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MEMOIRS PRESENTED TO THE CAMBRIDGE PHILOSOPHICAL SOCIETY, ON THE OCCASION OF THE JUBILEE OF SIR GEORGE GABRIEL STOKES, BART. HON. LL.D., HON. SC.D. LUCASIAN PROFESSOR CAMBRIDGE AT THE UNIVERSITY PRESS I900 THESE Memoirs are also issued as Volume XVIII. of the Transactions of the Cambridge Philosophical Society. IN HONOREM GEORGII GABRIELIS STOKES PHYSICAM ET MATHEMATICAM APVD CANTABRIGIENSES IAM QVINQVAGINTA ANNOS PROFITENTIS CONTENTS. PAGE Order of Proceedings at the formal celebration by the University of Cambridge of the Jubilee of Sir GEORGE GABRIEL STOKES, Bart., Lucasian Professor 1849-1899 vii The Rede Lecture: La théorie des ondes lumineuses: son influence sur la physique m oderne. By A LFRED CORNU.................................................................... xvii I. On the analytical representation of a uniform branch of a monogenic function. By G. MITTAG-LEFFLER................................................................... 1 II. Application oj the Partition Analysis to the study of the properties of any system of Consecutive Integers. By Major P. A. MACMAHON, R.A., D.Sc., F.R.S., Hon. M em. C.P.S...................................................... 12 III. On the Integrals of Systems of Differential Equations. By A. R. FORSYTIH, Sc.D., F.R.S., Sadierian Professor of Pure Mathematics......................................... 35 IV. Ueber die Bedeutung der Constante b des van der Waals'schen Gesetzes. Von L. BOLTZMANN und Dr M ACHE, in W ien........................................................ 91 V. On the Solution of a Pair of Simultaneous Linear Differential Equations, which occur in the Lunar Theory. By ERNEST W. BROWN, Sc.D., F.R.S., Professor of Mathematics at Haverford College, Philadelphia.......................................... 94 VI.. The Periodogram of iIagnetic Declination as obtained from the records of the Greenwich Observatory during the years 1871-1895 (Plates I, II). By ARTHUR SCIIUSTER, F.R.S., Professor of Physics at the Owens College, Manchester......... 107 VII. Experiments on the Oscillatory Discharge of an Air Condenser, with a Determination of "v." By OLIVER J. LODGS, D.Sc., F.R.S., Professor of Experimental Physics, University College, Liverpool, and R. T. GLAZEBROOK, M.A., F.R.S., Director of the National Physical Laboratory............................................. 136 VIII. The Geometry of Kepler and Newton. By Dr C. TAYLOR, Master of St John's C ollege........................................197 IX. Sur les Groupes Continus. Par' H. POINCARÉ................................................ 220 X. Contact Transformations and Optics. By E. O. LOVETT.................................... 256 XI. On a Class of Groups of Finite Order. By W. BURNSIDE, M.A., F.R.S., Professor of Mathematics, Royal Naval College, Greenwich.......................................... 269 XII. On Green's Fuinction jor a Circular Disc, with applications to Electrostatic Problems. By E. W. HOBsoN, Sc.D., F.R.S., Fellow of Christ's College........................... 277 VOL. XVIII. b vi CONTENTS. PAGE XIII. Demonstration of Green's Formula for Electric Density near the Vertex of a Riight Cone. By H. M. MACDONALD, M.A., Fellow of Clare College............... 292 XIV. On the Effects of Dilution, Temperature, and other circumstances, on the Absorption Spectra of Solutions of Didymium and Erbium Salts (Plates 3-23). By G. D. LIVEING, M.A., F.R.S., Professor of Chemistry................................. 298 XV. The Echelon Spectroscope. By A. A. MICHELSON............................................ 316 XVI. On Minimal Surfaces. By H. W. RICHMOND, M. A., Fellow of King's College, Cambridge........................................................................... 324 XVII. On Quartic Surfaces which admit of Integrals of the first kind of Total Differentials. By ARTHUR BERRY, M.A., Fellow of King's College, Cambridge' 333 XVIII. An Electromagnetic Illustration of the Theory of Selective Absorption of Light by a Gas. By HORACE LAMB, M.A., F.R.S., Professor of Mathematics at the Owens C ollege, M anchester................................................................................ 348 XIX. The Propagation of' Waves of Elastic Displacement along a Helical Wire. By A. E. H. LOVE, M.A., F.R.S., Sedleian Professor of Natural Philosophy in the U niversity of Oxford..................................................................... 364 XX. On the construction of a 3lodel showing the 27 lines on a Cubic Su:efiace (Plates XXIV, XXV). By H. M. TAYLOR, M.A., F.R.S., Fellow of Trinity College... 375 XXI. On the Dynamics of a System of Electrons or Ions: and on the Influence oj a Llfagnetic Field on Optical Pheno2mena. By J. LARMOR, M.A., F.R.S., Fellow of St John's College.............................................................................. 380 XXII. On the Theory of Functions of several Complex Variables. By H. F. BAKER, M.A., F.R.S., Fellow of St John's College.................................................. 408 I n d ex................................................................................................ 4 4 5 PLATES. Frontispiece. Portrait of Sir George Gabriel Stokes, Bart. I, 1I, illustrating Professor Schuster's paper, page 107. 3-23 (contained in 14 leaves) illustrating Professor Liveing's paper, page 298. XXIV, XXV, illustrating Mr H. M. Taylor's paper, page 375. ERRATA. Page 210, line 5 from the top, omit the equation AB. CD = kBC. DA. 332. In regard to ~~ 7-11, reference ought to have been made to the results of Lie, Math. Ann. xiv. pp. 373-378, or to Darboux ~ 325, from which the special type of surfaces considered might also be derived. ORDER OF PROCEEDINGS AT THE CELEBRATION OF THE JUBILEE OF SIR GEORGE GABRIEL STOKES, BART. vi1 a ORDER OF PROCEEDINGS AT THE FORMAL CELEBRATION BY THE UNIVERSITY OF CAMBRIDGE OF THE JUBILEE OF SIR GEORGE GABRIEL STOKES, BART., M.A., HON. LL.D., HON. Sc.D. Thursday, 1 June, 1899. In the evening the Vice-Chancellor was present at a Conversazione in the Fitzwilliam Museun. About one thousand guests accepted the invitation of the University. Lord Kelvin, on behalf of the subscribers to the marble busts of Sir G. G. Stokes by Hamo Thornycroft, R.A., offered one of them to the University, and the other to Pembroke College. The former was accepted on behalf of the University by the Vice-Chancellor, the latter on behalf of the College by the Rev. C. H. Prior. M.A. Friday, 2 June, 1899. A Congregation was held this day at 11 A.M. Sir G. G. Stokes sat on the right hand of the Vice-Chancellor. The Delegates sent by Universities, Academies, Colleges and Societies were presented to the Vice-Chancellor in the chronological order of the Institutions represented. The names of the Institutions and of the Delegates were announced by the Registrary, as follows: University of Paris Professor Gaston Darboux, Doyen de la Faculté des Sciences. University of Oxford Sir William Reynell Anson, Bart., M.P., and Robert Edward Baynes, M.A., Lee's Reader in Physics. University of Heidelberg Professor Quincke. b2 viii ORDER OF PROCEEDINGS. University of St Andrews P. R. Scott Lang, M.A., Regius Professor of Mathematics. University of Glasgow Very Rev. Robert Herbert Story, D.D., Principal, and Lord Kelvin, M.A., Hon. LL.D., G C.V.O. Academies of Upsala, Copenhagen, Helsingfors Professor Mittag-Leffler. University of Aberdeen Sir William Duguid Geddes, LL.D., Principal. University of Edinburgh George Chrystal, M.A., Professor of Mathematics, and G. F. Armstrong, M.A., Professor of Engineering. University of Dublin George Salmon, D.D., Provost, and Benjamin Williamson, M.A., D.Sc. Royal Society Lord Lister, Hon. LL.D., President. Alfred Bray Kempe, M.A., Treasurer. Michael Foster, M.A., Professor of Physiology. Arthur William Riicker, M.A. Secretaries. (Oxon.), Professor ofPhysics, Royal College of Science. Académie des Sciences, Paris Professor Becquerel. University of Pennsylvania } Professor G. F. Barker, Vice-President. American Philosophical Society Pr Gesellschaft der Wissenschaften zu Gottingen Edward Riecke, Professor of Physics. New York, Columbia University Robert S. Woodward, Ph.D., Professor of Mechanics and Mathenmatical Physics, Dean of the Faculty of Pure Science. Princeton University, New Jersey Professor Edgar Odele Lovett. Imperial Academy of Military Medicine, St Professor Egoroff. Petersburg Bataafsch Genootschap voor Physika, Rotter- Dr Elie van Rijckevorsel. dam Académie Royale des Sciences des Lettres et Professor Alphonse Rénard, Professor G. Van des Beaux Arts de Belgique der Mensbrugghe. Manchester Literary and Philosophical Society Reginald Felix Gwyther, M.A., Secretary. Royal Irish Academy Earl of Rosse, K.P., President, George F. FitzGerald, M.A., Professor of Natural and Experimental Philosophy, Trinity College, Dublin. Royal Society of Edinburgh Lord Kelvin, M.A., Hon. LL.D., President, and Sir John Murray, K.C.B., Hon. Sc.D. St Edmund's College, Ware Right Rev. J. L. Patterson, M.A. (Oxon.), Bishop of Emmaus. cole Polytechnique, Paris Professor Cornu and Professor Becquerel. cole Normale Supérieure, Paris Professor Borel. Royal Institution Sir J. Crichton Browne, M.D. (Edinb.), Treasurer. ORDER OF PROCEEDINGS. ix Philosophical Society of Glasgow Lord Blythswood. University of Bonn Professor Kayser. Cambridge Philosophical Society Joseph Larmor, M.A., President. Royal Astronomical Society George Howard Darwin, M.A., Plumican Professor of Astronomy, President. University of Toronto R. Ramsay Wright, M.A., B.Sc., Professor of Biology. St David's College, Lampeter A. W. Scott, M.A., Trinity College (Dubl.), Professor of Physical Science and Mathematics. Institution of Civil Engineers William Henry Preece, -C.B., President. King's College, London Archibald Robertson, D.D. (Durham), Principal. British Association Sir William Crookes, President. University of Durham Ralph Allen Sampson, M.A., Professor of Mathematics. Solar Physics Committee, Science and Art Prof. G. H. Darwin. Department Cambridge Ray Club Alfred Newton, M.A., Professor of Zoology and Comparative Anatomy. University of London Sir H. Roscoe. London Chemical Society Dr T. E. Thorpe. Queen's College, Belfast Thomas Hamilton, D.D., President. Queen's College, Galway Alexander Anderson, M.A., President. University of Sydney Philip Sydney Jones, M.D. (Lond.), Fellow of the Senate of the University of Sydney. Royal College of Science, London John Wesley Judd, C.B., LL.D., Dean; W. A. Tilden, Professor of Chemistry. The Owens College, Manchester Alfred Hopkinson, Q.C., M.A., Principal. University of Bombay Dr H. M. Birdwood, M.A., C.S.I. University of Madras Hon. H. H. Shephard, M.A., Puisne Judge of the High Court of Madras. London Mathematical Society Lord Kelvin, M.A., Hon. LL.D., President. University of New Zealand Edward John Routh, M.A., Sc.D. Durham College of Science, Newcastle-on- Henry Palin Gurney, M.A., Principal. Tyne University of Adelaide Horace Lamb, M.A., Professor of Mathematics in Owens College, Manchester. University College of Wales, Aberystwyth Robert Davies Roberts, M.A. Physical Society of Paris M. Henri Deslandres. Yorkshire College, Leeds Leonard J. Rogers, M.A., Professor of Mathematics. Physical Society of London Oliver J. Lodge, D.Sc., Professor of Physics, University College, Liverpool, President. Mason College, Birmingham John Henry Poynting, Sc.D., Professor of Physics. x ORDER OF PROCEEDINGS. Johns Hopkins University, Baltimore Simon Newcomb, Hon. Sc.D., LL.D., Professor of Mathematics and Astronomy; and Professor Ames. Firth College, Sheffield William Mitchinson Hicks, Sc.D., Principal. University College, Bristol Frank R. Barrell, M.A., Professor of Mathematics. City and Guilds of London Institute for Sir Frederick Abel, K.C.B. Advancement of Technical Education University College, Dundee John Yule Mackay, Principal. University College, Nottingham John Elliotson Symes, M.A., Principal. Victoria University Nathan Bodington, Litt.D., Vice-Chancellor. Royal University of Ireland Right Rev. Monsignor Molloy, D.D., D.Sc. Royal College of Science for Ireland Walter Noel Hartley, Professor of Chemistry. University College, Liverpool Richard Tetley Glazebrook, M.A., Principal. University of the Punjab Sir Charles Arthur Roe, M.A., late First Judge of the Chief Court, Punjab; late Vice-Chancellor of the University. University College of South Wales, Cardiff H. W. Lloyd Tanner, M.A. (Oxon.), Professor of Mathematics. University College of North Wales, Bangor Henry R. Reichel, M.A. (Oxon.), Principal. Royal Indian Engineering College, Coopers Prof. A. Lodge, M.A. (Oxon.), Professor of Hill Miathematics. University of Allahabad G. Thibaut, Ph.D., Principal of the Muir Central College, Allahabad. University of Wales J. Viriamu Jones, M.A., Vice-Chancellor. The following Institutions sent Addresses: Yale University. American Academy of Arts and Sciences, Boston. Royal Academy of the Netherlands. Imperial University of Tokio. Reale Accademia dei Lincei di Roma. A telegram was received from the Hungarian Academy, and a letter from Professor Pascal, in the name of himself and the University of Pavia. At 1.30 P.M. the Vice-Chancellor gave a luncheon at Downing College, at which the Chancellor, Sir G. G. Stokes, the Delegates, the invited guests of the University, and many members of the Senate were present. ORDER OF PROCEEDINGS. xi A second Congregation was held at 2.45 P.M. A Procession was formed at the Library at 2.35 P.M. in the following order: The Esquire Bedells SIR G. G. STOKES THE CHANCELLOR The Recipients of the Degree of Doctor in Science, honors cttusa: 1. Marie Alfred Cornu 2. Jean Gaston Darboux 3. Alfred Abraham Michelson 4. Magnus Gustaf Mittag-Leffler 5. Georg Hermann Quincke 6. Woldemar Voigt The Lord Lieutenant The Vice-Chancellor accompanied by the Registrary The Representatives in Parliament The Heads of Colleges Doctors in Divinity Doctors in Law Doctors in Medicine Doctors in Science and Letters Doctors in Music The Public Orator The Librarian Professors Members of the Council of the Senate The Proctors The Procession passed round Senate House Yard, and entered the Senate House by the South Door. Xii ORDER OF PROCEEDINGS. The following Address, as approved by the Senate, and sealed with the University seal, was read by the Public Orator, and presented to Sir George Gabriel Stokes by the Chancellor. Baronetto insigni Georgio Gabrieli Stokes Iuris et Scientiarum Doctori Regiae Societatis quondam Praesidi Scientiae Mathematicae per annos quinquaginta inter Cantabriqienses Professori S. P. D. Universitas Cantabrigiensis. QUOD per annos quinquaginta inter nosmet ipsos Professoris munus tam praeclare ornavisti, et tibi, vir venerabilis, et nobis ipsis vehementer gratulamur. Iuvat vitam tam longam, tam serenam, tot studiorum fructibus maturis felicem, tot tantisque honoribus illustrem, tanta morum modestia et benignitate insignem, hodie paulisper contemplari. Anno eodem, quo Regina nostra Victoria insularum nostraruin solio et sceptro potita est, ipse eodem aetatis anno Newtoni nostri Universitatem iuvenis petisti, Newtoni cathedram postea per decem lustra ornaturus, Newtoni exemplum et in Senatu Britannico et in Societate Regia ante oculos habiturus, Newtoni vestigia in scientiarum terminis proferendis pressurus et ingenii tanti imaginem etiam nostro in saeculo praesentem redditurus. Olim studiorum mathernaticorum e certamine laurea prima reportata, postea (ne plura commemoremus) primum aquae et immotae et turbatae rationes, quae hydrostatica et hydrodynamica nominantur, subtilissime examinasti; deinde vel aquae vel aëris fluctibus corporum motus paulatim tardatos minutissime perpendisti; lucis denique leges obscuras ingenii tui lumine luculenter illustrasti. Idem etiam scientiae mathematicae in puro quodam caelo diu vixisti, atque hominum e controversiis procul remotus, sapientiae quasi in templo quodam sereno per vitam totam securus habitasti. In posterum autem famam diuturnam tibi propterea praesertim auguramur, quod, in inventis tuis pervulgandis perquam cautus et consideratus,Y nihil praeproperum, nihil immaturum, nihil temporis cursu postea obsolefactum, sed omnia matura et perfecta, omnia omnibus numeris absoluta, protulisti. Talia propter merita non modo in insulis nostris doctrinae sedes septem te doctored. honoris causa nominavertunt, sed etiam exterae gentes honoribus eximiis certatim cumulaverunt. Hodie eodem doctoris titulo studiorum tuorum socios nonnullos exteris e gentibus ad nos advectos, et ipsorum et tuum in honorem, velut exenmpli causa, libenter ornamus. In perpetuum denique observantiae nostrae et reverentiae testimonium, in honorem alumni diu a nobis dilecti et ab aliis nomismate honorifico non uno donati, ipsi nomisma novum cudendum curavimus. In honore nostro novo in te primum conferendo, inter vitae ante actae gratulationes, tibi omnia prospera etiam in posterum exoptamius. Vale. Datumn in Senaculo mensis Iunii die secundo L-S A. S. MDCCCXCIX. ORDER OF PROCEEDINGS. xiii A Commemorative Gold Medal was presented to Sir G. G. Stokes by the Chancellor. Professor Cornu and Professor Becquerel presented the Arago Medal to Sir G. G. Stokes on behalf of the Academy of Sciences, Paris. The following degrees were conferred: Doctors in Science (honoris caused) Marie Alfred Cornu (Professor of Experimental Physics in the cole Polytechnique, Paris) Jean Gaston Darboux (Dean of the Faculty of Sciences in the University of France) Albert Abraham Michelson (Professor of Experimental Physics in the University of Chicago) Magnus Gustav Mittag-Leffler (Professor of Pure Mathematics, Stockholn) Georg Hermann Quincke (Professor of Experimental Physics in the University of Heidelberg) Woldemar Voigt (Professor of Mathematical Physics in the University of Gottingen) The Public Orator made the following speeches in presenting the several recipients of honorary degrees to the Chancellor. Primum vobis praesento artium plurimarum Scholae Parisiensis professorem, quem in hoc ipso loco die hesterno perspicuitate solita disserentem audivistis, virum non modo solis de lumine in partes suas solvendo, sed etiam orbis terrarum de mole metienda per annos plurimos praeclare meritum. Lucis in natura explicanda, quanta cum doctrinae elegantia, quanta cum experimentorum subtilitate, quam diu versatus est. Idem quam accurate velocitatem illam est dimensus, qua per aeris intervallum immensum lucis simulacra minutissima transvolitant, 'suppeditatur enim confestim lumine lumen, et quasi protelo stimulatur fulgere fulgur.' Lucis transmittendae in Xa/iraî7jcopta quam feliciter lampada a suis sibi traditam ipse etiam trans aequor Atlanticum alii tradidit. Duco ad vos ALFREDUM CORNU. VOL. XVIII. c xiv ORDER OF PROCEEDINGS. Sequitur deinceps vir insignis Nemausi natus, Parisiensium in Universitate illustri geometriam diu professus et scientiarum facultati toti praepositus. Peritis nota sunt quattuor illa volumina, in quibus superficierum rationem universam inclusit; etiam pluribus notum est, quantum patriae legatus deliberationibus illis profuerit, quae a Societate nostra Regia primum institutae, id potissimum spectant, ut omnibus e gentibus quicquid a scientiarum cultoribus conquiritur, indicis unius in thesaurum, gentium omnium ad fructum, in posterum conferatur. Incepto tanto talium virorum auxilio ad exitum perducto, inter omnes gentes ei qui rerum naturae praesertim scientiam excolunt, sine dubio vinculis artioribus inter sese coniungentur. Duco ad vos IOHANNEM GASTONUM DARBOUX. Trans aequor Atlanticum ad nos advectus est vir insignis, qui ea quae professor noster Lucasianus de aetheris immensi regione, in qua lux propagatur, orbis terrarum motu perturbata, olim praesagiebat, ipse experimentis exquisitis adhibitis penitus exploravit. Lucis explorandae in provincia is certe scientiarum inter lumina numeratur, qui olim fratrum nostrorum transmarinorum in classe non ignotus, lampade trans oceanum e Gallia sibi tradita feliciter accepta, etiam exteris gentibus subito affulsit, velocitatem immensam eleganter dimensus, qua ]ucis fluctus videntur (ut Lucretii verbis utar) 'per totum caeli spatium diffundere sese, perque volare mare ac terras, caelumque rigare.' Duco ad vos ALBERTUM ABRAHAM MICHELSON. Scandinavia ad nos misit scientiae mathematicae professorem illustrem, qui studiorum suorum velut e campo puro laudem plurimam victor reportavit. Idern Regis sui auspiciis, qui praemiis propositis magnum huic scientiae attulit adiumentum, etiam exterarum gentium ad communem fructum prope viginti per annos Acta illa Mathematica edidit, quae in his studiis quasi gentium omnium internuntium esse dixerim. Ipse Homerus (ut Pindari versus verbo uno tantum mutato proferam) a"yyeov o"Xov "'a 7Lràv iey[la-rav 7rparyuart 7ravTrb (épetv av feTat ical M a O7o-t s c àyyeXkia op0àâs. Duco ad vos MAGNUM GUSTAVUM1 MITTAG-LEFFLER. Universitatem Heidelbergensem abhinc annos quadraginta professorum par nobile spectroscopo invento in perpetuum illustravit. Adest inde discipulorum plurimorum in scientia physica praeceptor, qui et in instrumentis novis inveniendis sollertiam singularem et in eisdem adhibendis industriam indefessam praestitit. Ei qui in scientiae physicae ratione universa versati, viri huiusce inventis utuntur, etiam de sua scientia verum esse confitebuntur, quod de arte oratoria praesertim dixit Quintilianus:-' in omnibus fere minus valent praecepta quam experimenta.' Duco ad vos GEORGIUM HERMANNUM QUINCKE. ORDER OF PROCEEDINGS. xv Universitatem Goettingensem, a Rege nostro Hanoveriensi Georgio secundo conditam, vinculo non uno cum Universitate nostra coniunctar esse constat. Constat eandem etiam per annos prope quinquaginta Caroli Frederici Gaussii, scientiae mathemraticae et physicae professors celeberrimi, gloria esse illustratam, qui cum ingenio fecundissimno disserendi genus consummatum coniunxit. Iuvat inde professorem ad nos advectum excipere, qui scientiae eiusdein pulcherrirnam nactus provinciam, etiam lucem ipsam et crystalla ingenii sui lumine illustravit. Sex virorum insignium seriem consummavit hodie WOLDEMAR VOIGT. In the evening the CHANCELLOR presided at a dinner in the Hall of Trinity College (kindly placed at the disposal of the University by the Council of the College), at which Sir George Gabriel Stokes, the Delegates, and the invited guests of the University were entertained. JOHN WILLIS CLARK, Registrary. c2 LA THÉORIE DES ONDES LUMINEUSES: SON INFLUENCE SUR LA PHYSIQUE MODERNE*. PAR ALFRED CORNU, DE L'ACADEMIE DES SCIENCES ET DE LA SOCIÉTÉ ROYALE DE LONDRES, PROFESSEUR A L'ÉCOLE POLYTECHNIQUE. THE REDE LECTURE (1er JUIN 1899). Notre époque se distingue des âges précédents par une merveilleuse utilisation des forces naturelles; l'homme, cet être faible et sans défense, a su, par son génie, acquérir une puissance extraordinaire et plier à son service des agents subtils ou redoutables, dont ses ancêtres ignoraient même l'existence. Cet admirable accroissement de la puissance matérielle de l'homme dans les temps modernes est dû tout entier à l'étude patiente et approfondie des phénomènes de la Nature, à la connaissance précise des lois qui les régissent et à la savante combinaison de leurs effets. Mais ce qui est particulièrement instructif, c'est la disproportion qui existe entre le phénomène primitif et la grandeur des effets que l'industrie en