upon which its existence depends, then such a definition ought to coincide with the necessary conclusions deduced by those principles in their ordinary applications : but the question will at once present itself, whether such a definition is dependent or not upon the definition of the simple differential coefficient in this and in all other cases. In the first case it will be a proposition, and not a definition, merely requiring the aid of the principle of the permanence of equivalent forms for the purpose of giving at least an hypothetical existence to ^ ^^ for general, as well as for integral values of ft.. M. Liouville then supposes that all rational functions of x are ex- pressible by means of series of exponentials, and that they are consequently reducible to the form 2 A^ e*"^, and are thus brought under the operation of his definition. Thus, if x be positive, we have, — = / e-"*^' X .J a. and therefore. d^\ r^ REPORT ON CERTAIN BRANCHES OF ANALYSIS. 219 and have been chiefly du'ected to meet the difficulties connected with the estimation of the values of the coefficient of differen- tiation in the case of fractional and general indices. If we should extend those investigations to certain classes of tran- Tvhich is easily reducible to the form, I) ,.1 +M ^1 (-irr(i + ^) an expression which we have analysed in the note on p. 211. This part of M. Liouville's theory is evidently more or less included in M. Fourier's views, which we have noticed above. The difficulties which attend the complete / -1 \ M -r* ^1 I \ developement of the formula — for all values of u, which the principle of equivalent forms alone can reconcile, will best show how little progress has been made when the ^ttth differential coefficient of — is reduced to such a form. ^ M.Liouville adopts an opinion, which has been unfortunately sanctioned by the authority of the great names of Poisson and Cauchy, that diverging series should be banished altogether from analysis, as generally leading to false results ; and he is consequently compelled to modify his formulas with refer- ence to those values of the symbols involved, upon which the divergencj' or convergency of the series resulting from his operations depend. In one sense, as we shall hereafter endeavour to show, such a practice may be justified ; but if we adopt the principle of the permanence of equivalent forms, we may safely conclude that the limitations of the formulae will be sufficiently ex- pressed by means of those critical values which will at once suggest and re- quire examination. The extreme multiplication of cases, which so remark- ably characterizes M. Liouville's researches, and many of the errors which he has committed, may be principally attributed to his neglect of this important principle. It is easily shown, if /3 be an indefinitely small quantity, that p/Sx p— /3r pm/Ji _ „— n/3i iy — e e ^j. e e 2/3 (m + w) /3 and that consequently any integral function A -j- A x -)- . . A„ xP, involving integral and positive powers of x only, may be expressed by 2 A^ t"*^, where m is indefinitely small ; and conversely, also, 2 A^j^ e"''' may, under the same circumstances, be always expressed by a similar integral function oix. M. Liou- ville, by assuming a particular form, ^^^ 2/3 ' where C is arbitrary, and /3 indefinitely small, to represent zero, and differen- tiating, according to his definition, gets _ dx^ '^ 2V/3 2 ' but it is evident that by altering the form of this expression for zero we might show that was equal cither to sera or to ivjinity ; and that in the latter 220 THIRD REPORT — 1833. scendental functions, such as e"**, sin m x, and cos m x, we shall encounter no such difficulties, in as much as the differen- tials of those functions corresponding to indices which are ge- neral in form, though denoting integral numbers, are in a form case the critical value infinity might be merely the indication of the existence <P of negative or fractional powers of a; in the expression for , which were not expressible by any rational function of e^^ under a finite form and in- volving indefinitely small indices only. And such, in fact, would be the re- sult of any attempt to differentiate this exponential expression for x or its powers, with respect to fractional or negative indices. It has resulted from this very rash generalization of M. Liouville that he has assigned as the ge- neral form of complementary arbitrary functions, C + Ci a; + C2 a:2 + Cs x^ + &c., which is only true when the index of differentiation is a negative whole number. Most of the rules which M. Liouville has given for the differentiation of algebraical functions are erroneous, partly in consequence of his fundamental error in the theory of complementary arbitrary functions, and partly in consequence of his imperfect knowledge of the constitution of the formula r ( 1 + >»; . |.jjyg ^fjgy deducing the formula T{l+n — r) d'' • 7 VtT^ {-ly.a'-. r (n + r) ("'" + ^) = 1 .2...(«-l) (ax + b)n + r' dx'' which is only true when n is a whole number, he says that no difficulty pre- sents itself in its treatment, whilst n + r is > 0, but that T {n + r) be- comes infinite, when »i + r < 0, in which case he says that it must be transformed into an expression containing finite quantities only, by the aid of complementary functions ; whilst, in reality, T {n + r) is only infinite when n -\- r is zero or a negative whole number, and the forms of the com- plementary functions, such as he has assigned to them, are not competent to effect the conversion required. In consequence of this and other mistakes, dr 1 in connexion with the important case ' {ax -\- &)" , nearly all his conclu- dx'T sions with respect to the general differentials of rational functions, by means of their resolution into partial fractions, are nearly or altogether erroneous. The general differential coefficients of sines and cosines follow immediately from those of exponentials, and present few difficulties upon any view of their theory. In looking over, however, M. Liouville's researches upon this sub- ject, I observe one remarkable example of the abuse of the first principles of „, , „ d^ cos mx ... , reasoning m algebra. There are two values 01 , one positive and dx'^ the other negative, considered apart from the sign of m, whether positive or negative : but if we put cos m x = — cos m x + — cos m x, we get ' ,v li dr cosmx 1 «■ cos mx "i. d' cos m x _ dx- ^ d^^ ^ dx^ REPORT ON CERTAIN BRANCHES OF ANALYSIS. 221 which is adapted to the immediate apphcation of the general principle in question. Thus, ii' u = e'" *■, we get d x-"^^ ' d x-" "^ "" d X" ~ "* ^ ' when r is a whole number, and therefore, also, when r is any quantity whatsoever. it u = sm m X, -J— = m sm {-—■+mx), -z — -„ = 7n^ sin dx \2 / d X- d^ U ( T "K \ (w + m x), . . . . y—y = m'' sin ( —^ + m x j when r is a whole number, and therefore generally. In a similar manner if u = cos m X, or rather ti — cos m {Vf x, (introducing P as a factor in order to express the double sign of m x, if de- termined from the value of its cosine,) then we shall find fjh' -ft \ T TT I - — = (m a/ ly COS < —^ + {m \^ I) x >, whatever be the value of r. If u = e" * cos m x, we get, by very obvious re- ductions, making o = . - and fl = cos ' — , -, — = p'' e"'^ cos (m X + n fl). dx^ '^ ^ ' It is not necessary to mention the process to be followed in ob- and if we combine arbitrarily the double values of the two parts of the second member of this equation, we shall get four values of — ^°^ *" ^, instead of two ; and, in a similar manner, if we should resolve cos m x into any number of parts, we should get double the number of values of cos m x ^ j^ ^j^.^ dr principle of arbitrary combinations of algebraical values derived from a com- aJx mon operation was admitted, we must consider -^ — as having two values, and its equivalent series x^ + x^-\-x- + &c. as having an infinite number. But it is quite obvious that those expressions which involve implicitly or explicitly a multiple sign must continue to be estimated with respect to the same value of this sign, however often the reci- pient of the multiple sign may be repeated in any derived series or expression. The case is difteretit in those cases where the several terms exist indepen- dently of any explicit or implied process of derivation. 222 THIRD REPORT — 1833. taining the general differential coefficients of other expressions, such as (cos x)", cos tn x x cos n x, &c., which present no kind of difficulty. In all such cases the complementary arbitrary functions will be supplied precisely in the same manner as for the corresponding differential coefficients of algebraical func- tions. The transition from the consideration of integral to that of fractional and general indices of differentiation is somewhat starthng when first presented to our view, in consequence of our losing sight altogether of the principles which have been employed in the derivation of differential coefficients whose in- dices are whole numbers : but a similar difficulty will attend the transition, in every case, from arithmetical to general values of symbols, through the medium of the principle of the perma- nence of equivalent forms, though habit and in some cases im- perfect views of its theory, may have made it familiar to the mind. We can form distinct conceptions of m . m, m . m . m, m .m .7)1 , . . . (r), where m is a whole number repeated twice, thrice, or r times, when r is also a whole number ; and we can readily pass from such expressions to their defined or as- sumed equivalents m^, m^, ... tm'': in a similar manner we can rea- dily pass from the factorials * 1.2, 1.2.3,... 1 . 2 ... r, to their assumed equivalents r(3), r(4), . . . r(l + r), as long as r is a whole number. The transition from m^ and r (1 + r) when r is a whole nvimber, to 7n'' and r (1 + r) when r is a general symbol, is made by the principle of the equivalent forms ; but by no effort of mind can we connect the first conclusion in each case with the last, without the aid of the intermediate formula, involv- ing symbols which are general in form though specific in value ; and in no instance can we interpret the ultimate form, for values of the symbols which are not included in the first, by the aid of the definitions or assumptions which are employed in the establishment of the primary form. In all such cases the interpretation of the ultimate form, when such an interpre- tation is discoverable, must be governed and determined by a reference to those general properties of it which are inde- pendent of the specific values of the symbols. » Legendre has named the function r(l + r) = 1 . 2 . . . r, the function gamma. Kramp, who has written largely upon its properties, gave it, in his Analyse des Refractions Astronomiques, the name oifaculte nimierique; but in his subsequent memoirs upon it in the earlier volumes of the Annales des Ma- thematiques of Gergonne he has adopted the name oi' factorial function, which Arbagost proposed, and which I think it expedient to retain, as recalling to mind the continued product which suggests this creature of algebraical lan- suaffe. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 223 The law of derivation of the terms in Taylor's series, , d i( J cP u li^ cP u h^ , p '' = ^^ + ^ • ^' + ^- O + ^^ • TT^TS + ^^^ is the same as in the more general series and if we possess the law of derivation of —, — and of —, — ^, we (JL X (IX can find all the terms of both these series, whatever be the value of r. The first of these terms must be determined through the ordinary definitions of the differential calculus ; the second must be determined in form by the same principles, and gene- ralized through the medium of the principle of equivalent forms. Both these processes are indispensably necessary for d'' u . . the determination of -. — : but it is the second of them which dx^ altogether separates the interpretation of-r— ^ from that of -7—, d or rather of -7—^ when r is a whole number, unless in the par- ct oc ticular cases in which the symbols in both are identical in value. There are two distinct processes in algebra, the direct and the inverse, presenting generally very different degrees of dif- ficulty. In the first case, we proceed from defined operations, and by various processes of demonstrative reasoning we arrive at results which are general in form though particular in value, and which are subsequently generalized in value likewise : in the second, we commence from the general result, and we are either required to discover from its form and composition some equivalent result, or, if defined operations have produced it, to discover the primitive quantity froni which those operations have commenced. Of all these processes we have already given examples, and nearly the whole business of analysis will consist in their discussion and developement, under the infinitely varied forms in which they will present themselves. The disappearance of undulating and of determinate func- tions with arbitrary constants, upon the introduction of inte- gral or other specific values of certain symbols involved, is one •of the chief sources * of error in effecting transitions to equiva- * The theory of discontinuous functions and of the signs of discontinuity will show many others. 224 THIRD REPORT — 1833. lent forms, whethe, the process followed be direct or inverse. Many examples of the first kmd may be found in the researches of Poinsot respecting certain trigonometrical series, which will be noticed hereafter, and wliich had been hastily gene- ralized by Euler and Lagrange ; and a remarkable example of the latter has already been pointed out, in the disappearance of the functions with arbitrary constants in the transition from u to ■^—^, when r becomes a whole positive number. The gene- ral discussion of such cases, however, would lead me to an examination of the theory of the introduction of determinate and arbitrary functions in the most difficult processes of the integral calculus and of the calculus of functions, which would carry me far beyond the proper limits and object of this Re- port. I have merely thought it necessary to notice them in this place for the purpose of showing the extreme caution which must be used in the generalization of equivalent results by means of the application of the principle of the permanence of equivalent forms*. The preceding view of the principles of algebra would not only make the use and form of derivative signs, of whatever kind they may be, to be the necessary results of the same ge- neral principle, but would also show that the interpretation of their meaning would not precede but follow the examination of the circumstances attending their introduction. I consider it to be extremely important to attend to this order of succession between results and their interpretation, when those results belono- to symbolical and not to arithmetical algebra, in as much as the neglect of it has been the occasion of much of the con- fusion and inconsistency which prevail in the various theories which have been given of algebraical signs. I speak of deri- * Euler, in the Peiersburgh Acts for 1774, has denied the universality of this principle, and has adduced as an example of its failure the very remark- able series 1 _ am (1 _ a»») (1 — a™-i) (1 — C") (1 — a*"-!) (1 — C'-Z) T^^ + 1^^2 + 1 - «' "^ ' which is equal to m, when m is a whole number, but which is apparently not equal to m, for other values of m, unless at the same time a = 1 : the occur- rence, however, of zero as a factor of the (»n + 1)* and following terms in . the first case, and the reduction of every term to the fonn -^ in the second, would form the proper indications of a change in the constitution of the equi- valent function corresponding to these values of w and a, of which many ex- amples will be given in the text. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 225 vative signs as distinguished, from those p«^itive signs of ope- ration which are used in arithmetical algebra ; but such signs, though accurately defined and hmited in their use in one sci- ence, will cease to be so in the other, their meaning being de- pendent in symbolical algebra, in common with all other signs which are used in it, upon the symbolical conditions which they are required to satisfy. I will consider, in the first place, signs of affection, which are those symbolical quantities which do not affect the magnitudes, though they do affect the specific nature, of the quantities into which they are incorporated. Of this kind are the signs + and — , when used independ- ently ; or their equivalents + 1 and — 1, when considered as symbolical factors ; the signs (-}- 1)" and (— 1)", or their sym- bolical equivalents cos 2rmT-\- \/ — \ sin 2 r w tt and cos {2r -\- \) n w + -v/ — 1 sin (2 r + 1 ) w T ; 2r?nr-v/~l 1 (2j-+ 1) n!r\/— 1 or e and e^ The affections symbolized by the signs + 1 and — 1 admit of very general interpretation consistently with the symbolical conditions which they are required to satisfy,- and particularly so in geometry : and it has been usual, in consequence of the great facility of such interpretations, to consider all quan- tities aftected by them (which are not abstract) as possible, that is, as quantities possessing in all cases relations of exist- ence which are expressible by those signs. It should be kept in mind, however, that such interpretations are in no respect distinguished from those of other algebraical signs, except in the extent and clearness with which their conditions are sym- bolized in the nature of things. The other signs of affection, different from + 1 and — 1, which ai'e included in (1)" and ( — 1)", are expressible generally by cos fl + v^ — 1 sin fl, or by « + /3 V^ — 1 ,where a and |3 may have any values between 1 and — 1 , zero included, and where «^ -f- /3^ = 1. To all quantities, whether abstract or concrete, expressed by symbols affected by such signs, the common tei'm impossible has been applied, in contradistinction to those possible magni- tudes which are affected by the signs -|- and — only. If, indeed, the affections symbolized by the signs included under the form cos 9 -|- -/ — 1 sin fl, admitted in no case of an in- terpretation which was consistent with their symbolical condi- tions, then the term impossible would be correctly npplied to quantities affected by them : but in as much as the signs + and 1 833. Q 226 THIRD REPORT — 1833. — , when used indepefndently, and the sign cos 6 + V — I sm d, when taken in its most enlarged sense, equally/ originate in the generalization of the operations of algebra, and are equally in- dependent of any previous definitions of the meaning and extent of such operations, they are also equally the object of inter- pretation, and are in this respect no otherwise distinguished from each other than by the greater or less facility with which it can be applied to them. Many examples * of their consistent interpretation may be pointed out in geometry as well as in other sciences : thus, if + a and — a denote two equal lines whose directions are op- posite to each other, then (cos 9 + v^ — 1 sin fl) a may denote an equal line, making an angle 9 with the line denoted by -^ a ; and consequently a ^ — \ will denote a line which is perpen- dicular to -1- a. This interpretation admits of very extensive application, and is the foimdation of many important conse- quences in the application of algebra to geometry. The signs of operation 4- and — may be immediately inter- preted by the terms addition and subtraction, when applied to unaffected symbols denoting magnitudes of the same kind : if they are applied to symbols affected with the sign — , these signs, and the terms used to interpret them, become convertible. Thus a + {— h) — a — b, and a — {— b) = a + b; or the al- gebraical sum and difference of a and — b, is equivalent to the algebraical difference and sum of a and b : but if they are applied to lines denoted by symbols affected by the signs cos 9 + V —I. sin 6, and cos 9' + -/ — 1 sin 9', the results will no longer de- note the arithmetical (or geometrical) su?n and difference of the lines in question, but the magnitude and position of the dia- gonals of the parallelogram constructed upon them, or upon lines which are equal and parallel to them. Thus, if we denote the line A B by a, and the h ne A C at right angles to it by 6 V — 1 , and if we complete the parallelograms AB D C and AB C E, then a + b \/ - i will denote the diagonal A D, and a — b \/ — I will denote the other diagonal B C, or the equal and parallel line A E. It is easily shown that a + b \/ ^^ = V'Ca^ + *^ (cos 9 cos" ^ a + »/ —\ sin 9), (where 9 = — r ^ , J , and a — b i/ — I a: v'(«' + b^) {cos 9 — a/^^I sin 9} ; it follows, therefore, that * Peacock's Algebra, chap. xii. Art. 437, 447, 448, 449. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 227 a + b V — I and a — b ^Z — I may be considered as repre- senting respectively a single line, equal in magnitude to \/{a^ + b^) *, and aiFected by the sign cos 9 + -/ — 1 sin 9 in one case, and by the sign cos 9 — •i/ — 1 sin Hn the other ; or as denoting the same lines through the medium of the opera- tions denoted in the one case by +, and in the other by — , upon the two lines at right angles to each other, which are de- noted by a and b V —\. We have spoken of the signs of operation + and — , as di- stinguished from the same signs when used as signs of affection, and we have also denominated a -\- b \Z — 1, and a — b V' — 1, the sum and difference of a and b V —\, though they can no longer be considered to be so in the arithmetical or geometrical sense of those terms ; but it is convenient to explain the mean- ing of the same sign by the same term, though they may be used in a sense which is not only very remote from, but even totally opposed to f , their primitive signification ; and such a licence in the use both of signs and of phrases is a necessary consequence of making their interpretation dependent, not upon previous and rigorous definitions as is the case in arithmetical algebra, but upon a combined consideration of their symbolical conditions, and the specific nature of the quantities represented by the symbols. It is this necessity of considering all the re- sults of symbolical algebra as admitting of interpretation sub- sequently to their formation, and not in consequence of any previous definitions, which places all those results in the same relation to the whole, as being equally the creations of the same general principle : and it is this circumstance which jus- • The arithmetical quantity ^{a- -)- b^) has been called the modulus of a + 6<^ — 1 by Cauchy, in his Cours d' Analyse, and elsewhere. It is the single unaffected magnitude which is included in the affected magnitude a + 6 V — 1 : conversely the affected magnitude (cos 6 + V— 1 sin d) ^Ja^ + b"- is reducible to the equivalent quantity a -f- 6 a/— 1, if cos &=. „ , and therefore sin ^ = t The sum of a and — b, or a -\- (— b], is identical with the difference of a and b, or with a — b. The term operation, also, which is applied generally to the fact of the transition from the component members of an expression to the final symbolical result, will only admit of interpretation when the nature of the process which it designates can be described and conceived. In all other cases we must regard the final result alone. Thus, if a and b denote lines, we can readily conceive the process by which we form the results a + 6 and a — 6, at least when a is greater than b. But when we interpret a + 6 \/— 1 to mean a determinate single line with a determinate position, we are incapable of con- ceiving any process or operation through the medium of which it is obtained. q2 2^g THIRD REPORT— 1835. tifies the assertion, which we have made above, that quantities or their symbols affected by the signs +, — , or cos 9 + \^ — \. sin 6, are only distinguished from each other by the greater or less facility of their interpretation. The geometrical interpretation of the sign V — 1, when applied to symbols denoting lines, though more than once suggested by other authors, was first formally maintained by M. Buee in a paper in the Philosojihical Transactions for 1806*, which contains many original, though very imperfectly deve- loped views upon the meaning and application of algebraical signs. In the course of the same year a small pamphlet was pub- lished at Paris by M. Argand, entitled Essai sur une Maniere de repr^senter les Quanfites Imaginaires, dans les Construc- tions G^om^triques, written apparently without any knowledge of M. Buee's paper. In this memoir M. Argand arrives at this proposition. That the algebraical sum f of two lines |, estimated both according to magnitude and dii'ection, would be the dia- gonal of the parallelogram which might be constructed upon them, considei'ed both with respect to direction and magnitude, which is, in fact, the capital conclusion of this theory. This memoir of M. Argand seems, however, to have excited very little attention ; and his views, which were chiefly founded upon analogy, were too little connected with, or rather dependent upon, the great fundamental principles of algebra, to entitle his conclusions to be received at once into the great class of admitted or demonstrated truths. It would appear that M. Argand had consulted Legendre upon the subject of his me- moir, and that a favourable mention of its contents was made by that great analyst in a letter which he wrote to the brother of M. J. F. Fran9ais, a mathematician of no inconsiderable eminence. It was the inspection of this letter, upon the death of his brother, which induced M. Franpais to consider this subject, and he published, in the fourth volume of Gergonne's Annales des Mathematiqties for 1813, a very curious memoir upon it, containing views more extensive, and more completely developed than those of M. Argand, though generally agreeing with them in their character, and in the conclusions deduced from them. This publication led to a second memoir upon the same theory from M. Argand, and to several observations upon it, in the same Journal, from MM. Servois, Franpais, and Gergonne, in which some of the most prominent objections to it were proposed, and partly, though very imperfectly, an- * This paper was read in 1805. f -ta somme dirigee. I Lignes dirigies. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 229 swered. No further notice appears to have been taken of these researches before the year 1828, when Mr. Warren's treatise on the geometrical representation of the square roots of negative quantities * w as published. In this work Mr. Warren proposes to give a geometi'ical representation to every species of quan- tity ; and after premising definitions of addition, subtraction, multiplication and division, involution and evolution, which are conformable to the more enlarged sense which interpretation would assign to those operations when applied to lines repre- sented in position as well as in magnitude ; and after showing in great detail the coincidence of the symbolical results obtained from such definitions with the ordinary results of arithmetical and symbolical algebra, he proceeds to determine the meaning of the different symbolical roots of 1 and — I, when applied to symbols denoting lines, under almost every possible circum- stance. The course which Mr. Warren has followed leads almost necessarily to very embarrassing details, and perhaps, also, to the neglect of such comprehensive propositions as can only derive their authority from principles which make all the results of algebra which are general in form independent of the specific values and representation of the symbols : but at the same time it must be allowed that his conclusions, when viewed in connexion with his definitions, were demonstrably true ; a character which could not be given to similar conclusions when they were attempted to be derived by the mere aid of the arith- metical definitions of the fundamental operations of algebra. This objection to the course pursued by Mr. Warren will more or less apply to all attempts which are made to make the previous interpretations of algebra govern the symbolical con- clusions ; for though it is always possible to assign a meaning to algebraical operations, and to pursue the consequences of that meaning to their necessary conclusions, yet if the laws of combination which lead to such conclusions are expressed through the medium of general signs and symbols, they will cease, when once formed, to convey the necessary limitations of meaning which the definitions impose upon them. It is for this reason that we must in all cases consider the laws of com- bination of general symbols as being arbitrary and independent in whatever manner suggested, and that we must make our in- terpretations of the results obtained conformable to those laws, and not the laws to the interpretations : it is for the same reason, likewise, that our interpretations will not be necessary, though • A Treatise on the Geometrical Representation of the Square Moots of Ne- tfntive Quantities, by the Rev. John Warren, M.A., Fellow and Tutor of Jesus College Cambridge. J 828. 230 THIRD REPORT — 1833. governed by necessary laws, except so far as those interpreta- tions are dependent upon each other. Thus, if a be taken to represent a line in ynagnitude , it is not necessary that (cos 3 + -v/ — 1 sin 6) « should represent a line equal in length to the one represented by o, and also making , an angle 5 with the line re- presented by a ; but if (cos 3 + v^ — 1 sin 6) a, may, consistently with the symbolical conditions, represent such a line, without any restriction in the value of 9, then, if it does represent such a line for one value of 9, it must represent such a line for every value of 9 included in the formula. It is only in such a sense that interpretations can be said in any case to have a necessary and inevitable existence. It is this confusion of necessary and contingent truth which has occasioned much of the difficulty which has attended the theories of the interpretation of algebraical signs. It has been sui:>posed that a meaning could be transmitted through a suc- cession of merely symbolical operations, and that there would exist at the conclusion an eqvially necessary connexion between the primitive definition and the ultimate interpretation, as be- tween the final symbolical result and the laws which govern it. So long as the definitions both of the meaning of the symbols and of the operations to which they are required to be subject are sufficient to deduce the results, those results will have a necessary interpretation which will be dependent upon a joint consideration of all those conditions ; but whenever an operation is required to be performed under circumstances which do not allow it to be strictly defined or interpreted, the chain of con- nexion is broken, and the interpretation of the result will be no longer traceable through its successive steps. This must take place whenever negative or other affected quantities are introduced, and whenever operations are to be performed, either with them, or upon them, even though such quantities and signs should altogether disappear from the final result. This principle of interpretation being once established, we must equally consider — I, \/ — \, cos 9 + -v^ — 1 sin 9, as signs of impossibility, in those cases in which no consistent meaning can be assigned to the quantities which are aflPected by them, and in those cases only : and it must be kept in mind that the impossibility which may or may not be thus indicated, has re- ference to the interpretation only, and not to the symbolical result, considered as an equivalent form : for all symbolical results must be considered as equally possible which the signs and symbols of algebra, whether admitting of interpretation or not, are competent to express. But there will be found to be many species of impossibility which will present themselves in REPORT ON CERTAIN BRANCHES OF ANALYSIS. 231 considering the relations of formulae with a view to their equi- valence, and also under other circumstances, which will be in- dicated by such means as Avill destroy all traces of the equiva- lence which would otherwise exist. The capacity, therefore, possessed by the signs of affection involving -v/ — 1 of admitting geometrical or other interpreta- tions under certain circumstances, though it adds greatly to our power of bringing geometry and other sciences under the dominion of algebra, does not in any respect affect the general theory of their introduction or of their relation to other signs : for, in the first place, it is not an essential or necessary pro- perty of such signs ; and in the second place, it in no respect affects the form or equivalence of symbolical results, though it does affect both the extent and mode of their application. It would be a serious mistake, therefore, to suppose that such inci- dental properties of quantities affected by such signs constituted their real essence, though such a mistake has been generally made by those who have proposed this theory of interpretation, and has been made the foundation of a charge against them by others, who have criticised and disputed its correctness*. * This charge is made by Mr. Davies Gilbert in a very ingenious paper in the Philosophical Transactions for 1831, " On the Nature of Negative and Im- possible Quantities." He says that those mathematicians take an incorrect view of ideal quantities, — mistaking, in fact, incidental properties for those which constitute their real essence, — who suppose them to be principles of perpendicularity, because they may in some cases indicate extension at right angles to the directions indicated by the correlative signs + and — ; for with an equal degree of propriety might the actually existing square root of a quan- tity be taken as the principle of obliquity, in as much as in certain cases it indicates the hypothenuse of a right-angled triangle. In reply to this last observa tion, it may be observed, that I am not aware that in any case the sign /\/— I has had such an interpretation given to it. It is quite impossible for me to give an abridged, and at the same time a fair view of Mr. Davies Gilbert's theory, within a compass much smaller than the contents of his memoir. But I might venture to say that his proof of the rule of signs rests upon some properties of ratios or proportions which no arith- metical or geometrical view of their theory would enable us to deduce. In con- sidering, also, imaginary quantities as creations of an arbitrary definition, en- dowed with properties at the pleasure of him who defines them, he ascribes to them the same character as to all other symbols and operations of algebra ; but in saying "that quantities affected by the sign /^— 1 possess a. potential existence only, but that they are ready to start into energy whenever that sign is removed," he appears to me to assert nothing more than that symbols are impossible or not, according as they are affected by the sign ^ — 1 or not. Again, in examining the relation of the terms of the equation n(n— I) Q , „ , » (« — 1) (« — 2) „_« „ ,.! , - 232 THIRD REPORT— 1833. Signs of transition are those signs which indicate a change in the nature or form of a function, when considered in the whole course of its passage through its different states of ex- istence. Such signs, if they may be so designated, are gene- rally sero and infinity. Zero and infinity are negative terms, and if applied to desig- he denies the correctness of the reasoning by which it is inferred that the second term of the first, and the even terras of the second members of this equation are equal to oue anotlier (when x is less than 1), because they are the only terms whicli are homogeneous to each other, in as much as we thus ascribe real properties to ideal quantities ; and he endeavours to make this equality depend upon an assumed arbitrary relation between x and y, though it is obvious that if y = cos &, we shall find x = cos n 6, and that, therefore, this relation is determinate, and not arbitrary. A little further examination of this conclusion would show that it did not depend upon any assumed homogeneity of the parts of the members of this equation to each other, but upon the double sign of the radical quantity which is involved upon both sides. In arithmetical algebra, where no signs of affection are employed or recog- nised, both negative and imaginary quantities become the limits of operations ; and when this science is modified by the introduction of the independent signs + and — and the rule for their incorporation, the occurrence of the square roots of negative quantities, by presenting an apparent violation of the rule of the signs, becomes a new limit to the application of this new form of the science. The same algebraists who have acquiesced in the propriety of making the first transition in consequence of the facility of assigning a meaning to negative quantities, at the same time that they retained the definitions and principles of the first science, were startled and embarrassed when they came to the second ; for it was very clear that no attempt could be made to recon- cile the existence and use of such quantities, consistently with the main- tenance of that demonstrative character in our reasonings which exists in geometry and arithmetic, where the mind readily comprehends the nature of the quantities employed, and of the operations performed upon them. The proper conclusion in such a case would be that the operations performed, as well as the quantities employed, were symbolical, and that the results, though they might be suggested by the primitive definitions, were not dependent upon them. If no real conclusions had been obtained by the aid of such merely symbolical quantities, they would probably have continued to be re- garded as algebraical monsters, whose reduction under the laws of a regular system was not merely unnecessary, but altogether impracticable. But it was soon found that many useful theories were dependent upon them ; that any attempt to guard against their introduction in the course of the progress of our operations with symbols would not merely produce the most embarrassing limitations, when such limitations were discoverable, but that they would present themselves in the expression of real quantities, and would furnish at the same time the only means by which such quantities could be expressed. A memorable example of their occurrence under such circumstances presents itself in what has been called the irreducible case of cubic equations. In the Philosophical Transactions for 1778 there is a paper by Mr. Playfair on the arithmetic of impossible quantities, in which the definable nature of algebraical operations is asserted in the most express terms, and in which the truth of conclusions deduced b}^ the aid of imaginary symbols is made to depend upon the analogy which exists between certain geometrical properties REPORT ON CERTAIN BRANCHES OF ANALYSIS. 233 nate states of quantity, are equally inconceivable. We are ac- customed, however, to speak of quantities as infinitely great and infinitely small, as distinguished from finite quantities, whether great or small, and to represent them by the symbols 00 and 0. It is this practice of designating such inconceivable states of quantity by symbols, which brings them, in some de- of the circle and the rectangular hyperbola. It is -well known that the circle and rectangular hyperbola are included in the same equation y=^^{\ — sr), if we suppose x to have any value between + °° and — 00 : let a circle he described with centre C and radius C A = 1, and upon the production of this radius,; let a rectan- gular hyperbola be de- scribed whose semiaxis is 1, in a plane at right an- gles to that of the circle: iftf denote the angle AC P, then the circular cosine and sine (C M and P M) are expressed by and — = 2 2 -/— 1 respectively ; whilst the hyperbolic cosine and sine (to adopt the terms pro- posed by Lambert) corresponding to the angle 6 -v/— 1 (in a plane at right angles to the former) are expressed by -^ and V- 1 y ^ )' o"- by -^2~~ 2 ' if they be considered as determined by the following conditions ; namely, that (hyp. cosine)^ — (hyp- sine)- = 1, and that hyp. cos 6 = hyp. cos — ^, and hyp. sine ^ = — hyp. sine — ^. A comparison of these processes in the circle and hyperbola would show, says Mr. Playfair, that investigations which are conducted by real symbols, and therefore by real operations, in the hy- perbola, would present analogovs imaginary symbols, and therefore analogous imaginary operations in the circle, and conversely ; and that the same species of analogy which connects the geometrical properties of the circle and hyper- bola, connects the conclusions, of the same symbolical forms, when conducted by real and imaginary symbols. This attempt to convert an extremely limited into a very general analogy, and to make the conclusions of symbolical algebra dependent upon an insu- lated case of geometrical interpretation, would certainly not justify us in drawing any genera! conclusions from processes involving imaginary symbols, unless they could be confirmed by other considerations. The late Professor Woodhouse, who was a very acute and able scrutinizer of the logic of ana- lysis, has criticised this principle of Mr. Playfair with just severity, in a paper in the Philosophical Transactions for 1802, "On the necessary truth of certain conclusions obtained by means of imaginary expressions." The view which he has taken of algebraical equivalence, in cases where the connexion between the expressions which were treated as equivalent could not be shown to be the result of a defined operation, makes a very near approach to the principle 234 THIRD REPORT— 1833. gree, under the ordinary rules of algebra, and which compels us to consider different orders both ofinjinities and of zeros, though when they are considei*ed without reference to their symbo- lical connexion, they are necessarily denoted by the same sim- ple symbols oo and : thus there is a necessary symbolical di- stinction between (00)2, 00 and (oo ) , and between (0)2, and (0) ; though when considered absolutely as denoting infinity in one case and zero in the other, they are equally designated by the simple symbols 00 and respectively. Though the fundamental properties of and co , considered as the representatives of zero and infinity, are suggested by the ordinary interpretation of those terms, yet their complete in- terpretation, like that of other signs, must be founded upon the of the permanence of equivalent forms : thus, supposing, when x is a real quantity, we can show that «^ = 1 + ^ + rr2 + r:V^ + ^^•' but that we cannot show in a similar or any other manner that 1 . 2 1 . 2 . 3 ^^ °^*^"' then the equivalence in the latter case is assumed, by considering c* as the abridged symbol for the series of terms 1 + ^ V- 1 -Y72- TT2T3 + ^^- ' in other words, the form which is proved to be true for values of the symbols which are general in form, though particular in value, is assumed to be true in all other cases. It is true that such a generalization could not be considered as legitimate, without much preparatory theory and without considerable modifications of our views respecting nearly all the fundamental operations and signs of arith- metical algebra ; but I refer with pleasure to this incidental testimony to the truth and universality of this important law, from an author whose careful and bold examination of the first principles of analytical calculation entitle his opinion to the greatest consideration. Mr. Gompertz published, in 1817 and 1818, two tracts on the Principles and Application of Imaginary Quantities, containing many ingenious and novel views both upon the correctness of the conclusions obtained by means of ima- ginary quantities and also upon their geometrical interpretation. The first of these tracts is principally devoted to the establishment of the following position: "That wherever the operation by imaginary expressions can be used, the propriety may be explained from the capability of one arbitrary quantity or more being introduced into the expressions which are imaginary previously to the said arbitrary quantity or quantities being introduced, so as to render them real, without altering the truth they are meant to express ; and that, in consequence, the operation will proceed on real quantity, the introduced arbitrary quantity or quantities necessary to render the first steps of the reasoning arguments on real quantity, vanishing at the conclusion ; REPORT ON CERTAIN BRANCHES OF ANALYSIS. 235 consideration of all the circumstances under which they pre- sent themselves in symbolical results. In order, therefore, to determine some of the principles upon which those interpreta- tions must be made, it will be proper to examine some of the more remarkable of their symbolical properties. and from whence it will follow that the non-introduction of such can pro- duce nothing wrong." Thus, x^ -I- a x -\- b, which is equal to x{v('+Ty-AT-o} is also equal to whatever be the value of the quantity /3 ; a conclusion which enables us to reason upon real quantities and to make /3 = 0, when the primitive factors are required. Similarly, if mstead of ^ ^ y, we suppose ■ = V — "■ . and if mstead of ^ = x. 2 2a/-1 we suppose = x — R, we shall find, whatever /3 may be, 2 V/3— 1 g*V/3-i — j,_j^f_|_ ^^ — I (^ — R)^ a result which degenerates into the well known theorem e '^~ ^ = y -\- V — 1 x, if /3 = 0. Many other ex- amples are given of this mode of porismatizing expressions, (a term derived by Mr.Gompertz from the definition of porisms in geometry,) by which operations are performed upon real quantities which would be otherwise imaginary : and if it was required to satisfy a scrupulous mind respecting the correctness of the real conclusions which are derived by the use of imaginary expressions, there are few methods which appear to me better calculated for this purpose than the adoption of this most refined and beautiful expedient. The second tract of Mr. Gompertz appears to have been suggested by M. Buee's paper in the Philosophical Transactions, to which reference has been made in the text : it is devoted to the algebraical representation of lines both in position and in magnitude, as a part of a theory of what he terms func- tional projections, and embraces the most important of the conclusions obtained by Argand and Fran9ais, with whose researches, however, he does not appear to have been acquainted. I should by no means consider the process of rea- soning which he has followed for obtaining these results to be such as would naturally or necessarily follow from the fundamental assumptions of algebra : but it would be unjust to Mr. Gompertz not to express my admiration of the skill and ingenuity which he has shown in the treatment of a very novel subject and in the application of his principles to the solution of many curious and difficult geometrical problems. 236 THIRD REPORT — \833. If we assume a to denote a finite quantity, then (1.) a + = a, and a + oo = + oo . Consequently does not affect a quantity with which it is connected by the sign + or — , whilst co , similarly connected with such a quantity, altogether absorbs it. (2.) axO = 0, axoo =oo;--- = co and — = 0. 00 It is this reciprocal relation between siero and infinity which is the foundation of the great analogy which exists between their analytical properties. (3.) If these symbols be considered absolutely by themselves, without any reference to their symbolical origin, then we must consider ~ = \ and = 1. CO But if those symbols be considered as the representatives equally of all orders of zeros and infinities respectively, then ~ and may represent either I or « or or co , its final form and value being determined, when capable of determina- tion, by an examination of the particular circumstances under which those symbols originated. The whole theory of vanish- ing fractions will depend upon such considerations. Having ascertained the principal symbolical conditions which and CO are required to satisfy, we shall be prepared to con- sider likewise the principle of their interpretation. The exami- nation of a few cases of their occurrence may serve to throw some light upon this inquiry. Let us consider, in the first place, the interpretation of the critical values 0, oo and ~ in the formulae which express the values of x and y in the simultaneous equations, a X + b y = c \ a' X + b' y ■= c' J In this case we find ^ t:- and y = X '•4^-!} -'{I'-l.}' „' „ „' If -7- = -jT, — = — 7) and therefore -z- — -n, then a: = -^ b a a bo and y = -Q- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 237 In this case «' = in a, U = m a, and c' = m c, and the second equation is deducible from the first, and does not furnish, there- fore, a new condition: under such circumstances, therefore, the values of x and y are really indeterminate, and the occur- rence of -r- in the values of the expressions for x and y is the sign, or rather the indication of that indetermination. If 4- be not equal to -,-t, but if — be equal to —f, then x — co b o, a and y = 00 . In this case vfe have a! — m a,h' =■ mh, but c' is not equal to m c; and the conditions furnished are inconsistent, or more properly speaking impossible. In this case, the occur- rence of the sign oo in the expressions for x and y is the sign or indication of this inconsistency or impossibility: and it should be observed that no infinite values of x and y, if the infinities thus introduced were considered as real existences and identi- cal in both equations, would satisfy the two equations any more than any two finite values of x and y which would satisfy one of them. We may properly interpret go in this case by the term impossible. c e' b' b If -^ = ^7, but if —7 be not equal to — , then x is zero and y h V a a is finite, and therefore possible. It is in this sense that we should include %ero amongst the possible values of x or y, a use or rather an abuse of language to which we are somewhat familiarized, from speaking of the zero of quantity as an exist- ing state of it in the transition from one affection of quantity to another. If we should take the equations of two ellipses, whose semi- axes are a and b, a' and b' respectively, which are f! J. ^ - ] „2 -f- ^,2 - i' ^ ^ it - \ and consider them as simultaneous when expressing the co- ordinates of their points of intersection, then we should find X = //6^ W\ and y = /f^ ^ . V la^ ~ ci'^S V 16^ ~ b'^S If we suppose — = —7, or the ellipses to be similar, and at the same time b not equal to b', then or = op and y = cc , Avhich 238 THIRD REPORT — 1833. would properly be interpreted to mean that under no cir- cumstances whatever, whether in the plane of x y or in the plane at right angles to it, in which the hyperbolic portions* of curves expressed by those equations are included, would a point of intersection or a simultaneous value of x and y exist : or in other words, the sign or symbol oo would in this case mean that such intersection was impossible. If we supposed — = — ;; and also h = b',or the ellipses to be coincident in all their parts, then we should find o^ = -y- and y = ^, indicating that their values were indeterminate, in as much as every part in the iden- tical curves would be also a point of intersection, and would fur- nish therefore simultaneous values. If we should suppose b greater than b', a greater than a\ and -j- not equal to -77 , then we should find X = u and y = ^ \^—U or x = a. \/ — 1 and y = j3^ according as -j- is less or greater than -77. In this case, one ellipse entirely includes the other, but the hyperbolic portions at right angles to their planes, which are in the direction of the major axis in one case and in that of the minor axis in the other, will intersect each other at points whose coordinates are the values of x and y above given : it wovdd appear, therefore, that the impossible intersection of the curves would be indi- cated by the sign or symbol go alone, and not by \/—l. The preceding example is full of instruction with respect to the interpretation of the signs of algebra, when viewed in con- nexion with the specific values and representations of the sym- bols ; and there are few problems in the application of algebra to the theory of curve lines which would not furnish the mate- rials for similar conclusions respecting them : but it is chiefly with reference to the connexion of those signs with changes in the nature of quantities, and in the form and constitution of ex- pressions, that their interpretations will require the most care- ful study and examination. We shall proceed to notice a few of such cases. a;2 2/2 y * If in the equation — + -Tg- = 1, we suppose y replaced by y v — 1, and the line which it represents when not afifected by V — 1 to be moved through 90° at right angles to the plane of x y, we shall find an hyperbola included in the equation of the ellipse. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 239 The second member of the equation — ~~ + T2 + «3 + • • • • a — b a a'- a" preserves the same form, whatever be the relation of the values of a and b, and the operation, which produces it, is equally prac- ticable in all cases. As long as a is greater than b, a~b is po- sitive, and there exists, or may be conceived to exist, a perfect arithmetical equality between the two members of the equa- tion. If, however, a = b, we have -jr upon one side and the sum of an infinite series of units multiplied into — upon the other, and both the members are correctly represented by oo ; but if a be less than b, we have a negative and a finite value upon one side of the equation, and an infinite series of perpe- tually increasing terms upon the other, forming one of those quantities to which the older algebraists would have applied the term plus quam infinitum, and which we shall represent by the sign or symbol co . It remains to interpret the occurrence of such a sign under such circumstances. The first member of this equation — — -r is said to pass through infinity when its sign changes from + to — , or con- versely : its equivalent algebraical form presents itself in a se- ries which is incapable of indicating the peculiar change in the nature of the quantity designated by — — -y , which accompa- nies its change of sign. The infinite values, therefore, of the equivalent series (for in its general algebraical form, where no regard is paid to the specific values of the symbols, it is still an equivalent form,) is the indication of the impossibility of ex- hibiting the value of ^ in a series of such a form under such circumstances. Let us, in the second place, consider the more general series for {a — by, or „ f, b ^ n{n- 1) b^ n {n -I) in - 2) 1% & 1 The inverse ratio of the successive coefficients of this sei'ies S40 THIRD REPORT — 1833. approximates continually to — 1 as a limit, and tlie terms be- come all positive or all negative, according as the first negative coefficient is that of an odd or of an even power of — . It follows, a therefore, that if a be greater than b, the series will be conver- gent and finite in all cases ; if « be equal to b, it will be 0, 1, or 00 , according as n is positive, 0, or negative ; and if a be less than b, it will be infinite. The occurrence of tlie last of these signs or values is an in- dication generally that some change has taken place in the na- ture of the quantity expressed by {a — i)", in the transition from a > 6 to a <C 6, which is of such a kind that the correspond- ing series is not competent to express it : thus, if w = 4» then {a — by is affected with the sign V — I when a is less than b, whilst no such sign is introduced nor introducible into the equi- valent series corresponding to such relative values of a and b : and a similar change will take place, whenever a transition through zero or infinity takes place. In this last case [a — by would appear to attain to zero or in- finity, but not to pass through it, and no change would appa- rently take place in its affection corresponding to the change of affection of « — b; but the corresponding series will under the same circumstances change from being finite to infinite, a cir- cumstance which we shall afterwards have occasion to notice, and which we shall endeavour to explain in the course of our observations upon the subject of diverging and converging series. In the preceding examples the sign or symbol go has not presented itself immediately, but has replaced an infinite series of terms, whose sum exceeded any finite magnitude ; and it may be considered as indicating the incompetence of such a series to express the altered state or conditions of the quantity or fraction to which it was required to be altogether, as well as algebraically, equivalent. In the examples which follow, it will present itself immediately and will be found to be the indica- tion of a change in the algebraical form of the term or terms in which it appears, or rather that no terms of the form assigned can present themselves in the required equivalent series or expression. The integral I x^ d x = ^ + C is said to fail when n ■= — 1, in as much as it appears that under such circum- x"' ~ ' stances 7 becomes 00 , which is an indication that the va- ra + 1 REPORT ON CERTAIN BRANCHES OF ANALYSIS. 241 riable part of ^" d x is no longer expressible by a function under the form r' but by one which must be determined n + i by independent considerations. A knowledge, however, of the nature of its form in this particular case has enabled algebraists to bring it under a general form, by which the sign of failure or impossibility is replaced by the sign of indetermination A /r" + ' a;" "*" ' a" "*" ' jr-; for if we put — -— r + C = — — ^ + C, (borrowing from the arbitrary constant,) we shall get an expression -«" + ' n + 1 which becomes -^ when n = — 1, and whose value, determined according to the rules which are founded upon the analytical properties of 0, will be log a; + C. A more general example of the same kind, including the one which we have just considered, is given in the note to page 211, where it is required to determine the general form of — — — and of -, — . — (where n is a positive number) for all dx"" X dx'' x" ^ values of r : a formula is there constructed, from our knowledge of the form in the excepted case, which is capable of correctly expressing its value in all cases whatever. The cases in which the series of Taylor is said to fail are of a similar nature. Thus, if u = <f (x) = x + \f x — a, then du , , d^u W , d^u h^ ^ u' = <p (:c + h) = u + j-^h + j^, Y72 + J^' T7273 ^^- ' ft U and if we suppose x =i a, all the differential coefficients ^, -^—2, &c., become infinite, which is an indication that no terms of such a form exist in its developement, which becomes, under such circumstances, a + V'/'- The reasons of this failure in such cases have been very completely explained by Lagrange and other writers ; but it is possible, by presenting the deve- lopement which constitutes Taylor's series under a somewhat different and a somewhat more general form, that the series may be so constructed as to include all the excepted cases. There are two modes in which the developement of <p (.r + h) according to powers of h may be supposed to be effected. In the first and common mode we begin by excluding all those terms in the developement whose existence would be incon- 1833. R 242 THIRD REPORT — 1833. sistent with general values of the symbols : m the second we should assume the existence of all the terms which may cor- respond to values of the symbols, whether general or specific, and then prescribe the form which they must possess, con- sistently with the conditions which they are required to satisfy. If we adopt this second course, and assuming u ■= (^ {x) and 2^' = (p (a; + li), if we make u' = « + A /i" + B /^* + C h" + &c., the inquiry will then be, if there be such a term as A h", where A is a function of x or a constant quantity, and a is any quantity whatsoever, what are the properties of A by which it may be determined ? For this purpose we shall proceed as follows. It is very easy to show, from general considerations, that if iH be considered successively as a function of x and of h, -7—7 = -rrr. , for all values of r, whether whole or fractional, positive or negative : it will follow, therefore, (adopting the principles of differentiation to general indices which have been laid down in the note, p. 211,) that d'^u' _ r(i + a) ^ r(i + &) B,,_„ . . omitting the arbitrary complementary functions, which will in- volve powers of h. In a similar manner we shall get d'u' d^u d" K j„ d^B ,, dx" dx" dx" dx"' If these results be identical with each other, we shall find r (1 + a) y _ d^ rji) ' dx"' and, therefore, A = -=-?! — ; — n • -i-~^, since r(l) = 1. It is easy r{l + a) dx"' ^ ^ •' to extend the same principle to the determination of the other coefficients, and we shall thus find d" u h" # u //* o ,, . or, in other words, it follows that the coefficient of any power of h whose index is r will be 1 d' u r{l+r)'dx'^' REPORT ON CERTAIN BRANCHES OF ANALYSIS. 243 The next step is to adapt the series (1) to the different cases which an examination of the constitution of the function ?/ will present to us. If we suppose x to possess a general value, then u' and u will possess the same number of values, and no fractional power of h can present itself in the developement. In this case T(l + a) = I . 2 . . . a, and it may be readily proved that the successive indices a, b, c, &c., are the successive numbers 1, 2, 3, &c., and that consequently, , d u J cV^ u h^ cPu h^ o du dx^ 1.2 dx^ 1.2.3 It will also follow that the series for m' can involve no negative and integral power of h ; for in that case the factorial T (1 + «), which appears in its denominator, would become oo, and the term would disappear. If it should appear, also, that for spe- cific values of x any differential coefficient and its successive values should become infinite, they must be rejected from the developement, in as much as in that case the equation would no longer exist, which is the only condition of the intro- duction of the corresponding terms. In other words, those terms in the developement of u' must be equally obliterated, which, under such circumstances, become either or oo. If the general differential coefficient of u could be assigned, its examination would, generally speaking, enable us to point out its finite values wherever they exist, for those specific va- lues of the symbols which make the integral differential coeffi- cients zero or infinity. For all such values there will be a cor- responding term in the developement of u under those circum- stances. Thus, if we suppose u ■= x t/a — x, we shall find 1 .J-u (-')"'^(l) [ K«-.)-(|-r).r ] r{\ + r)r[^-rj 1^ {^--rya-xy-i J if we make x = a, this expression will be neither zero nor infinit?/ in two cases only, which are when r =i—, and when 3 »' = -g- : in the first case we get, rf* u , — - (1)-'-^ R 'Z 844 THIRD REPORT — 1833. and in the second we get, — — (t'u 2 ^ ' ■ (I) 3 \a — a) = >/•=!; dx^ Ar(0) X X (a -a) since r(l) =s 1 = r(0), and the symbol in the denominator 3 3 = — ^, is a simple zero. The corresponding developement of m' under such circumstances is ^/-«2.A^ + -/-I. A^ a result which is very easily verified. If we pay a proper regard to the hypotheses which deter- mine the existence of terms in the series for u' for specific values of the independent variable, we shall be enabled without difficulty to select the indices of the differential coefficients which can present themselves amongst the coefficients of the different powers of h in the developement. For, in the first place, /*», and the differential coefficient whose index is — , will possess the same number of values, and the same signs of affec- tion. If there be a term in u which = P (ar — «)», where P neither becomes zero nor infinity, when x — a, and where the multiple values of P, if any, are independent of those contained - .„ ,1 j.(J^ .V .{x — «)» in (a: — «)», then it will appear that the term ot — m dx» m which is independent of {x — «)» is P . — ^ > and that d X" m fill • fl# all the other terms of —, being either ssero or infinity when d x^ X ^= <t, or, if finite, introducing, through the medium of the factorial function by which they are multiplied, multiple values which are greater in number than those contained in u , must be rejected, as forming no part of the developement. It will of course follow, that the function P will become, under such cir- cumstances, a function of h, and if we represent it by P', and denote its values, and those of its successive differential coeffi- iiients, when h = 0, by p, p\ p", p"', &c., we shall find REPORT ON CERTAIN BRANCHES OF ANALYSIS. 245 none of which become ssero or injinity, in as much as P does not vanish when x ■= a. If there exist other terms in m of a similar kind, such as m' m" Q,{x — by, R (ar — cy^ , &c., the same observations will apply- to them. Such terms will correspond to values of x, which make radical expressions of any kind zero or infinity, and the form of the function u must be modified when necessary, so that such radicals may present themselves in single terms of the form V {x — «)». The same observations will apply to ne- gative as well as positive values of — , unless we suppose — a negative whole number. The principle of the exception in this last case may be readily inferred from the remarks in the note, d~^ 1 p. 211, on the subject of the values of -3 — zTr • — 7> when « is a whole number. If we suppose, therefore, u to involve terms such as P (a; — «)», Q (w — Z>)»', &c., the most general form under which its developement can be put, supposing all terms which become zero or infinity for specific values of x to be rejected, will be as follows : m m VI a — a p, d« {x — o)» h" (^^ rfj ■r(i + ^) m' m! m' h — h „, dn' {x — 6)"' Jin' d x^' (> - 5) or. , du J d^u h^ d^u .h^ ^ «=« + ^^' + ^^r:2 + rf^ 17273+ ^*^" m h — h ( Ifi \ Vi + &c. We have introduced the discontinnoiis sigtis or factors : a — a g46 THIRD REPORT — 1833. ^ ~ &c., which become equal to 1 when x = a or x =: b, &c., X -^ o but which are ^ero for all other values of x, to show that the terms into which they are multiplied disappear from the deve^ lopement in all cases except for such specific values of x. The existence of the terms of the series for u is hypothetical only, and the equation which must be satisfied, as the essential condition of the existence of any assigned hypothetical term, at once directs us to reject those terms which would lead to infi- nite values of the differential coefficients, as well as those which possess multiple values which are incompatible with those con- tained in «/. It is quite obvious that upon no other principle could we either reject such infinite values, or justify the con- nexion of a series of terms with the general form of ?/, which have no existence except for specific values of x. The con- clusion obtained is of considerable importance, in as much as it shows that the series of Taylor, if considered and investigated as having a contingent, and not a necessary existence, may be so exhibited as to comprehend all those cases in which it is commonly said to fail : and it will thus enable us to bring under the dominion of the differential calculus many peculiar cases in its different applications which have hitherto required to be treated by independent methods. Thus, if it was required to determine the value of the fraction (2 2\^ — " ^" , when x=sa,we should find it to be, x^ {x — ay fly . (x^ — a-f dx^ x^ {x — of or {x + af.-~-,'{x-nf d x^ a conclusion which would be justified by the developement of the numerator and denominator of this fraction by the complete form of Taylor's series, when x = a. Many delicate and rather obscure questions in the theory of maxima and minima, particularly those which Euler has deno- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 247 minated maxima and minima of the second species, and others also relating to the singular or critical points of curve lines, must depend for their dilucidation upon this more general view of Taylor's series, as connected with the consideration of ge- neral differential coefficients *. * Euler has devoted an entire chapter of his Calculus Differentialis to the examination of what he terms the differentials of functions in certain peculiar cases. It is well known that he adopted Leibnitz's original view of the prin- ciples of the differential calculus, and considered diiferentials of the first and higher orders as infinitesimal values oi differences of the first and higher orders. Such a principle necessarily excludes the consideration of differential coefficients as essentially connected with determinate powers of the increment of the inde- pendent variable, which may be said to constitute the essence of Taylor's theorem, and which must be the foundation of all theories of the differential calculus, which make its results depend upon the relation of forms, and not upon the relation of values. As long, however, as the independent variable continues indeterminate, the symbolical values of the differentials are the same upon both hypotheses. But when we come to the consideration of specific va- lues of the independent variable which make differential coefficients above or below a certain order, infinite or zero, then such a view of the nature of dif- ferentials necessarily confounds those of different orders with each other. Thus, ii y=. a^ -\- {x — a)^, Euler makes, when a; = a, d y = (d a;)^, instead of (5 \ = 3 = (d x)^. Iiy = 2ax — x"-\- a^/ (a^ — a;3), he makes, y) t ^"^ when X = a, d y = a \/ — 2 a . d x^, instead of ,j^ ff"^ (d «)■? These examples are quite sufficient to make manifest the inadequacy of merely arithmetical views of the principles of the differential calculus to ex- hibit the correct relation which exists between different orders of differentials, and, a fortiori, therefore, between different orders of differential coefficients. M. Cauchy, in his Lecons sur le Calcul Infinitesimal (published in 1823), has attempted to conciliate the direct consideration of infinitesimals with the purely algebraical views of the principles of this calculus, which Lagrange first securely established ; and it may be very easily conceded that no attempt of this able analyst, however much at variance with ordinary notions or ordinary practice, would fail from want of a sufficient command over all the resources of analysis. He considers all infinite series as fallacious which are not convergent, and that, consequently, the series of Taylor, when it takes the form of an indefinite series, is not generally true. It is for this reason that he has transferred it from the differential to the integral calculus, and exhibits it as a series with a finite number of terms completed by a definite integral. It is very true that M. Cauchy has perfectly succeeded in dispensing with the consideration of infinite series in the establishment of most of the great principles of the differential and integral calculus ; but I should by no means feel disposed to consider his success in over- coming difficulties which such a course presents as a decisive proof of the expe- diency of following in his footsteps. The fact is, that if the operations of algebra be general, we must necessarily obtain indefinite series, and if the symbols we employ are general likewise, it will be impossible to determine, in most cases, 248 THIRD REPORT — 1833. Signs of discontinuity are those signs which, in conformity with the general laws of algebra, are equal to 1 between given limits of one or more of the symbols involved, and are equal to zero for all their other values. If merely conventiofial signs were required, we might assume arbitrary symbols for this purpose, attaching to them far greater clearness as diventical marks, the limits of the symbol or symbols between which the sign of discontinuity was supposed to be applied. Thus, we might suppose ^T)a to denote 1, when x was taken between and a, to denote zero for all other values ; ■^Da+^j, to denote 1, when X was taken between a and a + b, and zero for all other values ; and similarly in other cases. Thus, if 2/ = a X + /3 and y =^ of x + /3' were the equations of two lines, and if we supposed that the generating point whose coordinates are x and y was taken in the first line between the limits and a, and in the second line between the limits a and b, then we should have generally, y = -D; (a ^ + ^) + -D/ {«.' X + /30 (1.) the couvergency or divergency of the series which result. It is only, therefore, when we come to specific values that a question will arise generally respecting the character of the series : and it is only when we are compelled to deduce the function which generates the series from the application of the theory of limits to the aggregate of a finite number of its terms, that its convergency or diver- gency becomes important as afi^ecting the practicability of the inquiry : in short, it must be an erroneous view of the principles of algebra which makes the result of any general operation dependent upon the fundamental laws of algebra to be fallacious. The deficiency should in all such cases be charged upon our power of interpretation of such results, and not upon the results themselves, or upon the certainty and generality of the operations which produce them : in short, the rejection of diverging series from analysis, or of such series as may become divergent, is altogether inconsistent with the spirit and principles of symbolical algebra, and would necessarily bring us back again to that tedious multipli- cation of cases which characterized the infancy of the science. A verj' instruc- tive example of the consequences of adopting such a system may be seen in the researches of M. Liouville, which have been noticed in the note at p. 217. Lagrange in his Theorie des Fonctions Analytiques, and in his Calcul des Fonctions, has given theorems for determining the limits between which the remainder of Taylor's series, after a finite number of terms, is situated : and the same subject has been very fully discussed in a memoir by Ampere, in the sixth volume of the Journal de I'Ecole Poly technique. Such theorems are ex- tremely important in the practical applications of this series, but they in no respect affect either the existence or the derivation of the series itself. It is a very common error to confound the order in which the conclusions of algebra present themselves, and to connect difficulties in the interpretation and appli- cation of results with the existence of the results themselves : and it is the in- fluence of this prejudice which has induced some of the greatest modem ana- lysts, not merely to deny the use, but to dispute the correctness of diverging series. Messrs. Swinburne and Tylecote, the joint authors of a Treatise on the true REPORT ON CERTAIN BRANCHES OF ANALYSIS. 249 Thus, if in the triangle A C B, we draw C D, a perpendicular from the '\f,, vertex to the base, and if we suppose A D = «, A B = 6, A the origin of m,.-' the coordinates, A B the axis of x, y =. a. X the equation of the line A C, >< and y = a' .r + /3' the equation of the ■* ^ ^ ^ line B C, then we should find that the value of y represented by the equation y = -^Da" . a ^ + ^Dj" (a' a; + /3') * (2.) would be confined to the two sides A C and B C of the triangle ABC, excepting only the point C, which cori-esponds to the common limit of the discontinuous signs. For if we suppose 'Do° and ■^Dj" to be true up to their limits, we shall find, when X =■ a, that ■^D«° + '^Dj" = 2. If we replace, however, ^D; by ^D; - ^, and ^D," by ^D*" - |^^, Developement of the Binomial Theorem, Avhich was published in 1827, have contended vigorously for the restriction of the meaning of the sign = to simple arithmetical equality, and would reject its use when placed between a function and its developement, unless its complete remainder, after a finite number of terms, should replace the remaining terms of the series ; or unless, when the indefinite series was supposed to be retained, the value or the generating func- tion of this remainder could be assigned. In conformity with this principle they have assigned the remainder in tlie series for (a -\- x)", which they exhibit under the following form : (^ + .,)n = «„ + ,,a„-l^.4.....«(«-_l)---(»->-+l).an-r;er + -«'-^'(« + ^)"{(^+V^ + ('• + 1) (r + 1) (r + 2) . . ■ (n - 1) T • • (a + xy + i ' 1 (a + xy + ^ Zl^X. 1 . U . . . (n — r—l) ■ (a + .r)"/ ' the remainder being (a + j;)« jC + 1 multiplied into n — r terms of the deve- 1 1 lopement of -77 — ; — ;^ •>..,, or of — r-;. ^ {(a -f x) — aj'^+i' a»- + i The method which they have employed for this purpose, which is extremely ingenious, succeeds for integral values of n, whether positive or negative, but fails to assign the law when the index is fractional. But my own views of the principles of symbolical algebra would, of course, induce me to attach very little value to results which were exhibited in such a form as to be incapable of being generalized, a defect under which the formula given above evidently labours. • The conventional sign '^Dj'* might be replaced, though not with perfect 1 /»o propriety, by the definite integral _ ^ / dx. 250 THIRD REPORT — 1833. and if we make, therefore, y=^'D."-^«^„+^W-i^*^(»'.+^') (3.) the equation will be true for the ordinate of every point of the sides A C and C B of the triangle ABC. More generally, if we suppose y = f^ x, y ■=. (p^x,y ■=■ (^^x, y = (p^x, &c., to be the equations of a series of curves, then the equation of a polylateral curve composed of the several portions of the separate curves corresponding to values of x, included between the limits a and b, b and e, c and d, &c., would be, + ('D/-f-^,)fcx + &c.; (4.) the value of the ordinate at each successive limit being replaced by that of the succeeding curve. In this manner, if we should grant the existence of the sign of discontinuity, we should be enabled to represent the equations of polygons, and of poly- lateral curves of every description. It remains to consider the nature of the expressions which are competent to express ^Dj". The expressions which have been generally proposed for this purpose are either infinite series, or their equivalent definite integrals. Le Comte de Libri, however, a Florentine analyst of distinguished genius, has proposed* a finite exponential ex- pression which will answer this purpose. The examination of the expression would readily show that its value is 1 when x is greater than a, and that it is when x is equal to or less than a. It will therefore follow that the product ^(logO) e^^^s")'^- ") ^ ^(logO)e^'^e*'>(*-^) is equal to 1 between the limits a and b, and is equal to at those limits, and for all other values. And, in as much as * Memoires de Mathematique et de Physique, p. 44. Florence 1829. The author has since been naturalized in France, and has been chosen to succeed Legendre as a member of the Institute : he has made most important additions to the mathematical theory of numbers. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 251 p(iogO) _ Q^ yfQ jyjay replace the preceding product by the equi- valent expression This expression, which is equivalent to ^Db" '■ j, has been applied by Libri to the expression of many important theorems in the theory of numbers *. The definite integral / — sin r x has been shown by It Eulerf and many other writers, to be equal to -^ when x is positive, to when x is 0, and to -^ when x is negative. It follows, therefore, that 2 r^dr . (b-a) C (a + b)l — / — sm ^^ — Tz — - r cos ■< x — ^^ — 7^ — - r > vJo r 2 L 2 J = — / ^ — sin (^ — a) r H / — sin (^ — 6) r vJo r ^ ' TTi/o »* is equal to 1 , when x is between the limits a and 6, to ^, when x is at those limits, and to zero, for all other values. If we denote the definite integral — / — sin — - — - r cos -s x — ^ — -- — '- > r ^ -K JQ r a L 2J by Cj", we shall get, xT\ « n«i ^ ^ 1 " — b ^' -^* +2(a:-a) + 2]F=T)' and consequently the equation of a polylateral curve, such as that which is expressed by equation (4.), will be, y = Cb . <P]X -{■ C/ . 9.2 -^ + Cd . ^3 X +&C., in as much as at the limits we have <Pi {b) = p^ (b), (p^ (c) s= a^ (<?), and consequently for such limits Cj" ^j (b) + C/ f^ (^) = f 1 (^) = ip^ (b), and not 2 (p^ (b). All definite integrals which have determinate values within given limits of a variable not involved in the integral sign, may be converted into formulae which will be equal to 1 within those • Cr elk's Journal for 1830, p. 67. t Inst. Calc. Integ., torn. iv. ; Fourier, Theorie de la Chaleur, p. 442. ; Frul- loiii, Memorie della Societa Italiana, torn. xx. p. 448. ; Libri, Memoires de Ma- thematique et de Physique, p. 40. 