a fait jaillir. Ainsi, ces formidables engins fondés sur l'électricité ou la vapeur ne dérivent ni de la foudre, ni des volcans; ils tirent leur origine de phénomènes presque imperceptibles qui seraient * En dehors de l'intérêt que présente un coup d'ceil se borne à décrire les faits, puis les résume dans un énoncé d'ensemble sur les progrès et l'influence de l'Optique, cette empirique, sans explications hypothétiques. Il se défend lecture offre les conclusions d'une étude approfondie du même de faire aucune théorie, quoique l'intervention des Traité d'Optique de Newton. On verra que la pensée du ondes excitées dans l'éther lui apparaisse comme fort progrand physicien a été singulièrement altérée par une sorte bable. De sorte que l'impression générale résultant de la de légende répandue dans les traités élémentaires où la lecture du Traité d'Optique, et surtout des "Questions" du théorie de l'émission est exposée. Pour rendre plus claire troisième livre, peut se résumer en disant que Newton, la théorie des accès, les commentateurs ont imaginé de loin d'être l'adversaire du système de Descartes, comme on matérialiser la molécule lumineuse, sous la forme d'une le représente généralement, est, au contraire, très favoflèche rotative se présentant alternativement par la pointe rable aux principes de ce système: frappé des ressources et par le travers. Ce mode d'exposition a contribué à qu'offrait l'hypothèse ondulatoire pour l'explication des faire croire que toute la théorie newtonienne de l'émission phénomènes lumineux, il l'aurait sans doute adoptée, si était renfermée dans cette image un peu enfantine; il n'en l'objection grave relative à la propagation rectiligne de la est rien. Nulle part, dans son Traité, Newton ne donne lumière, résolue seulement de nos jours par Fresnel, ne une représentation mécanique de la molécule lumineuse: il l'en avait détourné. xviii ALFRED CORNU, LA THÉORIE DES ONDES LUMINEUSES: demeurés éternellement cachés aux yeux du vulgaire, mais que des observateurs pénétrants ont su reconnaître et apprécier. Cette humble origine de la plupart des grandes découvertes dont l'humanité bénéficie montre bien que c'est l'esprit scientifique qui est adjourd'hui le grand ressort de la vie des nations et que c'est dans le progrès de la Science pure qu'il faut chercher le secret de la puissance croissante du monde moderne. De là une série de questions qui s'imposent à l'attention de tous. A quelle occasion le goût de la Philosophie naturelle, si chère aux philosophes de l'Antiquité, abandonnée pendant des siècles, a-t-il pu renaître et se développer? Quelles ont été les phases de son développement? Comment ont apparu ces notions nouvelles qui ont si profondément modifié nos idées sur le mécanisme des forces de la Nature? Enfin, quelle est la voie féconde qui, insensiblement, nous conduit à d'admirables généralisations, conformément au plan grandiose entrevu par les fondateurs de la Physique moderne? Telles sont les questions que je me propose, come physicien, d'examiner devant vous: c'est un sujet un peu abstrait, je dirai même un peu sévère; mais nul autre ne m'a paru plus digne d'attirer votre attention, à la fête que l'Université de Cambridge célèbre aujourd'hui, pour honorer le cinquantenaire du professorat de Sir George-Gabriel Stokes, qui, dans sa belle carrière, a précisément touché d'une main magistrale aux problèmes les plus profitables à l'avancement de la Philosophie naturelle. Ce sujet est d'autant mieux à sa place ici qu'en citant les noms des grands esprits qui ont le plus fait pour la Science, nous trouverons ceux qui honorent le plus l'Université de Cambridge, ses professeurs ou ses élèves, Sir Isaac Newton, Thomas Young, George Green, Sir George Airy, Lord Kelvin, Clerk Maxwell, Lord Rayleigh; et le souvenir de gloire qui se perpétue à travers les siècles jusqu'au temps présent rehaussera l'éclat de cette belle cérémonie. Cherchons donc, dans un rapide coup d'oeil sur la Renaissance scientifique, à reconnaître l'influence secrète, mais puissante, qui a été la force directrice de la Physique moderne. Je suis porté à penser que l'étude de la lumière, par l'attraction qu'elle a exercée sur les plus vigoureux esprits, a été l'une des causes les plus efficaces du retour des idées vers la Philosophie naturelle, et à considérer l'Optique comme ayant eu sur la marche des Sciences une influence dont on ne saurait exagérer la portée. Cette influence, déjà visible dès la création de la Philosophie expérimentale, par Galilée, a grandi dans de telles proportions qu'on prévoit aujourd'hui une immense synthèse des forces physiques, fondée sur les principes de la Théorie des ondes lumineuses. On se rende compte aisément de cette influence lorsqu'on songe que la voie par laquelle arrive à notre intelligence la connaissance du monde extérieur est la lumière. C'est, en effet, la vision qui nous fournit les notions les plus rapides et les plus complètes sur les objets qui nous entourent; nos autres sens, l'ouïe, le toucher, nous apportent aussi leur part d'instruction, mais la vue seule nous fournit une abondance d'informations simultanées, forme, éclat, couleur, qu'aucun des autres sens ne peut nous donner. SON INFLUENCE SUR LA PHYSIQUE MODERNE. xix Il n'est donc pas étonnant que la lumière, lien perpétuel entre notre personnalité et le monde extérieur, intervienne à chaque instant, par toutes les ressources de sa constitution intime, pour préciser l'observation des phénomènes naturels. Aussi chaque découverte relative à quelque propriété nouvelle de la lumière a-t-elle eu un retentissement immédiat sur les autres branches des connaissances humaines; souvent même, elle a déterminé la naissance d'une science nouvelle en apportant un nouveau moyen d'investigation d'une puissance' et d'une délicatesse inattendues. L'Optique est véritablement une science moderne; les anciens philosophes n'avaient pas soupçonné la complexité de ce qu'on appelle vulgairement la lumière: ils confondaient sous la même dénomination ce qui est personnel à l'homme et ce qui lui est extérieur. Ils avaient cependant aperçu une des propriétés caractéristiques du lien qui existe entre la source lumineuse et l'ceil qui perçoit l'impression: la lumière se meut en line droite. L'expérience vulgaire leur avait révélé cet axiome, en observant les traînées brillantes que le Soleil trace dans le ciel en perçant les nuées brumeuses ou en pénétrant dans un espace obscur. De là étaient résultées deux notions empiriques: la définition des rayons de lumière et celle de la ligne droite; la première devint la base de l'Optique; l'autre, la base de la Géométrie. Il ne nous reste presque rien des livres d'Optique des anciens; nous savons, toutefois, qu'ils connaissaient la réflexion des rayons lumineux sur les surfaces polies et l'explication des images formées par les miroirs. Il faut attendre bien des siècles, jusqu'à la Renaissance scientifique, pour rencontrer un nouveau progrès dans l'Optique; mais celui-là est considérable, il annonce l'ère nouvelle: c'est l'invention de la lunette astronomique. L'ère nouvelle commence à Galilée, Boyle et Descartes, les fondateurs de la Philosophie expérimentale; tous trois consacrent leur vie à méditer sur la nature de la lumière, des couleurs et des forces. Galilée jette les bases de la Mécanique, et, avec le télescope a réfraction, celles de l'Astronomie physique; Boyle perfectionne l'expérimentation; quant a Descartes, il embrasse d'une vue pénétrante l'ensemble de la Philosophie naturelle; il repousse toutes les causes occultes admises par les seholastiques; il pose en principe que tous les phénomènes sont gouvernés par les lois de la Mécanique. Dans son système du monde, la lumière joue un rôle prépondérant*; elle est produite par les ondulations excitées dans la matière subtile qui, suivant lui, remplit tout l'espace. Cette matière subtile (qui représente ce que nous appelons aujourd'hui l'éther), il la considère comme formée de particules en contact immédiat; elle constitue donc en même temps le véhicule des forces existant entre les corps matériels qui y sont plongés. On reconnaît là les fameux tourbillons de Descartes, tantôt admirés, tantôt bafoués aux siècles derniers, mais auxquels d'habiles géomètres contemporains ont rendu la justice qui leur est due. Quelle que soit l'opinion qu'on porte sur la rigueur des déductions du grand philosophe, on doit rester frappé de la hardiesse avec laquelle il affirme la liaison des grands problèmes cosmiques, et de la pénétration avec laquelle il annonce des solutions dont les générations actuelles s'approchent insensiblement. * Le Monde de IM. Descartes ozt le Traité de la Lumière. Paris, 1664. XX ALFRED CORNU, LA THÉORIE DES ONDES LUMINEUSES: Pour Descartes, le mécanisme de la lumière et celui de la gravitation sont inséparables; le siège des phénomènes qui leur correspondent est cette matière subtile qui remplit l'Univers et leur propagation doit s'effectuer par ondes autour des centres actifs. II Cette conception de la nature de la lumière heurtait les idées en faveur; elle souleva de vives oppositions. Depuis l'Antiquité, on avait coutume de se représenter les rayons lumineux comme la trajectoire de projectiles rapides lancés par la source radiante, leur choc sur les nerfs de l'ceil produisant la vision; leur rebondissement ou leur changement de vitesse, la réflexion ou la réfraction. La théorie cartésienne avait toutefois des aspects séduisants qui lui amenèrent des défenseurs: les ondes excitées à la surface des eaux tranquilles offrent une image si claire de la propagation d'un mouvement autour d'un centre d'ébranlement! D'autre part, n'est-ce pas par ondes que nous arrivent les impressions sonores? L'esprit éprouve donc une véritable satisfaction à penser que nos deux organes les plus précis et délicats, l'oeil et l'oreille, sont impressionnés par un mécanisme de même nature. Cependant, une grave différence subsiste; le son ne se meut pas nécessairement en ligne droite comme la lumière; il tourne les obstacles qu'on lui oppose et parcourt les routes les plus sinueuses presque sans s'affaiblir. Les physiciens se partagèrent alors en deux camps: les uns, partisans de l'émission, les autres, partisans des ondes. Comme chacun des deux systèmes se prétendait supérieur à l'autre, et l'était en effet sur quelques points, il fallait en appeler à d'autres phénomènes pour trancher entre eux. Le hasard des découvertes en amena plusieurs qui auraient dû décider en faveur de la théorie des ondes, ainsi qu'on le reconnut un siècle plus tard; mais les claires vérités n'apparaissent jamais sans un long labeur. Un compromis singulier s'établit entre les deux systèmes, à l'abri d'un nom illustre entre tous, et la victoire fut attribuée, pendant un siècle, à la théorie de l'émission; en voici l'étrange histoire: En 1661, un jeune élève plein d'ardeur et de pénétration entrait à Trinity College de Cambridge; il se nommait Isaac Newton; il avait déjà lu dans son village l'Optique de Kepler. A peine entré, tout en suivant les leçons d'Optique de Barrow, il étudie avec passion la Géométrie de Descartes; il achète sur ses économies un prisme pour étudier les couleurs et, entre temps, médite déjà longuement sur les causes de la gravité. Huit ans après, ses maîtres le trouvent digne de succéder à Barrow dans la chaire lucasienne, et il enseigne à son tour l'Optique. L'élève dépasse bientôt le maître et annonce une découverte capitale: La lumière blanche, qui semblait le type de la lumière pure, n'est pas homogène; elle est formée de rayons de diverses réfrangibilités. Et il le démontre par la célèbre expérience du spectre solaire, dans laquelle un rayon de lumière blanche est décomposé en une série de rayons colorés comme l'arc-en-ciel; chacune de ces couleurs est simple, car le prisme ne la décompose plus. Telle est l'origine de l'analyse spectrale. SON INFLUENCE SUR LA PHYSIQUE MODERNE. xxi Cette analyse de la lumière blanche amena Newton à expliquer les colorations des lames minces qu'on observe en particulier sur les bulles de savon; l'expérience fondamentale, dite des anneaux de Newton, est l'une des plus instructives de l'Optique, et les lois qui la résument sont d'une admirable simplicité. Il en exposa la théorie dans un discours adressé à la Société Royale sous le titre: Hypothèse nouvelle concernant la lumière et les couleurs. Ce discours provoqua de la part de Hooke une vive réclamation. Hooke avait antérieurement observé aussi les colorations des lanes minces et cherché à les expliquer dans le système des ondes: il avait eu le mérite (que Newton lui-même reconnut sans peine) de substituer à l'onde progressive de Descartes une onde vibratoire, notion nouvelle et extrêmement importante: il avait même aperçu le rôle des deux surfaces réfléchissantes de la lame mince, ainsi que l'action mutuelle des ondes réfléchies. Hooke eût été ainsi le véritable précurseur de la théorie moderne, s'il avait eu, comme Newton, la perception claire des rayons simples; mais ses raisonnements vagues pour expliquer la coloration ôtent toute valeur démonstrative à sa théorie. Newton fut très affecté de cette réclamation de priorité; il combat les arguments de son adversaire en rappelant que la théorie des ondes est inadmissible, parce qu'elle ne rend pas compte de l'existence du rayon lumineux et des ombres. Il se défend d'avoir constitué une théorie, il déclare qu'il n'admet ni l'hypothèse des ondes, ni celle de l'émission; seulement il est obligé, pour abréger le discours et faire image, d'avoir recours à l'une et à l'autre, comme s'il les admettait. Et, en fait, dans la XIIe Proposition, au IIe livre de son Optique*, qui constitue ce que l'on a appelé depuis la théorie des accès, Newton reste absolument sur le terrain des faits. Il dit simplement: "Le phénomène des lames minces prouve que le rayon lumineux est mis alternativement dans un accès de facile réflexion ou de facile transmission." Il ajoute, toutefois, que si l'on désire une explication de ces alternances, on peut les attribuer aux vibrations excitées par le choc des corpuscules et propagées sous forme d'ondes par l'éthert-. En résumé, malgré son désir de rester sur le terrain solide des faits, Newton n'a pas pu s'empêcher d'essayer une explication rationnelle; il a trop lu les écrits de Descartes pour n'être pas, au fond, comme Huyghens, partisan de l'universel mécanisme et pour ne pas désirer secrètement trouver, dans les ondulations pures, l'explication du beau phénomène qu'il a réduit en lois si simples. Son admirable livre des Principes porte la trace de ses profondes méditations sur la propagation des ondes, car on y trouve, pour la première fois, l'expression mathématique de leur vitesse, aussi bien pour les vibrations longitudinales des corps compressibles que pour les vibrations transversales des surfaces fluides. * Prop. XII.-Tout rayon de lumière dans son passage réfringente, et entre les retours, à être aisément réfléchie à travers une surface réfringente est mis dans un certain par elle. état passager qui, dans la progression du rayon, revient à (Sir Isaac Newton, Opticks or a Treatise of the Reintervalles égaux et dispose le rayon, à chaque retour, à flections, Refractions, Inflexions and Colours of Light.être facilement transmis à travers la prochaine surface London, 1718, second edition, with additions, p. 253.) + Loc. cit., p. 255. VOL. XVIII. d xxii ALFRED CORNU, LA THÉORIE DES ONDES LUMINEUSES: Mais c'est surtout le troisième livre de son Optique, qui témoigne le plus vivement de ses aspirations cartésiennes et surtout de sa perplexité. Ses fameuses "Questions" sont un exposé si complet des arguments en faveur de la théorie des ondes lumineuses que Thomas Young les citera plus tard comme preuve de la conversion finale de Newton à la doctrine ondulatoire. Newton aurait certainement cédé à ce secret entraînement si la logique inflexible de son esprit le lui avait permits; mais, après avoir énuméré toutes les ressources dont la théorie des ondes dispose pour expliquer la nature intime de la lumière, arrivé aux dernières questions, il s'arrête comme pris d'un remords subit et la rejette résolument. Et le seul argument qu'il donne, c'est qu'il n'y voit pas la possibilité de rendre compte du rayon lumineux rectiligne*. * Voici, d'abord, un extrait des "Questions" qui prouve les phénomènes de la lumière par de nouvelles modificala tendance des vues de Newton vers la théorie ondulatoire tions des rayons? et les idées cartésiennes. " Question 28.-Ne sont-elles pas erronées toutes les "Question 12.-Les rayons de lumière, en frappant le hypothèses dans lesquelles la lumière est supposée confond de l'oeil, n'excitent-ils pas des vibrations dans la sister en une pression ou un mouvement propagé à travers tunica retina? Ces vibrations, étant propagées le long des un milieu fluide? fibres solides des nerfs optiques dans le cerveau, causent la " Si elle (la lumière) consiste seulement en une pression sensation de la vision... ou un mouvement propagé instantanément ou progressive" Question 13.-Les diverses sortes de rayons ne font- ment, elle se courberait dans l'ombre. Car une pression elles pas des vibrations de diverses grandeurs, qui, suivant ou un mouvement ne peut pas se propager en ligne droite leurs diverses grandeurs, excitent les sensations des diverses dans un fluide au delà de l'obstacle qui arrête une partie couleurs, de la même manière que les vibrations de l'air, du mouvement; il y a inflexion et dispersion de tous côtés suivant leurs diverses grandeurs, excitent les sensations dans le milieu en repos situé au delà de l'obstacle... des divers sons? Et, en particulier, ne sont-ce pas les "... Car une cloche ou un canon peuvent s'entendre au rayons les plus réfrangibles qui excitent les plus courtes delà d'une colline qui intercepte la vue du corps sonore, et vibrations pour produire la sensation du violet extrême; les sons se propagent aussi bien à travers des tubes courbés les moins réfrangibles, les plus grandes, pour produire la qu'à travers des tubes droits. Tandis que l'on ne voit sensation du rouge extrême, etc.?... jamais la lumière suivre des routes tortueuses, ni s'in" Question 18.-La chaleur d'un espace chaud n'est-elle fléchir dans l'ombre." pas transmise à travers le vide par les vibrations d'un Devant cette objection, Newton se voit obligé de revenir milieu beaucoup plus subtil que l'air, qui reste dans le vide à la théorie corpusculaire. après que l'air en a été enlevé? " Question 29.-Les rayons de lumière ne sont-ils pas " Et ce milieu n'est-il pas le même que le milieu par de petits corps émis par les substances brillantes?... lequel la lumière est réfractée et réfléchie, par les vibra- " Question 30.-Les corps grossiers de la lumière ne sonttions duquel la lumière communique la chaleur aux corps ils pas convertissables l'un dans l'autre?... Le changement et est mise dans les accès de facile réflexion et de facile des corps en lumière et de lumière en corps matériels est transmission? très conforme au cours de la nature, qui se plaît aux trans" Et ce milieu n'est-il pas infiniment exceedinglyl) plus mutations." rare et subtil que l'air et infiniment plus élastique et actif? La logique le force à poursuivre l'hypothèse du vide et Et ne remplit-il pas tous les corps? Et (par sa force des atonies et même à invoquer (question 28, p. 343), à ce élastique) ne se répand-il pas dans tout l'espace céleste?" sujet, l'autorité des anciens philosophes de la Grèce et de Newton examine ensuite le rôle possible de ce milieu la Phénicie: on ne doit donc pas s'étonner de voir sa per(l'éther) dans la gravitation et dans la transmission des plexité se traduire par les paroles suivantes: sensations et du mouvement chez les êtres vivants (ques- " Question 31e et dernière.-Les petites particules des tions 19 à 24). Les propriétés dissymétriques des deux corps n'ont-elles pas certains pouvoirs, vertus ou forces, rayons du spath d'Islande attirent également son attention par lesquels elles agissent à distance non seulement sur les (questions 25 et 26). rayons de lumière pour les réfléchir, les réfracter ou les Puis arrive cette volte-face soudaine, cette espèce de infléchir, mais aussi les unes sur les autres pour produire remords d'avoir exposé avec trop de complaisance les une grande partie des phénomènes de la Nature?" ressources de la théorie cartésienne fondée sur le plein: il Mais il s'aperçoit qu'il va peut-être un peu loin et qu'il fait, en quelque sorte, amende honorable et continue ainsi: va se compromettre: aussi ses secrètes tendances, dévelopL" Question 27.-Ne sont-elles pas erronées toutes les pées dans la première question, reparaissent-elles un inhypothèses qui ont été inventées jusqu'ici pour expliquer stant: SON INFLUENCE SUR LA PHYSIQUE MODERNE. xxiii Considéré à ce point de vue, le troisième livre de 1'Optique n'est plus la discussion seulement impartiale de systèmes opposés; il apparaît comme la peinture des souffrances d'un génie puissant, tourmenté par le doute, tour à tour entraîné par les suggestions séduisantes de l'imagination et rappelé par les exigences impérieuses de la logique. Nous assistons à un drame, à l'éternel combat de l'amour et du devoir, et c'est le devoir qui a été le plus fort. Telle est, j'imagine, la genèse intime de la Théorie des accès, mélange bizarre des deux systèmes opposes; elle a été beaucoup admirée à cause de l'autorité du grand géomètre qui a eu la gloire de ramener l'ensemble des mouvements célestes à la loi unique de la gravitation universelle. Aujourd'hui, cette théorie est abandonnée; elle est condamnée par l'experimentum crucis d'Arago, réalisé par Fizeau et Foucault: on doit pourtant reconnaître qu'elle a constitué un réel progrès par la notion précise et nouvelle qu'elle renferme. Le rayon de lumière considéré jusque-là était simplement la trajectoire d'une particule en mouvement rectiligne: le rayon de lumière tel que le décrit Newton possède une structure périodique régulière, et la période ou longueur d'accès caractérise la couleur du rayon; c'est là un résultat capital. Il ne manque plus qu'une interprétation convenable pour transformer le rayon lumineux en une onde vibratoire; mais il faut attendre un siècle, et c'est le Dr Thomas Young qui, en 1801, aura l'honneur de la découvrir. III Reprenant l'étude des lames minces, Thomas Young montre que tout s'explique avec une extrême simplicité, si l'on suppose que le rayon lumineux homogène est l'analyse de l'onde sonore produite par un son musical; que les vibrations de l'éther, soumises aux lois des petits mouvements, doivent se composer, c'est-à-dire interférer, suivant l'expression qu'il propose pour exprimer leur action mutuelle. Quoique Young eût pris l'habile précaution de se réclamer de l'autorité de Newton*, l'hypothèse n'eut aucune faveur; son principe d'interférence conduisait à cette singulière conséquence que la lumière ajoutée à de la lumière pouvait, dans certains cas, produire l'obscurité; résultat paradoxal, contredit par l'expérience journalière. La seule vérification que Young apportât était l'existence des anneaux obscurs dans l'expérience de Newton, obscurité due, suivant lui, à l'interférence des ondes réfléchies aux deux faces de la lame; mais, comme la théorie newtonienne interprétait le fait autrement, la preuve restait douteuse; il fallait un experiment'um crucis, Young ne réussit pas à l'obtenir. " Comment ces attractions (gravité, magnétisme et élec- initiateur de la théorie de l'émission. En réalité, il hésite tricité) peuvent-elles se produire, je ne m'y arrête pas ici. entre les deux systèmes opposés dont il aperçoit claireCe que j'appelle attraction peut être produit par des impul- ment l'insuffisance et, dans cette discussion, il s'efforce de sions ou par d'autres moyens que j'ignore..." s'éloigner le moins possible des faits bien établis: voilà Il y aurait encore bien des remarques curieuses à faire pourquoi il ne formule aucune théorie dogmatique. Il sur l'état d'esprit du grand physicien, géomètre et philo- serait donc injuste de rendre Newton responsable de tout sophe, qui se révèle naïvement dans ces " Questions." Les ce que les partisans de l'émission ont abrité sous son courts extraits qui précèdent suffisent, je crois, à justifier autorité. la conclusion qui ressort de cette étude, à savoir, que * The Bakerian Lecture, on the Theory of Light and Newton n'avait pas, sur le mécanisme de la lumière, les Colours.