252 THIRD REPORT — 1833. limits and also including the limits, and to zero for all other values *. But the expressions which thence arise, though fur- nishing their results in strict conformity with the laws of sym- bolical combinations, possess no advantage in the business of calculation beyond the conventional and arbitrary signs of dis- continuity which we first adopted for this purpose : but though it is frequently useful and necessary to express such signs ea;- plicitly, and to construct formulfE which may answer any as- signed conditions of discontinuity, yet such conditions will be also very commonly involved implicitly, and their existence and character must be ascertained from an examination of the pro- perties of the discontinuous formulae themselves. We shall now proceed to notice some examples of such formulas. The well known series f X 1 1 . 1 . r TT + -3- = sin j; — ^ sin 2 j: + -?r sin 3 x — -p sin 4.r + &c. (1.) is limited to integral values of r, whether positive or negative, and to such values oirit •\- -^ as are included between -^ and — -^ : the value of r, therefore, is not arbitrary but condi- * If a definite integral (C) has n determinate values »i, «2, . . . un, within the limits of the variable a and b, and no others, the values at those limits being included, and if C be equal to zero for all values beyond those limits, then we shall find «D « = _ (C — eti) ( C — eta) • • • (C — »n) ^ j. oil X oCq X • . ■ cin thus in the case considered in the text, we get = _ 2 (C — 1) ^C — i-) + 1 = — 2 C2 + 3 C. t The principle of the introduction of r w in equation (1.) by which it is ge- neralized, will be sufficiently obvious from the following mode of deducing it : = log {i + e'^~' } - log {1 + e-' ^^} log 1 I '*' 1 +e , -X V-l ; ^ . - , ■ ( S'/^ -JfV-A = log e =. X a/— I + 2r t V — 1 = Ve — e / and, therefore, dividing by 2 V* — 1» and replacing the exponential expressions by their equivalent values, we get X . 1 1 . 1 . r IT + — = sin a; sin 2 a; + — sin 3 a; sin 4 a; + &c., 2 2 3 4 where x upon the second side of the equation may have any value between •+ 00 and — 00 . REPORT ON CERTAm BRANCHES OF ANALYSIS. 25S -re 2 tional. If we successively replace, therefore, a" by -^ + t and — — x,yvQ shall get •It X 1 . ^ 1 „ rir + -J- + "o = cos X + -^ &\a'ax — -^ cos 3 x — :j- sin 4 X + &c. /«• + -; 7r- = COS a: yr sin2x ^ cos 3 jr 4 2 » 3 H — J- sin 4 d7 + &c. Adding these two series together and dividing by 2, we get (f J- f^\ •B' 1 1 ^-^ — '- * + -J- = cos ^ IT- cos 3 a: + -^ cos 5 X — &c. {2.) It It If X be included between g- and —, then r = and r' = 0, and we get -r = COS X ^ cos S X + -p- cos 5 j: — &c. (3.) 4 3 5 ^ •' ■jt 3 iz* If J? be included between -g- and -^, then r = — 1 and r' = 0, and we get w 1 1 — -r = cos j: -^ cos 3 ^ + ^r cos 6x — &c. (4.) 4 3 5 ^ ' If the limits of x be -^ and -^, —5- and -^, -^ and — -3-, ^ and — '-^, we shall obtain values of the series nt Tt (2.), which are alternately -7- and — -r-. X . 1 . Again, if in equation (1.), or rir + -tj- = sin x — -^sinS^r + -q- sin S X -7- sin 4 t + &c., we replace x by tt — x, we shall get »•' ff -I -^ — = sin or + -^ sin 2 x + -q- sin 3 J? + -j- sin 4 j? + &c. Adding these equations together and dividing by 2, we get 254 THIRD REPORT — 1833. (r + r') IT 1 1 Tf" + -J- = sm X + -TT sin 3 z + — sin 5 jr + &c. (5.) 2 3 which may be easily shown to be equal to -j- and j- altera 4 4 nately, in the passage of x from to v, from ^ to 2 tt, from 2 it to 3 It, &c., or from to — tt, from — it to —2it, &c.: its values at those limits are zero. The series (2.) and (5.) have been investigated by Fourier, in his TMorie de la Chaleur *, by a very elaborate analysis, which fails, however, in showing the dependence of these series upon each other and upon the principles involved in the deduction of the fundamental series : and they present, as we shall now proceed to show, very curious and instructive examples of dis- continuous functions. The equation y 2 is that of an indefinite straight line, Q A P, making an angle with the axis of x, whose tangent is ■g-j and which passes through the origin of the coordinates : whilst the equation y sz sinx zr- &m2x + -jr- sin 3 a; ;i- sin 4 .r + &c. » o ^ is that of a series of terminated straight lines, d' c, dC,T> C, &c., passing through points a, A, A', &c., which are distant 2 It from each other : the portion d C alone coincides with the primitive line, whose equation is y == -^. Again, the line whose equation is y = ~^, is parallel to the 1 i IS G' d, C * From page IG/" to ipo ; also 267 and 3')6. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 255 It T 71" axis of X at the distance -j- above it : the line whose equation is TT — -J-, is also parallel to the axis of x, at the distance -j- below it : the line whose equation is y = cos X «- cos S X + -^ cos 5 x — &c. consists of disco7itinuous portions of the first and second of those lines, whose lengths are severally equal to tt. The values of y at the points B and b, corresponding to a- = -^ and — ^, are equal to srero, since the equidistant points D and C, c and d, are common to both equations at those points. It would appear, therefore, in the cases just examined, that the conversion of one member of the equation of a line into a series of sines and cosines would change the character of that equation from being continuous to discontinuous, the coinci- dence of the two equations only existing throughout the ex- tent of one complete period of circiUation of the trigonometrical series : and more generally, if, in any other case, we could ef- fect this conversion of one member of the equation of a curve into a series of sines or cosines, it is obvious that the second equation must be disco7iti7iuous, and that the coincidence would take place only throughout one period of circulation, whether from to tt or from — -^ to -^. It remains therefore to consider whether such a conversion is generally practicable. Let us take ti equidistant points in the axis of the curve whose equation is 7/ = 4> x, between the limits and tt, those limits being excluded : if we denominate the corresponding values of the ordinate by i/^, i/c^, . . . . i/„, and if it be proposed to express the values of these ordinates by means of a series of sines (of 7i terms) such as o, sin a; + ag sin 2 x + «3 sin 3 4? -f- . . . . + a„ sin 71 x, then we shall get the following n equations to determine the n coefficients a^, %> ^3 • • • • ^«* TT yj = ttj sm + «2 sm ^2 = «i sin -^ — ^ + O2 sin y^ = a^ sin - + flTg sm 277 , . 37r + «T n + \ ^ 71+1 .a„.m^^j, 47r . Gtt — ; — 7 + ff sm 7fi+ \ ^ n + \ + . ^nn: 71 + V + . 3M7r . ffnSin — — -,, n + V 256 THIRD REPORT — 1833. w TT , . 2 tn: . 3 tin . 1^-K If any assigned coefficient a^ be required to be determined from this system of equations, we must multiply * them seve- rally by n , m It ^ . 2mTr ^ . Smrr _. nmv 2 sm — —T, 2 sin — --r, 2 sin ^ , ... 2 sin r, n + r n + l n + I n + I when all the coefficients except a^ will disappear from the sum of the resulting equations : and we shall thus find 2 f . OTTT . . 2mTt . nrmr'X «», = — r~\\ Vi sin — --r + Vo sin — —^ + . • . -/„ sin — -— ;- >. M + 1 L n + \ ^^ « + 1 "^ n + IJ It would thus appear that it is always possible to determine a series of sines of n terms with finite and determinate coeffi- cients, which shall be the equation of a curve which shall have n points in common with the curve whose equation is ?/ =: <p x, within the limits corresponding to values of x between and v ; and it is obvious that the greater the number of those points, the more intimate would be the contact of these two curves throughout the finite space corresponding to those limits. If we should further suppose the number of those points to be- come infinitely great, then the number of terms of the trigono- metrical series would be infinite likewise, and the coincidence of the curve which it expresses with the curve whose equation is y =s <p X, would be complete within those limits only, producing a species of contact to which the texxn. finite osculation has been applied by Fourier f . Beyond those limits the curves would have no necessary relation to each other. It would follow, also, from the preceding view of the theory of finite osculations, that the curve expressed hy y = <p x might be perfectly arbitrary, continuous, or discontinuous. Thus, it might express the sides of a triangle, or of a polygon, or of a multi- lateral curve, or of any succession of points connected by any conceivable law ; for in all cases when the corresponding or- dinates of equidistant points are finite, we shall be enabled to determine values of the coefficients a^ which are finite or zero by the process which has been pointed out above. * This is the process proposed by Lagrange in his "Theorie du Son," in the third volume of the Turin Memoirs, as stated by Poisson in his memoir on Periodic Series^ &c., in the 19th cahier of the Journal de I'Ecole Polytech- nique. f TMorie de la Chaleur, page 250. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 257 The hypothesis of n being infinite would convert the series for «„ into the definite integral * 2 /»0 2 /»0 . , — / ax sin mx ax. COS a; if we make r = x and r = d x : or otherwise if we ?^ + 1 w + 1 assume the existence of the series (p X ■=■ a^ sin a: + «2 sin 2 x + . . . a,n sin m « + &c., it may be readily shown, by multiplying both sides of the equa- tion by sin m x d x, that a,n = — / (p X sin m X d X : and in a similar manner, if we should assume <p X = Oq cos .r + «i cos x + . . . a^ cos 7n x -\- &c., that cr^ = — / <p X cos mx a X -f . Thus, if we should suppose ip jp = cos x, we should find 4 C 2 4 . 6 1 = — \ Tj — ^c sin 2 ^ + H — p sin 4 x -f -p — = sin 6 a; + &c. I • »r[l.o o.o o./ J' a very singular result, which is of course only true between the limits and it, excluding those limits %. If we should suppose (p .r = a constant quantity — between the limits and a, and that it is equal to zero between a. and tt, we should find (1 — cos«) . (1 — cos2«) . _ (1 — cosoa) (p X = ^ ^ sin X + ^ ^ -' sm 2 ^ + ■ ^ ~^ x sin 3 X + &c., excluding the limiting value u, when the value of the series is only -^§. If we should suppose <p x =. 'Dj" . a-x + "D,^ . («' x + /3'), which is the equation of the sides of a triangle (excluding the • Poisson, Journal dc I'Ecole Polytcchniqve, cahier xix. p. 447. + Fourier, Th/'oric de la Chaleiir, pp. 235 & 240. \ IhUL, p. 233 ; Poisson, Joimial de I'Ecole Polytechmque, cahier xix. p. 418. § Fourier, Tlicorie de la Chaleur, p. 244". loutj. s 258 THIRD REPORT — 1833. limit X = h), whose base is represented by tt, then we shall find * 2 , / IX f 1 r . sin 2 .r sin 3 ^ o 1 (p X — — [a-n + [a. — c^) b) ismx ^ \- — g &c. |- 2 , ,s fsinZ>sinar , sin26sin2j: , sin3isin3j; ^ 1 + — («-«) I [2 + 2^ + 32 +&c.|. The trigonometrical series, in this last case, would represent a series of triangles placed alternately in an inverse position with respect to each other ; and a similar observation would apply to the discontinuous curves which are represented by any series of sines and cosines. Thus, \iy = <^ xho. the equation of the curve P C C" Q, and if we suppose 2/ = (p ^ = Oj sin X + ttg s''" 2x + a^ sin 3 x + &c., between the limits and w ; and if we make A B = w, A A' = 2 TT, A B' = 3 TT, &c., we sliall get a discontinuous curve, consisting of a series of similar arcs, C D, a C", C D', &c., placed successively in an inverse relation with respect to each other upon each side of the axis of x, of which one arc C D alone coincides with the primitive curve. If we should suppose the same curve to be expressed be- tween the limits and tt by a series of cosines or 2/ = ^ ar = «o + ^1 cos X + a^ cos 2 x + &c., and if we make A B = tt, A 6 = — tt, A A' = 2 tt, A B' = 3 w, &c., then the trigonometrical equation will represent a discon- tinuous curve c? C D C D', of which the portions C D and C d, • Fourier has given a particular case of this series, p. 246. UKPORT ON CERTAIN BRANCHES OF ANALYSIS. 259 C' D' and C D will be symmetrical by pairs ; but one portion only, C D, will necessarily coincide with the primitive curve. The theory of discontinuous functions has recently received considerable additions from a young analyst of the highest pro- mise, Mr. Murphy, of Caius College, Cambridge. In an admi- rable memoir on the Inverse Method of Definite Integrals *, he has given general methods for representing discontinuous func- tions, of one or a greater number of breaks, by means which are more directly applicable to the circumstances under which they present themselves in physical problems than those which have been proposed by Fourier, Poisson, and Libri. Mr. Murphy had already, in a previous memoir f , given a most remarkable extension to the theory of the application of Lagrange's theo- rem to the expression of the least root of an equation, which we shall have occasion to notice hereafter ; and he has shown that if (p {x) be an integral function of x then the coefficient of — in the developement of — log ^— will represent the least root of the equation <p x = 0. We thus find that the least of the two quantities « and (3 will be represented by the coeffi- cient of — m the series for log ^ ^-^ ~, which is («+iy + ot47G- (^-±^) and if we re])lace « and B by — and -^, the least of the two ^ cc p quantities — and -^, or the greatest of the tv/o quantities « and ^, will be represented by • Transactions of the Philosophical Society of Cambridge, vol. w. p. 374. t Ihid. p. 125. X If we represent the series (2.) by S, we shall get d«-> S 1 „ __ — or 0, (— l)»-i T{n) d«»-i according a,s « is greater or less than (i : thus — — - would represent the at- traction within and without a spherical shell, which is or — , where « is the distance from the centre. s2 260 THIRD REPORT — 1S33. Thus, if y — a .r — /3 = and y — «' .r — /3' = be the equa- A B p p- tions of two lines B C and D C, forming a triangle with a por- tion B D of the axis of x, then the system of lines which they form will be expressed by the product (y_«^_^) (2,_«/^-_^')^0. (3.) Now it is obvious that if common ordinates P M, P M' be drawn to the two lines, the least of them will belong to the sides of the triangle BCD; if we denote, therefore, P M and P M' by'^i and y,^, the equation y = y\ ya yi + ys + 1 . 1 2.4 • (H^f 1 2 J V 2 "will become the equation of the sides of the triangle BCD, when yi and y^ are replaced by their values ; for y will denote P M for one side and 2^ "* for the other. In order to express a discontinuous function <p, which as- sumes the successive forms ipj, ip,,, f^, &c., for different values of a variable which it involves between the limits « and j3, /3 and y, y and 8, &c., Mr. Murphy assumes S («i 2;), S (/Sj z), . . 1 . S (yi ss), &c., to denote the coefficient of — in the several series for and supposes log „ </ S («, «) , /> (/ S (/3, z) + /3 d S (y, ^) + &c. doL '-''' d^ '-''■'' dy If a. be less than z or 2 greater than «, then S (a, ^) = «, , , t. d S (u, z) 1 ■r■a^ 1 i.u 4-1 '■^ S (^, s) and therefore r"^— ^ = 1 = up be less than z, then ^-^ — - d « rf^ 1 : if Y be less than z, then injll = 1 and so on; con- ' d y REPORT ON CERTAIN BRANCHES OF ANALYSIS. 261 sequeiitly, in the first case we have <p = /i = <Pi : in the second, 9 =/, +/2 = ?2, and therefore/^ = (^2 - ^, : in the third, <P =fi +A+f3 = <P3> and therefore /a = (p^ - ^3. It appears therefore that is a formula which is competent to express all the required conditions of discontinuity *, Equivalent forms may be considered as permanent within the limits of continuity, and no further, unless the requisite signs of discontinuity, whether implicit or explicit, exist upon both sides of the sign = : thus, the equation 4 r ^ 4 6 "I •'■D/cos^ = —\ T-^ sin 2 a; + -^^— psin4^ + -=— =sm6a: + &c. \ 11.0 0.0 0,1 I is permanent within the limits indicated by the sign '*'D^° and no further, and similarly in most of the cases which have been considered above. The imprudent extension of such equivalent forms, which has arisen from the omission of the necessary signs of discontinuity, has frequently led to very erroneous conclusions : thus, the equation .r» 1 1 r.r2 - 1 1 r* - 1 1 ^^ - 1 . )t -D.« log a: = 2-|^^— I + -J . (^-.qpT)^+ y • (:,qri)a+ &«• \ which is true for all values of x between and co , has been extended to all values of x between — 00 and + co , and has thus been made the foundation of an argument for the identity of the logarithms of the same number, both when positive and negative. There are two species of discontinuity which we have consi- dered above, one of which may be called instantaneous and the oi\\ev finite : the first generally accompanies such changes of form as are consequent upon the introduction of critical values * These formulae would require generally a correction at their limits, in order to render them symbolically general. The nature of these corrections may in most cases be easily applied from the observations which we have made above. t This series is given by M. Bouvier in the 14th volume of Gcrgoniie's Aimalcs ties Malhemutifjiies. The conclusion referred to in the text assumes the identity of the logarithms of x" and of ( — .r)^, which is in fact the wholo (jucstion in dispute. 262 THIRD KEPORT — 1833. of the variables, when the corresponding equivalent form no longer exists, or when the conditions which determined its exist- ence no longer apply ; the second restricts the existence of the equivalent form to limits of the variable which have a finite dif- ference from each other. In neither case, if we suppose the con- ditions of the discontinuity to be implicitly involved, or if we suppose the explicit signs of discontinuity to be assumed con- formably to the general laws of algebra, can we consider the law of the permanence of equivalent forms to be violated. It is only when a continuous formula is assumed to be equivalent to a discontinuous formula, without the introduction of the requi- site sign of discontinuity to limit the extent of the continuous formula, that we can suppose this fundamental law to be vio- lated or the asserted equation between such expressions to be false. Many important errors have been introduced into ana- lysis from the neglect of those conditions. The identity of the values of powers of 1, whose indices are general whole numbers, and also of the sines and cosines of angles which differ from each other by integral multiples of 360°, is a frequent source of error in the generalization of equi- valent forms, when the symbols which express those indices or multiples are no longer whole numbers. A very remarkable example of both these sources of error has occurred in the for- mula {2 cos x)*" = cos 7n {2 r TT + x) + m cos (/« — 2) {2 rv + x) -\ \ — ^ — cos {m — 4') {2rTr + x) + ike. + i/^l { sin 7n {2rTf + x) + m sin {m — 2) (2 r if + x) -,. '^'1^-^ sin {m - 4) (2 r TT -}- x) + &c. (1.) If we suppose m to be a whole number, this equation degene- rates into {2 cos x)"" = cos mx + m cos {m — 2) x -\ ^j — -^ — -' cos (w — 4) X -I- &c. (2.) the series first discovered by Euler, and which he assumed to be true for all values of m. li, however, we should suppose tn to be a fraction of the form —> we should have g values of . 9 the first member of the equation (2.), and only one of the second. And if we should confine our attention to the arithmetical value (/} of the first, it would not be equal to the second, un- REPORT ON CEKTAIN BRANCHES OF ANALYSIS. 2G3 less m was a whole number ; for if we should denote the series of cosines cos {m2rir + x) + w cos (m — 2) (2 r * + .r) + &c., by Cr, and the series of sines sin m {2rTf + X) + m sin {tn — 2) (2rTr + x) + &c., by Sr, we should find, when cos x is positive, Cr ^r ^ "" COS 2 m rir ~ sin 2 m rv' and when cos x is negative. and ^ cos m {2 r + \) ir sin w (2 r + 1) ** It will follow, therefore, that when r is not a whole number, p will be expressible indifferently by a series of cosines or of sines, unless cos 2 mrie = or s'm 2m r if =■ 0, when cos x is positive, or cos m (2 r + 1) * = 0, or sin ?n {2r + 1) w = 0, when cos x is negative. In a similar manner, assuming 1 . 2 ... 6 r («2-P) 3 X = n < cos X — ^:j — g — ^ cos'^ j:* we shall find cos 7« (2 »• TT + x) = cos n ( 2 r + Y ) '^ • ^ + cos (w - 1) (2 r + 4") 'T . X'. If we suppose r to be equal to zero, this equation will become nif ^ (n — 1)* ^, cos n X = cos -^ . X + cos 5 . A', which is the form which has been erroneously assigned by La- grange* and Lacroixf as generally true for all values of w. Many other examples of similar undulating functions, ex- * Cakttl den Funciio)'s, chap. xi. t TVaite du Caktil Diff. ct lu/erj., torn. i. p. 261. 204 ^ THIRD REPORT — 1833. pressing the various relations between the cosines and sines of multiple arcs and the powers of simple arcs, whether ascending or descending, have been given by Lagrange * and other writers as general, which are either degenerate forms of the coi'rect and more comprehensive eqviations, or altogether erroneous. Poisson had pointed out some of the inconsistencies to which some of these imperfect equations lead, and had slightly hinted at their cause and their explanation ; and the discussion of such cases became soon afterwards a favourite subject of speculation with many writers in the Mathematical Journals of France f and Germany % ; but the complete theory and correction of these expressions was first given by M. Poinsot in an admirable me- moir which was read to the Academy of Sciences of Paris in 1823, and published in 1825. They form a most remarkable example of expressions extremely simple and elementary in their nature, which have escaped from the review and analysis of the greatest of modern analysts, in forms which were not merely imperfect, but in some cases absolutely erroneous. The difficulties which have presented themselves in the theory of the logarithms of negative numbers, as compared with those of the same numbers with a positive sign, have had a very similar origin. If we consider the signs of quantities as yflc/or,y of their arithmetical values, and if we trace them through- out the whole course of the changes which they undergo, we shall find many examples of results which are identical when considered in their final equivalent forms, but which are not in every respect identical when considered with respect to their derivation: thus (+ aY is identical with {— a)^, when consi- dered in their common result + a^, but not when considered with respect to their derivation. Let us now consider their se- veral logarithms, the common arithmetical value of the logarithm of a being denoted by p : Iog(+a)2 = log(l)2a2 = 4r7r \/'~i +2p (1.) log (- af = log (- ly^ a^- = (2r + 1)2 nf \/'^l + 2p (2.) log a^ = log 1 . o- = 2 /• Tf V' ^ + 2 p (3.) It thus appears that the values of log ( + «)^ and log ( — a)- are included amongst those of log «^, but not conversely; and also that the values of log (+ «)^ and log(— a)'^, the arithmetical value being excepted, are not included in each other. * Correajjottdence sur I'Ecole Fulyirvlin'iqHo, torn. ii. p. 212. ■\ In Gergoniie's-yJ«««/pi' des Matlu'inutiqncs, torn. xiv. xv. xvi. xvii. I In Creile's Journal fur die rcine uiid aiigewandie Muthemafik. Berlin. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 265 Again, if we consider — a" as originating from (— 1) (+ a) Me shall set nt \og — a"" = (2 r + 2 7)1 r' + I) Tf \/ — I + m p* : 2' if we suppose /k = — , r = and r' = — 1, we shall get log - x/a =^ p = ^ log « = log i/ a ; or the logarithm of a negative quantity will be identical with the logarithm of the same quantity with a positive sign. In a similar manner, if we suppose 7)i = i~-~, where p is prime to n, r' = — n and r = ^^-^^, then 2 r + 2 7n r' + 1=0, and the corresponding logarithm of — a™ will coincide with the arith- metical logarithm of a"'. We should thus obtain possible loga- rithms of negative numbers in those cases in which we should be prepared to expect them from the ordinary definition f of logarithms. In the absence of all knowledge of the specific process of de- rivation of quantities, such as a'" and — a"*, we should consider their logarithms as identical with those of l"", A and (—1) 1 *". A, where A is the arithmetical value of a"' : and in considering the different orders of logarithms which correspond to the same value of a"^ or of — «*", they will be found to diflfer from each other by the logarithms of V" and (— I) 1'" only, which are 2mr'K V — \ and {2 r -\- 2 m r' -{- 1) * -/ — 1 respectively. The logarithms in qviestion are Napierian logarithms whose base is e. If we should suppose the logarithms to be calculated to any other base, we should replace the Napierian logarithms of 1"* and (— 1) 1"' by the logarithms of those qviantities (or signs) multiplied by the inodiiliis INl : the same remarks will apply to such logarithms which have been made with respect to Na- . pierian logarithms. The question of the identity of the logarithms of the same number, whether positive or negative, was agitated between Leibnitz and Bernoulli, between Euler and D'Alembert, and has been frequently resumed in later times. The arguments in * Peacock's Algebra, p. 569- t The logarithm being defined to be the index of the power of a given base which is equal to a given number, it would follow, since ai = + n, that— ;j is eijually the logarithm of + n and — h. The same remark aj plies to all in- dices or luyarilhmv which are rational fractions with even dentrainatorb. 266 THIRD REPORT — 1833. favour of the affirmative of this proposition, which were for the most part founded upon the analytical interpretation of the pro- perties of the hyperbola and logarithmic curve, were not en- titled to much consideration, in as much as they were not drawn from an analysis of the course followed in the derivation of the symbolical expressions themselves and from the principles of interpretation which those laws of derivation authorized. A very slight examination of those principles, combined with a re- ference to those upon which algebraical signs of affection are in- troduced, will readily show the whole of the very limited nvmi- ber of cases in which such a proposition can be considered to be true *. * In the 15th volume of the Annales des Mathematujues of Gergonne, there is an ingenious paper by M. Vincent on the construction of the logarithmic and other congenerous transcendental curves. Thus, if y = e^^ there will be in the plane of x y a continuous branch such as is commonly considered, and a discontinuous branch corresponding to those negative values of y which arise from values of x, which are expressible by rational fractions with even deno- minators : thus, if we suppose the line between x z= and a- = 1 to be di- vided into an even number 2 p of parts, (where p is an odd number,) the values of* will form a series effractions, J_ ± 3 2 2p-l 2p p Zp p 2p which have alternately odd and even denominators, and which correspond therefore to values of y which are alternately single and double. If we may suppose, therefore, a curve to be composed of the successive apposition of points, the complete logarithmic curve will consist of two symmetrical branches, one above and the other below the axis of x, one of which, in cor- responding parts of the curve, will have double the number of points with the other. The inferior curve, therefore, may in this sense be considered as dis- continuous, being composed of an infinite number of conjugate points, forming, in the language of M. Vincent, une h-anche pointillee. 'J he same remark ap- plies to other exponential curves, such as the catenary, &c. It was objected to this theory of M. Vincent by M. Stein, another writer in the same journal, that every fractional index in this interval might be con- verted into an equivalent fraction with an even denominator, which would give a double possible value of the ordinate, which would be different from that given by the fractional index in its lowest terms ; and that consequently there would necessarily be a double ordinate for every point of the axis, and therefore also a double number, one positive and the other negative, corre- sponding to every logarithm. In reply to this objection, it is merely neces- m mp m mp sary to observe that the values of a" and o"^ or of 1 " and 1 "^ are in every respect identical with each other, the n p values in the second case consisting merely of p periodical repetitions of those in the first. In a paper in the Philosophical Transactions for 1829, Mr. Graves has given a very elaborate analysis of logarithmic formulee, and has arrived at some conclusions of great generality which it is difficult to reconcile with those which have been commonly received. Amongst some others may be men- tioned the formula which he has given for the Napierian logarithms of 1, REPORT ON CERTAIN BRANCHES OF ANALYSIS. 267 Convergency and Divergency of Series. — The subject of di- vergent series, their origin, their interpretation and their use in analysis, is one of great importance and great difficulty, and has been and continues to be the occasion of much controversy and doubt. I shall feel it necessary, for such reasons, to notice it somewhat in detail. If the operations of algebra be considered as general, and the symbols vi'hich are subject to them as unlimited in value, it will be impossible to avoid the formation of divergent as well as of convergent series : and if such series be consi- dered as the results of operations which are definable, apart from the series themselves, then it will not be very important to enter into such an examination of the relation of the arith- metical values of the successive terms as may be necessary to ascertain their convergency or divergency ; for, under such circumstances, they must be considered as equivalent foi'ms representing their generating function, and as possessing, for the purposes of such operations, equivalent properties. Thus, if they result from the division of the numerator of an alge- braical fraction by its denominator, then they will ^jrorfwce the numerator when multiplied into the denominator or divisor : if they result from the extraction of the square or cube root of an algebraical expression, then their square or cube will pro- duce that expression ; and similarly in other cases, no regard , 2r v which is not 2 tt v — 1, but ^z /^^., which, thouarh it includes the former, is not included by it. It appears to me, however, that there exists a fundamental error in the attempt which has been made by Mr. Graves to generalize the ordinary logarithmic formulie upon the same principles which have been applied by Poinsot to the generalization of the trigonometrical series which have been noticed in the text. He assumes / {&) ^= cos 6 + -v^l sin 3 := e and makes the series for/(^) and/"' (&), combined with the equa- tion/ (x &) — a. value of/ {&f, and therefore/"' / ^ = 2 r 5r + ^, the foun- dation of his logarithmic developements : in other words, he makes e '^~' a periodic quantity the base of his system of logarithms, an assumption which is essential to the truth of the formula/"'/^ = 2 ?• ar -f and to the gene- ralization of the series for/"' 6 by means of it; an hypothesis which is al- together at variance with our notions of logarithms as ascertained by the ordi- nary definition. The logarithms of + 1 and of (+ 1)"' alone, for very obvious reasons, can be considered as possessing such a character. Though I have felt myself called upon to state my objections to the fun- damental principle assumed in this memoir of Mr. Graves, and consequently to many of the conclusions which are founded upon it, yet I think it right at the same time to observe that it displays great skill and ingenuity in the con- duct of the investigations, and is accompanied by many valuable and ori- ginal obscrvationb upon the general principles of analysis. 268 THIRD REPOrwT — 1833. being jjaid in such cases to terms which are at an infinite di- stance from the origin. It is this last condition, which, though quite indispensable, is rather calculated to oftend our popular notions of the values of series as exhibited in their sums. We speak of series as having sums when the arithmetical values of their terms are considered, and when the actual expression for the sum of n terms does not become infinite when n is infinite, or when, in the absence of such an explicit expression, we can show from other considerations that its value is finite. In all other cases the sex'ies, arithmetically speaking, may be considered as di- vergent, and thei'efore as having no sum *, if the word sum be used in an arithmetical sense only, as distinguished from gene- rating function. We are in the habit of considering quantities which are in- finitely great and infinitely little as very difl^erently circum- stanced with respect to their relation to finite magnitude. We at once identify the latter with zero, of which we are accus- tomed to speak as if it had a real existence ; but if we subject our ideas of zero and infinity to a more accurate analysis, we shall find that it is equally impossible for us to conceive either one or the other as a real state of existence to which a mag- nitude can attain or through which it can pass. But it is the relation which magnitudes in their finite and conceivable state still bear to other magnitudes in their course of continued in- crease or continued diminution, which enables us to consider their symbolical relations when they cease to be finite ; and whilst quantities infinitely little are neglected as being absorbed in a finite magnitude, so likewise finite magnitudes are consi- dered as being absorbed in infinity, and therefore neglected when considered with relation to it. The principle, therefore, of neglecting terms beyond a finite distance from the origin, in converging series, is both safe and intelligible, whilst the case is very different with respect to the neglect of similar terms in a diverging series. Of such series it is said that they have no arithmetical sum ; but it may be said in the same sense of all algebraical series involving general symbols that they have no sum. But it is not the business of symbolical algebra to deal with arithmetical values, but with symbolical results only ; and such series must be considered with reference to the functions which generate them, and the laws of the operations employed for that purpose. The neglect, therefore, of terms beyond a * This would appear Cauchy's view of the subject : see the 6th chapter of his Cours d\biafyse. REPORT ON CERTAIN BRANCHES OF ANALYSTS. 2C9 finite distance from the origin would be perfectly safe as far as it does not influence the determination of the series from the generating function, or the generating function from the series ; and it is upon this principle that the practice is both founded and justified. A few examples may make this reasoning more plain. Let it be required to determine the function which generates the series a + a x + a x^ + a a^ + &c. (1.) Let s be taken to represent this function, and therefore s = a + a X + a •r'^ + a a:'^ + &c. = a + X {a + a X + a x^ + a x^ + &c. } = a + X s : consequently a s = . I — x If the arithmetical values of the terms of this series be con- sidered, and if x be less than 1, then , is the stim of the 1 — X series : in all other cases it is its generating function. We may consider, however, s (whether it expresses a sum or a generating function) as identical with *,, s^, s^, &c., in the several expi'essions s= a + X s^ s = a + a X + x^ s^ s = a + a X + a x^ + x^ Sg s = a + ax + ax'^+ . . . a x^"-^ + x"' s„, : for if the number of terms of the series * be expressed by n and if ?i be infinite, we must consider *,, s^, Sq, a . . . Sm as abso- lutely identical expressions ; for otherwise we must consider an infinite as possessing the properties of an absolute number, and must cease to regard infinities with finite differences as iden- tical quantities when compared with each other. It is for this reason that we assume it as a principle that no regard must be paid to terms at an infinite distance from the origin, whatever their arithmetical values may be. The sum of the series a — a-\-a — a-\- &c. was assigned by Leibnitz, upon very singular metaphysical considerations, to be — : the principle just stated would allow us to put ^70 THIRD REPORT — 1833. s = a — {a — a + a — a + &c.) = a — * ; and therefore s = -r-*, 2 * The same principle would show that the equation *• = « +/(« +f(a+f{a + . . .))) is identical with the equation x= a +f (X) ; and that * = «/(«/ (a/ (a...))) is identical with X = af(x). The example in the text is the most simple case of a class of periodic series, the determination of whose sums to infinity has been the occasion of much controversy and of many curious researches. The general property' of such series is the perpetual recurrence of the same group of terms whose sum is equal to zero : thus, if there should be p terms in each group, and if the num- ber of terms n = mp + i, their sum would be identical with that of the i first terms of the series ; and if we should denote those terms by aj, a.,, . . . o , and if we should take the successive values of this sum for all the values of i between 1 axidp inclusive, their aggregate value would be represented by pai + (p—1) a<^+ (p _ 2) a, + . . . a^. of which the average (A) or mean would be represented by pay + (p — I) a.2 + (p — 2) as + . . . a^ P If this periodic series was continued to infinity, it was contended by Daniel Bernoulli, in memoirs in the 17th and 18th volumes of Novi Commmtarii Petropolitani, for 1772 and 1773, that its sum would be correctly represented by the average (A), in as much as it was equally probable that any one of the p values would be the true one. Upon this principle it would follow, that of the apparently identical series 1 — 1 + 1 — 1 + 1 — &c. ... 1+0— 1 + 1 + 0— 1 + 1 + &c. ... 1-t-O + O— 1 + 1+0 + 0— 1+&C. 12 3 the first would be equal to — , the second to — -, and the third to — . In the A o 4 same manner we should find 1 + 1 — 1 — 1 + 1 + 1 — 1 — 1 +, &c. equal to 1, and 1+1+0 — 1 — 1 + 1 + 1+0— 1 — l+I + l +&C. equal to — -. The same observations would apply to the series 1 + cos X + cos 2 X -\- cos 3 .r + cos 4 a; + &c. and 1 + cos a- + + cos 2 X -\- cos 3 .r + + cos 4 .r + &c. where x is commensurable with 2 tt. These conclusions, however, though curious and probable, rested upon no REPORT ON CERTAIN BRANCHES OF ANALYSIS. 271 If we consider this principle of the identity of series, whose terms within a finite distance from the origin are identical, as established, we shall experience no difficulty in admitting the perfect algebraical equivalence of such series, and their gene- secure basis founded upon the general principles of analysis, and their truth •was not, therefore, generally admitted amongst mathematicians. In the year 1798, Callet, the author of the logarithmic tables which go by his name, pre- sented a memoir to the Institute for the purpose of showing that the sums of such periodic series were really indeterminate : thus, if we divide 1 by I + X and subsequently make x = 1, we get 1 — 1+1 — 1 + &C. (1.) the value of which is — . In a similar manner, if we divide 1 + a; by 1 + a; + a;2, we get for the quotient 1 — x" + x3 — x^ + x^ — x^ + &c., which becomes the same series (1.), though the value of the generating func- 2 tion under the same circumstances becomes — . The same remark applies to the result of the division of 1 + x + x^ + . . x"' by I + x + x- -\- . . x", which produces the same series (1.) when a; =: 1, though under such circum- stances its generatmg function becomes — . This memoir of Callet gave occasion to a most elegant Report upon this delicate point of analysis by Lagrange, who justified upon very simple prin- ciples the conclusion of Daniel Bernoulli. The series which results from the division of 1 + a; by 1 + .r + a^, if the deficient terms be replaced, becomes l+O.x — x^ + x^ + O.x'^— a^ + x^ + O.x'^ — x^ + &c., which degenerates, when x ■=■ \, into the series 1+0—1 + 1+0—1 + 1 + &c., and not into the series (1.). The same remark applies to the series which arises from the division of 1 + a; + . . a;*" by 1 + a; + . . . . x", n "Z m; which becomes, when x z= I, 1+0 + + + &c. — 1+0 + + &c. +1+0 + &c„ •which is equal, by Bernoulli's rule, to — . n But it is not necessary to resort to this expedient for the purpose of deter- mining the sums of such series ; for the series Oi + On X + 03 a- + . . a x^~ + a, a'' + &c. is a recurring series resulting from the developement of a, + Oo a; + 03 x- -\- . . a x^" 1 - xP which becomes — when a: = 1. If we replace x by —, this fraction will become 272 THIRD REPORT — 1833. rating functions. Foi* the same principle would justify us in rejecting remainders after an infinite number of terms, whatever their arithmetical values may be ; for such remainders can in- fluence no terms at a finite distance from the origin, and there- fore can in no respect affect any reverse operation, by which it may be required to pass from the series to any expression dependent upon the generating function. Thus, if 1 =a + ax + ax^+ ixc = *, we shall get a = (I — x) s = a, if we reject remainders after an infinite number of terms ; and similarly in other cases. It would thus appear that algebraical equivalence is not necessarily dependent upon the aritlametical equality of the series and its generating function. It is, however, an inquiry of the utmost importance to be able to ascertain when this arithmetical equality exists ; or, in other words, to ascertain under what circumstances we can determine the sum of the series, either from our knowledge of the law of formation of its successive terms, or approximate, to any required eti z^ + a^ z^~ + . . a z which becomes by the application of the ordinary rule of the differential cal- culus, when s; = 1 or .r = 1, p Oi + (p — I) a^ + . . Op ~ f P which is the average or mean value determined by Bernoulli's rule. The discussion of the values of these periodic series has-been resumed by Poisson in the twelfth volume of the Journal de I'Ecole Poly technique. He considers them as the limits of these series when considered as converging series, a view of their origin and meaning which is almost entirely coincident with that of Lagrange. Thus, the gum of the series sin q + p sin (x + q) + p" sin (2 .r + q) + &c. is equal to sin q -{- p sin (x — q) 1 — 2 p cos X -\- p^ ' when|) is less than 1, an expression which degenerates, when^ = 1, into — sinq + ~ cos q cot -j' which may be considered, therefore, as the limit of the sum of the series sin q + sin (.r + ?) -|- sin (2 x + </) + &c. in wfiii. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 273 degree of accuracy, to its value by the aggregation of a finite number of those temns. Many tests of the siimmability of series (considered as different from the determination of their gene- rating functions,) have been proposed, possessing very different degrees of certaintj' and apphcabiUty. The geometrical series which we have just been considering is convergent or divergent, that is, summable or not, according us x is greater or less than 1 ; and it is convenient, for this and for other reasons, to assume it as the measure of the convergency or divergency of other series. If it can be shown that a converging geometrical series can be formed whose terms within a finite and assignable distance from the origin become severally greater than those corresponding to them of the assigned series, then that series is convergent. And if it can be shown that a divergent geometric series can be formed whose terms within a finite and assignable distance from the origin are severally less than those corresponding to them of the assigned series, then that series is divergent*. Such tests are certain, as far as they are applicable ; but there may be many cases, both of divergent and convergent series, which they are not sufficiently delicate to comprehend. It would appear from the preceding observations that di- verging series have no arithmetical sums, and consequently • Peacock's Algebra, Art. 324, and following. Cauchy, Cours d' Analyse Algebrique, chap. vi. This last work contains the most complete examination of the tests of convergency with which I am acquainted. The measure of convergency mentioned in the text, which was first sug- gested and applied by D'AIembert, will immediately lead to the following: " If u^ represent the m"^ term of a series, it is co?ivergent (or will become so) if the superior limit of (ii )" be less than 1, when 7i is infinite; divergent in the contrary case." " If the limit of the ratio tr^, j to 7/^^ be less than 1, the series is convergent, and divergent in the contrary case." Many other consequences of these and other tests are mentioned by Cauchy in the work above referred to. M. Louis Olivier, in the second volume of Crelle's Journal, has proposed the following test of convergencj'. " If the limit of the value of the product n u^ \)B finite or zero when n is infinite, then the series is divergent in the first case, and convergent in the second." This principle, however, though apparently very simple and elementary, has been shown by Abel, in the same Journal, to be not universally true. Thus, the series 2 log 2 ^ 3 log 3 ^ 4 log 4 ^ ■ ■ ■ ■ ^ ?j log « may be shown to be infinite, though the product n n^ is equal to zero when w is infinite. The same acute and original analyst has shown that there is no func- tion of n whatever which multiplied into u^ will produce a re.sult which is zero or finite when « is infinite, according as the series is convergent or divergent. 1833. T 274 THIRD REPORT — 1833. admit of no arithmetical interpretation. And it will be after- wards made to appear that such series do not include in their expression, at least in many cases, all the algebraical conditions of their generating functions. Before we proceed, however, to draw any inferences from this fact, it may be expedient in the first instance to give a short analysis of some of the circum- stances in which such series originate. The series ] 1 6 62 + -2 + 73 + &C- a — b a a^ a^ is convergent or divergent according as a is greater or less than b. As this series is incapable, from its form, of receiving a change of sign corresponding to a change in the relation of a and b to each other, it would evidently be erroneous in the latter case if it admitted of any arithmetical value, in as much as it would then be equivalent to a quantity which is no longer arithmetical. In this case, therefore, the series may be replaced by the symbol go , which is the proper sign of transition, (see page 237,) which indicates a change in the constitution of the generating function, of such a kind as to be incapable of being expressed by the series which is otherwise equivalent to it. The same observations apply to the equation r, b n(n-\) b^ (a-bf = a'' < 1 - rt — + — ; ^ . -2 ^ ^ l_ a \ . 2 or ~ 1.2.3 • ^ + ^""'J' as we have already stated in our remarks upon signs of transition, in page 237. It will be extremely important, however, to examine, both in this and in other cases, the circumstances which attend the transition from generating functions to their equivalent series, in as much as they will serve to explain some difficulties which have caused considerable embarrassment. The two series If, 2b , 3b^ 4 63 -| a^ [^ a a'' a'' J I If, 2b . 3b'' 4*3 (« - bf and J _ If 2 a 3a^ 4 a^ \ - af~ b^X '^ b + ~b^ "^ 63 + *'^-/ (b will be divergent in one case, and convergent in the other, whatever be the relation of a and b, though they both equally REPORT ON CERTAIN BRANCHES OF ANALYSIS. 875 braically, as well as arithmetically, equivalent to each other. It might be contended, therefore, that in this instance the sign oo , which replaces one of the two series, is no indication of a change in the constitution of the generating function which is conse- quent upon a change of the sign of a — b or b — a. But though a^ — 2 a b + b'^is equal to (« — b)^, and b^ — 9, ab -\- c? to {b — (if ; and though a^ — 2 a 6 + Z>^ is identical in value and signification with b"^ — 2 ab + a^ when they are considered without reference to their origin, yet we should not, on that account, be justified in considering (a — hf and {b — af as algebraically identical with each other. The first is equal to (+ 1)^ (a — by, and the second to (— 1)^ (« — hf ; or the first to (- \f {b — af, and the second to (+ 1)^ {b — af. But the signs (+1)^ and (—1)^ are not algebraically identical with each other, though identical when considered in their common result, in as much as their square and other roots and logarithms are different from each other*. It follows, therefore, that there is a symbolical change in the quantity denoted by -; j-^ in its passage through infinity, which is indicated by the infinite value of the equivalent series, in as much as it is not competent to ex- press, in its developed form, the algebraical change which its generating function has undergone. The same remarks will apply to the series for {a — 6)" and (b — a)", in all cases in which w is a negative even number. When w is a negative odd number, the change of constitution of the generating function is manifest, and requires no explanation. The two series and If, b b^ b^ b^ „ -1 a L « « a a* J 1 _ 1 /, a a^ a^ a'' . \ correspond to the same generating function, though one of them is divergent, and the other convergent. But the divergent series, whose terms are alternately positive and negative, cannot be replaced by the symbol oo , in as much as it does not indicate • Thus, if a denote a line, (+ af and (— «)« can only be considered as identical in their common result a'. When (+ a)2 and ( — a)' are considered with reference to each other, they are not identical quantities, though equal to each other. T 2 ^76 THIRD REPORT 1833. t any change in the constitution of the generating function. They may both of them, therefore, be considered as representing the vahie of this function, though in one case only can we approxi- mate to its arithmetical value by the aggregation of any number of its terms *. Similar observations would apply to the series / , i\n « r I n b « (w — 1 ) 6^ . o ~1 (a + 5)" = «» |l -f - + -hV-^ + ^^-j when n is not a positive whole number. In all such cases, the developement will sooner or later become a series, whose terms are alternately negative and positive, and which will be di- vergent or convergent, according to the relation of a and b to each other. More generally we might assume it as a general proposition, " that divergent series which correspond to no change in the constitution of the generating function, will have their terms or groups of terms alternately positive and nega- tive :" and conversely, " that divergent series which correspond to a change in the constitution of the generating function, will have all their terms or groups of terms affected with the same sign, whether -f- or — , and the whole series may be replaced by the symbol oo ." In both these propositions the change of which we speak is that which corresponds to those values of the symbols which convert the equivalent series from convergency to divergency, and conversely. I am not aware of any proof of the truth of these important propositions which is more general than that which is derived from an induction founded upon an examination of particular cases. But such or similar conclusions might be naturally ex- pected to follow from the fundamental principles and assump- tions of symbolical algebra. If the rules of algebra be perfectly general, all symbolical conclusions which follow from them must be equally true : and those rules have been so assumed, that when the symbols of algebra represent ai'ithmetical quantitiesj the operations with the same names represent arithinetical operations, and become symbolical only when the correspond- ing arithmetical operations are no longer possible. It will be essential, therefore, to the perfection of algebraical language that it should be competent to express fully its own limitations. * The equations s = and « = -r 7- will equally give us s ;= in one case, and s = r in the other, whatever be the relation , a + b a + of and b. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 277 Such limitations will be conveyed by the introduction of signs of affection, of signs of transition, or of signs of discontinuity, which may be involved either implicitly or explicitly. It is for such reasons that all those signs must be considered in the interpretation of algebraical formulie, and their occurrence will at once suggest the necessity of such an examination of the circumstances of their introduction as may be required for their correct explanation*. We thus recognise two classes of diverging series, which are distinct in their origin and in their representation. The first may be considered as involving the symbol or sign co implicitly, and as capable, therefore, of the same interpretation as we give to the sign when it presents itself explicitly. The second re- presents finite magnitudes, which in their existing form are incapable of calculation by the aggregation of any number of their terms. Such series are in many cases capable of trans- formations of form, which convert them into equivalent con- verging series ; and in some cases, where such a transformation is not practicable, or is not eflPected, tlie approximate values of the generating functions may be determined, from indirect con- siderations, supplied by very various expedients. The well known transformation of the series ax — ba:"' + cx^ — dx^-\-ex^— fx^ + &c., which Euler has given f, into the equivalent series f "v** y'^ "j^ a — Ti—. — -To .^a + T^i— — To .A^a — 71:—. — vt • ^^ a + &c. 1 + x"' (l + x)2-""'-(l + ^)3-- " {l + xy would be competent to convert a great number of divergent series of the second class into equivalent convergent series, or into such as would become so. In this manner the Leibnitzian series 1 - 1 -h 1 - 1 -f &c. may be shown to be equal to -^. The series 1 - 3 + 6 - 10 + 15 - 21 + &c. ♦ The essential character ot arithmetical division is that the quotient should approximate continual)}' to its true value, and that the terms of the quotient which are introduced by each successive operation should be less and less con- tinually. In the formation, therefore, of the quotient of j and , the analogy between the arithmetical and algebraical operation would cease to exist, unless a was greater than b, or unless the several terms in the quotient went on diminishing continually. f Jnstitulioncs Calculi Dlfferenlialis, Pars posterior, cap. i. 278 THIRD REPORT— 1833. of triangular numbers to -5-. The series 1 - 4 + 9 - 16 + 25 - &c. of square numbers to 0. The series of tabular logarithms log 2 — log 3 + log 4 — log 5 + &c., would be found to be equal to •0980601 nearly. If we should suppose X negative and greater than 1, the original and the transformed series would become divergent series of the first class. The series , ,,,(«- \f {a -If {a- \y , 5 log a = (a - 1) - ^ g ^ + ^ 3 ^ - —^ + &C-- is divergent when a is greater than 2, and convertible by Euler's formula into the convergent series («-l) , 1 (« - If ,1 {a-\f \ {a- \f . or by the method of Lagrange into the series „ (^a _ 1) _ |. (^a _ 1)2 + |- (^_ 1)3 _ &c., which may be made to possess any required degree of con- vergency. But it is not necessary to produce further examples of such transformations, which embrace a very great part of the most refined artifices which have been employed in analysis. One of the most remarkable of these artifices presents itself in a series to which Legendre has given the name of demicon- vergent*. The factorial function 1^(1 + a:) is expressed by the continuous expression (^) '(2*a:)*R, where R is a quantity whose Napierian logarithm is expressed by A B_ C \ .2.x 3 . 4 . x2 ''■ 5 . 6 . ^i" - &c-> where A, B, C, &c., are the numbers of Bernoulli. The law of formation of these numbers, as is well known, is extremely • Fonetions ElUptiques, torn. ii. chap. ix. p. 425. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 279 irregular, and after the third term they increase with great rapidity. The series under consideration, therefore, even for considerable values of s, becomes divergent after a certain number of terms. But an approximate value of the series will be obtained from the aggregation of the convergent terms only : and it has been proved by a German analyst* that the error which is thus made in the value of the generating function will in this case be less than the last of the convergent, or the first of the divergent, terms. It has been usual amongst some later mathematicians of the highest rank to denominate diverging series, without any di- stinction of their class, as false, not merely when arithmetical values are considered, but also when employed as equivalent forms, in purely symbolical processes. The view of their ori- gin and nature which we have taken above would explain the sense in which they might be so considered in relation both to arithmetical processes and to the calculation of arithmetical values. It seems, however, an abuse of terms to apply the term false to any results which necessarily follow from the lavys of algebra. M. Poisson, perhaps the most illustrious of living analysts, has referred, in confirmation of this opinion, to some examples of erroneous conclusions produced through the me- dium of divergent series f ; and as the question is one of great importance and of great difficulty, I shall venture to notice them in detail. Let it be required to express the value of »+i d X =/: {{\-2ax + a^){\ -2bx + b^)Y { by means of series. Assuming K = (1 - 2 « x + «2)- 4 and K' = (1 - 2 5 r + b^ *, let us suppose K and K' developed according to ascending and descending powers of a and b respectively ; or, K = 1 + a X, + «'■' Xg f a^ X3 + &c. K' = 1 + 6 Xj + 62 Xg + b^ X3 + &c. K' = l + ^Xi + lx,+ lx3+&c. * Erchinger in Schrader's Commentatio de Siimmatione Seriei, Src. Weimar 1818. t Journal de I'Ecole Polylechnique, torn. xii. 280 THIRD REPORT 1833. The coefficients Xj, Xj, Xg, &c., are reciprocal* functions, pos- sessing the following remarkable property, that / X„ X„ rf .r = 0, in all cases, unless n = m, in which case / X„ X„ </ a; 1 ^-' ~2n + r . The knowledge of this property will readily enable us to de- termine the following four different values of s : «1 = 1 + -g- + -^ + -y- + &C. 1 (J/ fll ft "^2 ^ T + ST^ "^ 5^ "*■ 76^ + ^''• % = — + Q^2 + K-73 + i?^ + &C. "4 — a ' 3 a2 -1^ 5 a3 ^ 7 a^ 1 1 _1_ _1_ a 6 ^ 3 a2 62 -T- 5 ^3 ^3 -T^ ,j, ^4 ^4 Whatever be the relation of a and b to each other and to 1 , two of these four series are convergent, and two of them di- vergent. But it appears from the examination of the finite in- tegral / K K' c? a:, that one only of these two convergent series gives the correct value of z, being that which arises from the combination of the two convergent developements of K and K"*, whilst the incorrect value arises from the combination of a convergent developement of K with a divergent developement of K', or conversely. The conclusion which is drawn from this fact is, that the introduction of the divergent developement of K or of K' vitiates the corresponding value of z, even though that value is expressed by a convergent series. Let us now /» + ' examine how far the definite integral of / K K' c? a: will jus- tify such an inference. If we denote K K' by — , we shall easily find, * Functions which possess this property have been denominated reciprocal functions by Mr. Murphy, in a second memoir on the Inverse Method of Defi- nite Integrals, in the fifth vohnneof the Transactions of the Philosophical Society of Cumhridge, in which general methods are given for discovering all species of such functions, and where one very remarkable form of them is assigned. The functions referred to in the text were first noticed by Legendre, in his first me- moir on the Attraction of Ellipsoids, and subsequently, at great length, in the Fifth Part of his Exercices du Calcul Integral. Cauchy has used the term recipro- ea/ function in a different sense; see Exercices des Mathematiques, torn. ii. p. HI. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 281 p-^d^^ _^=.log {P-P- + 9,p ^Th) + const.; and if we denote by r and r' the extreme values of p, when a: = — 1 and X — + 1, we shall find, r + ^dx _ 1 ^ 2r\/ab + ^ab-{a + h){\ + ah )'} ^~J -\ "y ~ 4 s/ah ^^i2r^ //ah-^ah- (a + 6)(l+a6)y inasmuch as^-^ is 4 a 6 — (« + A) (1 + « 6) in one case, and a X — A^ah — {a + h){\ + « 6) in the other. It will appear like- wise that r and r' will have the same sign, whether + or — , in as much as p will preserve the same sign throughout the whole course of the integration. If, therefore, r' = + (1 + «) (1 + a), then r = + (1 - a) (1 - 6) ; and if / = - (1 + a) (1 + h\ then r = — (1 — a) (1 — h). It thus appears that (1 — «) (1 — h) must have the same sign with (1 + a) (1 + 6), and consequently if a 7 1, and h 7 1, we shall have, . = _L^ loff (<?-!) (^> - 1) j/^ + 4a 6 - (a 4- 6) (1 + «6) 4 ./a 6 (a + l)(6 + l) v'"«6-4a6-(« + 6)(l +«*) = - — =- log . Iv «_ + _j_ (stj.i]jinrr out the common divisor _ 1 - a/« 6 + 1 , 2^ah-a-h)^ ^^^ log ^^T^^^^Y = ± -4- If a 2: 1 and 6 l\, we shall find r = (1 - a) (1 - h), and « = === log I 7= ) = ± «i. 2 A/a 6 ^ Vl - A/a 6/ If o Z 1 and h 7 1, we shall find r = (1 — a) (1 — h), and 1 , ( Vh + a/«^ X = 1 ( Vh + A/a\ , log ( ~T ^ ) = + ^2. 2 -/afi If a 7 1 and 6 Z 1, we shall find r = (a — 1) (1 — h), and 2 \/ It would thus appear that the definite integral would furnish erroneous values of z if no attention was paid to those values of the factors of r and »•', which the circumstances of the inte- gration require : and it may be very easily shown that an atten- tion to the developements of K and K' will, with equal certainty, enable us to select the proper devclopement for ^r. Thus, if a 7 I 282 THIRD REPORT— 1833. and 6 7 1, we have r = {a — 1) (b — I) = {(a-lf{b-lf}-i- and the value of s (z^) is determined by the combination of the two last developements. In a similar manner, if a Z 1 and 6 Z 1, SI (^i) will be formed by the combination of the two first. If « Zl and 6 7 1, then r = (1 - «) (6 - 1) = ^~ ^— ^ : and the value of s (^g) is formed by the combination of the first and third developement. And if a 7 1 and 6 Z 1, then the value of s (sg) will be formed by the combination of the second and third developements : in other words, the selection of the de- velopements is not arbitrary, in as much as {{l — af}-i and {(a — 1)*}~* ought not to be considered, as we have already shown, as identical quantities. These combinations of the convergent and divergent series form all the four values of s, of which it appears that one value alone is correct for any assigned relation of a and b to \, being that which arises from the combination of the convergent series for K and K' only. The considerations, however, which deter- mine the selection of the correct developement of z are as de- finite and certain when the general series are employed as when that value is determined directly from the definite integral which expresses the value of ;£. It would appear to me, there- fore, that not only was the employment of divergent series necessary for the determination of all the values of z, but that when the theory of their origin is perfectly vmderstood they are perfectly competent to express all the limitations which are essential to their usage. The attempt to exclude the use of divergent series in symbolical operations would necessarily im- pose a limit upon the universality of algebraical formulae and operations which is altogether contrary to the spirit of the science, considered as a science of symbols and their combina- tions. It would necessarily lead to a great and embarrassing multiplication of cases ; it would deprive almost all algebraical operations of much of their certainty and simplicity ; and it would altogether change the order of the investigation of results when obtained, and of their interpretation, to which I have so fre- quently referred in former parts of this Report, and upon which so many important conclusions have been made to depend. Elementary Works on Algebra. — There are few tasks the execution of which is so difficult as the composition of an ele- mentary woi'k ; and very few in which, considering the immense number of such works, complete success is so rare. They re- quire, indeed, a union of qualities which the class of writers who usually undertake such works are not often competent to REPORT ON CERTAIN BRANCHES OF ANALYSIS. 283 furnish. Great simplicity in the exposition and exempHfication of first principles, a perfect knowledge of the consequences to which they lead, and great forbearance in not making them an occasion for the display of the peculiar opinions or original re- searches of their authors. There is, in fact, only one elementary work which is entitled to be considered as having made a very near approach to per- fection. The Elements of Euclid have been the text-book of geometers for two thousand years ; and though they labour under some defects, which may or may not admit of remedy, without injury to the body of the work, yet they have not re- ceived any fundamental change, either in the propositions them- selves, or in their order of succession, or in the principles of their demonstrations, in the propriety of which geometers of any age or country have been found to acquiesce. It is true that both the objects and limits of the science of geometry are per- fectly defined and understood, and that systems of geometry must, more or less, necessarily approach to a common arrange- ment, in the order of their propositions, and to common prin- ciples as the bases of their demonstrations. But even if we should make every allowance for the superior simplicity of the truths to be demonstrated, and for the superior definiteness of the objects of the science to be taught, and also for the superior sanction and authority which time and the respect and accept- ance of all ages have assigned to this remarkable work, we may well despair of ever seeing any elementary exposition of the prin- ciples of algebra, or of any other science, which will be entitled to claim an equal authority, or which will equally become a model to which all other systems must, more or less, nearly approximate. There are great difficulties in the elementary exposition of the principles of algebra. As long as we confine our attention to the principles of ai'ithmetical algebra, we have to deal with a science all whose objects are distinctly defined and clearly un- derstood, and all whose processes may be justified by demon- strative evidence. If we pass, however, beyond the limits which the principles of arithmetical algebra impose, both upon the re- presentation of the symbols, and upon the extent of the opera- tions to which they are subject, we are obliged to abandon the aid which is afforded by an immediate reference to the sensible objects of our reasoning. In the preceding parts of this Report we have endeavoured to explain the true connexion between arithmetical and symbolical algebra, and also the course which must be followed in order to give to the principles of the latter in their most general form such a character as may be adequate to justify all its conclusions. But the necessity which is thus 284 THIRD REPORT — 1833. imposed upon us of dealing with abstractions of a nature so complete and compreliensive, renders it extremely difficult to give to the principles of this science such a form as may bring them perfectly ^vithin the reach of a student of ordinary powers, and which have not hitherto been invigorated by the sevei'e dis- cipline of a course of mathematical study. The range of the science of algebra is so vast, and its appli- cations are so various, both in their objects and in their degrees of difficulty, that it is quite impossible to fix absolutely the proper proportion of space which should be assigned to the developement of its difterent departments. If a system of al- gebra could be confined to the statement of fundamental prin- ciples, and to the establishment of fundamental propositions only, it might be possible to approximate to a fixed standard, which should possess the requisite union of simplicity and of sufficient generality. But it is a science which cannot be taught by an exposition of principles and their general consequences only, but requires a more or less lengthened institution of ex- amples of many of its different applications, in order to produce in the student mechanical habits of dealing with symbols and their combinations. The extent also to which such develope- ments are necessary will vary greatly with the capacities of dif- ferent students, and it would be quite impossible to determine any just mean between diffuseness and compression which shall be best adapted to the wants of the general average of students, or to the systems of instruction followed by the general average of teachers. In the early part of the last century the Algebra of Maclaurin was almost exclusively used in the public education of this country. It is unduly compressed in many of its most essential elementary parts, and is also undvdy expanded in others which have reference to his own discoveries. It was written, however, in a simple and pure taste, and derived no small part of its authority as a text-book from the great and well-merited repu- tation of its author. It was subsequently, in a great measure, superseded, in the English Universities at least, by the large work of Sanderson, which was composed by this celebrated teacher to meet the wants of his numerous pupils. It was, in consequence, swelled out to a very unwieldy size by a vast number of examples worked out at great length ; and it laboured under the very serious defect of teaching almost exclusively arithmetical algebra, being far behind the work of Maclaurin in the exposition of general views of the science. At the latter end of the last century Dr. Wood, the present learned and venerable master of St. John's College Cambridge, in conjunc- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 285 tion with the late Professor Vince, undertook the pubHcation of a series of elementary works on analysis, and on the appU- cation of mathematics to different branches of natural philo- sophy, principally with a view to the benefit of students at the Universities. The works of the latter of these two writers have already fallen into very general neglect, in consequence partly of their want of elegance, and partly in consequence of their total unfitness to teach the more modern and improved forms of those different branches of science. But the works of his colleague in this undertaking have continued to increase in circulation, and are likely to exercise for many years a consi- derable influence upon our national system of education ; for they possess in a very eminent degree the great requisites of simplicity and elegance, both in their composition and in their design. The propositions are clearly stated and demonstrated, and are not incumbered with unnecessary explanations and illustrations. There is no attempt to bring prominently forward the peculiar views and researches of the author, and the dif- ferent parts of the subjects discussed are made to bear a proper subordination to each other. It is the union of all these qua- lities which has given to his works, and particularly to his Algebra, so great a degree of popularity, and which has se- cured, and is likely to continue to secure, their adoption as text-books for lectures and instruction, notwithstanding the absence of very profound and philosophical views of the first principles, and their want of adaptation, in many important particulars, to the methods which have been followed by the great continental writers. In later times a great number of elementary works on algebra, possessing various degrees of merit, have been published. Those, however, which have been written for purposes of in- struction only, without any reference to the advancement of new views, either of the principles of the science, or to the ex- tension of its applications, have generally failed in those great and essential requisites of simplicity, and of adequate, but not excessive, illustration, for which the work of Dr. Wood is so remarkably distinguished ; whilst other works, which have pos- sessed a more ambitious character, have been generally devoted too exclusively to the developement of some peculiar views of their authors, and have conseqviently not been entitled to be generally adopted as text-books in a system of academical or national education. There are, however, many private reasons which should prevent the author of this Report from enlarging upon this part of his subject, who is too conscious that there are few defects which he could presume to charge upon the ^86 THIRD REPORT— 1833. works of other authors from which he could venture to exempt his own. The elementary works on algebra and on all other branches of analytical and physical science which have been published in France since the period of the Revolution, have been very extensively used, not merely in this country, but in almost every part of the continent of Europe where the French lan- guage is known and understood. The great number of illus- trious men who took part in the lectures at the Normal and Polytechnic Schools at the time of their first institution, and the enlarged views which were consequently taken of the prin- ciples of elementary instruction and of their adaptation to the highest developement of the several sciences to which they lead, combined with the powerful stimulus given to the human mind in all ranks of life, in consequence of the stirring events which were taking place around them, at once placed the scien- tific education of France immensely in advance of that of the rest of Europe. The works of Lagrange, particularly his Calcul des Fonctions and his Theorie des Fonctions Analytiqiies, which formed the substance of lectures given at the Ecole Polytech- nique, exhibited the principles of the differential and integral calculus in a new light, and contributed, in connexion with his numerous other works and memoirs, which are unrivalled for their general elegance and fine philosophical views, to fami- liarize the French student with the most perfect forms and with the most correct and at the same time most general prin- ciples of analytical science. The labours of Monge also, upon the application of algebra to geometry, succeeded in bringing all the relations of space, with which every department of na- tural philosophy is concerned, completely under the dominion of analysis *, and thus enabled their elementary and other MTiters to exhibit the mathematical principles of every branch of natural philosophy under analytical and symmetrical forms. Laplace himself gave lectures on the principles of arithmetic and of algebra, which appear in the Stances de V Ecole Nor- male and in the Journal de V Ecole Polytechniqiie ; and there are very few of the illustrious men of science, of that or of a subse- quent period, who have done so much honour to France, who have not been more or less intimately associated with carrying * The developement of the details of this most important branch of analy- tical science, which has been so extensively and successfully cultivated in France, is greatly indebted to Monge's pupils in the Polytechnic School, many of whom have subsequently attained to great scientific eminence : their results are chiefly contained in the three volumes of Correspondance sur I' Ecole Polytechnique. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 287 on the business of national education in its highest departments. The influence of such men has been felt not merely in the very general diffusion of scientific knowledge in that great nation, but also in the form and character of their elementary books, which are generally remarkable for their precision and clear- ness of statement, for their symmetry of form, and for their adaptation to the most extensive developement of the several sciences upon which they treat. The elementary works of M. Lacroix upon almost every de- partment of analytical science have been deservedly celebrated : they possess nearly all the excellences above enumerated as characteristic of French elementary writers, and they are also remarkable for the purity and simplicity of the style in which they are written *. The Cours des Math^matiques Pures of M. Francoeur possesses merits of a similar kind, being too much compressed, however, for the purposes of self-instruction, though well adapted to foi'm a basis for the lectures of a teacher. The works of M. Garnier are chiefly valuable for their careful illustration of, and judicious selection from, the writings of Lagrange, and are well calculated to make the general views and principles of that great analyst and philosopher familiar to the mind of a student, li^he Arithmetic, Algebra-f, and Appli- cation of Algebra to Geometry, of M. Bourdon are works of more than ordinary merit, and present a very clear and fully developed view of the elements of those sciences. Many other works have been published of the same kind and with similar views by Reynaud, Boucharlat and other writers. I am too little acquainted with the elementary works which are used in the different Universities of Germany to be able to express any opinion of their character. Those which I have seen have been wanting in that precise and symmetrical form which constitutes the distinguishing merit of the French elementary writers ; but they are generally copious, even to excess, in their examples and illustrations. The immense developement which public instruction, in all its departments, has received in that country would lead us to conclude that they possess elementary mathematical works, which are at least not inferior to those which • Before the Revolution, the Cours des Mathematiques Pures et Appliquees of Bezout, in six volumes, was generally used in public education in France : it is a work much superior to any other publication of that period of a simi- lar kind which was to be found in any European language. t A part of the Algebra of Bourdon has been translated and highly com- mended by Mr. De Morgan, a gentleman whose philosophical work on Arith- metic and whose various publications on the elementary and higher parts of mathematics, and particularly those which have reference to mathematical education, entitle his opinion to the greatest consideration. 288 THIRD REPORT — 1833. exist in other languages : and the labours of Gauss, Bessel, and Jacobi, and the numerous and important memoirs which appear in their public Journals and Transactions upon the most difficult questions of analysis and the physical sciences, sufficiently show that the mathematical literature of this most learned nation is not less diligently and successfully cultivated than that which belongs to every other department of human knowledge. The combinatorial analysis, which Hindenburg first intro- duced, has been cultivated in Germany with a singular and perfectly national predilection *; and it must be allowed that it is well calculated to compress into the smallest possible space the greatest possible quantity of meaning. In the doctrine of series it is also frequently of great use, and enables us to ex- hibit and to perceive relations which would not otherwise be easily discoverable. Without denying, however, the advantages which may attend either the study or the use of the notation of the combinatorial analysis, it may be very reasonably doubted whether those advantages form a sufficient compensation for the labour of acquiring an habitual command over the use and interpretation of a conventional symbolical language, which is necessarily more or less at variance with the ordinary usage and ' meaning of the symbols employed and of the laws of their com- binations. These objections would apply, if such a conven- tional use of symbolical language was universally adopted and understood ; but they acquire a double force and authority, when it appears that they are only partially used in the only country f in which the combinatoi'ial analysis is extensively cultivated, and that, consequently, those works in which it is adopted are excluded from general perusal, in consequence of their not being written in that peculiar form of symbolical language with which our mathematical associations ai'e indis- solubly connected. Trigonometry. — The term Trigonometry sufficiently indicates the primitive object of this science, which was the determina- tion, from the requisite data, of the sides and angles of trian- gles : it was in fact considered in a great degree as an inde- * See Eytelwein's Grundlehre der hohern Analysis, avery voluminous work, which contains the principal results of modern analysis and of the theory of series exhibited in the language and notation of this analysis. \ Professor Jarrett, of Catherine Hal], Cambridge, in some papers in the Transactions of the Philosojihical Society of Cambridge, and in a Treatise on Algebraical Developement, has attempted to introduce the use of the lan- guage of the combinatorial analysis. The great neglect, however, which has attended those speculations, which are very general and in some respects extremely ingenious, is a sufficient proof of the difficulty of overcoming those mathematical habits which a long practice has generated and confirmed. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 289 pendent science, and not as auxiliary to the application of al- gebra to geometry. It is to Euler* that we are indebted for the emancipation of this most important branch of analytical science from this very limited application, who first introduced the functional designations sin z, cos z, tan ^, &c., to denote the sine, cosine, tangent, &c., of an arc ^, whose radius is 1, which had previously been designated by words at length, or by simple and independent symbols, such as a, b, s, c, t, &c. The intro- duction of this new algorithm speedily changed the whole form and character of symbolical language, and greatly extended and simplified its applications to analysis, and to every branch of natural philosophy. The angles which enter into consideration in trigonometry are generally assumed to be measured by the arcs of a cn-cle of a given radius, and their sines and cosines are commonly de- fined with reference to the determination of these arcs, and not with refei'ence to the determination of the angles which they measure. It is in consequence of this defined connexion of sines and cosines with the arcs, and not immediately with the angles which they measure, that the radius of the circle upon which those arcs are taken must necessarily enter as an element in the comparison of the sines and cosines of the same angle determined by different measures : and though they Vvcre ge- nerally, at least in later writers, reduced to a common standard, by assuming the radius of this circle to be 1, yet formulae were considered as not perfectly general unless they were expressed with reference to any radius whatsoever-}-. In the application, likewise, of such formulae to the business of calculation, the consideration of the radius was generally introduced, producing no small degree of confusion and embarrassment ; and even in the construction of logarithmic tables of sines and cosines the • Introductio in Analysirn Injinitoruw, vol. i. cap. viii. " Quemadmodum logarithmi peculiarem algorithmum vequinint, cujus in universa analyst summiis extat usus, ita quantitates circulares ad certain quoque algorithmi normam perduxi : ut in calculo aeqiie commode ac logarithmi et ipsae quantitates alge- braicae tractari possent." — Extract from Preface. t We may refer to Vince's Trigonometry, a work in general use in this country less than a quarter of a century ago, and to other earlier as well as contemporary writers on this subject, for examples of formulse, which are uni- formly embarrassed by the introduction of this extraneous element. Later writers have assumed the radius of the circle to be 1, and have contented themselves with giving rules for the conversion of the resulting formulae to those which would arise from the use of any other radius. It is somewhat remarkable that the elementary writers on this subject should have continued to encumber their formulae with this element long after its use had been abandoned by Euler, Lagrange, Laplace, and all the other great and classical mathematical writers on the Continent. 1833. u 290 THIRD REPORT — 1833. occurrence of negative logarithms was avoided by a fiction, which supposed them to be the sines and cosines of arcs of a circle whose radius was 10'". A very slight modification of the definition of the sine and cosine would enable us to get rid of this element altogether. In a right-angled triangle, the ratio of any two of its sides will determine its species, and conse- quently the magnitude of its angles. If we suppose, therefore, a point P to be taken in one (A C) of the two lines A C and A B containing the angle BAG (5), and P M to be drawn perpendicular to the other line (A B), then we may define the . PM sine of fl to be the ratio . p , and the cosine of 9 to be the AM ratio -J— Ti. By such definitions we shall make the sine and A P •' cosine of an angle depend upon the angle itself, and not upon its measure, or upon the radius of the circle in which it is taken : and upon this foundation all the formulae of trigonometry may be established, and their applications made, without the neces- sity of mentioning the word radius*. If we likewise assume the ratio of the arc which subtends an angle to the radius of the circle in which it is taken, and not the arc itself, for the measure of an angle, we shall obtain a quantity which is independent of this radius. In assuming, therefore, the angle 9 to be not only measured, but also repre- sented by this ratio, we shall be enabled to compare sin & and cos Q directly with 9, and thus to express one of them in terms of the other. It is this hypothesis which is made in deducing the exponential expressions for the sine and cosine, and the series which result immediately from them-j-. * See A Syllabus of a Course of Lectures upon Trigonometry, and the Appli- cation of Algebra to Geometry, published at Cambridge in 1833, in which all the formulae of trigonometi-y are deduced in conformity with these definitions. t If we should attempt to deduce the exponential expressions for sin 6 and cos 6 from the system of fundamental equations, cos" d -\- sin^ ^ = 1 (1.) cos 6 = cos (— 6) (2.) sin ^ = — sin (— 6) (3.) we should find, aoaAH , g-A('\/^ gA^A/irT_g-A^v'irT COS 6 = ^^^— ' and sin ^ = ;;== 2 2 V — 1 in which the quantity A, in the absence of any determinate measure of the REPORT ON CERTAIN BRANCHES OF ANALYSIS. 291 The sines and cosines and the measures of angles defined and determined as above, are the only essential elements in a system of trigonometry, and are sufficient for the deduction of all the important formulas which are required either in algebra angle 6, would be perfectly indeterminate. It is the assumption of the measure of an angle which is mentioned in the text which makes it necessary to re- place A by 1. The knowledge of the exponential expressions for the sine and cosine would furnish us immediately with all the other properties of these transcendents. Thus, if the sines and cosines of two angles be given, we can find the sines and cosines of their sum and difference ; and from hence, also, we can find the sine and cosine of any multiple of an angle from the values of the sine and cosine of the simple angle ; and also through the medium of the solution of equations the sine and cosine of its subraultiples. In fact, as far as the symbolical properties of those transcendents are concerned, it is altogether indifferent whether we consider them to be deduced primarily from the assumed functional equations (1.), (2.), (3.), or from the primitive geome- trical definitions of which those equations are the immediate symbolical con- sequences. X»x d X f*y d y If we should denote the integrals / . and / — (com- l/O VI— X' Jq ^/\—yi raencing from respectively) by d and d' respectively, then the integral of the equation dx dy would furnish us with the fundamental equation sin {& 4- ff) = sin 6 cos 6' + cos & sin &', (/3.) if we should replace x by sin 6, •v/l — x^ by cos 6, y by sin 6', and Vl — y* by cos i'. If the formulae of trigonometry were founded upon such a basis, they would require no previous knowledge either of circular arcs considered as the measures of angles, or of the geometrical definitions of the sines and cosines, except so far as they may be ascertained from the examination of the values and properties of the transcendents which enter into the equation (a.). In a similar manner, if we should suppose 6 and 6' to represent the integrals /*J" dx Py dy of the transcendents / —yr- — ; — ^r and / —,7t- — 9^» then the integral Jo V(i+*-) Jo Vii—y^) ^ of the equation d X , d y „ / ^ aAi + ■■') ^Ai + r) would be expressed by the equation h sin (i + 6') = h sin & X h cos &' + h cos d X h sin 6', (3.) if we should make x ■=: h sin 6 (the hyperbolic sine of 6), and V (1 4- «*) = h cos 6 (the hyperbolic cosine of 6), y ^ h sin $', and Vl -\- y^ =: h cos &', adopting the terms which Lambert introduced, and which have been noticed in the note in p. 231 ; and it is evident that it would be possible from equa- tion (S.), combined with the assumptions made in deducing it, to frame a svstem of hvperbolic trigonometry (having reference to the sectors, and not u 2 292 THIRD REPORT — 1833. or in its applications to geometry. The terms tangent, co- tangent, secant and cosecant, and versed sine, which denote very simple functions of the sine and cosine, may be defined by those functions and will be merely used when they enable us to exhibit formulae involving sines and cosines, in a more simple form. By adopting such a view of the meaning and origin of the transcendental functions, the relations and properties of which constitute the science of trigonometry, we are at once freed from the necessity of considering those functions as lines described in and about a circle, and as jointly dependent upon the magnitude of the angles to which they correspond and of the radius of the circle itself. It is this last element, which is thus introduced, which is not merely superfluous, but calculated to give erroneous views of the origin and constitution of trigono- metrical formulae and greatly to embarrass all their applications. to the arcs of the equilateral hyperbola), whose formulse would bear a very striking analogy to the formulae of trigonometry, properly so called. Abel, in the second volume of Crelle's Jotirnal, has laid the foundation, of a species of elliptic trigonometry, (if such a term may be used,) in connexion with a remarkable extension of the theory of elliptic integrals. If we denote the elliptic integral of the first species X '^ ^/(l-c2sin2,^) by 6, and replace sin \p by jc, we shall get -/ d X or more generally -X ^{(l-a'2){l-t2a'2)} d X V{(1 +e"*=) {l-<?x^)y If we now suppose a; = <p ^, V (1 — c^ «") =/ ^ and V (1 + e^-r") = F tf, it may be demonstrated that <p{i) + e) - r+e2^2-^2-^-7^2-^» . , fd.f6'—c-(p6.(pd'.Y6. Fd' f {0 + 6) — 1 + e2 c2 (p2 tf . (p2 tf' _ ¥d.¥6'-\-e^(p6.<pd' .fd.fd \ or if, for the sake of more distinct and immediate reference to these peculiar transcendents, we denote (p S by sin 6 (elliptic sine of ff), f dhy cos 6 (elliptic cosine of ff), and F ^ by sur 6 (elliptic sursine of 6), REPORT ON CERTAIN BRANCHES OF ANALYSIS. 293 The primitive signs 4- and — , when applied to symbols de- noting lines, are only competent to express the relation of lines which are parallel to each other when drawn or estimated in dif- ferent directions; but the more general sign cos fl + ■v^— i sin 3, which has been noticed in the former part of this Report, when applied to such symbols, is competent to express all the rela- tions of position of lines in the same plane with respect to each other. It is the use of this sign which enables us to subject the properties of rectilinear figures to the dominion of algebra : thus, a series of lines represented in magnitude and position by «o, (cos 9i + -/ — 1 sin flj) a^, {cos (di + 62)+ V — 1 sin (9i-l- ^2) }«2» . . . {cos (fli + ^2 + • • • ^»-i) + ^^^\ sin (51+ 9.2 + . . . 9„_,)} «„_„ will be competent to form a closed figure, if the following equa- tions be satisfied : then these fundamental equations will become sin & cos ^' surs ^' + sin ^' cos & surs 6 T ^^'^^'- 1 + e2 c2 sin2 6 sin^ $' ~' e e cos 8 COS 6' — c2 sin & sin 6' surs 8 surs i' f A \^ AW e e e e e e_ COS (» + ff; _ 1 +e2c2^2 ^-^ina ^' • e e surs d surs 6' + e'^ sin i sin & cos & cos d' surs C« + tf ) _ 1 + e2 c2 sin2 ^ sin2 d' e e If we add, subtract and multiply, the elliptic sines, cosines and sursines of the sum and difference of 6 and 6' respectively, reducing them, when necessary, by the aid of the fundamental relations which exist amongst these three tran- scendents, we shall obtain a series of formulae, some of which are very remark- able, and which degenerate into the ordinary formulae of trigonometry, when c =: and c = 1 : we shall thus likewise be enabled to express sin n 6, cos n 6, e e surs n 8, in terms of sin 8, cos 6, surs 6. The inverse problem, however, to express « e e e sin 6, cos i, surs 6, in terms of sin n 6, cos n d, surs n 6, is one of much greater e e e e e e difficulty, requiring the consideration of equations of high orders, but whose ultimate solution can be made to depend upon that of an equation of (w -|- 1) dimensions only. It is in the discussion of these equations that Abel has dis- played all the resources of his extraordinary genius. It would be altogether out of place to enter upon a lengthened statement of the various properties of these elliptic sines, cosines, and sursines ; their periodicity, their limits, their roots, and their extraordinary use in the trans- formation of elliptic functions. My object has been merely to notice the ru- diments of a species of elliptic trigonometry, the cultivation of which, even without the aid of a distinct algorithm, has already contributed so greatly to the enlargement of the domains of analysis. g94 THIRD REPORT 1833. Co + flfi cosfii + a2COs(9, +S^) + , . «„-, cos (3, + fi2 + . • fi«-i) = (1.) aisinfli + ff2sin(fii + fl2) + . . a„_, sin (9i + 92 + . . 9„-i) = (2.) 91 + ^2+ ... 9„_, . = (« - 2 r) TT (3.) The first two of these equations may be called equations of figure, and the last the equation of angles, and all of them must be satisfied in order that the lines in question may be capable of being formed into a figure, along the sides of which if a point be moved it will circulate continually. If the values of flj, ^2 ~ ^1' ^3 ~ ^2 • • ^«-i "" ^n-2 ^^ ^11 positive, aud if r = 1, then the equation of angles will correspond to those rectilineal figures to which the corollaries to the thirty-second proposition of the first book of Euclid are applicable, and which are contemplated by the ordinary definitions of rectilineal figures in geometry. If we should suppose r = 2 or 3 or any other whole number different from 1, the equation would correspond to stellated figures, where the sum of the exterior angles shall be 8, 12, or 4 r right angles. The properties of such *^t?Z/a^e(/ figures were first noticed by Poinsot in the fourth volume of the Journal de FEcole Polytechnique, in a very interesting memoir on the Geometry of Situation*. All equal and parallel lines drawn or estimated in the same direction are expressed by the same symbol aflfected by the same sign, whatever it may be : and it is this infinity of lines, geometrically different from each other, which have the same algebraical representation, which renders it necessary to con- sider the position of lines, not merely with respect to each other, but also with respect to fixed lines or axes, through the medium of the equations of their generating points. In other words, it is not possible to supersede even rectilineal geometry by means of affected symbols only. We are thus led to the consideration of a new branch of analytical science, which is specifically de- nominated the Application of Algebra to Geometry, and which enables us to consider every relation of points in space and the laws of their connexion with each other, whatever those laws may be. It is not our intention, however, to enter upon the discussion of the general principles of this science, or to notice its present state or recent progress. A great number of elementary works on trigonometry have been published of late years in this counti'y, many of which are remarkable for the great simplicity of form to which they have reduced the investigation of the fundamental formulae. Such works are admirably calculated to promote the extension of • See also Peacock's Algebra, p. 448. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 295 mathematical education, by placing this most important branch of analytical science, the very key-stone of all the applications of mathematics to natural philosophy, within the reach of every student who has mastered the elements of geometry and the first principles of algebra. We have before had occasion to notice the work of the late Professor Vince upon this subject, which was generally used in the Universities of England for some years after the com- mencement of the present century. Its author was a mathema- tician of no inconsiderable powers, and of very extensive know- ledge, but who was totally destitute of all feeling for elegance in the selection and construction of his formulae, and who had no acquaintance with, or rather no proper power of appreciating, those beautiful models of symmetry and of correct taste which were presented by the works of Euler and Lagrange. But though this treatise was singularly rude and barbarous in its form, and altogether inadequate to introduce the student to a proper knowledge either of the objects or of the powers of this science, yet it was greatly in advance of other treatises which were used and studied in this country at the period of its pub- lication. Amongst these may be mentioned the treatise on Tri- gonometry which is appended to Simson's Euclid, which was more adapted to the state of the science in the age of Ptolemy than at the close of the eighteenth century*. The Plane and Spherical Trigonometry of the late Professor Woodhouse appeared in 1810, and more than any other work contributed to revolutionize the mathematical studies of this country. It was a work, independently of its singularly oppor- tune appearance, of great merit, and such as is not likely, not- withstanding the crowd of similar publications in the present day, to be speedily superseded in the business of education. The fundamental formulae are demonstrated with considerable elegance and simplicity ; the examples of their application, both in plane and spherical trigonometry, are well selected and very carefully worked out ; the uses of trigonometrical formulas, in some of their highest applications, are exhibited and pointed • Similar remarks might be applied to treatises upon trigonometry which were published both before and after the appearance of Professor Wood- house's Trigonometry. The author of this Report well recollects a treatise of this kind which was extensively used when he was a student at the Univer- sity, in which the proposition for expressing the sine of an angle in terms of the sides of a triangle, was familiarly denominated the hlaclc triangle, in con- sequence of the use of thick and dark lines to distinguish the primitive tri- angle amidst the confused mass of other lines in which it was enveloped, for the purpose of obtaining the required result by means of an incongruous combination of geometry and algebra. 296 THIRD REPORT — 1833. out in a very clear and striking form; and, like all otlier woi'ks of this author, it is written in a manner well calculated to fix strongly the attention of the student, and to make him reflect attentively upon the particular processes which are fol- lowed, and upon the reasons which lead to their adoption. The circumstances attending the publication and reception of this work in the University of Cambridge were sufficiently re- markable. It was opposed and stigmatized by many of the older members, as tending to produce a dangerous innovation in the existing course of academical studies, and to subvert the pre- valent taste for the geometrical form of conducting investiga- tions and of exhibiting results which had been adopted hy Newton in the greatest of his works, and which it became us, therefore, from a regard to the national honour and our own, to maintain unaltered. It was contended, also, that the primary object of academical education, namely, the severe cultivation and discipline of the mind, was more effectually attained by geometrical than by analytical studies, in which the objects of our reasoning are less definite and tangible, and where the processes of demonstration are much less logical and complete. The opposition, however, to this change, though urged with considerable violence, experienced the ordinary fate of attempts made to resist the inevitable progress of knowledge and the increased wants and improving spirit of the age. In the course of a few years the work in question was universally adopted. The antiquated fluxional notation which interfered so greatly with the familiar study of the works of Euler, Lagrange, La- place, and the other great records of analytical and philoso- phical knowledge, was abandoned * ; the works of the best mathematical writers on the continent of Europe were rapidly introduced into the course of the studies of the University ; and the secure foundations were laid of a system of mathematical and philosophical education at once severe and comprehensive, which is now producing, and is likely to continue to produce, the most important effects upon the scientific character of the nation. Theory of Equations. 1 . Composition of Equations. — The first and one of the most difficult propositions which presents itself in the theory of equations is to prove " that all equations under a rational form, and arranged according to the method * The continental notation of the differential calculus was first publicly introduced into the Senate House examinations in 1817- Though the change ■was strongly deprecated at the time, it was very speedily adopted, and in less than two years from that time the fluxional notation had altogether dis- appeared. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 297 of Harriott, the significant terms forming one membei', and zero the other, are said to be resolvible into simple or quadratic factors." It is only another form of the same proposition to say, " that every equation has as many roots as it has dimen- sions, and no more; those roots being either real* or ima- ginary ;" that is, being quantities which are expressible by symbols denoting real magnitudes affected by such signs as are recognised in algebra. We have before said that it is impossible to assign before- hand an absolute limit to the possible existence of signs of affection different from those which are involved in the sym- bolical values of (1)" and ( — 1)"; and when it is said that every equation is resolvable into factors of the form a; — a, we presume that a is either a real magnitude, or of the form a + (3 v' — 1, where « and jS are real magnitudes. If we should fail in esta- blishing this proposition, it would by no means necessarily fol- low that there might not exist other forms of factors like x — a, where a denoted a real magnitude affected by some unknown sign different from +, — , or cos fi + V — 1 sin 9, which might satisfy the required conditions : at the same time its demonstra- tion will show that our recognised signs are competent to de- note all the affections of magnitude which are subject to any conditions which are reducible to the form of an equation. If we assume in the first instance the composition of equa- tions to be such as we have stated in the enunciation of the fundamental proposition, we can at once ascertain the composi- tion of the several coefficients of the powers of x in the equa- tion and we can complete the investigation of all those general pro- perties of equations which such an hypothesis would lead to. All such conclusions, when established upon such a foundation, are conditional only. It is not expedient, however, to make the fate of any number of propositions, however consistent with each other, and however unquestionable their truth may appear to be from indirect or from a posteriori considerations, depend- ent upon an hypothesis, when it is possible to convert this hypo- thesis into a necessary symbolical truth. Using such an hypo- thesis, therefore, as a suggestion merely, let us propose the * It is convenient in the theory of equations, for the purpose of avoiding repetition, to consider symbols denoting arithmetical magnitudes and affected with the signs + or — , unreal; and quantities denoted by symbols affecttd with the" sign cos + V— l sin 6, as imaginary. ^^S THIRD REPORT — 1833. following problem, and examine all the consequences to which its solution will lead. " To find n quantities x, x^, x^, . . . x„_i, such that their sum shall be equal to ^9,, the sum of all their products two and two shall be equal to p^, the sum of all their products three and three shall be equal to p^, and so on, until we arrive at their continued product, which shall be equal to p„." The quantities x, a-j, . . . x„^i, are supposed to be any quan- tities whatever, whether real or affected by any signs of affec- tion whether known or unknown. It is our object to show that the only sign of affection required is cos 6 + -/^ sin 9, taken in its most general sense. It is very easy to show that the solution of this problem will lead to a general equation, whose coefficients are p^, p^, . . . jo„ : for if we suppose the first of these quantities x to be omitted, and Pj, Pg, . . . P„_i to be the quantities corresponding to p^, pc^, . . . pn when there are (w — 1) quantities instead of n, then we shall get a; + Pj =/?„ a: Pi + Pa = p<i, XV^ + P3=^3, ea^ ^ P»-2 + P»-l — pn-l> ^ Pn-1 = Pn- If we multiply these equations from the first downwards by the terms of the series x"-', x"-^, . . . x'^, x, 1, and add the first, third, fifth, &c., of the results together, and subtract the second, fourth, sixth, &c., we shall get the general equation x"~23i a:"-' + j)^ x"-^ -... + (- 1)«^„ = 0. (1.) In as much as />,, p^, . . . p„ may represent any real magni- tudes whatever, zero included, it is obvious that we may consi- der this equation as the result of the solution of the problem in its most general form. And in as much as x may represent any one of the n quantities involved in the problem, we must equally obtain the same equation for all those n quantities : it also fol- lows that every general solution of this equation must compre- hend the expression of all the roots. By this mode of presenting the question we are authorized in considering the spnbolical composition of the coefficients of everi/ equation as known, though the ultimate symbolical form of the roots is not knoivn ; and our inquiry will now be properly limited to the question of ascertaining whether symbols repre- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 299 senting real magnitudes afFected by the recognised and Icnown signs of aiFection only, are competent, under all circumstances, to answer the required conditions of the problem. If the value of one root can be ascertained, and that root be real, the problem can be simplified, and the dimensions of the equation depressed by unity ; for the coefficients of the reduced equation Pj . P^ . P«-i, which are also real, can be successively determined. If more real roots than one can be found, the dimensions of the equation can be depressed by as many unities as there are real roots. If the root determined be not real, and if a similar process for depressing the dimensions of the equation be adopted, the coefficients of the new equation would not be real, and the conditions of the problem with respect to the re- maining roots would be changed. But if we could ascertain a pair of such roots, such that their sum = x + x^ and their pro- duct = X x^ should be real, then the dimensions of the equation might be depressed by two unities, without changing the con- ditions of the problem with respect to the remaining roots; for if we supposed Q^, Q^, Qg, &;c., to represent the coefficients of the reduced equation, we should find, .r + 0-1 + Qi = pi, xx^ + (x + x{) Qi + Q2 = P2, ^ ^1 Ql + (a- + X^) Q2 + Qg = jOg, X a-1 Q„_4 + {x + x^) Q„_3 + Q„_2 = p„_i, X X^ (:)J„_2 = Pnf from which equations we can determine successively rational values of Qj, Q^, . . . Q„_2. It remains to show, therefore, that in all cases we can find pairs of roots which will answer these conditions. If the number of quantities x, x^, . . . .r„, be odd, it is very easy to p)rove that there is always a real value of one of them, x, which will satisfy the conditions of the general equation (I.) *, and that consequently the dimensions of the equation may be depressed by unity, and our attention confined therefore to the case where the dimensions of the equation are even. If m, therefore, be any odd number, the form of n may be either 2 m, 2^ m, 2^ m, 2" ?«, and so on. Let us consider, in the first place, the first of these cases. The number of combinations of 2 m, things taken two and two together, is m {2 m — \,) and therefore an odd number : these * This may be easily proved without the necessity of making any hypothesis respecting the composition of the equation. See the Article ' Equations' in the Supplement to the Encyclopaedia Britannica, written by Mr. Ivory. 300 THIRD REPORT — 1833. combinations may be either the sums of every two of the quanti- ties, X, x^,... T„_i, such as X + x^, x + x^^, &c., or their products, such as X Xi, or other rational linear functions of those quanti- ties, involving two of them only, such as, x + x^^ + x x^, x + x^ •^ 2 X x^, ov X + x-^ + k X x■^, where k may be any given num- ber whatsoever. If we take any one of these sets of combina- tions, we can form rational expressions for their sum, for the sum of their products, two and two, three and three, and so on, in terms of the coefficients p^, p.^, . . . pn, of the original equa- tion (1.), by means of the common theory of symmetrical func- tions *, and consequently, we can form the corresponding equa- tions of m {2 m — 1) dimensions which will have rational and known coefficients. Such equations being of odd dimensions must have at least one real root ; or, in other words, there must exist at least one real value of one of the sums of two roots, such as a; + x^, of one of the products, such as x x^, of one of the functions, x + x^ + x x^, or a: + a;] + k x Xy If the symbols which form the real sum x -\- x^ are the same with those which form the real value of the product x x^ then, under such circumstances, x and x^ are expressible by real magnitudes af- fected with the ordinary signs of algebra f . We shall now pro- ceed to show that this must be the case. If we form the equations successively whose roots are x -\- x^ + k X x-^, corresponding to different values of k, we shall have one real root at least in each of them. If we form more than m (2 m — 1), such equations for different values of k, we must at least have amongst them the same combination of x and x^^ forming the real root, in as much as there are only m {2 m — \) such combinations which are different from each other. Let k and X'l be the values of k which give such combinations, and let «' and (6' be the values of the real roots corresponding ; then we must have X + x^ -\- k X x^ ^= u.' X + .r, + k^ X x^ = /3' * The formation of symmetrical combinations of any number oi symbolical quantities x, x^, . . . a-n-i, and the determination of their symbolical values in terms of their sums {px), their products two and two (pq), three and three (JO3), and so on, involves no principle which is not contained in the direct processes of algebra, and is altogether independent of the theory of equations. The theorems for this purpose may be found in the first chapter of Waring's Meditationes Algebraicce, in Lagrange's Traite sur la Resolution des Equations Ni/meriques, chap. i. and notes 3 and 10, and with more or less detail in nearly all treatises on Algebra. t If J + •'^1 = a and xx^ = /i . , where « and /3 are real magnitudes, then j; = -^- + s'' s " — /3 ? the values of which are either real or of the form (cos (I + \^ — 1 sin 6) ^^/3, where the modulus \//3 is real. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 301 and therefore X J-, k - X-i "*" + "^1 ~ ' k^-k ' There are therefore necessarily two roots of the equation or two values of the symbols .r, a:{, x^, . . . Xn-\, such that x + x^ and X Xj are real ; and therefore it is always possible, in an equation whose dimensions are impariter par, to depress them by two unities, so that the reduced equation may still possess rational coefficients. If the number of symbols involved in the original problem be 2^ m, then the number of their binary combinations must be 9, m (2^ m — 1 ) or impariter par. It will immediately follow, from what we have already proved, that there are two values of the sum and product of the same symbols, which are either real or of the form a -\- ^ V — \ \ and consequently the symbols them- selves will admit of expression under a similar form *. If the dimensions of the original equation be 2^ m or £'* m, or any one in an ascending series of orders oi parity, it may be re- duced down to the next order of parity in a similar manner : and under all circumstances it may be shown that there must be two roots which are reducible to the form « + /3 -/ — 1 , where « and /3 real or zero ; and also in any equation of even dimensions, we can reduce its dimensions successively by two unities, thus pro- ducing a series of equations of successive or decreasing orders of parity, in which we can demonstrate the existence of successive pairs of roots of the required form until they are all exhausted. This mode of proving the composition of equations differs chiefly from that which was noticed by Laplace, in his lectures to the Ecole Normals in 1795f , in the form in which the ques- tion is proposed. A certain number of symbols, representing magnitudes with unknoum affections, are required to satisfy • I-et X + x' = r (cos tf + V — 1 sin ^) X x' = J (cos <p + V — 1 sin (p) X -\- x\- — A xx' ^= R-(cos 2;/' + \^ — 1 sin 2 ^l>) or *■ — x' =i R (cos ^Z' + -v^ — 1 sirn^) r cos tf + R cos ip (jr sin ^ + R sin ip) i X — - • H 2 V — 1 = r' (cos ic + -v^ — 1 sin x) x' = r' (cos X — V — 1 sin x)- t Lemons de I'Ecole Normale, torn. ii. 302 THIRD REPORT — 1833. certain real conditions : those conditions are found to be iden- tical with those which the unknown quantity, or, in other woi'ds, the root in an equation of n dimensions, is required to satisfy. The object of the proof above given is to show that it is always possible to find n real magnitudes with known aiFections which ai*e competent to satisfy these conditions ; and those quantities, therefore, are of such a kind that the equation, whose roots they are, is always resolvible into real quadratic factors ; a most important conclusion, which the greatest analysts have laboured to deduce by methods which have not been, in most cases at least, free from very serious objections. There are two classes of demonstrations which have been given of this fundamental proposition in the theory of equations. The first class comprehends those in which the form of the roots is determined from the conditions which they are required to satisfy ; the second class, those in which the form of the roots is assumed to be comprehended under different values of p and fl in the expression f (cos 9 + V^ — 1 sin 6), and it is shown that they are competent to satisfy the conditions of the equa- tion. To the first class belongs the demonstration given above ; those given by Lagrange in notes ix. and x. to his Resolution des Equations Niim^riques ; the first of those given by Gauss in the Gottingen Transactions ior 1816*; and by Mr. Ivory in his article on Equations in the Supplement to the Encyclopcedia Britannica. To the second class belongs the second demon- stration given by Gauss in the same volume of the Gottingen Transactions; by Legendre in the 14th section of the first Part of his Theorie des Nomhres ; by Cauchy in the 18th cahier of the Journal de VEcole Polytechniqiie ; and subse- quently under a slightly different form in his Cours d' Analyse Alg4hrique. The first of the demonstrations given by Gauss, like many other writings of that great analyst, is extremely difficult to follow, in consequence of the want of distinct enunciations of the propositions to be proved, and still more from their not always succeeding each other in the natural order of investi- gation. It requires the aid likewise of principles, or rather of processes, which are too far advanced in the order of the re- sults of algebra to be properly employed in the establishment of a proposition which is elementary in the order of truths, though it may not be so in the order of difficulty. If we may * There is another demonstration by Gauss, published in 1799, which I have never seen. In his Preface to his Demonstratio Nova Altera he speaks of its being founded partly on geometrical considerations, and in other re- spects as involving very different principles from the second. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 303 be allowed, however, to consider it apart from such considera- tions, it would appear to be complete and satisfactory, and very carefully guarded against any approach to an assumption of the proposition to be proved, a defect to which most of the demonstrations of this class are moi*e or less liable*. It extends to equations whose dimensions involve different or successive orders of parities, nearly in the same manner as in the demon- stration which we have given above. The demonstration given by Mr. Ivory is different from any other, and the principles involved in it are such as naturally present themselves in such an investigation ; and it will be re- commended to many persons by its not involving directly the use, or supposing the necessary existence of, imaginary quan- tities. It is not, however, altogether free from some very serious defects in the form under which it at present appears, though most of them admit of being remedied without any injury to the general scheme of the demonstration, which is framed with great skill, and which exhibits throughout a perfect command over the most refined and difficult artifices of analysis. Lagrange has devoted two notes to his great work on the Resolution of Numerical Equations to the discussion of the forms of the roots of equations. In the first of these notes, after examining the very remarkable observations of D'Alem- bert on the forms of imaginary quantities, he proceeds to con- sider the case of an equation such as f (x) + V = 0, where y (x) is a rational function o£ x ; if for different values a and b of the last term of this equation, where a ^b,we may suppose a root which is not real for values of V between those limits, to become real at those limits, he then shows that for values of V between those limits, and indefinitely near to them, the corresponding root of the equation must involve -/— 1, or •v/ — 1, or \/ — l, and so on; or, in other words, that the roots of the equation in the transition of their values from real to imaginary (whatever may be the affection of magnitude which re nde rs them imaginary), will change in form from a to 7W -f- « -/ — 1 . He subsequently shoAvs that the same result will follow for any values of V between a and b, and consequently, * I do not venture to speak more decidedly; for though I have read it en- tirely through several times with great care, I do not retain that distinct and clear conviction of the essential connexion of all its parts which is necessary to compel assent to the truth of a demonstration. It is unfortunately fre- quently the character of many of the higher and more difficult investigations connected with the general theory of the composition and solution of equa- tions to leave a vague and imperfect impression of their truth and correctness even upon the minds of the most laborious and best instructed readers. 304 THIRD REPORT — 1833. that in every instance, when roots of equations cease to be real, ' they will assume the form m + ?? V^— 1. This demonstration is not merely indirect, but it does not arise naturally from the question to be investigated. It seems likewise to assume the existence of some algebraical form which expresses the value of the root in terms of the coefficients of the equation, an assumption which, as will afterwards be seen, it M'ould be difficult to justify by any a priori considerations. The illustrious author hmiself seems to have felt the full force of these objections, and he proceeds therefore in the following Note to prove that every polynomial of a rational form will ad- mit of rational divisors of the first or second degree. The de- monstration which he has given is founded upon the theory of symmetrical functions, and shows that the coefficients of such a divisor may be made to depend severally upon equations all whose coefficients are rational functions of the coefficients of the polynomial dividend. Whatever be the degree of parity of the number which expresses the dimensions of this polynome, he shows the possibility of the coefficients of this quadratic di- visor, which is the capital conclusion in the theory. It ought to be observed, however, that the whole theory of the compo- sition of equations is so much involved in the different steps of this investigation, or, at all events, that so little provision is made in conducting it to guard against the assumption of this truth, that we should not be justified in considering this demonstration as perfectly independent or as furnishing an adequate foundation for so important a conclusion. If we view it, however, simply with reference to the problem for exhibiting the nature of the law of dependence which connects the coeffi- cients of the polynomial factor with those of the original poly- nomial dividend, it must still be considered as an investigation of no inconsiderable importance, as bearing upon the general theory of the solution and depression of equations. The second of the proofs given by Gauss, the proof of Le- gendre, and both of those which have been given by Cauchy, belong to the second class of demonstrations to which we have referred above. Assuming the root to be represented by p (cos 6 + 'S/ — 1 sin 9), the equation is reduced to the form P + Q ^/^-i, or ^(P2 + Q2) . (cos <p + V^l sin (f); and the object of the demonstration is to show that there exist neces- sarily real values of p and 9, which make P^ + Q'^ = 0. This is effected by Gauss by processes which are somewhat syn- thetical in their form, and such as do not arise very natu- rally or directly from the problem to be investigated ; and the REPORT ON CERTAIN BRANCHES OF ANALYSIS. 305 essential part of the demonstration requires a double integra- tion between assigned limits, a process against which serious objections may in this instance be raised, independently of its involving analytical truths and principles of too advanced an order. The demonstration of Legendre depends upon the possible discovery, by tentative or other means, of values of g and &, which render P and Q very small ; and subsequently requires us, by the application of the ordinary processes of approxima- tion, to find other values of g and S, subject to repeated correc- tion, which may render P and Q smaller and smaller, and ulti- mately equal to zero. The objection to this demonstration, if so it may be called, is the absence of any proof of the necessary existence of values of § and & ; and if they should be shown to exist, it seems to fail in showing that the subsequent correc- tions of their values which this process would assign would really and necessarily increase the required approximations. The demonstrations of Cauchy are formed upon the general scheme of that which is given by Legendre, at the same time that they seem to avoid the very serious defects under which that demonstration labours : he shows that (P- + Q^) must ad- mit of a minimum, and that this minimtim value must be zero. The second of the demonstrations differs from the first merely in the manner of establishing the existence and value of this minimum : they both of them appear to me to be quite com- plete and satisfactory. It is not very difficult to estabhsh this fundamental propo- sition by reasonings derived from the geometrical representa- tion of impossible quantities. This was done, though imper- fectly, by M. Argand, in the fifth volume of Gergonne's An- nates des Mathe??iatigt(es*, and has been since reconsidered by M. Murey, in a very fanciful work upon the geometrical in- terpretation of imaginary quantities, which was published in 1827. It seems to me, however, to be a violation of propriety to make such interpretations which are conventional merely, and not necessary, the foundation of a most important symbo- lical truth, which should be considered as a necessary result of the first principles of algebra, and which ought to admit of de- monstration by the aid of those principles alone. General Solution of Equations. — The solution of equations in its most general sense would require the expression of its roots by such functions of their coefficients as were competent * In the fourth volume of the same collection there are demonstrations of this fundamental proposition, given by M. Dubourguet and M. Encontre, which do not appear, however, to merit a more particular notice. 1833. X 306 THIRD REPORT — 1833. to express them, when those coefficients were general symbols, though representing rational numbers. Such functions also must equally express all the roots, in as much as they are all of them equally dependent upon the coefficients for their value ; and they mvist express likewise the values of no quantities which are not roots of the equation. The problem, in fact, is the inverse of that for the formation of the equation which is required to satisfy assigned condi- tions. And as we have shown that there always exist quanti- ties expressible by the ordinary signs of algebra which will fulfil the conditions of any equation with rational coefficients, so like- wise we might appear to be justified in concluding that there must exist explicable functions of those coefficients which in all cases would be competent to represent those roots. A very little consideration, however, would show that such a conclusion was premature. In the first place, such a function must be irrational, in as much as all rational functions of the coefficients admit but of one value ; and they must be such ir- rational functions of the coefficients as will successively insulate the several roots of the equation, — for they must be equally ca- pable of expressing all the roots, — and they must be capable likewise of effecting this insulation without any reference to the specific values of the S3rmbols involved, or to the relation of the values of the roots themselves ; for otherwise they could not be said to represent the general solution of any equation whatever of a given degree. The question which naturally presents it- self, after the enumeration of such conditions, is, whether we could conclude that any succession of operations which are, pro- perly speaking, algebraical, would be competent to fulfil them. If it be further considered that those successive operations must be assigned beforehand for every general equation of an assigned degree ; that every one of these operations can give one real value only, or at the most two ; and that the result of these operations, which must embrace all the coefficients, must express the n roots of the equation and those roots only ; it will readily be conceded that the solution of this great pro- blem is probably one which will be found to transcend the powers of analysis. The solutions of cubic and biquadratic equations have been known for nearly three centuries ; and all the attempts which have hitherto been made to proceed beyond them, at least in equations in which there exists no relation of dependence amongst the several coefficients, and no presumed or presuma- ble relation amongst the roots, have altogether failed of success : and if we consider that this great problem has been subjected to REPORT ON CERTAIN BRANCHES OF ANALYSIS. 307 the most scrutinizing and laborious examination by nearly all the greatest analysts who have lived in that period, we may be justified in concluding that this failure is rather to be attributed to the essential impossibility of the problem itself than to the want of skill or perseverance on the part of those "-ho have made the attempt. But in the absence of any compete and uncontrovertible proof of this impossibility, the question cannot be considered as concluded, and will still remain open to spe- culations upon the part of those with whom extensive and well- matured knowledge, and a deep conviction founded upon it, have not altogether extinguished hope. The different methods which have been proposed for the resolution of cubic and biquadratic equations, and the conse- quences of the extension of their principles to the solution of equations of higher orders, have been subjected to a very de- tailed analysis by Lagrange, in the Berlin Memoirs for 1770 and 1771, and in the Notes xiii. and xiv. of his Traits sur la Resolution des Equations Numiriques ; and it would be diffi- cult to refer to any investigations of this great analyst which are better calculated to show the extraordinary power which he possessed of referring methods apparently the most distinct to a common principle of a much higher and more comprehensive generality. In the subsequent remarks which we shall make, we shall rarely have occasion to proceed beyond a notice of the general conclusions to which he has arrived, and to show their bearing upon some later speculations upon the same subject. A very slight examination of the principles involved in the solution of the equations of the third and fourth degrees will show them to be inapplicable to those of higher orders. A no- tice of a very few of such methods vdll be quite sufficient for our purpose. Thus, the ordinary solution of the cubic equation a^-Sqx + 2r-Q* is made to depend upon that of the following problem : " To find two numbers or quantities such that the sum of their cubes shall be equal to 2 r and their product equal to q." If we represent the required numbers by u and v, we readily obtain the equation of reduction u^ — 2r u^ + ^"^ = 0, • This equation may be considered as equally general with a^_Aa;2 + B«— C = 0, in as much as we can pass from one to the other by a very easy transforma- tion ; and the same remark may be extended to equations of higher orders. Such a change of form, however, will determine the applicability or inappli- cability of many of the methods which are proposed for their solution. x2 308 THIRD REPORT — 183^. which gives, when solved as a quadratic equation, ti' = r + >^ if — (f), and consequently, and therefore ^^q_^ 9' u {r + v/ {r^ — (f')Y If we call 1, «, o?, the three cube roots of 1, or the roots of the equation s^ — 1 = 0, and if we assvime a to represent the arithmetical value of u, we shall obtain the following three values of ?< + v, which are a + -^s a a ■] 2_ « «^ -| 2-. a a u aw' These values, though derived from the solution of an equation of six dimensions *, are only three in number, and form, there- fore, the roots of a cubic equation. A little further inquiry will show that they are the roots of the cubic equation x^ — 3qx + 2r=0: for it may readily be shown, in the first place, that their sum = ; that the sum of their products two and two =: — 3 q; and that their continued product = 2 r; or in other words, that they are the roots of an equation which is in every respect iden- tical with the equation in question-}-. * There are six values of u, in as much as the values of u and v are inter- changeable, from the form in which the problem was proposed ; but there are only three values of u + v. t Since q ^ i it is usual to express the roots of the equation x^ — 3q x -\- 2r = 0, by the formula x={r+ V(r^-q^)}^ + {'•- V(»-=-?^)}i (1.) which is in a certain sense incorrect, in as much as it admits of nine values instead of three. The six additional values are the roots of the two equations x^ — 3ctqx + 2r = 0, x^ — 3a'^qx + 2r=0, and the formula (1.) expresses the complete solution of the equation («« — 2 rf — 27 </3 a;3 = 0, which is of 9 dimensions. It is the formula « = M + -|-, where u= {r + V (r^ — q^) }i and has the same value in both terms of the expression, which corresponds to the equation x^ — 3qx + 2r — 0. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 309 This mode of effecting the solution of a cubic equation would altogether fail if the original equation possessed all its tex-ms : and though the absence of the second term of a cubic equation cannot be said, in a certain sense at least, to affect the gene- rality of its character, yet it would lead us to expect that the method which we had followed was of so limited a nature as not to be appliccible to general equations of a higher order. Thus, if it was proposed to find two quantities, u and r, the sum of whose «"^ powers was equal to 2 r, and whose product was equal to q, we should find u = {r + ^(r^ - ?")}«; 1 ^ where m + ^? is the root of the equation x^-nq ^"- + '^^^ f a;«- - n {n - ^A)^{n- ^) ^ ^„_,^ + &c. = 2 r*. The form of this equation is of such a kind as to prevent its being identified with any general equation whatever, beyond a cubic equation wanting the second term ; a circumstance which precludes all further attempts, therefore, to exhibit the roots of higher equations by radicals f of this very simple order : but it is possible that there may exist determinate functions of the roots of higher equations (not symmetrical functions of all of them, which are invariable as far as the permutations of the roots amongst each other are concei'ned,) which may admit of triple values only, and which will be expressible, therefore, by means of a cubic equation, and consequently by the general formula for its solution. Thus, if x^, x^, oTg, x^, were assumed to represent the roots of a biquadratic equation • This equation was first solved by Demoivre in the Philosophical Trans- actions for 1737, and it was readily derived from the theorem which goes by his name. It was afterwards shown to be true, by a process, however, not al- together general, by Euler, in the sixth volume of the Comment, ^cad. Petrop., p. 226. See also Abel's " Me'moire sur une Classe particuliere d'Equations r^solubles algebriquement," in Crelle's Journal, vol. iv. t Abel has used the term radicality to designate such expressions. To say, therefore, that the root of an equation is expressible by radicalities, is the same thing as to say that the equation is solvable algebraically. It is used in contradistinction to such transcendental functions, whether of a known or unknown nature, as may, possibly, be competent to express those roots, when all general algebraical methods fail to determine them. 310 THIRD REPORT — 1833. x'* — p x^ + q x^ — r X + s = 0, (1.) sixch functions would be x^ x^ + x^ x^ and (ic, + x^ — x^ — x^^, which admit but of three different values, and which may seve- rally form, therefore, the roots of cubic equations, whose coeffi- cients are expressible in terms of the coefficients of the original equation. Such a function also would be {x-^ + x^^, if we should suppose p or the coefficient of the second term of equation (1.) to be zero *. The function (Xj + ^2) (■'^3 + ^4) would give three values only under all circumstances. The functions x^ + ^2 + ^3 ^^^ ^1 ^2 •''^3 ^^^ capable of four different values, and therefore do not admit of being expressed by a determina- ble equation of lower dimensions than the primitive equation. Functions of the form x^ Xo admit of six values, and require for their expression equations of six dimensions, which are reduci- ble to three, in consequence of being quasi recurring equations -j-. Innumerable functions may be formed which admit of 12 and of 24 values, and one alternate function which admits of two values only J. The success of such transformations in reducing the dimen- sions of the equation to be solved, would naturally direct us to the research of similar functions of the roots of higher equa- tions than the fourth, which admit of values whose number is inferior to the dimensions of the equation. We may presume that, if such functions exist, they are rational functions, for if not, their irrationality/ would increase the dimensions of the reducing equation, and would tend to distribute its roots into cyclical periods ; and what is more, it has been very clearly proved that if equations admit of algebraical solution, all the algebraical functions which are jointly or separately in- volved in the expression of their roots, will be equal to rational * The first of these transformations involves the principles of Ferrari's, some- times called Waring's, solution of biquadratic equations ; the second that of Euler ; and the third that of Des Cartes. See the third chapter of Meyer Hirsch's Sammhmg von Aiifgaben aus der Theorie der algehraischen Gleickuyigen, which contains the most complete collection of formulae and of propositions relating to symmetrical and other functions of the roots of equations with which I am acquainted. The combinatory analysis receives its most advan- tageous and immediate applications in investigations connected with the theory of such functions. See also Peacock's Algebra, note, p. 6I9. ■f The form of its roots being u and — , they are reducible by the same me- thods as are applied to recurring equations. I See Cauchy, Cokvs d' Analyse, chap. iii. and noteiv. The use of such al- ternate functions in the elimination of the several unknown quantities from n simultaneous equations of the first order, involving « unknov/n quantities, will be noticed hereafter. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 311 functions of these roots ; and consequently, if irrational func- tions of those roots are employed in the formation of the re- ducing equation, the roots of the eqviation must enter into the final expression of the required roots, in a form where that ir- rationality has altogether disappeared *. If we assume, there- fore, that such functions are in all cases rational, the next ques- tion will he, whether they are discovei'able in higher equations than the fourth. This inquii'y was undertaken by Paolo Ruffini, of Modena, in his Teoria delle Equazione Algebraiche, published at Bo- logna in 1799, and subsequently in the tenth volume of the Memorie delta Societa Italiana, in a memoir on the impossibi- lity of solving equations of higher degrees than the fourth. He has demonstrated that the number of values of such func- tions of the roots of an equation ofw dimensions must be either equal to 1 , 2 . 3 . , . ??, or to some submultiple of it ; and that when n = 5, there is no such function, the alternate function being excluded, which possesses less than 5 values. The pro- cess of reasoning which is employed by the author for this pur- * This proposition has been proved by Abel, in his Beweis der Unmbglich- keit algebraische Gleichungen von hoheren Graden als dem vierten allgemein Aufzidosen, in the first volume of Crelle's Journal : the same demonstration was printed at Paris, in a less perfectly developed form, during his residence in that capital. This proof applies to algebraical solutions only, excluding the consideration of the possibility of expressing such roots by the aid of un- known transcendents. After defining the most general form of algebraical functions of any assigned degree and order ; and after demonstrating the pro- position referred to in the text, and analysing the demonstrations of Ruffini and Cauchy, and showing their precise bearing upon the theory of the solution of equations, he proceeds to show that the hypothesis of the existence of such a solution in an equation of five dimensions will necessarily lead to an equation, one member of which has 120 values and the other only 10 ; an ab- surd conclusion. It is quite impossible to exhibit this demonstration in a very abridged form so as to make it intelligible ; and though some parts of it are obscure and not perfectly conclusive, yet it is, perhaps, as satisfactory, upon the whole, as the nature of the subject will allow us to expect. It is impossible to mention the name of M. Abel in connexion with this subject, without expressing our sense of the great loss which the mathematical sciences have sustained by his death. Like other ardent young men, he com- menced his career in analysis by attempting the general solution of an equa- tion of five dimensions, and was for some time seduced by glimpses of an imagined success ; but he nobly compensated for his error by furnishing the most able sketch of a demonstration of its impossibility which has hitherto appeared. His subsequent discoveries in the theory of elliptic functions, which were almost simultaneous with those of Jacobi, have contributed most materially to change the whole aspect of one of the most difficult branches of analytical science, and furnish everywhere proofs of a most vigorous and in- ventive genius. He died of consumption, at Christiania in Norway, in 1827, in the 27th year of his age. 312 THIRD REPORT 1883. pose is exceedingly difficult to follow, being unnecessarily en- cumbered with vast multitudes of forms of combination, and requiring a very tedious and minute examination of different classes of cases ; and it was, perhaps, as much owing to the necessary obscurity of this very difficult inquiry as to any im- perfection in the demonstration itself, that doubts were ex- pressed of its correctness by Malfatti* and other contemporary writers. The subject, however, has been resumed by Cauchy in the tenth volume of the Journal de VEcole Pohjtechnique , who has fully and clearly demonstrated the following proposition, which is somewhat more general than that of Ruffini : " That the number of different values of any rational function of n quantities, is a submultiple of 1 . 2 . 3 . , . n, and cannot be re- duced below the greatest prime number contained in n, without becoming equal to 2 or to 1." If we grant, therefore, the truth of this proposition, it will be in vain to seek for the reduction of equations of higher dimensions than the fourth, by transfor- mations dependent upon rational functions of the roots. The establishment of this proposition forms an epoch in the history of the progress of our knowledge of the theory of equa- tions, in as much as it so greatly limits the objects of research in attempts to discover the general methods for their solution. And if the demonstration of Abel should be likewise admitted, there would be an end of any further prosecution of such in- quiries, at least with the views with which they are commonly undertaken. Lagrange, in liis incomparable analysis of the different me- thods which have been proposed for the solution of biquadratic and higher equations, has shown their common relation to each other, and that they all of them equally tend to the formation of a reducing equation, whose root is Xi + u jTg + «^ .Tg + 0,^X4+ &c. where x\, x^, x^, &c., are the roots of the primitive equation, and where « is a root of the equation a"-' + «»-2 ^ ^«-3 -f . . . a + 1 = 0, where n expresses the dimensions of the equation to be solved. He then reverses the inquiry, and assuming this form as correctly representing the root of the reducing equation, he seeks to determine its dimensions. The beautiful process which he has employed for this purpose is so well known f that it is quite unnecessary to describe it in this place ; and the result, * Mcmnrie della Sor. Hal., torn. xi. t Resolution des Eqiiafions Ntimeriques, Note xiii. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 313 as might be expected, perfectly agrees with the conclusions which are derived from more direct, and, perhaps, more ge- neral considerations. If n, or the number of roots ^j, .Tg, x.^, &c., be a prime number, then the dimensions of the final re- ducing equation will be 1 . 2 ... (w — 2) ; and if w be a compo- site number := mp, then the dimensions of the final reducing equation will be 1.2. ..w 1.2...ra or {m- 1) m . (1 . 2 . . . J})'" {p -\)p.{\.2... m)P' according as we arrive at it, by grouping the terms of the ex- pression A'l + U X^-\- 0^ Xq + &c. into m periods of^ terms, or into jo periods of »« terms. It thus appears, that for an equation of 5 dimensions, the final reducing equation is of 6 dimensions ; for an equation of 6 dimensions, the final reducing equation is of 10 dimensions in one mode of derivation and 15 in the other ; and the higher the dimensions of the equation are, the greater will be the excess of the dimen- sions of the final reducing equation. And in as much as there exist no periodical or other relations amongst the roots of these reducing equations, it is obvious that the application of this process, and therefore also of any of those primary methods which lead to the assumption of the form of the roots of the reducing equation, must increase instead of diminishing the difficulties of the solution which was required to be found. It was the imagined discovery of a cyclical period amongst the roots of this reducing equation which induced Meyer Hirsch, a mathematician of very considerable attainments, to believe that he had discovered methods for the general solution of equa- tions of the fifth and higher degrees. Amongst the different methods which Lagrange has analysed in the Berlin Memoirs is that which Tschirnhausen proposed in the Acta Eruditorum for 1683, It proposed to exterminate, by means of an auxiliary equation, all the terms of the original equation except the first and the last, and thus to reduce it to a binomial equation. Thus, in order to exterminate the second term of x^ -\- a x + 6 = 0, we must employ the auxiliary equation y -\- K ■\- x = 0, and then eliminate x. To exterminate simultaneously the second and third terms of the cubic equation a;^ -J- « x'^ + b X + c = 0, we must employ the auxiliary equation y + A + B X + x'^ = 0, and then eliminate x ; and more generally, to destroy all the intermediate terms of an equation of « dimen- sions, x" + «j .I'"-' + a^ x"-' -{- . . . a„ = 0, 314 THIRD REPORT — 1833. we must employ the auxiliary equation 7/ + A + AjX + Ac^x^ + . . . ;r«-' = 0, whose dimensions are less by 1 than those of the given equation. Such a process is apparently very simple and uniform and equally applicable to all equations ; and so it appeared to its author. But it will be found that the equations upon which the determination of A, Aj, Ag, depend, in an equation of the fourth degree, will rise to the sixth degree, which are subse- quently reducible to others of the third degree ; and that for an equation of the fifth degree, it will be impossible to reduce them below the sixth degree. Such was the decision of La- grange, who has subjected this process to a most laborious analysis, and who has actually calculated one of the coefficients of the final reducing equation, and shown the mode in which the others may be determined *. Meyer Hirsch, however, though fully adopting the conclu- sions of Lagrange to this extent, attempted to proceed further ; and, deceived by the form which he gave to his types of combina- tion, imagined that he had discovered cyclical periods amongst the roots of this final equation, by which it might be resolved into two equations of the third degree. If such a distribution of the roots was practicable in tlie case of the final equation cor- responding to equations of the fifth degree, it would be practi- cable in that corresponding to equations of higher degrees. But some consequences of this discovery, and particularly the multiplicity of solutions which it gave, would have startled an analyst whose prudence was not laid asleep by the excitement consequent upon the expected attainment of a memorable ad- vancement in analysis, which had eluded the grasp even of Lagrange. Its author, however, was too profound an analyst to continue long ignorant at once of the consequences of his error and of the source from which it sprung. In the Preface to his Integraltafeln, an excellent work, which was published in 1810, within two years of the announcement of his discovery, he acknowledges with great modesty and propriety, that he had not succeeded in effecting general solutions of equations in the sense in which the problem was understood by Euler, Lagrange, and the greatest analysts. The well known Hoene de Wronski, in a short pamphlet pub- lished in 181 1, announced a method for the general resolution of equations. He assumes hypothetical expressions for the roots of the given equation in terms of the n roots of 1, and of the {m — V) * In the Berlin Memoirs for 1771. p- 170 : it forms a work of prodigious labour, such as few persons would venture to undertake or to repeat. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 315 roots of a reduced equation of (w — 1) dimensions, and employs in the determination of the coefficients of this reduced equation if ~^ fundamental eqaaiions, designated by the Hebrew letter {^, and ?«"~^ others designated by the Greek letter i2. It is un- necessary, however, to enter upon an examination of the truth of processes which the author who proposes them has left un- demonstrated ; and in as much as the application of his method to an equation of 5 dimensions would require the formation of 625 fundamental equations of the class Aleph and 125 of the class Omega, and the determination of the greatest common measure of 2 polynomials of 24 and 30 dimensions respectively, it was quite clear that M. Wronski might in perfect safety retire behind an intrenchment of equations and operations of this formidable nature. And this was the position which he took in answer to M. Gergonne, who, in the third volume of the An- nales de Mathematiques, in the modest form of doubts, showed that the form of the roots which he had assumed was not essen- tially different from those which Waring, Bezout, and Euler, had assumed, and which Lagrange had shown to be incompa- tible with the existence of a final reducing equation of the di- mensions assigned to it*. The process given by Lagrange for determining the dimen- sions and nature of the final reducing equation has been the touchstone by which all the methods which have been hitherto proposed for the solution of equations have been tried, and will probably continue to serve the same purpose for all similar at- tempts which may be hereafter made. Its illustrious author, however, hesitated to pronounce a decisive opinion respecting the possibility of the problem, contenting himself with demon- strating it to be so, with reference to every method which had been suggested, or which could be shown to arise naturally out * The works of Hoene de Wronski were received with extraordinary favour in Portugal, where the Baron Stockier, a mathematician of considerable at- tainments, and other members of the Academy of Sciences became converts to his opinions. There is, in fact, a bold and imposing generality, and appa- rent comprehensiveness of views in his speculations, which are well calculated to deceive a reader whose mind is not fortified by the possession of an extensive and well digested knowledge of analysis. In the year 1817, the Academy of Sciences at Lisbon proposed as a prize, " The demonstration of Wronski 's formulae for the general resolution of equations," which was adjudged in the following year to an excellent refutation of their truth by the academician Evangelista Torriani : it chiefly consists in showing, and that very clearly, that the coefficients of the reducing equation of (« — 1) dimensions, assuming the form of the roots of the equation which Wronski assigned to them, can- not be symmetrical functions of those roots, and therefore cannot be expressed by the coefficients of the primitive equation, whatever be the number, nature and derivation of the fundamental equations }^ and -Q which arc interposed. 316 THIRD REPORT — 1833. of the conditions of the problem itself. But even if we should assume the impossibility of the problem, to the full extent of Abel's demonstration, it is still possible that there may exist solutions by means of undiscovered transcendents. It is, in fact, quite impossible to attempt to limit the resources of analysis, or to demonstrate the nonexistence of symbolical forms which may be competent to fulfil every condition which the solution of this problem may require. In conformity with such views, we may consider the numerical roots of equations as the only discover- able values of such transcendental functions ; but it is quite obvious that such values will in no respect assist us in deter- mining their nature or symbolical form, in the absence of any knowledge of the course of successive operations upon all the coefficients of the equation which were required for their de- termination. Though we may venture to despair, at least in the present limited state of our knowledge of transcendental functions, of ever effecting the general resolution of equations, in the large sense in which that problem is commonly proposed and under- stood, yet there are large classes of equations of all orders which admit of perfect algebraical solution. The principal pro- perties of the roots of the binomial equation ^"—1=0, had long been ascertained by the researches of Waring and La- grange, and its general transcendental solution had been com- pletely effected. Its algebraical solution, howevei*, had been limited to values of n not exceeding 10 ; and though Vander- monde in some very remarkable researches *, which were con- temporary with those of Lagrange, had given the solution of the equation x" — 1 = 0, as a consequence of his general me- thod for the solution of equations, and had asserted that it could be extended to those of higher dimensions, yet his solu- tion contained no developement of the peculiar theory of such binomial equations, and was so little understood, that his dis- covery, if such it may be termed, remained al^arren fact, which in no way contributed to the advancement of our analytical knowledge. The appearance of the Disquisitiones Arithmeticee of the * Memoires de I'Academie de Paris for 1771. The result only of this solu- tion was given, the steps of the process by which it was obtained being omitted. This result has been verified by Lagrange in Note xiv. to his Traiie szir la Resolution des Equations Nameriqties. Poinsot, in a memoir on the solution of the congruence x" — 1 := M {p), which will be noticed in the text, has at- tempted to set up a prior claim in favour of Vandermonde for Gauss's memo- rable discovery ; in doing so, however, he appears to have been more influ- enced by his national predilections in favour of his countrymen, than by a strict regard to historical truth and justice. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 317 celebrated Gauss, in 1801, gave an immense extension to our knowledge of the theory and solution of such binomial equa- x"—l tions. It was well known that the roots of the equation — ; — r- =0, where w is a prime number, could be expressed by the terms of the series r + r'^ -\- r^ + . . . r"~^, where r represented any root whatever of the equation, and where, consequently, the first term r might be replaced by any term of the series. But in this form of the roots there is pre- sented no means of distributing them into cyclical periods, nor even of ascei'taining the existence of such periods or of determin- ing their laws. It was the happy substitution of a geometrical series formed by the successive powers of a primitive root* oin, in place of the arithmetical series of natural numbers, as the in- dices of r, which enabled him to exhibit not merely all the dif- X - 1 ferent roots of the equation j = 0, but which also made manifest the cyclical periods which existed amongst them. Thus, if a was a primitive root of n, and n — I — mk, then in the series **, r , r , r , . , , r , . , . r , the m successive series which are formed by the selection of every k^^ term, beginning with the first, the second, the third, and so on successively, or the k successive series which are formed in a similar manner by the selection of every m^^ term, are periodical ; and if the number m or k of terms in one of those periods be a composite number, they will further admit of resolutions into periods in the same manner with the complete series of roots of the equation. The terms of such periods will be reproduced in the same order, if they are produced to any extent according to the same law, it being understood that the multiples of n which are included in the indices which succes- sively arise, are rejected, for the purpose of exhibiting their values and their laws of formation in the most simple and ob- vious form. If two or more periods also are multiplied together, the product will be composed of complete periods or of 1 , or of multiples of them, the rules for whose determination are easily • There are as' many primitive roots of n as there are numbers less than n — 1 ■which are prime to it. Euler, who first noticed such primitive roots as determined by Fermat's theorem, determined them by an empirical pro- cess. Mr. Ivory, in his admirable article on Equations, in the Supplement to the Encyclopccdia Britannka, has given a rule for finding them directly. 318 THIRD REPORT — 1833. framed * ; and it arises from the application of such rules that we are enabled to determine the coefficients of an equation of which those periods are the roots, and thus to depress the original binomial equation to one whose dimensions are the greatest prime number, which is a divisor of w — 1. It follows, therefore, that if the highest prime factor of » — 1 be 2, the resolution of the binomial equation a;" — 1 = will be made to depend upon the solution of quadratic equations only, and consequently to depend upon constructions which can be effected by combinations of straight lines and circles, and therefore within the strict province of plane geometry : this will take place whenever n is equal to 2* + 1 and is also a prime number. Thus, if A' = 4 we have « = 17, a prime number, and therefore the solution of the equation a:'^ — 1 = will be reducible to that of four quadratic equations. Similar observations apply to the equations ^2V 1 _ 1 = and x'^^^+ 1-1=0. The same principles which enable us to solve algebraically binomial equations, under the circumstances above noticed, will admit of extension to other classes of equations, whose roots admit of analogous relations amongst each other. Gauss f has stated that the principles of his theory were applicable to func- tions dependent upon the transcendent /^yj\ 4\> which de- fines the arcs of the lemniscata, as well as to various species of congruencies ; and he has also partially applied them to certain classes of equations dependent upon angular sections, though in a form which is very imperfectly and very obscurely deve- loped. Abel, however, in a memoir X which is remarkable for the generality of its views and for its minute and careful ana- lysis, has not merely completed Gauss's theory, but made most important additions to it, particularly in the solution of exten- sive classes of equations which present themselves in the theory of elliptic transcendents §. Thus he has given the complete * Symmetrical functions of these periods will be multiples of the sum ( — 1) of these periods and of 1. This conclusion follows immediately from the re- placement of the arithmetical by the geometrical series of indices, according to the general process of Lagrange, without any antecedent distribution of the roots into periods. See Note xiv. to the Resolution des Equations Num.e- riques. It follows from thence that the coefficients of the reducing equations will be whole numbers. ■f Disqiiisitiones Arithmetic(B, pp. 595, 645. I " Sur une Classe particuliere d'Equations resolubles algebriquement," — Crelle's Journal, vol. iv. p. 131. § Crelle's Journal, vol. iv. p. 314, and elsewhere. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 319 algebraical resolution of an equation whose roots can be repre- sented by X, 6 X, &^ X, . . . . 6^-' X, where &^ x = x, and where 6 is a rational function of 'x and of known quantities ; and also of an equation where all the roots can be expressed rationally in terms of one of them, and where, if 9 ^ and d^ x express any other two of the roots, we have like- wise It is impossible, however, within a space much less than that of the memoir itself, to give any intelligible account of the pro- cess followed in the demonstration of these propositions, and of many others which are connected with them. We shall con- tent ourselves, therefore, with a slight notice of their applica- tion to circular functions. o -J. If we suppose a = — , the equation whose roots are cos a, /*■ cos 2 a, cos 3 «, . . . cos /x a is ^^_|.^^-2 + ^./i^ii:_l)^^-4. .. =0 (1.) which may be easily shown to possess the required form and properties ; — for, in the first place, cos m a =■ ^ (cos or), where 9 is, as is well known, a rational function of cos « or a:* ; and, in the second place, if 9 a' = cos m a and 9, a; = cos m^ a, then likewise 9 9i a: = cos mm-^a =■ cos ^Wj m a = 9i 9 x, which is the second condition which was required to be fulfilled. Let us suppose /«, = 2 « + 1, when the roots of the equation (1.) will be Stt 47r 4<nit ^ of which the last is 1, and the ti first of the remainder equal to the n last. The equation (1.) may be depressed, therefore, to one o{ n dimensions, which is x" + ^ x"-' - -T- (« — 1) x"-2 — — (n — 2) x"-3 1 (n-2){n-3) l_ (»-3)(/»-4) _ + 16' 1.2 "" ^ 32' 1.2 "" i^c.