-By Thomas Young. Philos. Trans. of the Royal idées arrêtées qu'on lui prête en le considérant comme SocietyJ of London, 1802, p. 12. d2 xxiv ALFRED CORNU, LA THÉORIE DES ONDES LUMINEUSES: La théorie des ondes retombait donc encore une fois dans l'obscurité des controverses, et le terrible argument de la propagation rectiligne se dressait de nouveau contre elle. Les plus habiles géomètres de l'époque, Laplace, Biot, Poisson, s'étaient naturellement rangés à l'opinion newtonienne: Laplace en particulier, le célèbre auteur de la Mécanique céleste, avait même pris l'offensive; il était allé attaquer la théorie des ondes jusque dans le plus solide de ses retranchements, celui qui avait été élevé par l'illustre Huyghens. Huyghens, en effet, dans son Traité de la Lumière, avait résolu un problème devant lequel la théorie de l'émission était restée muette, à savoir, l'explication de la biréfringence du cristal d'Islande; la théorie des ondes, au contraire, ramenait à une construction géométrique des plus simples la marche des deux rayons, ordinaire et extraordinaire; l'expérience confirmait en tous points ces résultats. Laplace réussit, à son tour, à l'aide d'hypothèses sur la constitution des particules lumineuses, à expliquer la marche de ces étranges rayons. La victoire de la théorie particulaire paraissait donc complète: un nouveau phénomène arrivait même tout à point pour la rendre éclatante. Malus découvrait qu'un rayon de lumière naturelle, réfléchi sous un certain angle, acquiert des propriétés dissymétriques semblables à celles des rayons du cristal d'Islande; il expliqua ce phénomène par une orientation de la molécule lumineuse, et, en conséquence, nomma cette lumière, lumière polarisée; c'était un nouveau succès pour l'émission. Le triomphe ne fut pas de longue durée; en 1816, un jeune ingénieur, à peine sorti de l'École Polytechnique, Augustin Fresnel, confiait à Arago ses doutes sur la théorie en faveur et lui indiquait les expériences qui tendaient à la renverser; s'appuyant sur les idées d'Huyghens, il avait attaqué la redoutable question des rayons et des ombres et l'avait résolue; tous les phénomènes de diffraction étaient ramenés a un problème d'analyse et l'observation vérifiait merveilleusement le calcul. Il avait, sans les connaître, retrouvé les raisonnements de Young, ainsi que le principe des interférences; mais, plus heureux que lui, il apportait l'experimentzum crucis, l'expérience des deux miroirs; là, deux rayons issus d'une même source, purs de toute altération, produisent par leur concours, tantôt de la lumière, tantôt de l'obscurité. L'illustre Young fut le premier à applaudir au succès de son jeune émule et lui témoigna une bienveillance qui ne se démentit jamais. Ainsi, grâce à l'expérience des deux miroirs, la théorie du D' Young, c'est-à-dire l'analogie complète du rayon lumineux et de l'onde sonore, est solidement établie. En outre, la théorie de la diffraction de Fresnel montre la cause de leur dissemblance; la lumière se propage en ligne droite parce que les ondes lumineuses sont extrêmement petites; au contraire, le son se diffuse parce que les longueurs des ondes sonores sont relativement très grandes. Ainsi s'évanouit la terrible objection qui avait tant tourmenté l'esprit du grand Newton. Mais il restait encore à expliquer une autre différence essentielle entre l'onde lumineuse et l'onde sonore; celle-ci ne se polarise pas, comment se fait-il que l'onde lumineuse se polarise? La réponse à cette question paraissait si difficile que Young déclara renoncer à SON INFLUENCE SUR LA PHYSIQUE MODERNE. xxv la chercher. Fresnel travailla plus de cinq ans à la découvrir; elle est aussi simple qu'inattendue: L'onde sonore ne peut pas se polariser parce que ses vibrations sont longitudinales; la lumière, au contraire, se polarise parce que ses vibrations sont transversales, c'est-à-dire perpendiculaires au rayon lumineux. Désormais, la nature de la lumière est complètement établie; tous les phénomènes présentés comme des objections absolues s'expliquent avec une merveilleuse facilité, jusque dans leurs plus minutieux détails. Je voudrais pouvoir vous retracer par quel admirable enchaînement d'expériences et de raisonnements Fresnel est arrivé à cette découverte, l'une des plus importantes de la science moderne; mais le temps me presse. Il m'a suffi de vous faire comprendre la grandeur des difficultés qu'il a fallu vaincre pour l'accomplir; j'ai hâte d'en faire ressortir les conséquences. IV Vous avez vu, au début, les raisons purement physiologiques qui font de l'étude de la lumière le centre nécessaire des informations de l'intelligence humaine. Vous devez juger maintenant par les péripéties de ce long développement des théories optiques, quelle préoccupation elle a toujours causée aux puissants esprits qui s'intéressent aux forces naturelles. En effet, tous les phénomènes qui se passent sous nos yeux impliquent une transmission à distance de force ou de movement; que la distance soit infiniment grande, comme dans les espaces célestes, ou infiniment petite, comme dans les intervalles moléculaires, le mystère est le même. Or, la lumière est l'agent qui nous amène le mouvement des corps lumineux: approfondir le mécanisme de cette transmission, c'est approfondir celui de toutes les autres, et Descartes en avait eu l'admirable intuition lorsqu'il embrassait tous ces problèmes dans une même conception mécanique: voilà le lien secret qui a toujours attiré les physiciens et les géomètres vers l'étude de la lumière. Envisagée à ce point de vue, l'histoire de l'Optique acquiert une portée philosophique considérable; elle devient l'histoire des progrès successifs de nos connaissances sur les moyens que la Nature emploie pour transmettre à distance le mouvement et la force. La première idée qui est venue à l'esprit de l'homme, dès l'état sauvage, pour exercer sa force hors de sa portée, c'est le jet d'une pierre, d'une flèche ou d'un projectile quelconque; voilà le germe de la théorie de l'émission: cette théorie correspond au système philosophique qui suppose un espace vide où le projectile se meut librement. A un degré de culture plus avancé, l'homme, devenu physicien, a eu l'idée plus délicate de la transmission du mouvement par ondes, suggérée d'abord par l'étude des vagues, puis par celle du son. Ce second mode suppose, au contraire, que l'espace est plein: il n'y a plus ici transport de matière, les particules oscillent dans le sens de la propagation, et c'est par compression ou dilatation d'un milieu élastique continu que le mouvement et la force sont transmis. Telle a été l'origine de la théorie des ondes lumineuses; sous cette forme, elle ne pouvait représenter qu'une partie des phénomènes, ainsi qu'on l'a vu précédemment; elle était donc insuffisante. Mais les géomètres et xxvi ALFRED CORNU, LA THÉORIE DES ONDES LUMINEUSES: les physiciens avant Fresnel ne connaissaient pas d'autre mécanisme ondulatoire dans un milieu continu. La grande découverte de Fresnel a été de révéler un troisième mode de transmission, tout aussi naturel que le précédent, mais qui offre une richesse de ressources incomparable. Ce sont les ondes à vibrations transversales excitées dans un milieu continu incompressible, celles qui rendent compte de toutes les propriétés de la lumière. Dans ce mode ondulatoire, le déplacement des particules met en jeu une élasticité d'un genre spécial; c'est le glissement relatif des couches concentriques à l'ébranlement qui transmet le mouvement et l'effort. Le caractère de ces ondes est de n'imposer au milieu aucune variation de densité, comme dans le système de Descartes. La richesse de ressources annoncée plus haut provient de ce que la forme de la vibration transversale reste indéterminée, ce qui confère aux ondes une variété infinie de propriétés différentes. Les formes rectiligne, circulaire, elliptique, caractérisent précisément ces polarisations si inattendues que Fresnel a découvertes et à l'aide desquelles il a si admirablement expliqué les beaux phénomènes d'Arago produits par les lames cristallisées. L'existence possible d'ondes se propageant sans changement de densité a modifié profondément la théorie mathématique de l'Élasticité. Les géomètres retrouvèrent dans leurs équations ces ondes à vibrations transversales qui leur étaient inconnues; ils apprirent, en outre, de Fresnel la constitution la plus générale des milieux élastiques, à laquelle ils n'avaient pas songé. C'est dans son admirable Mémoire sur la double réfraction que le grand physicien émet l'idée que, dans les cristaux, l'élasticité de l'éther doit être variable avec la direction, condition inattendue et d'une extrême importance qui transformera les bases fondamentales de la Mécanique moléculaire; les travaux de Cauchy et de Green en sont la preuve frappante. De ce principe, Fresnel conclut la forme la plus générale de la surface de l'onde lumineuse dans les cristaux et retrouva (comme cas particulier) la sphère et l'ellipsoïde que Huyghens avait assignés au cristal d'Islande. Cette nouvelle découverte excita l'admiration universelle parmi les physiciens et les géomètres; lorsque Arago vint l'exposer devant l'Académie des Sciences, Laplace, si longtemps hostile, se déclara convaincu. Deux ans après, Fresnel, élu membre de l'Académie à l'unanimité des suffrages, était élu, avec la même unanimité, membre étranger de la Société Royale de Londres; ce fut Young lui-même qui lui transmit la nouvelle de cette distinction avec l'hommage personnel de son admiration sincère. V L'établissement définitif de la théorie des ondes impose la nécessité d'admettre l'existence d'un milieu élastique pour transmettre le mouvement lumineux. Mais toute transmission a distance de mouvement ou de force n'implique-t-elle pas la même condition? C'est à Faraday que revient l'honneur d'avoir, en véritable disciple de Descartes et de Leibnitz, SON INFLUENCE SUR LA PHYSIQUE MODERNE. xxvii proclamé ce principe et d'avoir résolument attribué aux réactions du milieu ambiant l'apparente action à distance des systèmes électriques et magnétiques. Faraday fut récompensé de sa hardiesse par la découverte de l'induction. Et, comme l'induction s'exerce même à travers un espace vide de matière pondérable, on est forcé d'admettre que le milieu actif est justement celui qui transmet les ondes lumineuses, l'éther. La transmission d'un mouvement par un milieu élastique ne peut pas être instantanée; si c'est vraiment l'éther luminitère qui est le milieu transmetteur, l'induction ne doit-elle pas se propager avec la vitesse des ondes lumineuses. La vérification était malaisée; Von Helmholz, qui tenta la mesure directe de cette vitesse, trouva, comme autrefois Galilée, pour la vitesse de la lumière, une valeur pratiquement infinie. Mais l'attention des physiciens fut attirée par une singulière coïncidence numérique: le rapport de l'unité de quantité électrostatique à l'unité électro-magnétique est représenté par un nombre précisément égal à la vitesse de la lumière. L'illustre Clerk Maxwell, suivant les idées de Faraday, n'hésita pas à voir dans ce rapport la mesure indirecte de la vitesse d'induction, et, par une série d'intuitions remarquables, il parvint à élever cette célèbre théorie électro-magnétique de la lumière, qui identifie, dans un même mécanisme, trois groupes de phénomènes en apparence complètement distincts: Lumière, Électricité, Magnétisme. Mais les théories abstraites des phénomènes naturels ne sont rien sans le contrôle de l'expérience. La théorie de Maxwell fut soumise à l'épreuve et le succès dépassa toute attente. Les résultats sont trop récents et trop bien connus, ici surtout, pour qu'il soit nécessaire d'y insister. Un jeune physicien allemand, Henry Hertz, enlevé prématurément à la Science, empruntant à von Helmholz et à Lord Kelvin leur belle analyse des décharges oscillantes, réalisa si parfaitement des ondes électriques et électro-magnétiques, que ces ondes possèdent toutes les propriétés des ondes lumineuses; la seule particularité qui les distingue, c'est que leurs vibrations sont moins rapides que celles de la lumière. Il en résulte qu'on peut reproduire, avec des décharges électriques, les expériences les plus délicates de l'Optique moderne: réflexion, réfraction, diffraction, polarisation rectiligne, circulaire, elliptique, etc. Mais, je m'arrête, Messieurs; je sens que j'ai assumé une tâche trop lourde en essayant de vous énumérer toutes les richesses que les ondes à vibrations transversales concentrent aujourd'hui dans nos mains. J'ai dit, en commençant, que l'Optique me paraissait être la Science directrice de la Physique moderne. Si quelque doute a pu s'élever dans votre esprit, j'espère que cette impression s'est effacée pour faire place à un sentiment de surprise et d'admiration en voyant tout ce que l'étude de la lumière a apporté d'idées nouvelles sur le mécanisme des forces de la Nature. Elle a ramené insensiblement à la conception cartésienne d'un milieu unique remplissant l'espace, siège des phénomènes électriques, magnétiques et umineux; elle laisse xxviii ALFRED CORNU, LA THÉORIE DES ONDES LUMINEUSES, ETC. entrevoir que ce milieu est le dépositaire de l'énergie répandue dans le monde matériel, le véhicule nécessaire de toutes les forces, l'origine même de la gravitation universelle. Voilà l'oeuvre accomplie par l'Optique; c'est peut-être la plus grande chose du siècle! L'étude des propriétés des ondes envisagées sous tous leurs aspects est donc, à l'heure actuelle, la voie véritablement féconde. C'est celle qu'a suivie, dans sa double carrière de géomètre et de physicien, Sir George Stokes, à qui nous allons rendre un hommage si touchant et si mérité. Tous ses beaux travaux, aussi bien en Hydrodynamique qu'en Optique théorique ou expérimentale, se rapportent précisément aux transformations que les divers milieux font subir aux ondes qui les traversent. Dans les phénomènes varies qu'il a découverts ou analysés, mouvement des fluides, diffraction, interférences, fluorescence, rayons Rôntgen, l'idée directrice que je vous signale est toujours visible, et c'est ce qui fait l'harmonieuse unité de la vie scientifique de Sir George Stokes. Que l'Université de Cambridge soit fière de sa chaire Lucasienne de Physique mathématique, car, depuis Sir Isaac Newton jusqu'à Sir George Stokes, elle contribue pour une part glorieuse aux progrès de la Philosophie naturelle. MEMOIES PRESENTED TO THE CAMBRIDGE PHILOSOPHICAL SOCIETY. e I. On the analytical representation of a uniorm branch of a monogenic function. By G. MITTAG-LEFFLER. [Received 25 April, 1899.] LET a denote a point in the plane of the complex variable x, and associate with a an unlimited array of quantities F), (a), F() (a), (a., F) (a),.......................... (1), where each quantity is completely determinate when the position which it occupies in the array is known. Suppose that, as is possible in an infinite number of ways, these quantities F are chosen so that Cauchy's condition*, that the series P(l|a)= S! (i) (a) (x - a)b.......... (2), shall have a circle of convergence, is satisfied. In the theory of analytic functions constructed by Weierstrass, the function is defined by the series P (x a) and by the analytic continuation of this series. The function is completely determinate provided the elements F(a), F(l (a), F(2) (a),..., F() (a),... are given. We denote generally by F(x) the function in its totality which is defined by these elements. If K is a continuum formed by a single piece, which nowhere overlaps itself and encloses the point a, and if it is such that the branch of the function F (x) formed by P (x a) and by its analytic continuation within K remains uniform and regular, I shall denote this branch by FK(x). The problem to be discussed here is that of finding * Cauchy, Cours d'Analyse de l'École royalepolytechnique, is a finite magnitude. It is known that, if this finite 1lre partie, Analyse Algébrique, Paris 1821, chapitre 9, ~ 2, 1 theorème I, p. 286. Expressed in modem phraseology, magnitude be denoted by -, the quantity r is the radius of theorème i, p. 286. Expressed in modern phraseology, Cauchy's condition would be formulated thus: The upper the circle of convergence of the series (2). limit of the limiting values of the moduli.!F( (a)XI 1 VOL. XVIII. 1 2 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION ca analytical representation of c branch FK(x) which is to be chosen as extensive as possible. Merely from the definition of the analytic function F(x) and from that of the branch FK (x), there follows at once a kind of analytical representation of the branch FK (x) in question. In effect, such a representation is always given by an enumerable number of analytical continuations of P (xa). But as the radius of the circle of convergence of such an analytical continuation is given only by Cauchy's criterion already quoted, this mode of representing FK (x) becomes extremely complicated and rather unworkable. The analytical continuation ought rather to be regarded as the definition of the function than as a mode of representation. There is another mode of representation which arises immediately from the principles upon which Cauchy's theory of functions is based. Such a representation is given by the formula K ()= K ) dz.................................... (3), where the integral is taken along a closed contour S within K. By the definition of an integral, it is clear that the integral (3) can be replaced by an infinite sum of rational functions of x, the coefficients of which are expressed by special values of x (there being an enumerable number of these) and the corresponding values of FK1(x). This observation was the point of departure of the investigation of M. Runge* as well as of the subsequent investigations of MM. Painlevé, Hilbert and others. The analytical representation thus obtained accordingly requires a knowledge of the value of FK (x) at an infinite and enumerable number of points. Now in the customary problems of analysis these values are not known. In general it is, on the contrary, the series of values F(a), F( (a), F2 (a),... which is given. Adopting the usual point of view, it is thus for instance in the problem of the integration of differential equations. When, then, we have to find the analytical representation of FK (x), it must be drawn from the elements (1) and, by means of those elements alone, a formula must be constructed to represent the branch FK(x) completely. Let C denote the circle of convergence of the series (2). The expression e 1! F ) (a) (x - ac) 1 then gives the analytical representation of FC(x), the equality FC (x) - 1 F) (a) (x - a) holding for all points within C. This expression is constructed by means of the elements F (a), F() (a), F' (a),.... * " Zur Theorie der eindeutigen analytischen Functionen," ~ 1, pp. 229-239, Acta Mathematica, tome 6. OF A LTNIFORM BRANCH OF A MONOGENIC FUNCTION. 3 and of the rational numbers independent of the choice of the elements: and it is to be remarked that the expression is formed without any a priori knowledge of the radius of the circle C. This radius is determinate, in connection with the elements, by Cauchy's theorem, and there are various rnethods of obtaining it from them; but it does not enter explicitly into the expression. Thus Taylor's series is formed simply by the elements F (a), F(1) (a), F(2' (a),..., when these are the derivatives of the function. The following question may therefore be proposed: Is it possible to obtain for a branch FK(x) with the greatest range possible an analytical representation of this nature? As I have shewn in various notes, published in Swedish by the Stockholm Academy of Sciences during the past year, the reply is in the affirmative, and consequently it is possible to fill an important lacuna in the theory of analytic functions. In fact, hitherto it has been impossible to give for the general branch FK(x) an analytical representation similar to that found from the very beginning of the theory for the branch FC(x). For a fundamental treatment of the question which has been proposed, it is first necessary to define a domain K which shall be as great as possible. This I shall do by the introduction of a new geometrical conception-a Star-figure. In the plane of the complex variable x, let an area be generated as follows. Round a fixed point a let a vector I (a straight line terminated at a) revolve once: on each position of the vector, determine uniquely a point, say a1, at a distance from a greater than a given positive quantity, this quantity being the same for all positions of the vector. The points thus determined may be at a finite or at an infinite distance from a. When the distance between a, and a is finite, the part of the vector from a, to infinity is excluded from the plane of the variable. The region which remains after all these sections (coupures) in the plane of x have been made is what I call a Star-figure. Manifestly the star is a continuum formed of a single simply-connected area. Associate with a the elements F(a), F() (a), F(2) (a),..., F( (a),... satisfying Cauchy's condition; and form the series P (x a)= 1! F (a) (x -a)-t. Construct the analytical continuation of P(x a) along a vector from a. It may be the case that every point of this vector belongs to the circle of convergence of a series which itself is an analytical continuation of P(xla) obtained by proceeding along the vector; but it is also possible that, on proceeding along the vector, a point is met not situated within the circle of convergence of any analytical continuation of P(xla) along the vector. In the latter case, I exclude from the plane of the variable that part of the vector comprised 1 — 2 4 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION between the point thus met and infinity. On making this vector describe one complete revolution round a, a Star-figure (as defined above) is obtained. This star being given in a unique manner as soon as the elements (1) are assigned, I call it the Star belonging to these elements, and I denote* it by A. In defining the star, straight lines have been used as vectors: it is easy to see that curved lines, suitably defined, might have been chosen for the purpose. - In accordance with the phrase the star belonging to the elernents (1), I speak of the function F (x), as well as of the functional branch FA (x), belonging to these elements. These preliminaries being settled, my main theorem is as follows:Denote by A the star belonging to the elements F(a), Fl)(a), F(2)(a)..... and by FA (x) the corresponding functional branch belonging to the same elements; let X be any finite domain within A; and let o- denote a positive quantity as srall as we please. Then it is always possible to find an integer n such that the modulus of the difference between FA(x) and the polynomial gn (=x) = /c(l)Fv(6a) (x - a)v v for values of n greater than Tn, is less than o- for all the values of x belonging to X. The coefficients c() are chosen a priori and are absolutely independent of a, of F(a), F() (a), F(2)(a),..., and of x. It is very important to observe that the explicit knowledge of the star is not necessary for the construction of the function gn (x). When the elements F (a), F() (a), F(2)(a),... are once given, the star belonging to them is definite; but it does not enter explicitly into the expression gn(x). The case is precisely the same as for Taylor's series where the radius of the circle of convergence does not enter explicitly into the expression. The theorem can be proved by very elementary considerations, using especially the fundamental theorem established by Weierstrass in his memoir Zur Theorie der Potenzreihen, datedt 1841. Passing from the same theorem for functions of several variables, we can easily obtain a generalisation of my main theorem which includes the case of any finite number of independent variables. The coefficients denoted by c() are given a priori. They are quite independent of the special function to be represented just as are the coefficients in Taylor's series. But the choice of these coefficients is not unique. On the contrary it can be made in an infinitude of ways; and when conditions are given, the problem arises of making a choice which is the best adapted to these conditions. * As the first letter of the word da-rjp. + Ges. ~Werke, Bd. I, p. 67. OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 5 The formula ~n2 z42 n2nF 1 /+++ ( hl+h2+...+ha ) gn (x)=::_ o... h h)(, (............. (4) h,=O h2=O h,l= h 2!....... gives an expression for gn (x) which perhaps is the simplest of all as regards the mere form. There are other forms in which the coefficients c(V are rational numbers, or are numbers depending in a special manner upon the transcendents e and 7T, and which are of great simplicity. Upon this I shall not dwell: but I enunciate another theorem which is an almost immediate consequence of my main theorem. Denote by A the star which belongs to the elements F (a), F()(a), F(2)(a),....... and by FA(x) the corresponding functional branch belonging to the same elements. This branch FA(x) can always be represented by a series o=o where the quantities G,(x) are polynomials of the form G, (X) -= cv()F()(a) (X - a), each coefficient t) being a determinate number (which can be taken as rational) depending only upon jk and v. The series =o converges for every value of x within A, and it converges uniformly for every domain within A. For all values within A we have 00 Gx,() = Lim gn(x), iz=O ln= o where gn(x) is the polynomial in my main theorem. In what precedes, a definition has been given of the star belonging to the elements F (a), F "()(a), F (2)(a),.........................................(1). In accordance with this terminology, we can speak of the circle belonging to the elements (1) which, in fact, is the circle of convergence C of the series P (x | a) = I1 F(1)(a) (x - a). /x=O j!. It is evident that this circle is inscribed in the star which belongs to the same elements. The circle may be regarded as a first approximation to the star. To the circle C corresponds an analytical expression P (x a) which has the property of representing FA (x) within C, of converging uniformly for any domain within C, and of ceasing to converge outside 6 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION C. Between the circle and the star, intermediary domains C(0), (u= 1, 2, 3,...), exist, unlimited in number; each of them in succession includes the domain that precedes it; and they can be chosen so that, corresponding to each domain C(G), there is an analytical expression representing FA (x) within C(w) which converges uniformly for every domain within C(0) and ceases to converge outside C(G). On this question there is an interesting study to be made which I have merely sketched in my Swedish memoirs; to it I shall return on another occasion. The only writer who, so far as I know, has found a general representation of FA (x) valid outside the circle belonging to the elements (1) is M. Borel. In two important memoirs*, M. Borel is concerned with what he calls the summability of a series. It appears to me that the chief interest of this investigation of M. Borel is that the author really finds an expression valid for a domain which in general includes the circle C. The domains which I have called C(G) can easily be chosen so that C(1) becomes this domain K: so that M. Borel's domain K becomes the second approximation to the Star, the circle being the first as already indicated. But M. Borel has discussed the same class of ideas in another publication. In his book-t published without any acquaintance with my Swedish Notes of the same year, the author says+:"Pour résumer les résultats acquis sur le problème de la représentation analytique "des fonctions uniformes, nous pouvons dire~ que nous en connaissons deux solutions "complètes; l'une est fournie par le théorème de Taylor, l'autre par le théorème de "M. Runge l. Ces deux solutions ont une très grande importance à cause de leur "généralité; mais chacune d'elles a de graves inconvénients dont les principaux sont, pour "la série de Taylor, de diverger en des régions où la fonction existe; et, pour la repré"sentation de M. Runge et celles de M. Painlevé, d'être possibles d'une infinité de " manières ~.................................................................................. "Le but idéal à atteindre, c'est de trouver une représentation réunissant les avantages "de la série de Taylor et des séries de M. Runge ou de M. Painlevé, sans avoir aucun "de leurs inconvénients", et le but immédiat, c'est de trouver une telle représentation "pour des classes de fonctions de plus en plus étendues +-t." * Journal de Mathématiques, 5-e Sér., t. ii. (1896), require the knowledge of an enumerable number of values "Fondements de la théorie des séries divergentes som- of the function which correspond to points that approach mables," pp. 103-122; " Sur les series de Taylor admettant indefinitely near the limit of existence of the function. leur cercle de convergence comme coupure," pp. 441-454. ** It will be seen that I have achieved this aim, not only + Leçons sur la théorie des fonctions, Paris, 1898. for uniform analytic functions but also for the functional + pp. 88 ff. branch FA (x). It might be asked whether it would not be ~ All that follows on the analytical representation of possible to achieve the same aim for the function F (x) in uniform functions can be applied, mutatis mutandis, to the its totality. It is not so: such a question is too general. functional branch FA (x). The problem was mainly that of limiting the question so as I I have indicated above that, in M. Runge's theorem, to make a solution possible without diminishing the there is nothing which is not already in principle contained generality more than was necessary. I believe that this in the representation by Cauchy's integral. problem is solved by the introduction of the star and of the ~T In what precedes, I have pointed out what appears to functional branch FA (x). me a graver inconvenience, viz. that these expressions ++ It appears that M. Borel has not regarded his own OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 7 There exist a certain number of other investigations having relations with my theorems but belonging to a range of ideas quite different from M. Borel's. I have already spoken of the representation which follows from Cauchy's integral FA (x) = 2 -f - - dz. With M. Runge, we can transform this integral into a series every term of which is a polynomial in x. But in order to construct these polynomials, it is necessary to know not only the star A but also the values of the function for an enumerable number of points approaching indefinitely near the boundary of A. Investigations have been carried out in which the elements F(a), F(i) (a), F(2> (a),... are substituted for these values of the function. But these investigations always abut, in a manner more or less direct, upon the conformal representation of the circle of convergence on another figure known beforehand: and they still require that we should know, as to the function which is to be represented, that it is regular within the domain represented on the circle. The most interesting and the most significant theorem in this range of ideas appears to me to be that of M. Painlevé: Given a convex domain D and an internal point a, a set of polynomials - n0 (x), ni (,),..., Hn (x); (, = 1, 2, 3,...), can be constructed such that any function F (x) holomnorphic in D is developable in that domain in the form F(x) = I {F, (a),, (x) + F 1' (a) HU (x) +... + F(, (a) I, (x)}. z=O The resemblance between M. Painlevé's formula and mine is obvious. Writing IIuv (x) = t() (x - a) in M. Painlevé's formula, mine follows. Yet the resemblance is entirely formal, because the formation of the polynomials II,, (x), T,j (x),..., HI, (x) requires the a priori establishment of the domain D and the knowledge of the function F(x) that it is holomorphic in D: whereas with me the formula of representation, so far from supposing any a priori knowledge of the star A, gives on the contrary the means of determining the starF. In other publications, it is my intention to develop other theorems in the same range of ideas as well as to return to the numerous applications that can be made of my theorems: I restrict myself in this place to the following indications. I have just explained that, besides the circle C and the star A, there is an infinite number of other investigations on the summability of series from the point appeared in the Comptes Rendus (23 May, 1899). In the of view just indicated so clearlyin his book. Otherwise he same number of the Comptes Rendus, there is a note by rather might have said: that the immediate aim was to M. Borel related to my investigations. The reader is also find a general representation valid for a domain still more referred to an addition to the "mémoire sur les séries extensive than this domain K (that is, C()). divergentes par E. Borel" (Ann. de l'Éc. Norm., 1899), and * Comptes Rendus, t. cxxvi (24 Jan., 1898), pp. 320, 321. to two important notes by M. Picard (Comptes Rendus, + While the present note was passing through the 5 June, 1899) and M. Phragmén (Comptes Rendus, press, a new and interesting note of M. Painlevé's, dis- 12 June, 1899): all of them are connected with these cussing the relation of these investigations to my own, has investigations. 8 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION stars C(I), C(2), C),... each of which is circumscribed to that which precedes it and is inscribed* to that which follows it; to these there correspond expansions PC() ( la), PC(2) (x a), PC() (x a),... which preserve all the principal characters of the Taylor's series PC(x a). The expression PC() (x a) is merely a (/ + l)-ple series with limited convergence. There is another method of generalising Taylor's series as follows: Denote by A a star with its centre at a, and by A () an associate star, concentric with A and inscribed in A, defined with reference to A in some suitable manner. This star A(') is to be such that it becomes a circle when S = 1 and that it encloses in its interior every domain within A when the quantity S is sufficiently small. Now suppose that A is the star belonging to the elements F(a), F(W> (a), F(> (a),..., and construct the series P (l a)=F(a) + Z {h, (> (<) F(' (a) (x - a) + h,>() (S) F (2) () (x - a)2 +... + h() () F() (a) (a) - a)}..(5). À=l The coefficients h(À) (:),') ( =1, 2,..., X)' can be assigned a priori, independently of a, of F(a), F(>) (a), F(2) (a),... and of x, so that the series possesses the following properties: it converges for every point within A (8 and converges uniformly for every domain within A( ). If convergence takes place for any value, the value necessarily belongs to the interior of A(8) or is a point of the star A( ). When 8=1, the series becomes Taylor's series. The equality FA (x) = P3 (x1 a), exists throughout the interior of A (). Among other differences between the two generalisations of Taylor's theorem, this may be noted: that in the first the stars 0(l), C(2), C(3>,... form, so to speak, a discontinuons sequence of domains of convergence, while in the second there is a continuous transition from the circle C (== A)) to the star A (= A(~). The star which belongs to the elements F (a), F('> (a),... is given at the same time as these elements, just as the circle which belongs to the elements also is given. But in order actually to construct the star on the circle, we must in the first case know the points of the star (it is thus that I describe the points formerly denoted by a,) and in the second case the distance between a and the nearest point of the star. It might be difficult to deduce this knowiedge simply by the study of the elements F(a), F(') (a), F(2) (a),.... But in some problems the points of the star are directly given: e.g. the determination of the general integral of a differential all of whose critical points are fixed, being finite in number. In this case, we can construct the star directly and can obtain an analytical expression for the integral valid over the whole plane except * A star is inscribed in another which circumscribes it if the whole of the first star belongs to the second and if the two stars have common points such as a2. OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 9 at a finite number of determinate sections. Notwithstanding the remarkable researches of M. Fuchs and M. Appell and others, this problem of finding a representation, which at once is unique for the whole plane and is sufficiently simple, has not hitherto been solved. The beautiful researches of MM. Fabry, Hadamard, Borel and other French writers, which have their origin in M. Darboux's memoir "Sur l'approximation des fonctions de très-grands nombres " and which aim at the development of the criteria whether a point on a circle belonging to the elements F(a), F(1> (a), F(2) (a),... is a singularity of the function or not, are well known. My theorems make it possible to study this problem from a more general point of view than these writers and to find the criteria which distinguish the points of the star belonging to the elements F(a), F(1) (a), F(2) (a),... from other points. It can be stated that, to each selection of the coefficients called c(), there corresponds a special system of criteria. For these investigations, the following theorem can serve as the point of departure:If x is a point within the star A belonging to the elements F (a), F(') (a), F(2) (a),... and if e is a positive quantity sufficiently small, it is always possible to choose a positive number S so that, o being a positive quantity as small as we please, a positive integer X exists such that \h,1() ()FP(1) (a) (1 + e) (x - a) + h,2() () F(2) (a) ( (1 + e) (x-a)}2... +~hx)(a)FÀ(a){(l +e) (x-a)} < a, provided X\X. If on the contrary, x does not lie within A, this property does not hold. M. Poincaré has pointed out a certain substitution which is of great value in the study of certain mechanical problems, particularly in that of n bodies. When this substitution is used, a development of the function in powers of the time can be obtained which is valid for real values of the time as far as the first positive or negative singularity nearest the origin. But the mechanical problem requires in general a knowledge of the first positive singularity, and not merely the nearest singularity, positive or negative. It is obvious that the resolution of this problein can be brought within my theorem. In fact, knowing the elements F(to), F(1) (to), F(2) (to),... at a given epoch to, we can obtain a development which represents the function and is valid for all real values of t > to up to the first singularity of the function. Recently I had an opportunity of giving an account of a portion of my investigations before the Academy of Sciences of Turin. My friend M. Volterra then made the following interesting communication. If in any dynamical problem the unknown functions be regarded as analytic functions of the time, the problem will be solved completely froin the analytical point of view when it can be shewn that the real axis of the time falls completely within the stars of the ' Liouville, Journ. de MIath., 31e Sér., t. iv. (1875), t The quantities 6 and h(À) (6) have the same significance pp. 5-54. n as in the formula (5). VOL. XVIII. 2 10 PROF. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION unknown functions, the centre of the stars being the initial value of the time. In fact, it is sufficient to employ M. Mittag-Leffier's expansions to obtain the unknown functions for any value of the time. The coefficients in the expansions will be determined by the initial conditions of motion. 1~. A very extensive class of dynamical equations can be reduced to the integration of differential equations of the type = SK PPrK, 1 1 where a(r + a_)= 0. Since in this case a finite strip enclosing the real axis is contained in the stars of the functions Ps, the centre being t= O, new forms of the integrals of these equations can be derivable by M. Mittag-Leffler's expansions*. 2~. Passing to the problems of attraction, it may be remarked that the problem of the motion of a point attracted by fixed points placed in a straight line, the force being according to Newton's law, has not been resolved when the number of attracting points is greater than two. Let us consider the general case and suppose that the moment of the initial velocity of the moving point m, with reference to the axis x of fixed points, is not zero. Then e being the angle which the plane mx makes with a fixed plane through x, and r being the distance of m from the axis x, we have the areal integral r2% = C = constant, and the integral of vis viva T- P=h= constant, where T= 'm( 2r2 + 2 x2), = Mim T being the vis viva and P the potential: in the latter expression the masses of the fixed point are denoted by Mi and their distances from m by ri. It is at once obvious that r cannot vanish. In effect, if for t =t0, r can become indefinitely small, let us take this quantity as an infinitesimal of the first order. On account of the areal integral, à would be infinitely great of the second order, and consequently r232 (= C) would also be of the second order: T therefore would be infinitely great of the second order. But P if it become infinitely great, can be so only to the first order because the quantities ri are greater than r; hence if r could become infinitely small, the integral of vis viva would not be verified. It therefore is to be inferred that the real axis of the time is contained in the stars of the unknown elements: and consequently these elements are expressible by Mittag-Leffler's series. 3~. Given n points repelling one another according to the Newtonian law of force, the integral of vis viva may be written ei is,+ i + s) + m * I have studied this class of equations in three Notes class can be still further extended so as to include many published by the Academy of Turin in 1898 and 1899. The of the classical problems in dynamics. OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 11 where x, Yi, zi are the coordinates of the moving points, mi their masses, ris their distances, and h is a constant quantity. By noting that in this equation all the terms are positive, we infer that the points cannot collide and that their velocities are finite. Hence in this case also, the real axis of the time lies within the stars. But we can pass from the case of repulsion to that of attraction by changing t into t - 1. Through this transformation, the components of the velocities become imaginary if they were real, and vice versa. But if at the beginning of the time they were zero, the transformation leaves them zero. Hence we deduce the very curious theorem: Consider the problem of n bodies in the most general case, with the sole condition that the initial velocities of the bodies are zero: then taking the origin at the beginning of the time, the real axis is not included within the stars of the coordinates, but the imaginary axis is always completely included. That is to say, M. Mittag-Leffler's expansions will be valid for imaginary values of the time even if they are not so for all real values. 4~. Finally it may be remarked that M. Mittag-Leffler's expansions can be used for the motion of straight and parallel vortices. Reference may be made to Lecture XX. in Kirchhoff's Mechanik for the differential equations of the motion. The interest of this development is manifest. I remark, however, that the main importance of my theorems so far as concerns mechanics appears to me to be that they provide a means of finding a real and positive point of my star, and of determining whether this point is at infinity or not. M. Volterra on the contrary supposes as always known beforehand that this point is at infinity. My principal theorem also provides in this case a means of representing the function, with any approximation desired for any real domain whatever, by a polynomial into which there enter no elements taken from the function other than a limited number of the quantities F(to), F(') (to), F() (to),.... It appears to me that this point of view may become useful in applications to mechanics. PERUGIA, April, 1899. 