-U{4.) whose roots are 2 TT 4 TT 2 M cos ;z r, cos p: — ^, .... COS TT 2n + I 2/2 + 1 2« + 1 320 THIRD REPORT 1833. •r/. 2v , 2mn /. If cos 7i T = X = cos a, and cos ^ -^ = 8 j; = cos »? a, then these roots are reducible to the form xjxj'^x, . . . 6"-' X, or, cos a, cos m a, cos m^ a, . . . cos m" ' « : and if we suppose m to be a primitive root to the modulus 2n + I, then all the roots cos a, cos m a, cos m^ a, . . . cos m"~' « will be different from each other, and cos m" a = cos a; con- sequently it will follow, since the roots of the equation (2.) are of the form x,6a:,&^a:, . . . 6""' x, where $^x = x, they will admit, in conformity with the preceding theorems, of algebraical expression. Abel has given the general form of the expression for these roots, which in this case are all real ; and their determination will involve the division of a circle into 2 n equal parts, the division of an assigned or assignable arc into 2 n equal parts, and the extraction of the square root of 2 w + 1 ; a conclusion to which Gauss had also arrived, though he has not given the steps of the process which he followed for obtaining it*. If we suppose 2n = 2", we shall get the case of regvilar polygons of 2"+i 4- 1 sides, which admit of indefinite inscription in circles by purely geometrical means. It will follow from the same re- sult that the inscription of a heptagon will depend upon that of a hexagon, the trisection of a given angle, and the extraction of the square root of 7. Poinsotf has given a very remarkable extension to the theory of the solution of the binomial equation x" — 1 = 0, by showing that its imaginary roots may be considered in a certain sense as the analytical representation of the whole numbers which satisfy the congruence or equation ^« - 1 = M ip), whose modulus (a prime number) is p: thus, the imaginary cube roots of 1, or the imaginary roots of ^ — 1 = 0, are — 1 + V— 3 — 1 — V — 3 ^^ ^jjg whole numbers 4 and 2, 2 ' , 2 ' * Disquisitiones Arithinetic<E, p. 651. t Journal de I'Ecole Polytechnique, cahier 18. 1 REPORT ON CERTAIN BRANCHES OF ANALYSIS, 321 which satisfy the congruence ,r3 _ 1 = M X 7, - 1 + 7 + ^ — 3 + 7 whose modukis is 7; are expressed by 2 and ~ — — , which arise from adding 7 to the parts without and beneath the radical sign. The principle of this transition from the root of the equation to that of the congruence is sufficiently simple. We consider the roots of x" — 1 = as resulting from the expression for those of the congruence or" — 1 = M (/>), when M = ; and we thus are enabled to infer, in as much as M (/>), its multiples and powers, are involved in those formulse, whether without or beneath the radicals, and disappear, therefore, when M = 0, that some such multiples, to be determined by trial, or other- wise, are to be added when M (/?) is restored, or when 1 is replaced by 1 + M (p). When the congruence admits of in- tegral values of jr, which are less than p, then they can be found by trial : when no such integral values exist, then, amongst the irrational values which thus arise, those values will present them- selves which will satisfy the congruence algebraically, though they can only be ascertained by a tentative process. The equation of Fermat, ^p-' — 1 = M (^), where /> is a prime number, will be satisfied by all the natural numbers 1, 2, 3, . . as far as (^ — 1) : and it follows, therefore, that all the rational roots of the equation a7» — 1 = M (j9) will be common to the equation x^-i - 1 = M (^), the number of them being equal to {d), the greatest common divisor of w and oi p — \. If <^ be 1, then all the roots except 1 are irrational. If we suppose the equation to be tp - 1 = M (p), then all the roots will be equal to each other and to 1. It is unnecessary, however, to enter upon the further examination of such cases, which are developed with great care and sin- gular ingenuity in the memoir referred to. These views of Poinsot are chiefly interesting and valuable as connecting the theory of indeterminate with that of ordinary 1833. Y 322 THIRD REPORT — 1833. equations. It has, in fact, been too much the custom of analysts to consider the theory of numbers as altogether separated from that of ordinary algebra. The methods employed have generally been confined to the specific problem under consideration, and have been altogether incapable of application when the known quantities employed were expressed by general symbols and not by specific numbers. It is to this cause that we may chiefly attri- bute the want of continuity in the methods of investigation which have been pursued, and the great confusion which has been occasioned by the multiplication of insulated facts and propositions which were not referable to, nor deducible from, any general and comprehensive theory. Libri, in his Teoria del Nmneri, and in his Memoir es de Mathdmatiqtie et de Physique, has not merely extended the views of Poinsot, but has endeavoured to comprehend all those conditions in the theory of numbers, by means of algebraical or transcendental equations, which were previously understood merely, and not symbohcally expressed. He has shown that problems which have been usually considered as indeter- minate are really more than determinate, and he has thus been enabled to explain many anomalies which had formerly embar- rassed analysts, by showing the necessary existence of an equa- tion of condition, which jKiust be satisfied, in order that the problem required to be solved may be possible. By the aid of such principles the solutions of indeterminate equations, at least within finite limits, may be found directly, and without the necessity of resorting to merely tentative processes. A great multitude of new and interesting conclusions result from such views of the theory of numbers ; but the limits and object of this Report will not allow me to discuss them in de- tail, or to point out their connexion with the general theory of equations, and with the properties of circular and other func- tions. The reader, however, will find, in the second of the memoirs of Libri above referred to, a general sketch of the nature and consequences of these researches, which is unfor- tunately, however, too rapid and too imperfectly developed to put him in full and satisfactory possession of all the bases of this most important theory. On the Solution of Numerical Equations, — The resolution of numerical equations formed the subject of a truly classical work by Lagrange, in which this problem, one of the most im- portant in algebra, is not only completely solved, but the imper- fections of all the methods which had been proposed for this purpose by other authors are pointed out with that singular distinctness and elegance which always distinguish his reviews REPORT ON CERTAIN BRANCHES OF ANALYSIS. 323 of the progress and existing state of the diiFerent branches of the mathematical sciences. In the following report we shall commence by a general account of the state in which the pro- blem was left by him, and of the practical difficulties which attend the use of his methods, and we shall then proceed to notice the important labours of Fourier and other authors, with a view to bring its solution within the reach of arithmetical processes which are at once general and easy of application. The resolution of numerical equations involves two principal objects of research : the first of them concerns the separation of the roots into real and imaginary, positive and negative, and the determination of the limits between which the real roots are severally placed ; the second regards the actual numerical approximation to their values, when their limits and nature have been previously ascertained. Many different methods have been proposed for both these objects, which dilFer greatly from each other, both in their theoretical perfection and in their practical applicability. We shall begin with a notice of the first class of me- thods, which have been proposed for the separation of the roots. If the coefficients of an equation be whole numbers or rational fractions, their real roots will be either whole numbers or ra- tional fractions, or otherwise irrational quantities, which will be generally conjugate* to each other and which will generally pre- sent themselves, therefore, in pairs. The method of divisors which Newton proposed, and which Maclaurin perfected, will enable us to determine roots of the first class, and they are also determined immediately and completely by nearly all methods of approximation. It will be to roots of the second class, there- fore, that our methods of approximation will require to be ap- plied, though such methods will never enable us to assign them under their finite irrational form, nor would our knowledge of their existence under such a form in any way aid us, unless in a very small number of cases, in the determination of their ap- proximate numerical values. The equal roots of equations, if any exist, may be detected by general methods ; and the factors corresponding to them may be completely determined, and the dimensions of the equar * An irrational real root may be conjugate to the modulus of a pair of im- possible roots ; and there may exist, therefore, as many irrational real roots which have no corresponding conjugate real roots as there are pairs of im- possible roots in the equation. It is not true, therefore, generally, as is some- times asserted, that such irrational roots enter equations by pairs. It would not be very diflScult to investigate the different circumstances under which roots present themselves, and the different conditions under which they can be conjugate to each other ; but the inquiry is not very important, in as much as the knowledge of their form would not materially influence the application of methods for approximating to their values. y2 324 THIRD REPORT — 1833. tion depressed by a number of units equal to the number of such factors. We might suppose, therefore, in all cases, that the roots of the equation to be solved were unequal to each other ; but if it should not be considered necessary to perform the previous operations which are required for the detection and separation of the equal roots, the failure of the methods of approximation or other peculiar circumstances connected with the determination of the limits of the roots, would indicate their existence, and at once direct us to the specific opei'ations upon which their determination depends. If we svqjpose, therefore, the equal roots to be thus separated from the equation to be solved, and if we assume a quantity J which is less than the least difference of the unequal roots, then the svibstitution of the terms of the series k A,{Ji — \) A, . . . , 2 A, A, 0, - A, -2 A, ... . - k^ A, where ^ J is greater than the gi*eatest root, and — /'i A less than the least root *, will give a series of results, amongst which the number of changes of sign from + to — and from — to + will be as many as the number of real roots, and no more ; and v/here the pairs of consecutive terms of the series of multiples of A which correspond to each change of sign are limits to the seve- ral real roots of the equation. This is the principle of the me- thod of determining the limits of the real roots which was first proposed by Waring, and which has been brought into practical operation by Lagrange and Cauchy. It remains to explain the different methods which have been proposed for the purpose of determining the value of J. Waring first, and subsequently Lagrange, proposed for this purpose the formation of the equation whose roots are the squares of the differences of the roots of the given equation. If we subsequently transform this equation into one whose roots are the reciprocals of its roots, and determine a limit I greater than the greatest root of this transformed equation f, then— ;^ * A negative root is always considered as less than a positive root, unless the consideration of the signs of affection is expressly excluded. t Newton proposed for this purpose the formation of the equation whose roots are x — e, and v/here e is determined by trial of such a magnitude that all the coeflScients of the equation may become positive. In such a case e is the limit required. Maclaurin proved that the same property would belong to the greatest negative coeflBcient of the equation increased by 1. Cauchy, in his Cours d' Analyse, Note iii., and in his Exercices ties Mathematiques, has shown that if the coefficients of the equation, without reference to their sign, be Ai A2, . . Am, and \f n be the number of such coefficients which are different from zero, then that the greatest of the quantities \_ REPORT ON CERTAIN BRANCHES OF ANALYSIS. 325 will be less than the least difference of any two of the real roots of the primitive equation, and will consequently furnish us with such a value of A as will enable us to assign their limits. The extreme difficulty, however, of forming the equation of dif- ferences, which becomes nearly impracticable in the case of equations beyond the fourth degree*, renders it nearly, if not altogether, useless for the purposes for which this transforma- tion was intended by the illustrious analysts who first proposed it ; in other words, it is only in a theoretical sense that it can be said to furnish the solution of the problem of determining the limits of the real roots of an equation. Cauchy has succeeded in avoiding the necessity of forming the equation of the squares of the differences of the roots, by showing that a value of A may be determined from the last term of this transformed equation, combined with a value of a limit greater than the greatest root of the primitive equation. If we suppose H to represent this term, k to be the superior limit required, and a and b to represent any two roots of the equa- tion, whether real or imaginary, then he has shown that their difference a — b, or the modulus of their difference, will be will be a superior limit to the roots. An inferior limit (without reference to algebraical sign) may be readily found by the same process by the formation of the equation whose roots are the reciprocals of the former. M. Bret, in the sixth volume of Gergonne's Avnales des Mathematiques, has investigated other superior limits of the roots of equations, which admit of very easy application, and which likewise give results which are generally not very remote from the truth. One of these limits is furnished by the following theorem : " If we add to vnity a series of fractions whose numerators are the successive negative coefficients, taken positively, and whose denominators are the sums of the positive coefficients, including that of the first term, the greatest of the resulting values will be a superior limit of the roots of the equation." Thus, in the equation 2 x^ + 11 x^ — 10 aS _ 26 a;-! + 31 a.3 + 72 a;2 — 230 x — 348 = 0, the number 4, which is equal to the greatest of the quantities 1+2^,1 + ^.1 + ^,1 + ^. 13 13 11(3' 116 is a superior limit required ; and if we change the signs of the alternate terms, we shall have 1 + -— , or 7, a superior limit of the roots of the resulting equation : it will follow, therefore, that all the real roots of the first equation will be included between 4 and — 7- Other methods are proposed in the same memoir which are not equally new or equally simple with the one just given, and which I do not think it necessary to notice. * Waring, as is well known, gave the transformed equation of the 10th de- gree, whose roots were the squares of the differences of the roots of a general equation of the fifth degree, wanting its second term : it involves 94 different combinations of the coefficients of the original equation, many of them with large numerical coefficients. 326 THIRD REPORT — 1833. greater than »(»-i) :, if n denote the dimensions of the equation ; and in as much as H is necessarily, when the coeffi- cients are whole numbers, either equal to or greater than 1, it 1 will follow that «(n-i) , will furnish a proper value of J, where k has been determined by the methods described above, or in any other manner. The chief objection to the use of a value of J thus determined arises from its being generally much too small, and from the consequent necessity of making a much greater number of trials for the discovery of the limits of the roots than would otherwise be necessary. Lagrange has proposed different methods of determining the value of J, which, though much less laborious, at least for equations of high orders, than the equation of the squares of the differences, are still liable to great objections, in conse- quence of their being indirect, difficult of application, and likely to give values of J so small and so uncertain as greatly to mul- tiply the number of trials which are necessary to be made *. It is for this and other reasons that such methods have never been reduced to such a form as to be competent to furnish the re- quired limits by means of processes which are expressible in the form of arithmetical rules, like those which are given for the extraction of the square and cube root in numbers. In this re- spect, therefore, they have failed altogether in satisfying the great object proposed to be attained by their author, who con- sidered the resolution of numerical equations as properly consti- tuting a department of common arithmetic, the demonstration of whose rules of operation must be subsequently sought for in the general theory of algebraical equations +. The basis of all methods of solution of numerical equations must be found in the previous separation of the roots ; and the efforts of algebraists for the last two centuries and a half have been directed to the discovery of rules for this purpose. The methods, however, which have been proposed have been chiefly directed to the separation of the roots into classes, as positive and negative, real and imaginary, and not to the determination of the successive limits between which they are severally placed. The celebrated theorem of Des Cartes J gave a limit to the number of positive and negative roots, but failed in deter- * Resolution des Equations Numeriques, Note iv. t Jhid., Introduction. X The proper enunciation of this theorem is the following : "Every equa- tion has at least as many changes of sign from + to — and from — to + as it has real and positive roots, and at least as many continuations of sign REPORT ON CERTAIN BRANCHES OF ANALYSIS. 327 mining the absolute number either of one class or of the other, in the absence of any means of ascertaining the number of ima- ginary roots. If the roots of the equation were all of them real, and could be shown to be so by any independent test, it would be easy to determine the limits between which the roots were severally placed ; for the number of changes of sign which are lost upon the substitution ofx + e for x would show the number of roots which are included between and e ; and if, therefore, we should assume a succession of values of e, whether positive or negative, such as to destroy one change of signs and rio more, upon the substitution of any two of these successive values, we should be enabled to obtain the limits of every root of the equation. It was chiefly with a view to this consequence of Des Cartes's theorem that De Gua investigated and assigned the conditions of the reality of all the roots of an equation. If we suppose X = to be the equation, and X', X", X''', X'% X\ &c., to denote the successive differential coefiicients of X, then, if all the roots of X = be real, the roots of the several derivative equations X' = 0, X" = 0, X"' = 0, &c., must be real like- wise ; and if the roots of any one of these equations X^*"' = be substituted in X^''"'' and X'''+'', it will give results affected with different signs. If we form, therefore, a succession of equations in y by eliminating successively x from the equations t/ = XW . XC-^' and X'»-'' = 0, 1/ - X(»->^ . X(» 3) and XC- ) = 0, y = Xi X"' and X" = 0, y = X X'' and X' = 0, the coefficients of all these equations must be positive, forming from + to + and from — to — as it has real and negative roots." It is very doubtful, notwithstanding the assertions of some authors, whether Des Cartes himself was aware of the necessary limitation of the application of this theorem, which is required by the possible or ascertained existence of imaginary roots. The demonstration which was given by De Gua of this theorem in the Me- moires de I'Academie des Sciences for 1741, founded upon the properties of the limiting equation or equations, has been completed by Lagrange with his usual fullness and elegance, in Note viii. to his Resolution des Equations Nu- meriques. The most simple and elementary, however, of all the demonstra- tions which have been given of it, and the one, likewise, which arises most naturally and immediately from the theory of the composition of equations, is that which was given by Segner in the Berlin Memoirs for 1756. The few im- perfections which attach to this demonstration, as far as the classification of the forms which algebraical products may assume is concerned, have been completely removed in a demonstration which Gauss has published in the third volume of Crelle's Journal. This theorem is included as a corollary to Fourier's more general theorem for the separation of the roots, as we shall have occasion to notice hereafter. 328 THIRD REPORT — 1833. a collection of conditions of the reality of the roots of an equa- tion of n dimensions which are „ in number *. These speculations of De Gua were well calculated to show the importance of examining the succession of signs of these derivative equations, with a view to the discovery of their con- nexion with the nature of the roots of the primitive equation. The changes in the succession of signs of the coefficients of the equations which resulted from the substitution of a: + « and X + b, gave no certain indications of the nature and number of the roots included between a and b, unless it could be shown that all the roots of the primitive equation were real, a case of comparatively rare occurrence, and which left the general pro- blem of the separation of the roots, as preparatory to their actual calculation, nearly untouched. It was the conviction that all attempts to effect the solution of this problem by the aid of Des Cartes's theorem would necessarily fail, which led Fourier, one of the most pi'ofound and philosophical writers on analysis and physical science in modern times, to the examination of the * Resolution des Equations Numeriques, Note viii. The equation of the squares of the differences of the roots of an equation will indicate the reality 71 \7l ^^ 1 \ of all the roots, if its coefficients have ^ changes of sign, or be alter- nately positive and negative. The succession of signs of the coefficients very readily furnishes the indications of the number of impossible roots in all equa-- tions as far as five dimensions, as has been shown by Waring and Lagrange. The number of conditions of the reality of the roots of an equation of five dimensions would appear from the formula in the text to be 10 ; but some of these conditions, as Lagrange has intimated, may, and indeed are, included in the system of the others, so as to reduce them to a smaller number. La- grange has assigned two conditions (not three) of the reality of the roots of a cubic equation ; but the first of these is necessarily included in that of the second, so as to reduce the essential conditions to one. Similar consequences are found to present themselves in the examination of these conditions for an equation of the fourth degree, which are three in number, and not six, as the formula would appear to indicate. Cauchy, in the 17th cahier of the Journal de I'Ecole Polytechnique, has suc- ceeded, by a combined examination of the geometrical properties of the curve whose equation is y = X (where X is a rational function of x of the form «" + p^ .t«-i + . . . . pn)y and of their corresponding analytical charac- ters, in the discovery of general methods, not merely for the determination of the number of real roots, but likewise of the number of positive and negative roots, as distinguished from each other. These methods are equally appli- cable to literal and numerical equations. He has applied his method to ge- neral equations of the first five degrees, and the results are in every respect, as far at least as they have been examined in common, equivalent to those which are derived from the equation of the squares of the differences. It is impossible, however, in the space which is allowed to me in this Report, to give any intelligible account of this most elaborate and able memoir, and I must content myself, therefore, with this general reference to it. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 329 succession of signs of the function X and its derivatives, upon the substitution of different values of x. The conclusions which have resulted from this examination, which we shall now proceed to state, have completely succeeded in effecting the practical solution of this most difficult and important problem, as far, at least, as real roots are concerned. If we suppose X = .r™ + a I a:""-' + a^ x"'-^ + . . .a^ = 0, and if we write X and its derivatives in the following order, XW X^'"-'\ x(™-2), . . . X", X', X, then the substitution of yr and ^, will give two series of re- sults, the terms of the first series being all of them positive, and those of the second being alternately positive and negative. The same will be the case if, in the place of -^, we put any limit («) greater than the greatest root of the equation X = 0, and if in the place of — ^ we substitute any negative value of a' (— /3) (to be determined by trial or otherwise) which will make the first terms of X, X', X", &c., considered with regard to numerical value only, severally greater than the sum of all those which follow them. In the course of the substitution of values of x intei'mediate to those extreme values — /3 and a, all the ?« changes of sign of X and its derivatives, from + to — and from — to + , will disappear, in conformity wath the fol- lowing theorems, which are capable of strict demonstration. 1st. If, upon the substitution of any value o{ x, one or more changes of signs disappear, those changes are not recoverable by the substitution of any greater value of x. 2nd. If upon the substitution of two values a and b of .r, one change of signs disappears, there is one real root and no more included between a and b. If under the same circum- stances an odd number 2 p + \ of changes of sign have disap- peared, there must be at least one, and there may he 2 p' + I (where p' is not greater than 2^) I'eal roots between a and b ; but if an even number 2 p of signs have disappeared in the in- terval, there ma?/ be 2p — 2p' real roots, and p' pairs of ima- ginary roots corresponding to it, where p' is not greater than p. If no change of sign disappears, upon the successive substi- tution of a and b, then no root whatever of the equation X = can be found between the limits a and b. 3rd. If the substitution of a value a o{ x makes X = 0, then a is a root of the equation. If the substitution of the same value of X makes at the same time X = and X' = 0, then 330 THIRD REPORT 1833. there are two real roots equal to a ; and generally, as many of the final functions X, X', X", &c., will disappear, under the same circumstances, as there are roots equal to a. 4th. If the substitution of a value of a makes one intermediate function X'*"' equal to 0, and one only, and if the result be placed between two signs of the same kind, whether + and + or — and — , then there will be one pair of imaginary roots corresponding to this occurrence ; but if be placed between two unlike signs, + and — or — and +, then there will be no root corresponding to it, unless at the same time X = 0. If the substitution of a makes any number of consecutive derivative functions equal to 0, then, if there be an even number 2/> of consecutive zeros, there will be^ or (^ — 1) pairs of imaginary roots corresponding, according as they are placed between the same or different signs ; and if there be an odd number 9,p -\- \ of consecutive zeros, then there will be^ + \ o\ ]i pairs of imaginary roots corresponding, according as they are placed between the same or different signs *. The preceding propositions may be easily shown to include the theorem of Des Cartes ; for it is obvious that the substitution of for or in X and its derivatives will give a succession of signs identical with those of the successive coefficients of X, deficient terms being replaced by 0. If the extreme values « and — /3 be substituted, there will be m permanences in one case and m changes in the second ; it will follow therefore that the number of real and therefore positive roots between a. and cannot ex- ceed the number of changes of sign corresponding to o^ = 0, or amongst the successive coefficients of the equation ; and that the number of real and therefore negative roots between — /3 and cannot exceed the number of permanences corresponding to a: = 0, or of changes between and — /3, which is also identical with the number of successive permanences of sign amongst the coefficients of the equation. * I have stated this rule differently from Fourier, whose rule of the double sign appears to me to be superfluous. If we consider the zeros as possessing arbitrary signs, the nature and extent of the ambiguity which they produce will always be determined by the circumstances of their position with respect to the preceding and succeeding sign. The rule of the double sign, when one of the derivative functions X', X", X'", &c., becomes equal to zero, is made use of in a memoir by Mr. W. G. Horner, in the Philosophical Transactions for 1819, upon a new method of solving nu- merical equations. This memoir, though very imperfectly developed, and in many parts of it very awkwardly and obscurely expressed, contains many original views, and also a very valuable arithmetical method of extracting the roots of affected equations. It makes also a very near approach to Fourier's method of separating the roots of equations. It is proper to state that Fourier's proposition was known to him as early as 1796 or 1797, as very clearly appears from M. Navier's Preface to \\\s Analyse des Equations Deter- minees, a posthumous work, which appeared in 1831. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 331 In order to render the preceding propositions more easily in- telligible, we will apply them to two examples. Let X = x'* — 4<r^ — 3x+ 23 = 0, and underneath X'", X'", X", X', X, let us write down the signs of the results of the substitution of 0, 1, 2, 3, 10, in the place of x, in conformity with the following scheme : X', X, - + - + X-, X"', X" (0) + — (1) + — (^) + + (3) + + + (10) + + + + + For X = 0, there is a result placed between two similar signs ; there is therefore a pair of imaginary roots correspond- ing to it. Every value of x less than will give results alter- nately + and — , and there is therefore no real negative root. For X = I, there is a result placed between two dissimilar signs : there is therefore no pair of imaginary roots corre- sponding ; and since there is no loss of changes of sign in pass- ing from to 1, there is no real root between those values. For X = 2, there is a result placed between two dissimilar signs ; there is therefore no pair of imaginary roots correspond- ing, and there is no root between 1 and 2. For X = 3, there is a loss of one change of sign, and there is therefore one real root between 2 and 3. For jr = 10, there is a loss of one change of signs and all the resulting signs are positive ; there is therefore one real root between 3 and 10. The limits of the real roots are thus completely determined, and the substitution of the successive whole numbers, from 3 upwards, will show the nearest whole numbers 3 and 4, between which the greatest root is situated. Let X = ^6 _ 12 > + eOx'^ + 123 .r^ + 4567 x - 89012 = x^S x^ x^ X"S X", x\ X, (-10) + — + — + ~ + (-1) + — + — + — — (0) + — + + + — 0) + — + + + + — (10) + + + + + + + All tlie real roots of the equation are included between the extreme values — 10 and 10. 332 THIRD REPORT — 1833. • One change of sign is lost in the transition from — 10 to — 1, and there is therefore one real root between them ; the sign of the last term is therefore necessarily changed from + to — . For X = 0, there is a result laetween two similar signs ; there is therefore a pair of imaginary roots cori'esponding, and consequently a loss of two changes of sign. There is no root of the equation between and 1 . There is a loss of three changes of sign in the transition from 1 to 10, and therefore there are three roots corresponding, one or all of which may be real : the application of a subsequent rule will show that two of them are imaginary. It is obvious, in a series of derivatives, X^™^, X^"*"'', . . . X'*"' ... X, that X'*"', X^"*"'^ may be considered as the derivatives of the {m — r —ly^ and {m — r — S)* order from X'*"^, as well as the wj"' and {tn — 1)* derivatives from X, and that the same rules may be applied to the separation of the roots of tliese de- rivatives when they become equations, whether they be consi- dered as belonging to the inferior or to the superior order. The substitution, therefore, of a and b successively for x, will show the number of roots of the successive derivative equations which are found in this interval, which will be equal successively to the number of changes of sign which have disappeared in the transition from one value of x to the other. If we now place under the several results of the substitution of a and b, a series of zeros or numbers as indices to signify that no change, or an indicated number of changes of signs, have disappeared, then in passing from the left to the right, we shall find first zero, and sub- sequently, whether immediately or not, the numbers, 1, 2, &c., which will indicate the number of roots which must be sought for, in that interval, in the derivative or other functions, consider- ed as equations, which are severally placed above them. Thus, i{ X. = x'^ — x^ + 4< x^ + X — 4) — 0, then from the scheme X'% X"', X", X', X, (-10) + - + _ + 1 (-1) + — + — — 1 (0) + — -1- + — 1 2 2 3 (I) + + + + + we infer that there is one root of X = 0, and no root of any of the several derivative equations situated between — 10 and — 1 j REPORT ON CERTAIN BRANCHES OF ANALYSIS. 333 that there is one root of X' = 0, and no root of X =: 0, between — 1 and ; that there is one root of X"' = 0, two roots of X" = 0, two roots of X' = 0, and"three roots of X = 0, situated between and 1. It remains to determine whether these three roots are all of them real, or two of them imaginary, and also to assign the limits, in the first case *, between v/hich they are placed. In the first place, if imaginary roots exist in the derived, they will exist also in the primitive equation. The converse of this proposition is not necessarily true. If the succession of indices be 0, 1, 2, then the succession of signs corresponding to X", X\ X, or X('- + ^), X('^ X^'--'), will be (a) + - + or - + - 12 12 (b) + + + - _ _ There will be one real root between a and b in the equation X' = or X^*"^ = 0, and two roots, whether real or imaginary, corresponding to this interval, in X = or X^'"'^ = 0. In the first case, if there be two real roots between a and b, then the curve whose equation is y = X =y (a;), where o a = a, o b = b, a n =f(a), b m = f (b), will cut the axis at the points a and ^ between a and b. The curve will have no point of inflection between a and b, ^ ^ since X" preserves the same sign, whether + or — ; and there will be a point t, where the tangent is parallel to the axis, since X', in the same interval, changes from + to — , or con- versely, and therefore becomes equal to ^ero between those limits. In this case, the sum of the subtangents (considered without regard to algebraical signs) will be necessarily less than a b ; and if the interval a 6 be subdivided sufficiently, so as to furnish new limits a' and b', then one or both of these points will sooner or later be found between the points of intersection « and /3, and therefore/ («') and/ (6') will one or both of them change their signs. The analytical expression of those geo- metrical conditions, and therefore of the existence of two real roots, will be, that the sum of the subtangents or quotients *4-7^ ./ («) * We seek for the limits of the real roots only ; we have no concern with those of the imaginary roots or of their moduli. 834 THIRD REPORT — 1833. f (h\ + fufi ^b — a, when no regard is paid to the sign of/' (a) and f (b). In this case new Kmits must be taken successively, intermediate to a and b, until f («') and J" (b') one or both of them change their sign. In the second case, if there be two imaginary roots cor- responding to the interval p- „ between a and b, then the ^ curve whose equation is 2/=X though similar in its other ge- ometrical properties to fig. 1, will not cut the axis between a and b. In this case the sum of the subtangents a n' and h nil will either exceed the interval a b, or will ultimately ex- ceed it, when the interval a 6 is sufficiently diminished. The corresponding analytical character will be that ^^r^. + ^irrr: is either greater than 6 — a, or that it may ultimately be made to exceed it *. Thus, in the example refen*ed to above, p. 332, write down the following scheme : X", X-, X", X', X, 6 8 (0) + _ + + _ 13 2 3 (1) + + + + + 18 14 and place above and below the indices 1 and 2, in the succes- sion of indices 0, 1,2, the values of X"' and X" respectively, without regard to sign, corresponding to a: = and x ^=\; then 8 V . . 8 14 we shall find -^ 7 1 and, a fortiori, therefore -pr + rn, also greater than 1, which is the interval between which the roots required are to be sought for : it consequently follows that two of the roots corresponding to this interval are imaginary, and there remains, therefore, only one real root between and 1. If we suppose X=a;5 + a;4 + a^-2x2-|-2jr-l = 0, * The new values a' and 6' of a and h may be made a 4- ., \~{ and b — \^ ,J . ' f'(a) f{b)' which are n' and m' respectively : a second trial will generally succeed. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 335 the corresponding scheme will be as follows : X, X, x\ x\ 96 x"s 42 X", X- (-1) + 36 + 9 — + (-4) + — + — + 1 2 2 2 (0) + + + — + 24 6 4 2 1 2 a) + + + + 36 + 10 (1) + + + + + + If we take the interval from (— 1) to 0, we find two roots in- 42 6 eluded within it ; but since q^ + ht, is less than the interval, no certain conclusion can be drawn with respect to the nature of the corresponding roots. If we now consider the interval from H" to 0, which includes the same roots, we shall find 9 6 1. ^ + ^ = -^, a quantity equal to the whole interval, and we are consequently authorized in concluding that the correspond- ing roots are imaginary. In a similar manner, we find the in- dication of the existence of two roots between and -^ ; and 2 1. in as much as -j- = -^ = the whole interval, we at once con- clude that the two roots in question are imaginary*. It thus appears that we are enabled, by the processes just described, to separate all the real roots of an equation and to • When we speak of the existence of imaginary roots between two limits, we do not mean that such limits comprehend the moduli of these roots, but merely that the real roots which would be found between those limits, if cer- tain conditions were satisfied, are wanting, and that there are as many ima- ginary roots of the equation which may be said to correspond to them which are sufficient to complete the required number of changes of sign which are lost. The theory of Fourier as given in his work, determines nothing con- cerning the values or limits of the moduli, or of the peculiar nature of the signs of affection, of such imaginary roots. 