2-2 II. Ap plication of the Partition Analysis to the study of the properties of any system of Consecutive Integers. By Major P. A. MACMAHON, R.A., D.Sc., F.R.S., Hon. Mem. C.P.S. [Received 15 cMay, 1899.] INTRODUCTION. THE object of this paper is to solve a problem, concerning any arbitrarily selected set of consecutive integers, by the application of a new riethod of Partition analysis. I will first explain the problem, and afterwards the analysis that will be used. In the binormial and multinoinial expansions, the exponent being a positive integer, every coefficient is an integer. This fact depends analytically upon the circumstance that the product of any rn consecutive integers is divisible by factorial m; we have n+ (n+ i 2)( n+3) n + m) \~ 2 3 '" ' ^ an integer for all values of n. The present question is the determination of all values of ac, a, a3,... ar for which the expression (nI +1)ai (f + 2)2 n +3 3 (fn +,n )M ( V~) 2 3 ) ~ 3) (~m ) is an integer for all values of n; in particular the discovery of the finite number of ground or fundamental products of this form, from which all the forms may be generated by multiplication. There is a parallel theory connected with the algebraic product /1 _ xl+lai1 _ Xnl+2)a2 1X _ 1+3)a _ (l-t n+\ arn \ \-X [ 1 _ X2 ) [ 1-,3) '" ' X m where al, a2, a,... am have to be assigned so that the product is finite and integral for all values of n. This has been discussed by me in the 'Memoir on the Theory of the Partitions of Numbers, Part II.' Phil. Trans. R. S. 1899. It will be observed thatthe algebraic product merges into the arithmetical product for the particular case w= 1, so that ail algebraic products which are finite and integral produce in this manner arithmetical products which are integers. This, however, is as rauch as can be said, for MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS, ETC. 13 otherwise the theories proceed on widely divergent lines; as might be expected the arithmetical products form a more extended group than the algebraical. Denote, for brevity, 1 - -xns,n + s and 1 - xS s by Xs and Ns respectively. The principal X theorem, that has been obtained loc. cit., is to the effect that constructing any X rectangle X1 X2 X3 X4... X X2 X3 X4 X5... Xl+ X3 X4 X... X+2 X4 X5 X6 X7... Xi+3 Xm Xm+l X+,2 Xm~+3... X +-1, i and m having any values, with the law that any X has a suffix one greater than the X above it or to the left of it, the product X1X22X33... Xl+n-, obtained by multiplying all the X's together, is finite and integral for all values of the integer n. There are other forms as well, e.g. the product XlX,XX3X4X, which are not expressible in the rectangular lattice form, the theory of which is not yet complete. We see therefore that the product of N's contained in the rectangle N1 N2 N,...- N N, NV3 N,4... N1+1 V3 N4 N5... lY1+2 ATm N,,m~ Nrn~3 N... NI+n _ is an integer for all values of n. It will appear moreover that no product exists which is free from NI, so that all these products, being irreducible, are fundamental solutions of the problem. The method of partition analysis is concerned with the solution of one or more relations of the type XiCat + X.2 + X33 +... + Xsas ~ k11 + /13232 + 13/33 +... + t3t, the coefficients X and,u being given positive integers, and it is required to find the general values of Ca, a2.../3,,32,..., being positive integers, which satisfy the one or more relations. 14 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO This is accomplished by constructing the sum xlx2 x3a,... Xsas yW y2y3... yt.t for all sets of values of a2, a2.../3, /32,... which satisfy the relation. The expression obtained is found to indicate the ground solutions of the relations and the syzygies that connect them. The sum is expressible in the crude form Q 1 > 1 - rn'zx. 1 -rx2.m.... 1 - lw^ n Xs. 1 - m-ly y. 1 - y2... 1 - -nt~ yt where the symbol of operation Q is connected with the auxiliary symbol m in the following manner:The fraction is to be expanded in ascending powers of x1, x2,... y, y2,...; all terms containing negative powers of m are to be then deleted; subsequently, in the remaining terms, m is to be put equal to unity. Slight reflection will shew that the conditional relation will be satisfied in all products which survive this operation, and that if we can perform the operation so as to retain the fractional form we shall arrive at a reduced generating function which will establish the ground solutions and the syzygies which connect them. As a simple example of reduction which is of great service in what follows take al > 1,; Q 1 this leads to, 1- rx1. 1 - and observing that 1 1-,~ - 1 x, 1 - x~y 1 m Qi2M 1 1 we find =; >1 -x,. 1 - x y 1 - l m Q mS X1S also 1 — mx-. 1 — - yi which is the solution of aL, /, + s; THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 15 so that the solution of a, > fil Q is given by - mx,. 1 Again, if a - ii + 2, f 1 we have - -mr.-n.-1 1 -m-x,. 1 - - Yi. 1 Y2- - QZ 1 mm 1 - xm. 1 - x yl. I - xy the solution. Also the solution of a, > fi + fi2 is XI S -1 x. 1 - xy. 1 - x-Y2 Lastly, ai + a2 > f gives, by repeated application of the above simple theorem, 1 - xix2yi i - x -. - 1. - xyl. 1 - xy In general the subsequent work merely involves processes easily derivable from these cases. Particular theorems will be given as they become necessary, and for the general theory, which is here not needed, the reader is referred to Part III. of the Memoir on Partitions which may appear shortly in Phil. Trans. R.S. To come to the object of the paper I commence with ORDER 2. (n+ 1 )Cn+2 )a2 this product is an integer when n is even, but when n is uneven we must have ai > a,; and $NalNa2= Q i a 1- _,. i- - N,' 16 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS T shewing that the ground products are N1, N1N2, or (ala2)=(1, 0); (1, 1). ORDER 3. (1z + la) (nn + 2)2 (f + '' al When n is of form condition is 4m + 1, a, + 2ac3 > C (a), 4m, + 3, 2a1+ a3, c 3m + 1, a2 a3 (b), 3m + 2, a > ac2 (c). We may omit the second of these as being implied by the first and fourth and introducing the auxiliaries a, b, c in the relations marked (a), (b), (c) respectively we write down the fI function Q 1 b a,. 1 - acN. I -b. - - a bo as the expression of the sum NallNl2aN3%. It must be observed that the operating symbol Qf, has reference to each of the three auxiliaries a, b, c. These must be dealt with in the most convenient order, so that unnecessary labour may be avoided; this order is not always obvious without some preliminary experiments. In the present instance it is clearly advisable to commence with b because a occurs to the second power, and operation upon c will introduce a3. It should be remarked that operation upon one letter nmay cause two letters to vanish; this would indicate that the relations associated with these letters are not independent members of the system of relations. It does not follow conversely that if the relations are not all independent two letters must vanish as the result of operation upon some one letter. This does follow for a certain order of operation upon the letters, but not for all orders. Eliminating b we obtain 2 1 1 a 1 -acN. 1 - N2. 1 -_ 2N Ca C Observe that this expression would have presented itself if for the two relations a1 + a3 a2 ao > a3, we had constructed the sum N1ciala22 (j\T<N3)a. THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 17 The fact is that we can reduce the three relations (a), (b), (c) to two by writing a2 + a for a2, a tranformation that the relation (b) permits, and then we have to write N2NVs for N23 in the sum We next eliminate c, obtaining Q 1 i1 > 1-a aN. 1-1- N2. 1 - a2NlN2lAT3 a an expression that would have presented itself if we had been summing YN' iN22a (N N2T33)a3 for the single relation a, + 2a3 > a2, obtained from the relation al + aO3 a2, by writing a + a3 for al, a transformation permitted by the relation al > a3. The process employed is therefore equivalent to a gradual reduction in the number of the conditional relations associated with a proper transformation of the product to be summed. To eliminate a we require the subsidiary theorem IQ i__ 1 ___ 1 + 1 + xy- xyz - xyz2 i 1 - x. 1 - y. 1 - Y.yz. 1 - xz2 1 - a2x. 1- ay. I -z a and thence we derive 1i + N1N22N3 - N12N2T3 - N1N23n, 1 - N. 1 - NN,2. 1i - N1N2N3. 1 - NlV23N3 i - N2N1Y - N2N23N 1 - N12N24AT32 + Nl3N24V 39 + N1325N32 - N.1i N. 1 - N. 1 -NN - 1-NNiV 3.1 - 1-N1323NT3 In this result the denominator indicates the ground products, and the numerator the simple and compound syzygies which connect them. It is manifest that the ground products are N-, NN2,, N-N2N3, N1AT22N3, N1AT231T3 connected by the simple syzygies (A) = (NV) (N1N22N3) - (TÇ17T2) (N1N2N3) = 0, (B) = (N1) (N1N2N33) - (VN2,) (NV,2N3) = O, (C) = (NN,2N) (NN2,3N3)- (NN22N3)2' = o; VOL. XVIII. 3 18 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO and the compound syzygies (Nr) (C) - (1NT2N3) (B) = O, (NN22NV3) (B) - (NNV231 ) (A) = O; indicated by the numerator terms: -N12N22N3, - N2N1N3V, - IV12N24z2, + N3Y2V4NA32, + N13V25V32 respectively. The generating function takes also the suggestive form:1 - N ON:l3NV 1 - N. 1 - N1N,. 1 - NWN2N3. 1 - NXN23N + 1 - NN. 1 -N113.T2N23 1-nN. 1-N1N2N3. 1 -Nrl3N231N3 By proceeding in this manner we not only obtain the new ground products appertaining to the order but also those of lower orders previously obtained. It would be desirable to exclude the latter, and in the case before us we see cà posteriori that this could have been secured by impressing the additional condition al = 33; but no method, similar to this, seems to be available for an order higher than 3, as no equation invariably connects the indices of the ground products. ORDER 4. +)l (n 2a2 (f + a23) (n + 2aa I+4)~ N 4 N V N3aN When n is of the form condition is 4m + 1, a, + 2a3 > 4- 2a4, 4m + 2, a2 >a, 4m + 3, 2ax + a3 > a2 + 2a4, 3m + 1, 2 ass 3m + 2, ai + a4 > ~3 The Q function which can be at once written down is somewhat troublesome to deal with, so that I find it appropriate to divide the generating function into two parts according as ai >C3 a3 > al. Case 1. ai > a3. The conditions reduce to cal a3 (a), cl + 2a3 > a2 + 2a4 (b), C2 > a4 (c), a > a3 (d), THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 19 and it is convenient to add the implied condition ~l > (4 (e). We obtain, for ETalN2cN3a3N4a4, Q i-.i.4.-1 1 - abeNi. 1 - N 1 Y1b - N4 b ad b2ce and, eliminating d and e, this is Q 1 b1 a b-c which, eliminating c, is 1-\ N1N22N3N4 Q1 b 1-abN. 1 — N2.1 - NN.1-b N1N2N4. 1 - NN2N3N4 and, eliminating a, this becomes - i1 - N-12N22N3N4 the term 1 - N- NN22N34 disappearing. This is equal to I- V'2N'22AI3N4.1 - _2 b 1- NN2 T3N4 AT-YbN. 1 - N1N2 1 (:3 1 2 N1N2N ^ 1l b2NN2N3] - A12N2N3N41- - N 4N N N12N2'N3N4 1 -,. - N11\. 1 - N1NN2. 1 - N1N2N3N4 N1N2N f b2 1 1 - b2NIN2,3.1 - bN 1 - N - +>1 -bN.l-.12N1NN4 l-N1N2*1-NN2NN4 3-2 20 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO i i 1 - N1. 1 - VV 1 2NN. 1 - N1N2N3N4 i N1N22N3 + N1N23N3 1 - NV2. 1 - NN23.1 - AlN3N4. 1 - NT23N3 +n7Rj3N2N4 + 1- N. 1 - NN2. 1 - NNIV2N34. 1 - N13,N,2 Case 2. c<3> ai. The conditions become a3 > ca (a), 2al + a3 a2 + 2a4 (b), a2 > a4 (c), a2 a3 (cd), l + a4 %> 3 ((e); to which it is convenient to add the implied conditions al > a4 (f), 3 > a4 (g); the fi function is 1 _i ct b2e cd ab b e a 1 b Nl 1 cc 1 b"N 1 ~ bd~- N-T^2 *1N - NiN 34 g11__________ __de N d bNNN3 cl b A1- -VNN i N 2N. i- 1 1 -. 1b - L. 1 -.-eNNI b i 3N2N32NN 1bN21 b d 1 - NîNININ 4> ~ - b V 3. i - i I2. i - NNi N i NN2N N1N2N32N4 (i + N2N2N) 1 - N1N3.1 - N2N2N2. i - N,1N2NN4. i - N NNN THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 21 Hence the complete sum s N a, NZa,3a3N4a4, 1 - N1. 1 - NT2. - N1V2N3f. 1 - 1N vN4 N+ N N1N22N3 + NIVNN3 1-IVN. NiN1 2N3. i - N1Ni 2N3N4. 1 - NXNiV2N3 + XlYN13N2N4 - N1,.l -N1N2,.- NlN2NN, 1-N NN,4 N1N2NX32N4 (1 + N1N-i-13) + - N1,N2. - NiN23N3. 1 - NN2N3N4. 1 - NN221iV32iV ' and we have three ground products of order 4, viz.:N1VN32N4, and every product of order 4 can be compounded of these and of ground products of lower orders. I pause to observe that the form LN13N2N4 is one of a kind that always presents itself for an even order. The system is NJalNNa2Ni4a4vI66... NA2Sas, and may be separately examined. For the order 6 the ground products IV,2N2,N,, N14N2 3NTN, N62V, IV42N,, N18s2N,3N6, and for the order 8 NI7N2N4N6N8, NS1N2N4 T62N8, N18N23NNs8, N1ll0NT2N42N62l8, jl122N22N43N65JN3, are easily obtained. ORDER 5. We now come to a very complicated system of forms, which includes no fewer than 13 ground products of order 5. These I find to be N1N2N3 N4 5, N,NN2,32N4VN,, VN1,1V1N3N4N5, NVlN22N3NT4N5, N1N22V32N42lN5, XN1XN2Y33N42N5, NN23N32NJN5, N1N23N33N42N5, NN24N2N4N5, N12WVT2Ni7N45N5,A N13N22N3N42N,, N13N 3N,J\N 2 ll3L.24L33N^ 44F5î3. 22 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO The complete generating function can be obtained without difficulty, but, on account of its great length, I restrict my endeavours to the establishing of the 13 ground products. I find it necessary to adopt abridged notations, and in future, where it is convenient, I denote NalV,2a2i3a31AT4a4N5a, by (alC2a3a4a5). Further, if a portion of the generating function presents itself, which involves rnerely ground products already obtained in the previous work, I enclose it in brackets [ ] and thenceforward omit it. For example, I write A = [B] + C C = [D] + E E; and so on. ^ (n +1\^ fn +- ^2 fn<~a + 34 In a 5\^ For (+ 1) (-t 2) (- S ) ( 4) ( ) \-2 3 4:5 = laN2aiVyN44Na s = (ala ). When n is of form condition is 4p + 1, al + 2a3 + a5 > a2 + 2a4, 4p + 2, 2> a4,, 4p + 3, 2al + -a3 + 2a > a2 + 2x4, 3p + 1, a2 + a, > as,,3p + 2, a1 + a4 > a3, 5p + 1, a4 > as, 5pr +2, a3 > a5, 5p + 3, a2 > ac omit, 5p + 4, aa > a.; the eighth of these conditions may be omitted as being implied by the second and sixth. I separate the generating function into six portions corresponding to Case 1. a, > a, a2, 3; Case 2. ai a2, a3 > a,; Case 3. a2> a, Ca2> a, aI+a5 a3; Case 4. a2 >al, a2,>3 3 >, + a5; Case 5. a,>a, a3 >a2, alc+a,5 a; Case 6. a2 > a, cas>a2, a3 > a> + -5. THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 23 For Case 1. The conditions become i + 2a3 + a5 > 2 + 2~.......................................(a), a 2...............................................(b), 2 > 3...............................(................ ), 4 > 5............................................. (d ), a3 a5................................................(e), a > a.............................................( ), for which the generating function is Q i1 > - Cf a2e ci a - abY,.l-c N,. 1 - N,.1- N4.- 1- N aNb c a2c de which, eliminating b and c, is ~Q 1 i - a.1 -fN1N2. 1 - a2efNlN2N 3. 1 - i4. 1 - N5 and eliminating d, e, and f, 2 1i-(2211) 1-N1N2. 1 - a2NN2N3. 1 -- VN4. 1-alN. 1-a(1i111). 1-(1111) a2 Now f 1 - a2x. 1- ay.1- az. 1 - - w C 2 (f~ w }~w ^1 i - 1 a2 1- x > 1- a12x. 1- ay.1- az + 1-ay. l-az. 1 -1 + z2W + yzw 1-x.1- y. -z. 1- xw 1- xw. 1-. 1- z2w g wy2 +,>.w a 1- ay. I- az. 1 — 2- 1-xw _1 + z2W + yzw 1-x. -y. 1 -z. 1- xw 1- xw. 1 -z. - z2w + _ y îy___~2w. îy2W (y? y2 yZw) -. 1 -z. 1 - z2w.1- W 1 - y. 1 - Y2W. - _ w. 1 - XW 24 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO Hence, putting x=NN2N3, =N, z=, z=(11111), w=N1N2Nl4, we have xw = (2211), y2w = N3N2,N, z2w = (33232), yzw= (32121); and we arrive at the three ground products (11111), (32121), (33232), which, as far as this case is concerned, are irreducible. Case 2. al > a2, ca > a2. The system of conditions reduces to aI > a................................................(a ), 3 > 2...............................................(b), + a 4................................................ + a >+ a................................................( ) 4 > 5................................................(e), and 2NVlalN2aN3a3N4a^7a5 1N -Z cd b e cn c 1 - aN. 1- N4. 1 - NN4 N45 a a c -N. -c3. 1 -N4. 1 - (-. 1 N23. -11421). -4 1-N1.1-N N2N3. 1- N3N. 1(11111). 1-(11211) yielding the new ground product (11211). THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 25 Case 3. a 2> a, a2 a, c 1+a5 c3. The reduced conditions are > a................................................... (a), >2 a 3...................................................(b), Ca + 5 > c 3...................................................(c), + 2a3 + 5 2 + 2c 4.............................................(d), a, + ~4................................................... (e), 2 4................................................... (f ), 4 > g5................................................. (g ), a3 > a5...................................................(h ),................................................... ), of which the generator is Q ca I cdei l.1abf d2h eg 1 cd 1- _~ 4 1 _- AT __ a d bce 3 d1 _ 2f ghi5 bf r Q___ d2____________ d2 e 1 - bcefNn. 1 - N,. 1 - d 3.1 N4. 1 - ced N N2N3N4N l-bcefN1N2. bce d2f the result of eliminating a, g, h and i. It might be thought advisable at this stage to eliminate b or f. but experiment shews an advantage in proceeding with d. Consider P Q2 d 1 - d2. 1 - Y 1 -dzl.1 d2 d p y w > 1-y.-ZW d t _ }\ y ) -dz I w ) Q d d d p p 1 - 1 -xy. 1- zw.1- dx.1-d w 1 - xx. 1 - zw. 1 - ddz. 1 - Vd VOL. XVIII. 4 26 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO px + pz 1- x.- xy.11-zw 1- x. - z. 1- xy.1-zw pxw (1 + 0x) Pi/z x -. l - xy. 1 - z. 1 --- x2 -z. 1 -y. 1-zw. 1 - yz2 Hence the generator is:(putting p bfN, x- = N, y=.N4, z=ced (11111)) gce ce. 4. J N2 3T Q I ____ce_3 > 1 b-N fN3N,. 1 - ce(12111). -bcefNN, __+ bîeef(1-_bcef(12111) 1 1 1 - N3. 1 -ce (11111). 1 - b- N3,. 1 - bcef(l2111). 1 - bcefN, V, ____________+ __bc3 (34343) 1 c2 1 (11111) - c b Nf îV4. 1 - bcf(12111). 1 - - (22232). 1 - bcfNN,2 cf.f bfN (1 + f N2 N, ce \ ce + 1 2 1 -1 3. 1 -b -V3N4. 1 - bcef(12111). 1 - Nf23.1 -bcefNiVN2 bce bcf ce =A + B + C + D, suppose. Now A= ce 1 1 - bceNN2. 1 - bce (12111). 1 - e (12221). 1 - N bce +_____Q _ bc2eN2NV3N 4 1 bce 1 - bceNl. 1-e (1111).1 -e (12221)-.1 -1 b^ ce N2N3 {1 - N1NNl-, (11111)} > 1 - ceNN,. 1-ce (12111).1 - e (12221). 1 - VNrN. 1 - (12211) 1_Q2_ obe (2321) + > 1 - beN,2. 1-e (1111).1 - e (12221). 1 - N1N2N3 (13211) f - (121)(11111)} - 1 - (111) 111). 1 -(12211) 1 - (12221) ____+ _ (121) {1 - (121} (11111)} 1 - (11). 1- (111) - 1-(12111 ).1 (12211 ). 1 - (12221) 1 -(11). 1- (1111). 1- (111). 1 -(12221)' THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 27 a result which indicates the new ground forms (12211), (12221), (13211). B is easily shewn to have the expression (12111) (12211) (31-(1111).1-(11111) 211). B is easily shewn to have the expression 1-(11).1 —(11]).1-()1 - 1 -(11111). 1-(12111) + 1l-(lll). l-(11l). 1- (11111). 1- (12111). 1 -(122232) C, by elimination of b and c (in one operation), becomes (343) (3434 1-(4343) 1 1-(1111). 1 - (11111) 1 -_y(12111 ). 1-(l) - 1 (22232) (34453) QI./ 1 —(1111).1-(11111).l-(12221) ~ l _^(12in). -_ (232) __(34343) {1 - (415343)}.1 1 - (11).1 -(1111). 1 - (11111). 1 -(12111). 1 - (34343). 1- (33232) (46564) 1-(1111).1-(. -(11111).1 1-(1.1 11). 1-(34343) ' wherein observe that (45343)= (11) (34343), (46564) = (12221) (34343); so that (34343) is the only new ground product that emerges. Separating the numerator terms of D it can be written D1 + D2. For D1 we require the result w Q e ^ - - - z w 1- ex. 1 -ey. -. 1 - e e yw X+w X 2w2 1-y.1-xz. 1 -.1-y1- yw 1 - x.-. x - xw. - yw 1-. -. yw y2zw 1 -y. 1 - xz. 1 - yz. 1 - yw 4-2 28 MAJOR MACMAHON, APPLICATION 0F THE PARTITION ANALYSIS TO which (putting =xbcf(12111), y=bcf(ll), z =b1, w= -f N2N brings it to nQ b2f3 (131) ~~ I >1- N34. -f(12211). 1- bf3 (131). 1 - bf (11) bcf + b2f3 (14211) 1 1- N3 V4. 1-f/(12211). 1 - bf3 (1:31). 1 - bcf (1). 1 - bcf(12111) bcf Q b4f6 (28422) i - - NN. 1 -f (12211). 1 - b2f3 (131). 1 - bf (12111). 1 - b2f3 (14211) b2f4 (121)2 1 1 - b-fN 4 1-f(12211). 1 - b2(131). 1 bcf(ll). 1 -f(lll) (131) 1 - (11).1 - (11).1 - (131). 1 - (12211) + (1421{(14211) {1- (11) (12221)} + 1- 1.1 - (1111). 1 - (131). 1 -(12111). 1 - (12211). 1 - (12221) (14211)2 - (131). 1 - (12111). 1 - (12211). 1 - (12221). 1 -(14211) +1 -(11). î. __(121)2 - (11). 1 - (111). 1 - (1111). 1 - (131).1 - (12211)' yielding the single new ground product (14211). For D2 we require the result p Qi e2 1 - ex. 1 -ey.1- -z.1 - e e __p _ 2y x2 + x3w + y32z 1-xz.1-yw 1 -y 1 -y 1 -. 1-y 1-.1 w 1 -. -yz and putting p = n23N32, x=bcf(12111), y=bcf(ll), anbc N22N3, bo o~~ THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 29 and observing that we may put b=f = 1 and that moreover xz = (12111), yw = (131), py2 = (121) (131), pxy = (121) (14211), px2 = (13211)(14211), pxw = (13211)(14211)2, py3z= (121)3, xw = (14211), yz = (111), while operation upon the remaining letter c produces no new form, it is clear that no new form arises. Case 4. a2>, a 2 > a, a3, > a + a5. The reduced conditions are a2 > a,................................................. (a ), a3. >...........................................(b), 3 > ai + a4.............................................(c), 2al + 2a 5 ) 1 + 2a4............................................ (d), ai + a4 > a3..................................................(e), a, 3 a 04.................................................. (/ ), o........... (ye), a > a5.........................................( ), a >..................................................( ); leading to NV1 12Ta,3N3Na,-/4Nj5 1 IIn ______ ac = - d2eh la JT 1 bf c dN a1 N1.1 d-b 2 1 _ e3 1 e d2 N' ac d be d-f 4cgh and this by elimination of a, b, c, e, g and h becomes n d N2N3N4 Q 1 d 1 -^/(lll). i 1 1 - d2f. 1 - 1 5N, N, and since > 1 1 -fx. 1 -fy. 1 -f w 1 + yw 1-. 1-y. 1-Z.1-zw 1-y. 1-yw. 1-zw xw x 2w2 _ 1- x.1- y.1 - yw. 1 - zw _1- x. 1- xw. - yw. 1 - zw 30 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO this becomes: putting x=d2(111), y=d2(12211), z=-N2, w=2N4 NIV2NY3N4 1 1 3 1 \ d C i 2 1-dNN 3NN [1-d2 (111). 1-d2 (12211)1. i-.l-d3 _______+ _ (12221) - d2 (12211). 1 -- N (12221) (1111),1L d3 I -d2 (111). 1 - d2 (12211). 1 -d N' 4. 1 -(12221) the fourth fraction being omitted as obviously contributing nothing new. Now writing x=(111), y=(12211), =, z= w=N,2N3V4, P = 32 Y4, w n d_ _yw (1 + y2p) >1 i yi w p 1-y.l-yw.1 y3p, w 1-d2y. 1-d.1 - _ y. 1 - cyu. 1 - yx p Q_ d_ 2 _ yw (y 1 + Y2p) K xw + 1 - 3y. 1 - x. 1 y 1 - y where K1T = 1 + x +. + x2 + x + y. l+ xy2 + xy21 + (X2 + XY + y2 + x2y + xy2 + x22/2)p + ( +y +) y2 + p22) x2 d' d2 d3 + - x. 1. -x3p2. 1 -yp' 1 —xw yw ( y( 1 + y p) (1+ ) 'w -y.i + - 1-Y. 1 -yz2 - y. 1 -.1 -K xw + 1- x3p2. 1.-3 2p -1 —x-w where K= 1 +x+ y + x2+xy+y+ y + xy9+ x2y+ (x2 + xy + y, + %2y + xy2 + x2y2) p + (x2y + xy2 + X2y2) p2; d' 1- d2C 1-d. - 2y. 1 - d2. 1d THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 31 x w (1 + xz) (1 + y2p) xw (yz + i/2zp) x 1- y. 1 -xw.1- xz2.1- y3p 1 -y.1 - xw. 1 - yp2 xw (xyp + xyzp + x3p2 + x2yp2 + X4zp2 + x3y2p3 + X2y2zp2 + x_~yz2p2 + 3y2zp3) - x.1 - xw. 1 - xz2.1 - x3p2. 1 - y3p2 In verifying these laborious calculations the relations 1i_ _ 1 + dyz + d z 1-yz. - d2y. z 1- -dy. - - - 1-d2y.l d ci p + +p 1 1 + dy2 p d3 d2 d 1- -d'.1 p 1 - y32p. l - dy 1 2 1.P d3 y -Ys.l-j will be found useful. On examining these results we find that yw = (13321) is a new ground form, and that every other term is expressible by means of it and of ground forms previously reached. Case 5. a2>a, a3> a2, a+>5a c3. The reduced conditions are ~2 > cl (a), a3 > a, (b), ai + a5 > a3 (c), >1 a~4 (d), a4 a5 (e), al a a% (f); from which 1 IIQ c___________ab _ cf.1 ab b e c. 1 -1- - N. i -_ 1 —. - N4 - ' 1. - c (11111). 1 - 2. 1 — 3. 1 - d. -d(lll).l -V4 a dc cite Q dd (23422) >1 - d (12211. 1 _ d.111. 1 _ 1. 1 _. 11111).. 1 _. (11211) 32 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO (12211) (11211) f (12211) 1 - (11111). 1 - (11211) 1- (12211). 1 -(111). 1 - (llll) - (12221) ) + 1 - (12211).1 - (12221). 1 - (llll) so that new forms do not arise. Case 6. -2>1 >, c3 > c, a3 > a, + -. The reduced conditions are a3 > a2 (a), c>a+a< (b), a3 > Cl + 5 (b), 2a1 + a3 + 2a5 > ~2 + 2a (c), a2 > a4 (f), a2 + 05 3 0(3 (a, Ck~a4^c03 (f), a1 3 a, (g); leading to 1 nQ ab > 1 cf N. 1 _de 1- abc f ' c. b ac ef c2cd b4 which is readily thrown into the form eN1N3N (i-b AW2- 2N3N5) 1 - C (111). 1 -22N34. 1 _ -c N1.1. 1 - IN c b b b 62 and eliminating b this is c- c5~~~~~~c ~n c (12321) - c3 (11)(12321)2 Q c(12321) c 2 + Cf3(12211)(12321) Now Q c i 1 - C3. 1 - C2y. 1 z c2 1 iX + XJ2 + o223 1- x. -y. - - + 1- x. 1-. -. 1 - x2z3 THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 33 and Q ___________________ 1 - x.. 1 - c2. 1 - z i z - + 1-x.l-y.1 -w 1-x.-y. 1 -w. 1-wz XZ2 Z2 -+ ___ +yz 1-x.1-y.l- zz. 1-yz 1- y. 1-wz. 1-yz Xz3 + X2Z4 + 3xz5 1-. 1 -wz. 1yz. 1- x ' Putting now x=(11211), y=(111), w= (12211), z=(0111), we can examine the generating function. It is clear that xz= (12321) is a ground product. Also x2z = (12321) (13431) = (25752) is a ground product, (13431) not being a solution of the conditions. Further (12211) (12321)z = (12321) (13321), wz = (13321), (12211) (12321) yz2 = (121) (25752), (12211) (12321) xz3 = (13321) (25752); so that there are no more ground products. We have therefore in Case 6 obtained the new fundamental forms: (12321), (25752). The investigation that has been given does not establish that the 13 forms obtained are ground products quz the whole of the six cases, but it does prove that all the ground products are included amongst these 13. But it is clear that all forms in which ac=l are necessarily ground products. This accounts for 9 of the 13 and it is easy by actual experiment to convince oneself that the remaining 4, viz.:(25752), (32121), (33232), (34343), are, in fact, irreducible. VOL. XVIII. 5 34 MAJOR MACMAHON, APPLICATION OF THE PARTITION ANALYSIS, ETC. Hence the 13 ground products of order 5 are established. Finally, to resume the foregoing, it has been shewn, in respect of the arithmetical function /T + 1l /n + 2 a(n + 3 \a3 'Z + 4a4 +5a (+1) ( fl2) ') ( ~ ~) (J 5 - (al, a2, a3, (4, a5)' n being any integer whatever, that all integral forms are expressible as products of {(1) order 1, {(11) order 2, {(111) (121) order 3, ((131) ((1111) (3101) order 4, (1221) (11111) (11211) (12111) (12211) (12221) (12321) (13211) order 5. (13321) (14211) (257.52) (32121) (33232) `(34343) III. On the Integrals of Systems of Differential Equations. By Professor A. R. FORSYTH. [Received, 28 July, 1899.] INTRODUCTORY. THE present paper deals with the character of the most general integral of a system of two equations of the first order and the first degree in the derivatives of a couple of dependent variables with regard to a single independent variable, the integrals being determined with reference to assigned initial values. It will be seen that corresponding results can be established for a system of n equations, of the first order and the first degree in the derivatives of n dependent variables. When the equations are given in the form dy / \ dz dx = f=y(dx = z) then Cauchy's existence-theorem shews that, if x = a, y = b, z = c be an ordinary combination of values for the functions f and g, so that f and g are regular in the vicinity of x = a, y =b, z= c, there exist integrals y and z of the equations, which are regular functions of x and which acquire values b and c respectively when x = a; these solutions are the only regular functions satisfying the assigned conditions; and it may be (but it is not necessarily) the case that they are the only solutions of the equation (whether regular or non-regular functions of x) determined by the assigned conditions. If however a, b, c be not an ordinary combination of values, then the character of the integrals of the equations depends upon the character of the functions f and g in the vicinity. One important form, which includes a large number of cases, occurs when a, b, c is an accidental singularity of the second kind for both f and g, that is, the two functions are each of then expressible in the form P(x - a, y-b, z - c) Q( - a, y - b, z - c)' where P and Q are regular functions of their arguments, each of them vanishing when x = a, y = b, z = c. It is necessary to obtain an equivalent reduced form of the equations: and one method is the appropriate generalisation of Briot and Bouquet's method as applied to a single equation of the first order. This has been carried out in the case of 5-2 36 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF n variables by Kônigsberger*, and in the case of two variables by Goursatt. For our system, the most important reduced equivalent form is t d =aU+teV+ yt+...=01(U, V,t) /........................(A), t = U+3, + t +.. = ( U V, t)j u/t aU+/V~yt~O(UVt ~.(A), where 0 and 0 are regular functions of their three arguments each of which vanishes with U, V, t. The relations between the variables are x-a=t', y-b=(b1+U) t, z-c=(c1+V)t4, where 0, b,, are positive integers with no factor common to all three, and b1 and c, are appropriately determined constants. The new conditions attaching to the dependent variables U and V are that U = O and V = 0 when t = O; these correspond to the initial conditions that y = b and z = c when x = a; and the matter to be discussed is the determination of integrals of the equations (A) subject to the condition that U= O and V= O when t = O. The integrals, so determined, are either regular or non-regular functions of t: their existence and their character are affected by the nature of the roots of (e- 01) (- - ~&)-221 = O, which may be called the critical quadratic. Various theorems have been from time to time enunciated in various investigations. Thus Picard+ proved that the equations possess integrals, satisfying the required conditions and expressible as regular functions of t provided neither root of the critical quadratic is a positive integer; and Goursat shewed~ that, if the real parts of each of the roots of the critical quadratic are negative, then the equation possesses no integrals other than the regular functions of t satisfying the required conditions. Also Poincaré|l and, following him, Bendixson~, have discussed the integrals of n equations of the form t d = 0r (ul, 2,..., Un), (r= 1,..., n), the functions 0r being regular functions of their arguments and vanishing when ul = O, 2=O,..., u = 0: these can be made to include the system (A) by writing n= 3, and taking the third equation in the form du3 t dt = 3 with the condition that ua, u2, t13 all vanish with t. In this case, there is a critical * Lehrbuch der Theorie der Differelitialgleichtungen, 743-745; see also his Cours d'Analyse, t. II, ch. i. Leipzig (1889), pp. 352 et seq. ~ Amer. Joura. Math., vol. XI, p. 342. t Amer. Journ. Math., vol. xi (1889), pp. 340, 341. Il Inaugural Dissertation, 1879. + Comptes Rendus, t. LXXXVII (1878), pp. 130-432, ~T Stockh. Ofver., t. LI (1894), pp. 141-151. DIFFERENTIAL EQUATIONS. 37 cubic corresponding to the critical quadratic specified above; one root of the cubic being unity. But all the alternative possibilities for the general equation are not set out in detail in the memoirs specified, so that all the possibilities for the limited cubic would have to be considered independently. Again, a considerable portion of Chapter v. of Konigsberger's treatise, already cited, is devoted to the corresponding discussion for n equations; some difficulties as regards adequacy of proof of the theorems, independently of the general statement, prevent me from thinking the investigation entirely satisfactory, that is, if I understand it correctly *. Some papers by Hornt may be consulted: further references will be found in them. My intention in this paper is to take account of the different general cases that can arise owing to the various possibilities of the form of the roots of the critical quadratic. For this purpose, the method used by JordanJ for the corresponding discussion of a single equation is adapted to the system of two equations. The different cases are:I. The quadratic has unequal roots:(a) neither root being a positive integer: (b) one root being a positive integer, the other not: (c) both roots being positive integers. II. The quadratic has equal roots:(a) the (repeated) root not being a positive integer: (b) the (repeated) root being a positive integer. It should be added that a further assumption will be made for the present purpose, viz. that the critical quadratic has not a zero-root. As a matter of fact, the existence of a zero-root would imply (as for a single equation of the first order) that the reduced form of the system belongs to a type different from that here considered. * The investigation seems to imply (p. 397) that, taking are n =2, the non-regular integrals of bB cB2 1'2 + x 1dY2 ' \ X=A A 2 + -- h B' 21 —h,'2 + B-s + fz2 + '" x d=X2Y2+[x, Y, Y2]+ À 22 the unexpressed terms being terms of higher order in when the real parts of X1 and XN are positive, arend, denoting z respectively. Th ' Z, t', 12, and l, ~'2 denoting z 1, z 2 respectively. The Y_=- lx ic xl111+îvî+Àv4 12 only way in which z l can be a factor of x is by having Y2=x a zc'x2+vî + 21v j22 B=0, and then za2 is not a factor of y; and similarly as that is, x;À is a factor of Y, and xÀ2 a factor of Y2. But regards 22 and zl. the integrals of t Crelle, t. cxvI (1896), pp. 265-306, ib., t. cxvII (dx (1897), pp. 104-128, 254-266. d = Xlx + azx + bzy + cy2 + Cours d'Analyse, t. III, ~~ 94-97. z = + x + y + yy2 38 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF It is convenient to transform the variables. When the roots of the critical quadratic (- a0) (f-/3)- c1 = 0 are unequal, say e and e2, we introduce new variables u and v, such that u=XU+pUV, v=X'U+p'V, where (al - ()X + c/ar = 0 (a, - 2) X' + a21 = /1x + (a3- l) =o' ~ lx' + (2 - )2) '= O the ratios X: /L and X': L' are unequal, and consequently the new variables u, v are distinct. The equations become dv t - = 2V + 02 (u, v, t) dt = t2 +, (u t) where <^, çf are regular functions of their arguments, vanishing with them; except for a term in t, they have all their terms of the second or higher orders in u, v, t combined. When the roots of the critical quadratic are equal, having a value s, say, we introduce a new variable u such that u = xU+ V, where (al- ) X + a2u/ =, /1X + (2- ) = 0 -Then we have tdt eu-t+01(U' v, t), dt U udV- a2 t t = fu + + 0 (u, y, t), =,u~+ fV+p.,(u, V, ), say. It therefore appears that the equations corresponding to the cases I(a), I (b), I(c), are t- = u+ i(, t) dv = e.V + b2 (', V, t)) where el and 2 are unequal to one another: and that the equations corresponding to the cases II (a), II(b), are dv,, dt = fu++ cb2u v 4t)i DIFFERENTIAL EQUATIONS. 39 In both forms, the functions b, and 02 are regular functions of their arguments and vanish with them; and the only term of the first order in qb and 52 is possibly a term in t. For both forms, the initial conditions are that u= O, v = O, when t =0. For brevity, integrals, which are regular functions of t, will be called regular integrals: and integrals, which are not regular functions of t but are regular functions of quantities that themselves are not regular in t, will be called non-reqular integrals. The results are obtained for the transformed equations in u and v; since U and V are linear homogeneous combinations of u and v, the results apply to the original equations. REGULAR INTEGRALS. CASE I (a): the critical quadratic has unequTal roots, neither being a positive integer. 1. If the equations du dv A t - = e6 + 01 (u, v, t), tdt =- v + 02 (u, v, t), possess regular integrals vanishing with t, these integrals must have the form 00 00 U = Y ant, V = V bnt. n=l n=l That they may have significance, the power-series must converge; that they may be solutions, they must satisfy the equations identically. Accordingly, substituting the expressions and comparing coefficients of tn, we have (n - e1) an =fn, (n - 2) bn = gn, where f, and g, are the coefficients of tn in 01 and 02 respectively after the expressions for ut and v are substituted. From the forms of <j and b2, it is clear that f, and g, are linear combinations of the coefficients in 0j and n,, that they are rational integral combinations of the coefficients a,, a.,..., b,, b2,..., and that they contain no coefficient a after a,,_ and no coefficient b after bn,_ in the respective sets. Since neither e nor:2 is a positive integer, the equations can be solved in succession for increasing values of n, so as to determine formal expressions for all the coefficients. In particular, a,, and b, are obtained each of them as sums of quotients; the numerators of these quotients are integral algebraical functions of the coefficients in b, and 2,, and the denominators are products of powers of the quantities 40 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF To discuss the convergence of the power-series, we introduce an associated set of dominant equations. The functions Qb1 and 02 are regular in the vicinity of u = 0, v=0, t= 0; let their domain of existence include a region It j r, ulp'v <p"': of the two quantities p' and p", let p denote the smaller, so that 0i and 42 are certainly regular in a region t \i r, \ lp, v l p. Within that region, let M' denote the greatest value of | i1 and M" the greatest value of 21: of the two quantities M' and M", let M denote the larger, so that Si^ ^M, qb2 M, within the region specified, and M is a finite magnitude. Then if fijk and gijk are the coefficients of uivjtk in b1 and q, respectively, it is known that M M /fk pi+j rk' gijk < pi+j rk Further, no one of the quantities n - i, m - 2 for integer values of m vanishes; there is therefore a least (and non-zero) value of m -:i, lm- 21, for the various values of qn; let it be denoted by e. Now consider the equations EX =-t + X tk lr pi+j r1k Xi eY=-t+Y Xiyjt r p+J r" where the summation is for integer values of i, j, k such that i +j + k 2. Clearly X = Y; and each of them is given by M M_ Xi~itk EX = -t + X Xi+j tk V' pi+j rk Ir\ pI and therefore X (E+2 ( -X = M -M (1 - X) pA p/ t p r In this cubic equation, the term independent of X vanishes when t= O, and the term involving the first power does not vanish because e is not zero. Hence when t=O, the cubic equation has one root and only one root which vanishes. It therefore follows, from the continuity of roots of an algebraical equation, that the cubic has one root which vanishes with t and which is a regular function of t for values of 1 t less than the least modulus of a root of the discriminant, that is, for a finite range. DIFFERENTIAL EQUATIONS. 41 To obtain the expansion of this root as a regular function, it is sufficient to determine the coefficients in the power-series X = Ait + A2t2 +... + Antn +..., so that the equation EX = -t t Xi+jtk r pi+jrk is identically satisfied; because the root which vanishes with t is the only root of the cubic of that type, and the series for X is known to converge within the finite range indicated. Clearly we have Fn where Fn is the coefficient of tn on the right-hand side of the equation for eX. When this value of An is used for successive values of n, and the new expressions for A1,..., An_ are substituted in Fn, the ultimate formal expression obtained for Fn is the M quotient of an integral algebraical expression in the coefficients i+ k by a power of e. Pi+jrk Comparing the quantities \fin and Fn, we note that a quantity greater than fnl is obtained when in its numerator every term is replaced by its modulus; that this greater quantity is further increased when the modulus of the coefficient of uivjtk in 01 or in cp2 is replaced by i+j k; and that this increased quantity is still further appreciated when every factor of the type i - 1 in the denominator is replaced by e. But, on comparing the two coefficients an and An, it is clear that these three changes turn fn into Fn; accordingly fif, <E,. Similarly for gn and Fn, so that Ign < Fn. Also |n - f | e - e31 > e; hence ia n < An, ibn <A,,. The series At + At2 + At3 +... converges absolutely within a finite region round t=O; therefore also the series a1t + at2 + a at3 +..., blt + b2t2 + b3t8 +..., converge absolutely within that region. Hence the differential equations possess regular integrals which vanish with t. It is not difficult to prove that they are the only regular integrals which vanish with t. VOL. XVIII. 6 42 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF CASE I (b): the critical quadratic has unequal roots, one of which is a positive integer and the other of which is not a positive integer. 2. The equations may be taken in the form dt = mu + at + (u, v, t)) dv td = v + 't + y (, V, t) where m is a positive integer, e is not a positive integer, 0 and 0 are regular functions of their arguments, vanishing with u, v, t, and contain no terms of dimensions lower than 2. If regular solutions exist, which vanish with t, we can take u = t (\ + '11), v = t (p~ + vi), choosing the constants X and ui so that ul and v, vanish with t. Then t2 is a factor of 0 and b after this substitution is made, say 0 (u, v, t) = t2'1O (u,, v1, t), b (u, v, t) = t21 (u,, yI, t); but 0,' and b,' do not necessarily vanish when t, u,, v, vanish. The equations for the new variables are t l = (m - 1)X+a + (n- l) u, + t'(ul,, v, t) dvl dt tc = ( - 1)f++t (- 1)v,+tV,' (t,,,1, t) Now as u,, v, are regular functions of t, the expressions on the left-hand side vanish when t=O; hence (m- )X+a= 0, (-1)~ +/ = 0. If 01'(0, O, 0)=a, 0,'(0, 0, 0)=,, the equations are t dt' = (m-1) 4 + at ^ + toi (u, yv,, t) dv t d= ( - 1), + 3,t + to,(l,, v,, t) where 0, and (, are regular functions of their arguments and vanish when u =O, v =O, t = 0. The equations are, in form, the same as before, except that the coefficients of the first power of the dependent variables on the right-hand side have been reduced by unity; and the relation between the two sets of dependent variables is t6=t - 1+ ), v=t -.l+V).V mz - 1 ~:e-1 It is manifest that the equations in ul and vy can be subjected to a similar transformation with a corresponding result; and that, as m is a positive integer while t is DIFFERENTIAL EQUATIONS. 43 not, the process can be carried out m-1 times, but not more. Denoting the dependent variables after all these transformations have been effected by u', v', we have equations in the form du' t t= u + at+h (', v, t) dv' t -v = /v' + bt + k (u, ', t) where /c, = - m +1, is not a positive integer; h, k are regular functions of their arguments, vanishing with t and containing no terms of order less than 2. The relation between the variables uz, v and u', v' is of the form U = Pm-i + tm-U', V = Qn-i + tn-Vl, where Pm_- and Qm-_ are algebraical polynomials of degree m -1 vanishing with t; and u'= 0, v'= 0 when t= O. The coefficients a and b are algebraical functions of the original coefficients. The equations can possess regular integrals only if a is zero. For regular integrals must be of the form u =p pt + pt+..., v' = qt + qt2+...; substituting these, remembering that h and k are then of the second order at least in t, and equating coefficients of t in the first of the equations, we must have Pi = pl + a, which is possible for non-infinite values of pi only if a is zero. Suppose now that a is zero. Since u' and v' (if they exist as regular functions of t) vanish with t, we can assume Zf'= tvl,, = t%2, the sole transferred condition being that w7l and %2 are regular functions of t, which now need not necessarily vanish with t. We have t2 =h (tl1, tq2, t)=t2H (wl, 7; t), t2 d= (, - 1) tr2 + bt + k (tW,, tq2, t) t^^ t^t^+bt+2^,,t) =(,c-l ) tw72 + bt + t2K (w1l, w)2, t), where HI and K are regular functions of their arguments. The second equation shews that, when t = O, then (c - 1), + b = 0; accordingly taking b 72 = i- K+ 2 6- 2 44 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF we have r2 vanishing when t =. As regards r, there is, as yet, no restriction upon its value when t= O; denoting it by A, we take = A + gi, where i vanishes when t =. Both, and r2 are regular functions of t. When the values are substituted, A remains undetermined by the equations; and therefore an arbitrary (finite) value can be assigned to A. The equations for r; and 2 now are t d- tH / ~A+, t) A)f t d = -1) 2+tK (A+, i + b tj dt- = with the condition that ~ and ~2 must be regular functions of t vanishing with t. Let them, if they exist, be denoted by = SE antn, 2= E bStn; n=1 n=1 substituting in the equations which must be satisfied identically, and equating coefficients, we have relations nan =f/, (n- Kc + 1) bn = gz, similar to those in the Case I (a). These equations are treated in the same way as in the Case I (a). Since fc is not a positive integer, no one of the coefficients of bn vanishes; and thence it is easy to see that the whole of the treatment in I (a) subsequent to the corresponding stage can, with only slight changes in the analysis, be applied to the present case. It leads to the result that the power-series for ~i and 2 converge absolutely for a finite region round t=O; and from the form of the equations for ~ and ~, it is clear that the coefficients in the power-series will involve the arbitrary constant A. Hence it follows that, unless the condition represented by a= O be satisfied, the equations do not possess regular integrals vanishing with t = O. If that condition be satisfied, the equations possess regular integrals vanishing with t =O and involving an arbitrary constant: in other words, they possess a single infinitude of regular integrals vanishing with t = O. The condition represented by a = can be obtained from the original equations du =mu+ot+0(u, y, t td- = u + at + 0 (u, v, t) pcii as follows. Let m - 1pt+ p=l v= E cp tP + lt V; p=l DIFFERENTIAL EQUATIONS. 45 substitute in the equations, and determine (by comparison of the coefficients) the values of fi,. _l, g_,..., gl. Then the condition is that the coefficient of tm in /m-1 m-1 at + 0 ( fptP, g tP, t) -p=l p==l g shall be zero. This statement can easily be verified. CASE I (C): the critical quadratic has unequal roots, both of which are positive integers. 3. Let m and n be the two unequal roots, of which m is the smaller, so that the equations may be taken in the form dcv td = nv + ct + 0 (u, v, t), tdt = ~' + fit + ~b (uyt These equations can be transformed, as in the Case I (b), by m - 1 substitutions in turn; and ultimately they acquire the form t d = + at + h (u,, v' t) t -= CUv + bt + k (W, v, t) where K, = n-m + 1, is a positive integer greater than unity, u' and v' are regular functions of t vanishing when t = 0, and the functions h, k have the same signification as in I (b). If the equations possess regular solutions, the latter must be of the form -'= 2; pltl, v'= qitl; 1=1 I=1 substituting these values and equating coefficients, we have pl=p +a, q = cq + b, (l - 1)pi = coefficient of tl in h (u', v', t), (l- c) ql=............. tl in k (O', v', t). It is clear that, if a is different from zero, the first equation cannot be satisfied; and therefore as one condition for the possession of regular integrals, a must be zero. Assuming this satisfied, we see that p' is left undetermined: let a value a, provisionally arbitrary, be assigned to it. Solving now the remaining equations for the values 1= 1, 2,..., c - 1 in successive sets, each set being associated with one value of l, we have the values of pi,..., p~-_, q,q..., q"_c; all these in general involve a. In order that q" may have a finite value 46 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF so that (i - Kc) q vanishes for I = fc, we must have the coefficient of tK in k (u', v', t) zero. If this coefficient be zero, the value of qK is undetermined; let a value f/, provisionally arbitrary, be assigned to it. For the remaining values of l, the equations determine formal expressions for the remaining coefficients, involving a and /: and no further formal conditions need to be imposed. When the values of _p,..., p,_, q,..., q,- are inserted in k (u', v', t), the coefficient of tK in that quantity may be an identical zero; in that case, the functions u', v' involve two arbitrary constants a and f3 so that, if the functions actually exist, there is a double infinitude of regular solutions vanishing with t. Or the coefficient may be zero only if some relation among the constants of the original equations be satisfied; if the relation is not satisfied, there are no regular integrals of the original equations vanishing with t: if the relation is satisfied, there is a double infinitude of regular integrals. Or the coefficient may be zero only if some relation among the constants of the original equations and a be satisfied; this relation is then to be regarded as determining a, and then for each value of a so determined, there is a single infinitude of regular solutions vanishing with t. These results are stated on the assumption that the power-series, as obtained with the coefficients p and q, converge: the assumption can be justified as follows. Let AK._ = p1t + p2t2 +... + p,-lt —1, BK_ = qt + q2t2 +...