336 THIRD KEPORT — 183^. assign their limits, and thus to prepare them for the certain ap- plication of methods of approximation. They constitute a most important element in the theory of numerical equations ; and though they do not enable us to assign the limits of the moduli of the pairs of impossible roots nor to determine their signs of affection, yet they at once indicate both their existence and their number, and thus form the proper pi'eparation, at least for the application of methods, whether tentative or not, for the deter- mination of their values. Lagrange, in the fifth chapter of his Resolution des Equa- tions Numeriques, has shown in what manner the equation of the squares of the differences may be apjilied to the deter- mination of these imaginary roots ; and the methods which thence arise are equally complete, in a theoretical sense, with those which are made use of, by the aid of the same equation, for the determination of the limits of the real roots ; and Le- gendre, also, has furnished tentative methods of approximating to then' values. But all such methods are more or less nearly impracticable for equations of high orders ; and the invention of a ready and certain method of separating the imaginary roots of equations, as the basis of processes for approximating to their values, must still be considered as a great desideratum in algebra. The method of approximating to the roots of numerical equa- tions, when their limits are assigned, which Lagrange has given, by means of continued fractions, is so well known that it is quite unnecessary to enter upon a detailed examination of its princi- ples. If there is only one real root, included between two con- secutive whole numbers, there will be only one positive root in the several transformed equations, which is greater than 1 , and methods which are certain and sufficiently rapid may be applied to the determination of the several quotients which form the converging fractions. If, however, there are two or more roots included between two consecutive whole numbers, there will be two or more roots of the first transformed equation, and possi- bly, likewise, of the transformed equations which follow which are greater than 1, and which may be placed between two con- secutive whole numbers. The separation of such roots may be effected by the methods of Fourier, which have been explained above ; but when we have once arrived at a transformed equa- tion which has two or more roots greater than 1, no two of which are included between two consecutive whole numbers, then we shall find the same number of sets of successive transformed equations, which will furnish the several sets of quotients to the continued fractions, which represent the roots of the primitive REPORT ON CERTAIN BRANCHES OF ANALYSIS. 337 equation, which are included between two Hmits which are con- secutive whole numbers. The formation, however, of these transformed equations, and the determination of the next infe- rior integral limit of their roots, even when no further separation of the roots is required, is excessively laborious, and Lagrange has pointed out methods by which the operations required for both these objects may be greatly simplified. Legendre also, in the 14th section of the first part of his Theory of Numbers, has given a considerable practical extension to these methods of Lagrange. If we combine their processes for finding the neai'est inferior limit of the root with the theorems of Budan * for the formation of the transformed equations, we shall proba- bly have arrived at the greatest simplification which the practi- cal solution of numerical equations, by means of continued frac- tions, is capable of receiving. Lagrange has pointed out the principal defects of the me- thod of approximation to the roots of numerical equations which was given by Newton f. It is only under particular conditions that it is competent to attain the object proposed, and in no case does it immediately furnish a measure of the accuracy of the approximation. But notwithstanding these objections to this method, in the form under which it has been commonly applied, it is unquestionably that which most naturally arises out of the analytical conditions of the problem, and which is also capable of the most immediate and most simple application in almost every department of analysis. Lagrange had demon- strated that this method could only be applied with safety to find the greatest and least roots of an equation, and in those cases only in which the moduli of the imaginary roots, if any ex- isted, were included in value between such roots. But Fourier has shown, by considering the superior and inferior limits of every real root, and by a proper examination of certain condi- tions which those limits may be made to satisfy, and by insti- tuting the approximation simultaneously with respect to both those limits, that all sources of ambiguity may be removed and the accuracy of the approximation determined J. We shall now proceed to give a short notice of these researches. * Nouvelle Methode pour la Resolution des Equations Numiriques. It con- tains the exposition of exceedingly simple and rapid rules for the formation of the transformed equation whose unknown quantity is a; — e, where e is any integral or decimal number. In other respects, however, this publication, though announced with great pomp and circumstance, is a very superficial production, and is only remarkable for having received the charitable notice and approbation of Lagrange. t Resolution des Equations Numeriques, Note v. X Analyse des Equations determinees, livr. ii., Calcul des Racines. 1833. z 338 THIRD REPORT — 1833. 1. If /(^) = 0, or X = be the equation, /' (x), f" {x\ or X', X'' its first and second derivatives, then the limits a and h of one of the roots will be sufficiently near for the application of this method of approximation, if the three last indices (p. 332) be 0, 0, 1 . If this be not the case, the interval between a and h must be further subdivided until this last condition is satisfied. Under such circumstances there will be no root of the equa- tions/' {x) — and/^' {x) = 0, included between a and h : and if we suppose y — f ix) to be the equation of a paraboHc curve CAB, where O a=- a,Oh =^ h,am =f{a), b n ^f{b), then there will be no point of inflection between a and b, and no tan- gent parallel to the axis. The analytical conditions above men- tioned would show that /(«) and f{b) must necessarily have difi'erent signs. S. If we suppose b to represent the superior limit of the root (a), then the Newtonian approximation gives us the new su- perior limit b' = b — jrjj\ ; a new inferior limit will be found to be a = « frnl '• these limits are still superior and inferior limits of the root a, and are both of them nearer to it than the primitive limits b and a. If the same operation be repeated by replacing b and a in f{b) and/(«) by b' and a', nearer limits will be obtained, and it is obvious that the same process may be repeated as often as may be thought necessary. And in as much as we obtain both the inferior and superior limits corresponding to each operation, the difference between them will always be greater than the error of each approximation. If we refer to the above figure, and suppose n b' to be a tangent to the curve at n, and a' m to be drawn parallel to n b', then b b' = f'{by ^^^ « «' = f^Ty since/' {b) = tan w 6' 6 = tan m a a. It follows, therefore, that REPORT ON CERTAIN BRANCHES OF ANALYSIS. 339 O h' and O a' are the new limits h' and d : and if ordinates h' n' and a' m' be drawn to the curve, and n' b" be drawn a tangent, and m' a" parallel to n' b", then O b" and O a" will be the new values b" and a" of b' and a'. The progress of the approxima- tion, upon the continued repetition of this process, will now be sufficiently manifest. 3. If we consider the different arrangements of the signs of f (x), f (x), f{x), in the transition from the inferior limit a to the superior limit b, they will be found to be the following, it being kept in mind that the sign oi f{x) alone changes from + to — , or conversely. fix) fix) fix) + + - (2){ (3){ (4){ + + + a — — + b _ _ _ + - + + - - a - + — — + + In the first two cases, the formulae of approximation are b — >, ,,, and a — ^4-Wj and commence therefore with the su- perior limit. In the last two cases, the formulas of approxima- tion are a — 4rW and b — ^-7-Kjandcommence therefore with the inferior limit. In other words, that limit must in all cases be selected which gives the same sign to /" (x) andy (^), whe- ther + or — . The construction of the portions of the corre- sponding parabolic curves included between a and b in these several cases, will at once make manifest the reason of the selec- tion of the superior or inferior limit and likewise the progress of the approximation itself*. • If, in the figure p. 338, we join the extremities m and n of the ordinates a m and 6 « by the chord m N n, which cuts the axis of x in the point N, we shall cjr^TvT f{a){b — a) , f{h){h—a) ... . find O N= a — •' .)' ^ ., { = 6 — %)' „/ ( » "^^^^^ g'ves a new ap- proximate inferior limit in the first two cases considered in the text, and a new superior limit in the last two. Other constructions are noticed by Fourier, which give similar results. In the M^moires de I'Academie Royale de Bruxelles for 1 826, there is a memoir on the resolution of numerical equations by Dandelin, in which the analytical conditions which must be satisfied bvthe limit, towards which the z 2 •1 'Q 340 THIRD REPORT — 1833. 4. In tlie application of these rules some precautions are oc- casionally necessary. Thus, if/" (x) a.ndf{x) have a common measure 9 (or), and if a root (a) of ip (x) = be included between a and b, then there is a point of inflection of the parabolic arc between a and b at the point of its intersection with the axis. Under such circumstances, the method of approximation must be applied to the equation f (x) = 0, and not to tlie primitive equationy(a:') = 0, for the purpose of determining the value of «. Again, if there exists a common measure of^"' {x) andf{x), which becomes equal to zero, for a value of x between a and b, then there ai-e two or more equal roots of f{x) = in that interval, and the final succession of indices is no longer 0, 0, 1. Other precautions connected with the subdivision of the interval b — a are sometimes required, which the limits of this Report will not allow me to notice in detail. It remains to add a few remarks upon the rapidity of the ap- proximation, and upon the means by which it may be ascer- tained. If we express the primary and secondary intervals b — a and b' — «' by i and i', it may be very easily proved that 2f'{b) ' wherey" {a. . .b) denotes some value whichy" (x) assumes when we substitute for x a quantity between a and b : and if we form the quotient (C) which arises from dividing the greatest value of/" («) and/" (6)* by the least value of 2f' («) and 2/' (6), and suppose k the order of the greatest articulate or subarticu- approximation in Newton's method must be made, are established by a com- bination of analytical and geometrical considerations, and in which also the new limits h' and a' are respectively found by what he terms the rule of tan- gents in one case, and by the rule of chords in the other. Tlie first is the subtraction of the subtangent b b' or ^ from b, as involved in the or- dinary Newtonian approximation when the proper limit is selected. The second is the determination of the value of O N, or a — f\V. ~f (\ > °^ f (b\ ~ f( \ ' ^^ ^^^ method taught at the beginning of this Note. It is evident that these conclusions involve all that is important in Fourier's re- searches upon this part of the subject. This memoir of M. Dandelin, which contains a very full and a very clear exposition of the whole theory of the Newtonian method of approximation, preceded by five years the publication of M. Fourier's work. • Since no root of/'" {x) = is included between a and b, it follows that either/" (a) or /" (b) will be the greatest value of /" (n . . .h): the same remark applies likewise to/' (a) and/' {l). REPORT ON CERTAIN BRANCHES OF ANALYSIS. 341 late number * immediately greater than this quotient, and n the order of the articulate or subarticulate number which is not less than the difference of the limits b — a, then if we divide f{b) by/' {b), and continue the operation as far as the {2n + kf^ decimal, and increase the last digit by 1, the quotient which arises being subtracted from or added to, b, according as f{b) and/' (b) have the same or different signs, will give a result which will differ from the true value of the root by a quantity less than ( tt^ ) • And if the same operations be repeated, forming successively new limits by means of the results thus obtained, we shall obtain a series of limits which are correct as far as the {4>n + 3 kj^, the {8n + 7 kf^, &c., decimal place f. The processes of approximation which have been described above, as well as those which belong to all other methods, re- quire divisions and other operations with numbers which are sometimes beyond the reach of logarithmic tables, and which it is extremely important to abbreviate as much as possible, con- sistently with the determination of the accurate digits of the results which are required to be found. Such processes were taught by Oughtred and other algebraists of the seventeenth century, but both their theory and applications have been greatly and, perhaps, undeservedly, neglected in later times. The consideration, however, of such methods has been partially revived by Fourier and some other writers, the first of whom has given examples of what he terms ordinate division {division ordonn^e,) the principle of which is to conduct the division by the employment of a small number of the first digits of the divisor only, and to correct the successive remainders, augmented by the successive digits of the original dividend, in such a manner as to bring into operation the successive digits of the divisor when they are required for the determination of the correct digit of the quotient, and not before. Such processes, however, are incapable of being briefly described, and we can only refer to the original work| for the developement of the rule and for ex- amples of its application. * An articulate number is one of the series 1,10, 200, 7000, &c., where the first digit is followed by zeros only. A subarticulate number is one of the series '1, '02, '003, &c., and the number which designates the place of the first significant digit is supposed to be negative, t The course of the approximation, in order to be perfectly regular and rapid, would require that 2 n + h should be greater than n, or that n should be greater than — ^, a circumstance which might occur if A: or w was negative. In such a case it will be necessary, or rather expedient, to subdivide the in- terval b — a, until the difference of the two limits does not exceed ( — - ) , where « is equal to, or greater than, 1 — k. I Analyse des Equations determinecs, livr. ii. p. 188. 343 THIRD UEPOIIT 1833. Similar processes, also, have been investigated and applied with remarkable ingenuity and success by Mr. Holdred *, Mr. Horner f, and Mr. Nicholson J. The first of these writers, a mathematician in humble life, vpho had formed his taste upon the study of the older algebraical writers of this country, gave very ingenious rules for finding the roots of numerical equa- tions. The method proposed by Mr. Horner was founded upon much more profound views of analysis and of the relation which exists between the processes of algebra and arithmetic, and he has not only succeeded in making a very near approximation to the true principles upon which the limits of the roots of numerical equations are assigned, but by considering the rules for extract- ing the roots of numbers and of aflPected numerical equations as founded upon common principles, he has reduced the rules for these purposes to a form which admits of very rapid and ef- fective, though not perhaps of very easy, application. Mr. Ni- cholson, by a combination of the methods of Mr. Holdred and Mr. Horner, has greatly simplified them both, and reduced them to the form of practical rules, which are not much more compli- cated than those which are commonly given for the extraction of the cube and higher roots of numbers. The Newtonian method of approximation, which we have hitherto considered, may be termed linear, in as much as the equations of a straight line combined with the general equation of the parabolic curve are competent to express all the circum- stances which characterize it. But methods of approximation of higher orders than the first, involving the second or higher powers of the unknown quantity to be determined, have likewise been considered by Fourier and other writers. That of the se- cond order, viewed with reference to the properties of curve lines, may be said to result from the contact of arcs of a conical parabola. The superior and inferior limits, thus determined, converge with great rapidity, the error corresponding to each operation being the product of a constant factor with the cube of the preceding error. Such methods, however, if viewed with reference to the facility of their practical applications, are incom- parably less useful than those which are founded upon linear approximations ; but there is much which is instrvictive in their theory, and particularly as furnishing the means of determining immediately the nature of two roots of an equation included in a given interval, which the application of the methods for the * This method is particularly noticed in Mr. Nicholson's Essay on Involu. tion and Evolution. I have never seen the original tract published by Mr. Holdred. f Philosophical Transactions for 1819. I Essay oti Involution and Evolution. 1820. REPORT ON CERTAIN BRANCHES OF ANALYSIS. 343 separation of the roots which we have previously described may have left in the first instance uncertain. We refer to the end of the second book of Fourier's Analyse des Equations determin^es, for a very complete examination of the theory of such approximations*. It has been a question agitated on more than one occasion, whether the tests of the reality of the roots of equations of finite dimensions which De Gua established, or rather the principles of the much more general theorem of Fourier, were applicable likewise to transcendental equations. In a discussion of the transcendental equation 2/ = 1 - ^ + 22 - WTS'^ "*■ ^~7WT¥ ~ ^^■' which presents itself in the expression of the law of propagation of heat in a solid cylinder f of infinite length, Fourier ventured to apply the principles in question to show that all its roots were real ; but M. Poisson J has disputed the propriety of such an application, both in this case and in others : thus, if we suppose X = e'^— be"'', we shall find • The rule for the determination of the nature of two roots included in a given interval, which is given in page 333, is merely the expression of a con- sequence of the application of the method of linear approximation to the di- stinction of those roots ; and whatever difficulties in certain extreme cases may attend the successful application of that rule, will necessarily present themselves likewise in the application of the linear approximation under the same circumstances. This character, however, is not confined to the Newto- nian or linear method of approximation. If the interval of the roots be deter- mined, by the application of Fourier's theorem of the succession of signs of the original function X and its derivatives, so that no more than two roots may be said to exist in that interval, whose nature is unknown, whether real or imaginary, then the application of the method of continued fractions, as well as of other equivalent modes of approximation, will be competent to de- termine the values of those roots when real, and their nature, when imaginary. Such, at least, is the assertion of Fourier, who refers to the third book of his work on equations for its demonstration. It is unfortunate, however, that only two books of this work, which is full of such remarkable researches upon the theory of equations, were fully prepared for publication at the time of his death. Our knovi^ledge of the contents of the other five books, which were left unfinished, is derived from an Expose Synoptique prefixed to those which are published, and which contains a general review and analysis of their prin- cipal contents. It is to be hoped, however, that the materials which he has left behind him will be found to be sufficient at least for their partial, if not for their complete restoration. t Theorie de la Chaleiir, p. 372. t Journal de I' Ecole Poly technique, cahier xix. p. 381; Memoires de I'ln- st'Uut, torn. ix. p. 92. 344' THIRD RErORT — 1833 d'+'X d''+^ X dx = e' — b a" e"*, = e*' — 6a''+' e"'. .71+2 = e'' — b a"+2 e"'^, where w is any whole number, or zero. If we now suppose </"+^X _ and eliminate, by means of this equation, e", we shall get </»X dx"" ~ d»+^X and therefore d'^X d"+^X dx" ■ c?a;"+2 : - 6 (1 - a) a» e"^, = 6 (1 -a)a»+'e«*, = - J2(l _^)3^2»+lg2«^^ a quantity which is negative for every real value of x. The conclusion which should be drawn, in conformity with Fourier's principles, is, that all the roots of the equation e^ — be"'' = are real, as well as those of its successive derivatives; whilst the fact is, that each of those equations has one real root, and an infinite number of imaginary roots, which are included un- der the formula _ log 6 a" + 2 e TT '/^A X — ^ ■ In reply to this objection, it has been urged by Fourier that Poisson has not very accurately stated the terms of the propo- sition in question as applicable to such a case *, and also that he has neglected to take into consideration all the roots of the equation. For if we suppose that the substitution of two limits a and b, in a function f (x) and its derivatives, gives results which present the same succession of signs between y*("+'') (x) andy^"' (x), then those extreme derivative functions, and those * This inaccuracy of statement is rather chargeable upon Fourier himself than upon Poisson, who has certainly failed to notice the necessary limitation of this proposition upon the occasion which gave rise to its application in page 373 of the T/ieorie de la Chaleur. REPOKT ON CERTAIN BKANCHES OF ANALYSIS. 345 also which are included between them, when considered as equations, will contain the same number of roots, or none, be- tween those limits. This proposition is true, whether the num- ber of derivative functions be finite, as in the case of algebraical equations, or infinite, as in the case of transcendental equations. In the first case, however, it admits of absolute application, in consequence of our arriving at a final derivative, from which the comparison of the signs of the two series of results com- mences. In the second case we can draw no conclusion, in the absence of any difference in the signs of the series of results, in the transition from one derivative function to another, with respect to the number of roots of any of those functions which are included betw'een the given limits: thus, \i f{x) = sin x, we shall have the same series of signs of sin x and of its deri- vatives, however far continued, upon the substitution of the limits a and a + 2 ■n-, although it is manifest that there are two real roots of sin ^ = between those limits. The general pro- position, therefore, will, in such a case, authorize us in con- cluding merely that whatever number of roots the equation sin a; = includes between the limits a and « + 2 tt, will be possessed likewise by all its derivative equations between the same limits *. There is another point of view, likewise, in which the objec- tion advanced by Poisson may be considered as not altogether applicable to the example which he puts forward. In considering the roots of the derivative functions , „ , ^ — -p, -r — r^ , he has not included those of the factor e'^, which those functions 1 + — ) = 0, it follows that there are an infinite number of equal roots (where a: = — oo ) of e*' = 0, which equally reduce three or any number of conse- cutive derivative functions to zero, and to which, therefore, the test of De Gua is no longer appHcable. It would follow, therefore, that the existence of imaginary roots in the equation X = is no longer contradictory to Fourier's proposition, even • If the transcendental function denoted by / (j;) be a determinate function, it will always be possible to assign an interval 3, such that the derivative function/" {x) = contains no root, or a determinate number of roots, be- tween a and a + S. If such an interval or succession of intervals can be de- termined for any one derivative function, such as /(") (.i), it will become a point of departure for the determination of the number and nature of the roots corresponding to the same interval or intervals for all the other derivative functions which form the superior or inferior terms of the series. In the case of algebraical functions, the point of departure is that derivative function which ie a constant quantity. 346 irilKi) niijfoRT — luutj. admitting the correctness of that form of it which Poisson ha^ assigned*. * If we transform e*' by replacing a; by pi, we shall get the expression c •* , which may be easily shown as above, and also by other means, to be equal to zero when x' is equal to zero, and equal to 1 when x' is equal to in- finity. Professor Hamilton of Dublin, in a paper in the Irish Transactions for 1830, 1 has quoted the expression e •* as possessing some very peculiar properties, which are inconsistent with the universality of a very commonly received principle of analysis. It is commonly assumed that " if a real function of a positive variable x approaches to zero with the variable, and vanishes along with it, then that function can be developed in a real series of the form A a;" + B a^ + C ;r'' + &c. (1.) where ct, /3, y, &c., are constant and positive. A, B, C, &c., constant, and all those coefficients different from zero : but if we put the equation under the form y.-«. e~^^ = A + B xP'-"' + C xy-''+ &c„ supposing 86 the least of the several indices a, /3, y, &c., then if x =: 0, we 1 -~2 . 1 shall find x~'^ e *'=OorA equal to zero ; for if we replace — by y, we shall get 1 i. _i. =a;«e^' = y-«ey^ 4— a 6— a = 3,-« + /-« + y + _! + &c., ^ " ^1.2 1.2.3 all whose terms are positive, and which, when a;=: or y = oo , will necessarily become equal to infijiity : it follows, therefore, that the function e * is not ca- pable of developement in a series of the assumed form (1.), The same ex- pression, as has been remarked by Professor Hamilton, has been noticed by Cauchy as an example of the vanishing of a function and of all its differential coefficients, for a particular value of the variable, without the function va- nishing for other values of the variable, thus forming an exception to another principle generally received in analysis. In his Lemons snr le Calcul Infinite- simal, Cauchy has produced this last anomaly as a sufficient reason for not founding the principles of the differential calculus upon the developement of functions, as effected by or exhibited in, the series of Taylor. It is possible that more enlarged views of the analytical relations of zero and ivfinity, and of the interpretation of the circumstances of their occurrence, as well as of the principles and applications of Taylor's series, may enable us to explain these and other anomalies, and to show that they arise natu- rally and necessarily out of the very framework of analysis ; but it must be confessed that there are many other difficulties, which are yet unexplained, which are connected with the developement of e" when x is negative or ima- REPORT ON CERTAIN BRANCHES OF ANALYSIS. 347 Another method of approximation to the roots of equations by means of recurring series was proposed by Daniel Ber- nouUi *, and very extensively illustrated and applied by Euler f . If we write down m ai-bitrary numbers to form the first m terms of the series, and if we assume, for the scale of relation, the coefficients of an equation of m dimensions, and form by means of it and the assumed terms the other terms of the series which may be indefinitely continued, and if we also form a series of quotients by dividing each succeeding term (after the arbitrary terms) by that which precedes it, then the terms of the se- ries of quotients which thence arise, will converge continually towards the value of the greatest root of the equation ; and if we form the equation whose roots are the reciprocals of those of the original equation, and proceed in a similar manner, we shall obtain a series of quotients which will converge to the greatest root of this equation, whose reciprocal will be the least root of the original equation, considered without reference to its algebraical sign. Lagrange, in the 6th Note to his Resolution des Equations Numeriques, has analysed the principles of this method, and has shown that its success will depend upon the greatest real root, without reference to algebraical sign, being greater than the modulus of any of the imaginary roots. If this condition be not satisfied, the quotients will not approximate to the value of any root of the equation, a consequence which Euler had also pointed out. The recurring series which is formed by dividing the first derivative function /' {x) by / {x), which is equal to ginary. Some of these have been noticed in the note to p. 267, in connexion with our observations upon Mr. Graves's researches upon the theory of loga- rithms ; another is noticed by M. Clausen of Altona, in the second volume of Crelle's Journal, p. 287; it is stated as follows :— Since e^'"' '^"^ = 1, when ra is a whole number, we get g^ + ^ " '^ '^^ = e and therefore consequently e""*" '^' = 1, whenever w is a whole number, — a conclusion which M. Clausen characterizes as absurd. Its explanation involves no other 2»i sr \/ —\ . , difficulty than that which is included in the equation e =1, and must be sought for in the circumstances which accompany the transition from a function to its equivalent series, when a strict arithmetical equality does not exist between them. It must be confessed, however, that these difficulties arc of a very serious nature, and are in every way deserving of a more care- ful examination and analysis than they have hitherto received. * Comment. Acad. Petrop., vol. iii. t IntroducHo in Analysim Infinitorum, vol. i. cap. xvii. 348 THIRD REPORT — 1833. + &C., when a, /3, y, &c., are the roots of the equation, whether real or imaginary, has been shown by Lagrange to be the series fur- nished by this method which is most easily formed, and to be likewise that which converges most rapidly and certainly to a geometrical series in the case of equal roots. In every case the terms of the series of quotients are alternately greater and less than the root to be determined, and consequently furnish a mea- sure of the accuracy of the approximation. This method of approximation is generally less rapid and certain than those which have already been considered, and, as commonly stated, is extremely hmited in its application. It is true, as has been shown by Lagrange, that a knowledge of the limits of the roots would enable us to apply it to the deter- mination of all the real roots by means of a series of transformed equations equal to their number, such as is required in the New- tonian method of approximation, and also in that of Lagrange ; but under such circumstances, and with such data, it is more convenient and more expeditious to employ those methods in preference to the one which we are now considering. Fourier has shown in what manner this method may be ap- plied to determine all the roots of an equation, whether ima- ginary or real. Let us suppose a, b, c, d, e, &c., to represent the roots of the equation arranged in the order of magnitude,, the magnitudes of imaginary roots being estimated by the mag- nitudes of their moduli; and let A, B, C, D, E, &c., be the terms of the recurring series, whose quotients furnish the value of the greatest root, when that root is real. Form, in the se- cond place, a series whose terms are AD — BC, BE — CD, C F — D E, &c., which is also a recurring series, whose quo- tients may be easily shown to approximate to the sum of the two first roots a + b. Again, form a series whose terms are A C - B2, B D - C2, C E - W, &c., which is also a recurring series, whose successive quotients will approximate to the value of the greatest product a b. In a similar manner, we may de-' duce from the primitive recurring series three other recurring series, the terms of the convergent series formed by whose quo- tients will form, in the first series, the sum a + 6 + c of the three first roots ; in the second, the sum of their products two or two, ox ab + a c -\- be; and in the third their continued product a b c : and similarly for four or a greater number of roots. If, therefore, we suppose the first root a to be imaginary, the first series will give no result ; but the values of a + 6 and REPORT ON CERTAIN BRANCHES OF ANALYSIS. 349 of a b, which are given by the two first recurring series de- rived from the primitive recurring series, will enable us to de- termine their separate values : in both cases the series of quo- tients is convergent. If the third root be real, the third series of derived quotients is convergent ; if not, the fourth series will be so, and so on as far as we wish to proceed. These propositions have been merely announced by Fourier in his Introduction. The chapter of his work, which contains the demonstrations, has not yet been published. If the root of an equation be determined approximately, the equation may be depressed, and the general processes of solu- tion or of approximation may be applied to find the roots of the quotient of the division. Thus, in the equation ^3 _ 3 ^ _,. 2-0000001 = 0, one of the roots is very nearly equal to 1, if we divide the equation by a; — 1, and neglect the small remainder which re- sults from the division, we shall get the quotient x^ -X -2= {x -\){x + 2) = 0, whose roots are 1 and — 2 ; or we may suppose one of the roots to be 1*000 1, the second -9999, and the third —2; or we may suppose two of the roots to be imaginary, namely, 1 + -0001 V — \. All these roots are approximate values of the roots of the equation, which different processes, whether tentative or direct, may determine : and it is obvious that when two roots are equal, or nearly so, an inaccuracy of the approxi- mation to those roots which are employed in the depression of the primitive equation may convert real roots into imaginary, or conversely. Such consequences will never follow when the limits and nature of the roots are previously ascertained, and every root is determined independently of the rest ; but it is not very easy to prevent their occurrence when methods of ap- proximation are applied without any previous inquiries into the nature and limits of the roots, though the resulting conversion of imaginary roots into real, and of real roots into imaginary, may not deprive them of the character of true approximations to the values of the roots which are required to be determined. If the limits of the roots of an equation F ^c = be assigned, and if the Newtonian method of approximation be applied con- tinually to one of these limits a, we should obtain, for the value of the root, the series* «" ,^ .o «"' a-a'Ya+ ^ (F «)^ - ^--^ (F a)^ + &c. • Lagrange, li^sQlution des Equations Numeriques, Note xi. 350 THIRD REPORT — 1833. where «' =^, a" = - 1_ Fa a! F" a F" (F' of ~ (F' a)3 „, _ _ cH F" a 3 a' (F" of Wa 3 (F'^ af ~ (F «)4 + (F a)^ • This series was first assigned by Euler, and the observations which we have had occasion to make in the preceding pages upon hnear approximations will at once explain the circum- stances under which it may be safely applied : it cannot be viewed, however, in any other light than as the analytical ex- pression for the result of the application of such linear approxi- mations, repeated as many times as there are terms of the series succeeding the first. The celebrated theorem of Lagrange, which is so extensively used in the solution of the transcendental equations which pre- sent themselves in physical and plane astronomy, will enable us to assign, likewise, a series for the least root, or for any function of the least root of an equation in terms of its coeffi- cients. Mr. Murphy, in a very able memoir in the Transac- tions of the Philosophical Society of Cambridge for 1831, has shown the mode in which such series may be determined, by means of a very simple rule, which admits of very rapid and very extensive application. The rule is as follows : " To find the series for the least root of the equation ip {x) — 0, divide the equation by x, and take the Napierian logarithm of the quotient which arises ; then the coefficient of — with its sign changed is the series which expresses the least root re- quired." Thus, to find the series for the least root of the quadratic equation a;2 4. a a; + 6 = 0, find the coefficient, with its sign changed, of — in log ^^ (b\ 1 + X \, which is \ b b^ 4 63 Q.5 b^ 8.7.6 ^ , o \ KEPORT ON CERTAIN BRANCHES OF ANALYSIS, 351 and therefoi-e identical with that which arises from the deve- lopement oi — -^ + V\-r ~ ^)- If^be greater than -^, the roots of the equation are impossible, and the series becomes divergent, and gives no result- Any function / {x) of the least root of an equation <p (a;) = may be found " by subtracting from / (0) the coefficient of — m f (x) log ?-LJ," This more general theorem evidently includes the former. " The sum of any assigned number {ni) of roots of the equa- tion <p {x) = is equal to the coefficient, with its sign changed, of — m log ^-^. X ^ x"" The expression for the sum of m roots of an equation which is thus obtained gives the arithmetical value of the sum of the m least roots. In estimating the order of magnitude of such roots no regard is paid to their signs of affection. Mr. Murphy has shown in what manner the same general proposition which is employed in the deduction of the results just given may be applied to the investigation of some of the most general theorems which have been employed in analysis for the developement and transformation of functions. Amongst many others the following very remarkable theorem seems to merit particular notice. If x^^, Xci, ^3, . . ^TO be the m least roots of the equation (x - a)"' - hF (x) = 0, d--^{f(a)F{a)} = "^f^""^ + ^' l.2..{m-\)da--^ Jl_ d^"^-^ {f {a) {¥ {a)Y} *■ 1 .2' 1.2 {2m-\)da^'"-' "^ ^' If in this very general theorem we make w = 1, it becomes the theorem of Lagrange ; and if we make m equal to the di- mensions of the equation, or greater than ^y power of x in- volved in F {x), then it becomes the theorem which Cauchy has given, without demonstration, in the ninth volume of the new series of the Memoirs of the Institute, for the expression of the sum of the different values oi f{x), when x is succes- sively replaced by every root of the equation. The preceding conclusions, so very remarkable for their great generality, and for the very simple means employed in then, 352 THIRD REPORT — 1833. their derivation, will be sufficient to direct the attention of the reader to the other contents of this very original and valuable memoir. There ai-e some other most important departments of the general theory of equations which it was my intention to have noticed, and without which no report upon the present state and recent progress of algebra can be said to be complete. Amongst these may be particularly mentioned the theory of elimination and the solution of simultaneous equations, and also the theory of the solution of literal and of implicit equations. The very undue length, however, to which this Report has already extended, and the arrangements which have been made connected with the publication of this volume, compel me, though most reluctantly, to omit them. I venture to indulge a hope, however, that I may be allowed upon some future oc- casion to add a short supplemental Report upon this extensive department of analysis, in which I may be enabled to supply some of the numerous deficiencies of the preceding sketch. ERRATA IN THE FOREGOING REPORT. Page 197, line 21, dele not — 215. — 3, /o»-r(r) r(l»-) reorfr(r) r(l — r) — 215, — 11 from the bottom,*. jfor a;^:^- 2 + a; 1 read a:^ + 2 a; + 1 71 "' 71 — 221, — 15, /or cos i — read cos- 1 — e e cos-i a , , a — 226, — 2 from the bottom, for / , , ,, read cos-i , „ , ,„ Va^ + o2 Va^ + 62 — 234, — 7, for (0) read (0)» — 240, — 18, for In this last case (a — 6)" read If we suppose n to be an even whole number or a frac