+ q-_lt-l, the coefficients p and q being known; then if functions u' and v' exist of the specified form, we can assume u; = A_-1 + t-1 U', v' = BK-1 + t<-1V', where U' and V' are regular functions of t that vanish with t. Thus, assuming a=0, we have t dt- + tK dt + (K - 1)tK-1U' = A,, + tK-1U' + h (A _- + t-1,U BK- + tK- V t) Now the quantity dA _ t dt 1-AK-1 dt is equal to the aggregate of the terms involving t, t2,..., tK-l in h (AK1, BK-, t). Also in h (u', v', t) there are no terms of dimensions lower than 2 so that, in h (AK_1 + tK-1 U, BK-_ + tK- V', t) -h (AK1, BK-1, t), the coefficients of tK- U', tK- V' are of dimension at least unity, and therefore this expression may be taken as equal to tK- Hi (U', V', t), DIFFERENTIAL EQUATIONS. 47 where H1 is a regular function of its arguments, which vanishes with them and contains no terms of dimension lower than 2. Also let the terms in h (A,_1, B,_,, t) of order higher than K - 1 be cKtK + cK+1 t+ +...; then dU' t - + (c - 2) t-U = cKt" + c,,+,t"+ +... + t-1 H ( U', V' t), and therefore au' dt = (2 - ) + + H(UI, V', t), on absorbing the other powers of t into H1 and denoting by H the new function which has the same character as H1. Similarly d V' t += V'+ bKt+K( U, V',t), where the terms in k (AK_1, BK_i, t) of order higher than K - 1 are bKtK +..., and K is a function of the same character as H. As K is a positive integer > 1, 2 - c is not a positive integer 1. Thus the coefficient of U' is not a positive integer, while the coefficient of V' is unity; and thus the two equations for U' and V' are a particular instance of the general form discussed in I(b). There is no regular integral vanishing with t unless b" = O; the significance of this condition, either as an identity, or as a relation among the constants of the original equations, or as an equation determining a, has already been discussed. Assuming the condition bK = O satisfied, it is known from the preceding result that the equations in U' and V' possess regular integrals, which vanish with t and involve an arbitrary constant that does not appear in the differential equations. The inferences stated earlier are therefore established. It appears from the investigation that two conditions must be satisfied in order to the possession of regular integrals: one of them is a relation among the constants of the equation represented by a = 0: the other of them is the relation represented by b, = O. To obtain them directly from the original forms, we can proceed as follows. Let n = pIt, v= qltl, 1=1 1=1 be substituted in the original equations: and determine p,..., pi-_, q1,..., q_. The first condition is that the coefficient of tm in / (-1 -1 \t 48 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF shall vanish. Take pm =a; and then from the original equations determine p,+i,., n-n, qm,..., qn-. The second condition is that the coefficient of tn in 1n-1 n-1 \ 4, -plt\, Yqi t\t) 1=1 1=1 shall vanish. It is not difficult to verify these statements. Summarising the results, it appears that, unless one condition be satisfied, the equations possess no regular integrals vanishing with t. When the condition is satisfied, another relation must be satisfied. If this relation determines a parameter, the equations possess a single infinitude of regular integrals; if it involves only the constants in the differential equations, then, when it is not satisfied, there are no regular integrals vanishing with t: and, when it is satisfied, there is a double infinitude of such integrals. CASE II (a): the critical quadratic has equal roots, not a positive integer. 4. The equations are du t d = eu + 01 (u, V, t) dv t dv= KU+ ev+ 2, (u, v, t) where e is not a positive integer; the functions <0 and 02 are regular and (with the possible exception of a termin t) contain no terms of order lower than 2. If they possess regular integrals vanishing with t, they must have the forms U =, pntn, v = 2 q.tn. n=l n=l Substituting these expressions and equating coefficients, we find (n-_)pn=fn (n - ) qn = gn + KPn where f, and gn are the coefficients of t" in <0 and <k2 respectively, when the series for u and v are substituted. It is clear that fn and gn are linear in the coefficients of ~< and (p, that they are integral algebraical combinations of pi, p,,..., q 2, q,..., and that they contain no coefficient p or q in the succession later than pn- and qn-. As e is not an integer, the foregoing equations, taken for successive values of n, determine formal expressions for the whole set of coefficients p and q; in particular, p, and q,, are obtained as sums of quotients, the numerators of which are integral functions of the coefficients in q, and b2,, and the denominators of which are products of powers of the quantities 1-r, 2-,..., n-. To discuss the convergence of the power-series for u and v with these coefficients, we DIFFERENTIAL EQUATIONS. 49 introduce an associated set of dominant equations. Let a common region of existence of fb and P2 be determined by the range | u\<p, |v |<o-, | |t r; within this region let the greatest value of i be M, and that of 1 21 be N, so that within the region 1 + 1 I, l -YlN, M and N denoting finite magnitudes. Also, let e denote the smallest value of | m- for values of the integer m; and let c = c. Then we consider the dominant equations given by * eX Mt M eX =-M t + if XiYjtk eY = cX + - t + i X YJtk, r pi J rk where the summations on the right-hand side are for integer values of i, j, k such that i+j +k2. The general course of the argument is similar to that in I (a). In the first place, X and Y can be determined by the simultaneous equations eX = M -- X-Y ( r) ( p( ) P (l- E(\ r) (\ p) ) (i N N N Y=cX+- - N- -— Y -- y P From these we have NX =M(eY- cX), so that Y=X ( e when this value is substituted for Y in either equation, say in the first, we have / M N cM\^(. X\ N X N c e+- +-+ X - 1 — )1 -- + -} p -a 6e P - M --— r-M t p r a cubic equation in X. The term independent of X vanishes when t = 0; and the term involving the first power of X does not vanish when t = 0, because e is not zero. Hence the cubic has one (and only one) root vanishing when t=0. VOL. XVIII. 7 50 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF It follows, as before, that this root of the cubic vanishing with t can be expressed as a regular function of t in the form of a power-series, which converges absolutely for values of 1t1 less than the least modulus of a root of the discriminant, that is, for a finite range. When the power-series for X has been obtained, the power-series for Y is given by say X = Pt + P2t2 +... + Pntn +... Y= Qt t+ Qt2t2+.. + Qtn'+ In the second place it can, as before, be shewn that the analysis, effective for the determination of pn and qn in connection with the original equations, is effective for the deterinination of P, and Qn in connection with the dominant equations by merely making literal changes, and that these literal changes are such as to give Pn < Pn, I qn < Qn for all values of n. It therefore follows that the series plt+p2t + p3t +.... qlt + q2t2 + q3t3 +..., converge absolutely within a not infinitesimal region round t = 0. Consequently the equations possess regular integrals vanishing with t: and it is not difficult to prove that these regular integrals are unique as regular integrals with the assigned conditions. CASE II (b): when the critical quadratic has equal roots, the repeated root being a positive integer. 5. The equations are du t Ku = m v + + v + (t + (,,t) tdt =ilu+mv+j3t+ (U, v, t)j where mn is a positive integer, the functions 0 and b are regular, vanishing with u, v, t, and containing no terms of dimensions lower than 2. We transform the equations as in I (b) by successive substitutions, each of which leads to new equations of a similar form with a diminution by one unit in the coefficients of u and of v after each operation. We take u = t (X + U1), v = t (,/ + vI), choosing X and pU so that u1 and vI vanish with t: then ul and vI are regular functions of t, if the equations possess regular integrals. To secure this form of transformation, we must have (m- 1)X +a=0, kX + (w - 1) f + /3= 0, DIFFERENTIAL EQUATIONS. 51 so that q(-1 ',(m 2 n-1; and the new equations are t -= (m- 1) u, + a't + 01 (u1, vI, t) dt dvI t = 'czul + (m - 1) v + 3't + c (u1, V1, t) A similar transformation can be effected upon this pair, with a similar result; and the process can be carried out m- 1 times in all, leading to equations du' tt -= u+ at + h (u', v', t) dt t -T- = Icu + v' + bt + k (u', v', t) where h, k are regular functions, vanishing with u', v', t, and containing no terms of dimensions lower than 2; also u', v' are to vanish with t. There are two sub-cases to be considered, according as K is zero and Kc is different from zero. First, let Kc be O; so that the equations are du' t - t = + at + h (u', v', t) dv' t =v' + bt + k (u', v', t) It is easy to see, by substituting expressions of the form u' = plt + t2+..., v'= q1t + q2t2 +... that the equations cannot possess regular integrals vanishing with t unless a=O, b=O. Assume, therefore, that a = O, b = O. If the equations then possess regular integrals vanishing with t, we can take u'=tU', v'=tV', where now the only transferred condition to be imposed upon U' and V' is that they are to be regular functions of t. Substituting these values, we find dU' t2 -= h (t U', t V', t) = tH ( U', V', t), dV' t2 d = k (tU', tV', t)= t2K ( U', V', t), 7-2 52 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF so that dU' dV' dt= H (U',, t), = K (U', V', t), where H and K are regular functions of their arguments. To these equations, Cauchy's general existence-theorem can be applied; it shews that they possess integrals which are regular functions of t and assume assigned (arbitrary) values when t= O. Accordingly, the equations in u' and v', in the case when the conditions a=O, b=O are satisfied and when the constant K is zero, possess a double infinitude of regular integrals which vanish when t = O. Secondly, let K be different from zero. If the equations possess regular integrals, they are expressible in the form u'= aLt + a2t2 +..., v' = b+t + b2t2 +...; substituting these, and taking account of the first power of t on the two sides of both equations, we have aL = a, + a, b1 = ca, + b + b. Hence we must have a =; then b1 is undetermined, and b al = - - a finite quantity because K is not zero. Assuming that the condition a= O is satisfied, and assigning an arbitrary value A to b1, let (=t ( —+ ), v'=t(A +2), /C so that?,7 and r2 are to be regular functions of t vanishing with t; the equations for?, and 2 are -td = h -t- + t,, tA+tq, t) = t2H (i, 2, t), t2 2= Ktvl+k(- t-+tgl, tA + t2, t) dat \ /C tdc~ / = tctr + t2K(7q1, 2, t), that is, they are t = tH(l1, 2, t) dt t d= 71q + tK(7l, 12, t) where H, K are regular functions of their arguments and involve the arbitrary constant A. DIFFERENTIAL EQUATIONS. 53 These equations are now the same as in the Case II (a) when e is made zero. Accordingly, all the analysis of that earlier discussion applies when in it e is taken equal to unity. The equations in Wq and %2 possess regular integrals vanishing with t, and their expression involves A, the arbitrary constant; and therefore the original equations in u and v possess no regular integrals vanishing with t unless the condition represented by a = O be satisfied; but if that condition be satisfied, they possess a simple infinitude of regular integrals vanishing with t. The conditions represented by a= O and b = O in the sub-case when K is zero, and the condition represented by a = O in the sub-case when K is different from zero, can be expressed as before. For the former sub-case, we determine coefficients a and b so that u= ait +... + a- tm-l+...) v = bit +... + b,_ tm-l +...J satisfy the equations du7 t = mu + at + 0 (u, v, t) dv t - = mv + /3t + (u, v, t) and the conditions are that the coefficient of tm in (m-1 m-1 \ at. 0 Y. ait\, 2 bltl, t, I=1 1=1 and the same coefficient in (m-l m-1 t it + E ait, 2 bitl, t), 1=1 l=1 l shall vanish. For the latter sub-case, we determine the 2 (mn-l) coefficients in u and v so that the equations du t = mu + at+ (u, v, t) dt dv t = KCu + mv +,Bt + b (u,,t) are satisfied; and the single condition is that the coefficient of ttn in /m-l -l m at+k 2 altl, bitl, t) 1=1 1=1 J shall vanish. This completes the discussion of the regular integrals vanishing with t, with the respective results as enunciated in the various cases. 54 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF NON-REGULAR INTEGRALS. 6. It has been seen that, either in general or subject to certain conditions, the equations du tddt = al U + l,1 +C t Y t..*= 01 ( U V, z) t d= aU +/3,V+yt +... = 0(U, V, z) possess regular integrals which vanish with t: and these are unique as regular integrals. Denoting them by ul, v1, let U=u + x, V=v + y; so that if functions x and y exist, different from constant zero, they are non-regular functions of t, and they must vanish with t because U, u1, V, v1 all vanish with t. Then dx t = (ul + x, v1 +y, t)- 01 (ul, v, t) =! Â^y aw ^(l,- ) V,Iï n(iau,~ av) y t n i (X +y a) 02 (1, VI, t) are equations to determine x and y. On the right-hand sides there are no terms involving t alone; the only terms of the first order are alx + /y, a2x + 2y respectively; and the coefficients of the other powers of x and y are functions of t and of u1, v1, that is, after substitution of the values of u1, vy, these coefficients are regular functions of t. Hence we may take the equations in the form tdt = alx + /31y + à (x, Y, t) dy t=dYa2x-4- /32y + s1(xy, t) ) where àI, and 42 are regular functions of x, y, t, vanishing when x = 0, y = 0, and containing no terms of dimensions lower than 2 in x, y, and t. The dependent variables x and y, if they exist as other than zero constants (which manifestly satisfy the equations), are to be non-regular functions of t which vanish when t = 0. It is convenient to transform the equations by linear changes of the dependent variables, as was done for the discussion of regular integrals: the new forms depending upon the roots of the critical quadratic (F- a) ^ - Pz)- aAP1 DIFFERENTIAL EQUATIONS. 55 When the roots of the quadratic are unequal, say e and e2, we take new variables t, = Xx + xy, t2 = 'x + /y, where (ai-^+ap ==0l (a,- ~)X' + ~/'= O0 (2l-1) X + a22/ X2 + a,/~,} /1X + (,2- 1 =) O ' AX + (1 - ) / = oJ ' the equations become dt, t = lti + 1 (t,4 t2, t) Ct t t-2 =2t2 + 02 (tl, t2, t) where the regular functions b, and b2 vanish when tl=O, t2 =O, and contain no terms in ti, t2, t of dimensions lower than 2. When the roots of the quadratic are equal, the common value being:, the corresponding forms are t- = et, + 0 (ti, t,,t) ) tdt = t2+b +(ti, t2, t) dt with the same characteristic properties of the functions qb2 and 02 as for the former case; here t2= y and tl= x -+ y, where (a, - ) x + a2 = 0, flx + (8, - ) = o, and the constant Ic is given by cX = a2. We proceed to deal with the various alternative cases, as for the regular integrals: merely remarking that, for those instances of the original equations which do not possess regular integrals because the appropriate condition is not satisfied, it will be necessary to return to those original equations for the discussion of the non-regular integrals. 7. Some indication of the character of the solutions may be derived from the consideration of two simple examples, one of each form. A simple example of the case when the roots of the critical quadratic are unequal is dt\ t dt = Xt2 + att2 cdt dt2 =,Lt2 + b1tt, integrals (if they exist) are required which vanish when t= O. The solution of these equations, which are linear, can be made to depend upon that of a linear equation 56 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF of the second order having t=O for a singularity: it appears that the integrals are normal in the vicinity of t=O. Their full expression is t, f + abt2 - (abt2)2 2 A ( + p) 2.4(1 + p) (3 + p) + a Btb+l i1 + abt2 (abt2)2 -P ( 2(3-p) + 4(3 -p)(5 p)+- p t- At~II{ 1]+ abt2 (abt)2 ) 1 2(3 + p) 2.4(3p)(5 + p )(5+P)- } tr = + —7 +abt { (abt2)2 ~ + Bt 1 + 2 (1 - p) 2.4(1 -p) (3 - p) where p = X-: in order that the solution may be satisfactory, it is manifest that p may not be an integer, positive or negative. For the present purpose, the general integrals must be chosen so that they vanish with t; and consequently the most important terms in the immediate vicinity of t= O are t, = At' +a Bt+l i-p [^ t2 =1 At^+ +Btij the quantities A and B being arbitrary. If the real part of X and the real part of tu be both positive, then, when the variable t approaches its origin not making an infinite number of circuits round that origin, t, and t2 ultimately vanish when t= O, that is, as X and /L are not integers, there is a double infinitude of non-regular integrals vanishing with t. If the real part of X be positive and the real part of /u be negative, then, when t tends to zero as before, t2 can tend to zero only if B be zero: and if B =, then t, and to ultimately vanish when t= O, that is, there is a single infinitude of nonregular integrals vanishing with t. Similarly, if the real part of X be negative and the real part of IL be positive, there is a single infinitude of non-regular integrals vanishing with t. If both the real part of X and the real part of /u be negative, then t, and t2 vanish with t only if A = O, B= 0: that is, non-regular integrals vanishing with t do not then exist. This last result is in accordance with Goursat's result already quoted in the introductory remarks. It will be noticed that the parts depending upon t\ alone, when they exist, are of the form tl = tApl, t2 = tp2 DIFFERENTIAL EQUATIONS. 57 where pl is an arbitrary finite quantity and p2 is zero when t =O; and that the parts depending upon t' alone, when they exist, are of the form tl = t'7l, t2 = t[12, where a-2 is another arbitrary finite quantity, and o- is zero when t= O. These particular results are general and, in this form, can be established by an appropriate modification of Goursat's argument (I.c.). They are included in the more general theorems that will be considered immediately. A simple example of the case when the roots of the critical quadratic are equal is t X- tiî + attI dt t d= Kct + Mt2 integrals (if they exist) are required which vanish when t = O. The solution of these equations can, as for the preceding example, be made to depend upon the solution of a linear equation of the second order, having t = 0 for a singularity; and their expressions can be obtained in the form tl= Aat"+l(1 + ~aKt +...)+ Ba aKttX+ (1 + aaKt +...) log t+ (1 - ia22t2 -...) tJ}, t = At (1 + aKt +...) + B {actÀ (1 + agt +...) log t + (ac - a2t-...)tx}. When the real part of X is positive, these integrals vanish with t; and there is a double infinitude of then. When the real part of X is negative, then it is necessary that A and B both vanish: that is, the integrals do not exist if they are to vanish with t. When B is zero, then the integrals become of the form ti = t"pi, t2 = txp2, where p2 is an arbitrary finite quantity, and pl is zero when t = O. This result is general. There is no corresponding simple inference from the parts that depend solely upon B: the complication is caused by the term /Kt, in the second equation. The special results here obtained are included in the theorenis relating to the equations in their general form: they suggest that integrals exist which are regular functions of t, tx, and t" log t, when the real part of X is positive. VOL. XVIiI. 8 58 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF CASE I (a): the critical quadratic has unequal roots, neither of them being a positive integer. 8. It has been proved that the original equations in this case possess regular integrals vanishing with t: and therefore, in order to consider the non-regular integrals (if any) that vanish with t, we transform the equations as in ~ 6, and we study the derived system t - = ti + l(t, t2, t) dt2J t = 2t2 + 2 (tl, t2, t) where 01 and ^0 are regular functions of their arguments, vanishing when t =O, t2 = O, and containing no terms of dimensions less than 2 in t1, t2, t. The integrals t1 and t2 are to be non-regular functions of t, required to vanish with t. The main theorem is as follows:When the roots of the critical quadratic e and e2 have their real parts positive, and are such that no one of the quantities (X-1)l +/z2+ V, +x+(g - 1) 2+ V, vanishes for positive integer values of X, i, v such that X + ~+ v 2, then the equations possess a double infinitude of non-regular integrals vanishing with t, these integrals being regular functions of t, t6e, t$2. Immediate corollaries, when once this theorem is established, are as follows:If the real part of e be positive and that of e be negative, there is only a single infinitude of non-regular integrals vanishing with t: they are regular functions of t and tal. Likewise, if the real part of, be positive and that of e be negative, there is only a single infinitude of non-regular integrals vanishing with t: they are regular functions of t and te2. If the real part both of el and of e2 be negative, there are no non-regular integrals of the equations that vanish with t. These results (the last of which is due to Goursat in the first instance) will be found sufficiently obvious to dispense with any proof subsequent to the establishment of the main theorem. 9. In discussing the equations, it will be convenient to replace te' and tt2 by new variables, say t t= bz, t2= Z2, so that, by the general theorem, regular functions of Z1, Z2, t are to be established as DIFFERENTIAL EQUATIONS. 59 solutions of the equations. Accordingly, regarding t, and t2 as functions of these three arguments, assume tl = - m'np zlmZ2ntP' t2 = Zbnnp zjmZ2LtPJ where the summation is for all positive (and zero) values of the integers m, n, p, with the conventions a000 = O, boo0 = O. Moreover d a a a tdt at azx, a+ 2 Hence the differential equations are at at, at, t l + lZ + e2 | = ^tl + Sb0(t0, t2, t) at _z, az, at + lZ1at + 2Z2 - 2t2 + b2(tl, t2, t) Substituting the assumed values of t, and t2, and afterwards equating coefficients of zlmz2ntP, we have {(m- 1) i + ne2 + } amnp = =amnp. {n., + (n - 1) 2 +p} bmînp = 3'mp.mp where 'mnp is a rational algebraical function of the coefficients in 01, of the coefficients am',,,,p in ti such that m' m n, n' < n, p' p, rm' ++ n' + p' < n + n + p, and of the coefficients brn'np, in t2 with the same restrictions: and likewise for 'mn7p in relation to b2. As there is no term in bl (t1, t2, t) of dimension unity in t, t 2, t2, there can be no term of dimension unity in zl, z2, t after substitution of the values of t1 and t2: hence {(m - 1) el + n2 + p} tmnp = O, when m + n + p = 1. Accordingly a010 =, acOO = 0; but there is no limitation upon a100, so that it can be taken arbitrarily: we assume a100 = A. For similar reasons {mel + (n - 1) 2 + p} bn2p = 0, when m+n+p=l; and we infer that b100 = O, boo0 = O, bo0o = B, where B is arbitrary. 8-2 60 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF Suppose now that no one of the quantities (m-l)el+re+,+p, ml+ (n- 1) 2+p, for positive integer values of m, n, p such that mn + n +p > 2, vanishes. Then when the equations (m - 1) el + ne2 + p} acmnp = mnp {me, + (n - 1) e2 + p} bmnp = / mnp J are solved in groups for the same value of m+n -+p, and in successive groups for increasing values of z +n+p beginning with 2, they lead to results of the form amnp = ~mnp, brnnp = 3mnp> where annp, l3mnp are rational integral functions of the coefficients that occur in jb and 02, these functions being divided by a product of factors of the forms (m - )8 ++ n9+p, mi +(n - 1) 2 + p, for m n + p >2. It has been seen that aol,=0, bool= 0: we easily see that aoop=0, boop= for all values of p. For every term in b1 (t1, t2, t) and every term in 2 (t2, t2, t) involve ti, or t2, or both: and the equations for aoop, boop are (p - e1) aoop = Aop, (p - 2)) boop = Bop, where Aoo, Boo are integral functions of the coefficients in (l and 2,, and of coefficients a,,,, b,,, such that p' <p, these integral functions being divided by factors of the form p'- e, p'- 2. No term occurs either in Aoop, Boop independent of aop,, boop, because there is no term in b1 or in 02 independent of ti and t2. Hence if all the coefficients aop,, boop, vanish when p'< p, then aoop, boop also vanish. But a,0 = 0, boo0 = 0: hence a,02 = 0, b002=0: and so on with the whole series. Consequently in the expressions for t1 and t2, there occur no terms that involve t alone without either z1, or z2, or z1 and z,: which is therefore one general characteristic of the non-regular integrals if they exist. From t, and t,, let all the terms which do not involve z, be gathered together. By what has just been proved, there are no terms which involve t alone: hence the aggregates of the selected terms contain zl as a factor, and the aggregates of the remainders contain z, as a factor, so that we can write t, = z1p + Z201, t2 = Z17 + z2092, where p and T are regular functions of t and z1, which will be proved to be such that p=A, 7=0, when t=O, A being an arbitrary constant: and (), 02 are regular functions DIFFERENTIAL EQUATIONS. 61 of t, OZ, z2, which will be proved to be such that 01=0, 02=B when t=0, B being an arbitrary constant. The first stage of the proof will establish the existence of the parts zp, z-r: the second stage will establish the existence of the parts z^1, z2^2. It may be added that, had it been deemed desirable, a selection from t, and t2 of terms that do not involve z, might first have been made: the forms of ti and t2 would then have been tl =- 2pi + Zii, t2 = 2Ti7 + Z,19, where pl-=0, -T1 = B when t = O, and pl, ri are regular functions of t and z,2: also '1 = A, *.2 = 0 when t=0, and Ti, I2 are regular functions of t, zl, z. Further, it will be seen from the forms of the functions that p, T, i, T3 all vanish when A =0: and that 01, 03, pl, rT all vanish when B=O. 10. It is clear that if the equations under consideration possess integrals of the form tl = pzX, t2 = TZ1, where p and T are to be regular functions of z and zl, then, taking account of the forms of 1i and 02, the quantities p and T must satisfy the equations ta ap = t d pt) aT ar dr t + el - =t (2- )T + (p 1 t) The functions k1i '2 are regular in their arguments: both of them vanish when p =0, =0: in each of them, every term, which is of dimensions X in p and T combined, possesses a factor z,-1: and no term is of dimensions less than 2 in p, T, t combined. Because p and r are to be regular functions of t and zl, they will be expressible in the forms p = ZEkmnzitn, T = lmnnzf, ntîl; substituting these values and equating coefficients on the two sides of both equations, we find (n + mlf2) cn,, = Fmn ) {n + (m + 1) ei - e2} mln = l mn where k'n, and l'mn are linear in the coefficients of frî and 42 respectively, and are rational integral functions of the coefficients km',', I,' in p and T such that m'ém, n'én, m' + n'< m + n. From the forms of the functions ~1 and f2, we have k'oo=, 1'oo=0. Hence when m = O0, n = 0, the first of the coefficient-equations leaves k00 undetermined: we therefore make it an arbitrary (finite) quantity A: the second of the coefficient-equations gives 100 = 0, for 1 and:2 are unequal. 62 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF Since no one of the quantities (m - l) +n +p, m l+(n- l) +p vanishes for integer values of rn, n, p such that m + n + p ~ 2, it follows that no one of the quantities n+ m,, n + (m +1)- 2 vanishes for integer values of m and n such that m + n > 1. Hence when the coefficientequations for k and 1 are solved in groups for the same value of m + n, and in successive groups for increasing values of m + n beginning with 1, they lead to results of the form kmn == rnnn, mn == Xmn where y and X are integral functions of the coefficients that occur in fi and f,, each divided by a product of factors of the forms n + mi,, n + (m + 1)-. Moreover each of the coefficients k and 1, thus determined, contains A as a factor. It now is necessary to prove that the series for p and r, the formal expressions of which have been deduced, are converging series. For this purpose, we construct dominant equations as follows. Let a region of common existence of the functions Jr and p2 be defined by the ranges tl r, Iz l ri, p l a, 1 r /3: so that "1 and f2 are regular functions of their arguments within these ranges. In this region, let M1 be the greatest value of [ i1 and M2 the greatest value of 1 1: let M denote the greater of the two quantities M1 and M2. Further, since the quantities n + mnl, n + (m + 1) f - 2 do not vanish for integer values of m and n such that m + n>1, there must be a least value for the moduli of the quantities for the various combinations of m and n; let this value be v, so that n 4 men!??, | n+ (m + 1) ti- 2 1, in all instances. Also let A =A'. Then the dominant equations are chosen to be (P-A') = ST 1M M______ zP ZPMZT, Z, - i ^V - ti rio rl > (l r,(1 o r Clearly P-A'=T': their common values are given as the roots of the cubic equation (+ -/+ - T+ 1 - 1 _ T (1 _ 1A _ T (V rl le r^S- r+ o r r )T ri aa rla 1 r \r r,3 raa 7' DIFFERENTIAL EQUATIONS. 63 When t=O and z = O, the term in this equation independent of T vanishes: but the term in the first power of T does not vanish because q is not zero. Hence there is one root, and only one root, of the cubic equation which vanishes when t= O and z = 0; it is a regular function of t and zl in the immediate vicinity of t= O over a region which is not infinitesimal. Actually solving the equation for this root, we find MAI At A' \ I T= A- - + - zl + higher powers of t and z1; raGl or r1 ac and then P = A + -a + r + higher powers of t and zl.,7r1 \r ria, Now knowing that such a solution of the dominant equation exists, we can obtain its formal expression otherwise. Let P = A' + Y, zlîmtnpr n 1T- 2x2 -zlint rn i substitute these values in the dominant equations, expand their right-hand sides in the form of regular series, and equate coefficients of zntn on the two sides. We find rmn = K'<inn Instead of actually evaluating K'mn, the analysis used to determine ymn can be adopted. To this end, construct the value of y,,/, and, in its expression, effect the following changes in succession:i. Replace every modulus of a sum by the sum of the moduli of its terns: ii. Replace each denominator-factor I n + me, i and | n + (m + 1) b - 2 by r: iii. Replace the coefficients of pl, rl, zP, tq, in gb and 0 by 1M m a/3n1 r lrlq, for all values of mi, ni, pi, q1: iv. Replace A I by A'. The final expression, so modified, is K'mn. But the effect, upon the initial expression for I Ymn, of each of these changes is to appreciate the value: hence, taking the cumulative result, we have I Ymn I < rmn. Similarly | mn I < rmn. But the series A' + EZ zlm tn Fmn converges for a finite region round the origin t= O; hence the series p = A + Z 7ymnzm tn ) = ZZXmnZimtn) 64 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF converge absolutely: that is to say, the formal expressions p and r have significance, being regular functions of zl and t. The equations accordingly have integrals tl = pZl, t2 = T1, of the characteristics indicated. This completes the first stage of the proof. 11. For the second stage, let ti = pzl + Ti, t = 7z + T2; the equations for T1 and T2 are dT1 dt = 1 T1 + 01 (pz, + T1, 71 + T2, t) -, (pzl, Tz1, t) = Ti + T1 (T1, T2, zl, t) dT2 tdt 2 22 + 2. (T1, T2, z1, t) after substitution for p and r. Here 1f and f2 are regular functions of their arguments vanishing when T1 = 0, T2 = 0; they contain no terns of aggregate dimensions lower than 2 in T1, T2, z1, t. In accordance with the statement in ~ 9, it has to be proved that these equations possess solutions of the form T1 = z2 e1, T2 = z02, where ~1 and 20 are regular functions of t, zl, z2' it will appear that e2 =B (an arbitrary constant) and 1-=0, when t=O. Substituting these values for T1 and T2, we find t o + elzi al + e2z2a + (e2 - e1) e, =f (1, @2, z1, Z2, t)) at az, az, a~2 a- CH)2 a0i 02 ^ / f t + )z, + 2z =f2 (H1, OH zl, Z2) t) the functions fi and f1 are regular in their arguments, every term involves (, or e2 or both, and a term involving,1 and 0, in the form 61A ~0, has also a factor z2^+-*. If quantities e1, 20 exist, being regular functions of t, zi, z, and satisfying these equations, the substitution of expressions of the form,1 = Y= Zplmn Z1lz2 mt, e, = S 2qln ZllZ2ntn, in these equations must lead to identities. Accordingly, equating coefficients of zl1z2MXt on the two sides of both equations, we have {n + (I - 1) I + (m + 1) 2} Plmn = Ttlmnn, (n + ^l + m^2) Çlmen = KImn, DIFFERENTIAL EQUATIONS. 65 where 7rlmn, ti'mn are linear functions of the coefficients in fj and f2, and are integral functions of the coefficients pl'm'n' and qi'm'n', such that 1'.l, m'm, tn'n, +I'^+m'+n< l+m, +n. Owing to the forms of f and f,, we have 7qr000 = 0 'oo 000 = 0. Hence pooo = 0, and qooo is left undetermined; we take qooo = B, where B is an arbitrary constant. Moreover, no one of the quantities n +(l-1) f +m, m, n+li +(m-1),, vanishes for values of 1, m, n such that n + + m > 2; hence in the equations for Plmn, qim, no one of the coefficients of plnn, qlmn vanishes when n + m 1. Hence these equations can be solved for all the coefficients p and q after po00, q000. They are most conveniently solved in groups for the same value of n + + m, and in succeeding groups for increasing values of n +1 + m, beginning with 1; the results are plmn = '7lm,,n qmn = Klmn, where 7rmn, tclmi are sums of integral functions of the coefficients in fi and f,, each divided by products of factors of the types n +(l- l) e + (m + 1)e2, n+ I + mf. Expressions thus are obtained as formal solutions of the equations: it is necessary to establish the convergence of the infinite series. As before, we construct dominant equations for this purpose, as follows. Let a common region of existence of the functions f, and f2, which are regular in their arguments, be defined by the ranges t < r, | zl |< pi, |Z2 < p2, |~l|1 O 2|, 2 and within this region, let N denote the maximum value of Ii and jg', so that N is a finite quantity. Also let V denote the least among the values of |n+(l- 1)l+(m+1)el, | n + le +me2\, for the various combinations of the integers 1, m, n such that I + m + n >l 1; and let B = B'. Then the dominant equations to be considered are = vr (q(2 - B') i 0 N N Nz, _, Nz2(i)_ (l _ t)( _ ) p2o'1 P2;2 j VOL. XVIII. 9 66 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF The common value of (<P and?2,-B' is determined as a root of the cubic equation { Çi+ N +N + NB'( Z2 ) _ Z B' _z _) P2îf1 P22 P2 2) \ P2o'1/ P2'2 p272 B i + i - B2) -P2 i, t1 _ t) p(_ - \p2o' l p21'2 p22o-0'/ p202-1-2 (1 - - )1 {- p \2i - poeo When t = O, z = O, 2= O, the term in this equation independent of <Di vanishes: but the term in the first power of <>l does not then vanish, because q is different from zero. Hence there is one root, and only one root, of the cubic which vanishes when t = O, zl=0, 2 = O: and it is a regular function of t, zi, z, in the immediate vicinity* of t =O. Actually solving the equation for this root, we find NB' (t + \ z, +, P = +- - + + terms of higher orders; p2o'2 r Pl p20o/ and then we have 2 = B'+ B' -+ + terms of higher orders. vP2î7 \r Pl P2l 2 As in the preceding stage of proof of the main theorem, we can obtain the expression of these particular quantities 1DI and caD otherwise. Knowing that i)p and D2-B', equal to one another, are regular functions of t, zl, Z2, let = 4, - B' = Pimnzlz,2tn; substitute in the dominant equations, expand the right-hand side in the form of regular series, and equate the coefficients of zlz2ntn on the two sides. We find Plmn = lImn. But instead of actually deriving IImn from the equations so obtained, we can utilise the analysis that leads to the quantities 7rlm, clm,, as follows. Construct 7rlmn and, in its analytical expression, effect the following changes in succession:i. Replace every modulus of a suin by the sum of the moduli of the terms: ii. Replace each denominator-factor ]n + (1 - 1) el + (m + 1) 2 and |n + l, + m,| by 7: iii. Replace the coefficient of O,m,( m2z,1lz'n2tP in fi and f, by N- cra-m -2mp1,p22rP, for all values of m1, m, nl, n,, p: iv. Replace I B by B'. The final expression, after all these modifications have been made, is ITImn. But the * It remains a regular function so long as ct is less than the least of the moduli of the roots of the discriminant of the cubic. DIFFERENTIAL EQUATIONS. 67 effect, upon the initial expression for jIrlj, of each of the modifications is to appreciate the value; hence taking the cumulative effect, we have 7rmn1 < hImne Similarly I lmn I < rIlmn. Now the series for <P2, when Pmn is replaced by Hlmn, converges for a finite region round the origin; hence the series e,1 -= E rlmnZl'22mtn) 2, = B + C5Klmn. Zllz2mtn also converge for that region. Consequently the modified equations have integrals of the character T1 = Z201, Tg = z2O2 and therefore the original equations have integrals ti = pzi + z21, t2 = TzI + Z2~2, where p and r are regular functions of t and zl: and e1, e) are regular functions of t, Z3, Z2. This completes the proof of the main theorem with the specified conditions. CASE I (b): one root of the critical quadratic is a positive integer, the other is not a positive integer. 12. Let the integer root be denoted by m, the non-integer root by e; the equations can be taken in the form du t d = mu + at + O (u, v, t) tdt = v t+ (t + v, t)0 where 0 and Q are regular functions of their arguments, vanish with u, v, t and contain no terms of dimensions lower than 2. The sanie transformations as were used in ~ 2, viz. u=t(-m-1 +u) v=t( IV can be applied m - 1 times in succession: and ultimately we have equations dti t d = tl + at +fi (t9, t2, t) 9 2 68 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF where c, = -m+l, is not a positive integer, the functions f, and f2 are regular functions of their arguments of the same type as 0 and b above, and the integrals t, and t2 are to vanish with t. It has been proved that there are no regular integrals of the equation vanishing with t unless a is zero: and that, if a= O, 'there exists a simple infinitude of regular integrals satisfying the equations. We proceed, not in the first place to the complete theorem but only to a partial theorem, by shewing that when a is not zero, there exists a simple infinitude of non-regular integrals vanishing with t, these integrals being regular functions of t and t log t: and when a is zero, these non-regular integrals do not exist. To establish this result, we proceed from equations dx t d = ox + at + 01 (x, y, t) dt t d- = cy + bt + 02 (x, y, t) where a is taken to be a real positive quantity, a little less than 1 initially and equal to 1 ultimately: and, as the explicit forms of 0, and 02 are required, we suppose 0(x, y, t)= =22aijxiyJtP, (i +j + p > 2). 0O (x, y, t) = Sbj xyJttP, With these equations, we associate a set of dominant equations. Let laijpl = Aijp, bijp l= Bijp, a = A; then the dominant equations are dX t — -j X + At = (X, Y, t)| dt t - Y + Bt = 2 (X, Y, t) where )1(X, Y, t)=SY;AijjXjy Yjtp (2 (X, Y, t)-= SX BijpXiyjtpJ If K be real, not being a positive integer, we choose that sign for the term + Bt, which makes B K-i a positive quantity; if Kc be complex, we choose a term + Bt, such that B K-l is a real positive quantity and \B bj. DIFFERENTIAL EQUATIONS. 69 By the theorem of ~ 10, we know that solutions exist, which vanish with t and are expressible as regular functions of t and t'. Let a new variable 0 be introduced, defined by the equation t- t = (1 - -) 0; and, in the solutions indicated, replace t' by t + (1 - ) 0; they then become regular functions of t and 0, expressed as converging power-series. To obtain their coefficients in this form directly, let X = E a,m Omtn, Y = - bmn Omtn, where ao,=O, boo= O; then since dO t = a-O-t, we have dX dt = 22 a~, {nOmtn + m-1 t (a- - t)} = 2$ {(n + a-m) Omtn - mn-l1tn+1} amn, and tdY dt = 2- {(n + o-m) omtn - mo-1tn+1} b. Substituting in the differential equations and comparing coefficients, we have (n + o-m - o-) a,, - (m + 1) a,+i, n-i = Hm, n (n + -rnm - c) bmn - (m + 1) bm+i, n-l = Km, n where H., and K, n are sums of terms of the form Imn = NAYip,v jamn... amn,, bmi'n/'..* bmnj', and similarly for Km, n, such that i-+j+p>2 m n+... +n mi + m+... + m' = m p + ni +... + ni +... + nj = n N being a numerical quantity, representing the number of integer solutions of the last two equations. As regards the initial coefficients, we have the following expressions. For m + n =, so that m = 0, n = 0; then aoo = 0, boo = 0. For m+n=1, so that m=1, n=0: and m=O, n=l; then O.a = O, (o —t)b0 = O; (1 - or) cxoi - aio = - A, (1 - K~) 601 - 610 = î 5; 70 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF so that a10=(1 - ) a + A, (1 - c)bol =+B, bo= 0; thus ao, is undetermined and therefore can be taken arbitrarily, say = C, where C is positive. Thus ao0, ao, bol are positive. For m+n=2, so that m=2, n=0: m=1, n=l: and m=0, n=2; then oa20 = A 200 a2o 1 (2- - K) b20 = B200a210 a 11- 2a20 = 2A200 o,,, a, + A-o11a10 bo1 A101 ao, (1 + - ) b - 2b26o = 2B2,oaioao + B1l0al0bol + BlOI al0 (2 - o) a02 - ai = Aoa201 + o ao b + oao o 1 b2oo + Ao2o a2o + Alol + bol + Aoo2 2 (2 - c) b02 - b6n = B2oo a2o + B,, ao bo B2b2 + Ba, B Boo 2 And so on, taking in succession the groups of terms for increasing values of m + n, and taking, in each group, the equations for increasing values of n beginning with zero. The result is to give amn = 0mn, bmn = mmn, where m,, and omn are sums of a number of terms; each term is a quotient, the numerator being a positive integral function of the coefficients of 01 and 0. and containing a,0m as a factor, and the denominator being a product of quantities of the form n +- -m -o, n + cm- K. It can be proved, by an argument precisely similar to that in Jordan's Cours d'Analyse, t. iii, ~ 97, that the number of quantities entering into the denominator product for each of the terms in 09m and fmn is < m + 2n-1. On account of the theorem of ~ 10, establishing the existence of the integrals as regular functions of t and ta, it follows that the series 2amn O6tnn, Ybmn, Omtn converge absolutely. Now proceed to the limit in which o- increases to, and ultimately acquires, the value unity; then 0 becomes -tlogt, the differential equations become dtd X + At = o(X, Y, t) dY -dt- /Y + Bt = 2(X, Y, t) and the integrals change to ae mn mtn, a b/ mn om tn, where amn and b/n are the values of amn and bmn when o is replaced by 1. DIFFERENTIAL EQUATIONS. 71 In 8O,, let T be any one of the terms, and let T' be the value of T when o is replaced by 1. As regards the numerator in T, it is the sum of a series of positive quantities: and it is unaffected by the change of o, except that a10 is replaced by A, that is, by a diminished quantity; hence the numerator of T' is less than that of T. As regards the numerical denominator, each factor n + om- is replaced by n + m-1, which is a greater quantity than the factor it replaces, unless m vanishes; but when m = 0, then - 2 - 7, 'n - o n-1 because then n > 2. Also every factor n + om - K is replaced by n + m - c; the imaginary portions (if any) of these two are the same, but the real part of the new factor is greater than that of the old except when m =, and then they are the same. The number of factors in the denominator is not greater than m + 2n- 1: hence nI n + ~ crrm,- n + <m- 1 < (2 _ 7)f+n-i n+ m — n+-m- c < (2 - o)2m+2n The changes made have diminished the numerator of T; thus T' n + o-m -a- n + m - T n+m -1 n +m-K/ < (2 - o)2m+2' Remembering that 08m is a sum of terms T and bearing in mind the character of T, we have a nn <(2 _ ) j2+2. Similarly b'mn < (2 - ()2m+2n Now the series amn,,mtn, Z$bmn Omtn converge absolutely for a finite region round the origin. Let this be defined by t tlr, |0 s; and let -M1, M2 be the respective maximum values of the moduli of the series within that region. Then b <M2 amn < mrn bmn < smn; and therefore an < S fr M (2 - a)2} (2- a)2 b'mn< I<-Is M2 n (2-a)2} {(2-a)2} 72 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF Consequently the series $,Ea'mnOmtn, Z1b'mn fmtn converge absolutely for a finite region round t= O. If the original equations dx t -= x+ at + i (x, y, t) t dy= cy + bt+ 02 (x, y, t) possess integrals vanishing with t in the form of regular functions of t and t log t, these integrals rnay be assumed to be = 2fmn Smtn Y = YS;fn 0M t' when substituted, they must satisfy the equations identically. Choose fol so that If1o I = c, where C is the arbitrary constant in the integrals of the preceding equations. When the relations that arise from the comparison of the coefficients are solved so as to give fmn, gmn, it is easy to see that the same results are obtained as would be given by changing, in a'mn and b',,, A into -a, B into + b, Aij into aop, and Bijp into bijp, for all values of i, j, p. Bearing in mind that lal=A, |b|<J|lB, aijp l=Aijp, [b&jp=Bgjp, it is manifest that the real positive quantities | a',n and b'mn are superior limits for IfnI and gmn I, that is, jfrmn I < I a`mnl, I Igmn I< b' mn. But the series:,a'mn 6tn, -b'mn mtn converge absolutely: hence also the series yfmn O tn, ygmn o tn also converge absolutely, and the equations accordingly possess integrals as stated in the theorem. Note. If a is zero, then a',, = O; a'0= O, a'/ = O; and it is immediately obvious that a mn = O, for all values of m > 0 and all values of n. Similarly b'mn = 0, for the same combinations of m and n. In this case, 0 disappears entirely from the expressions f, fOn' m vt, sygmn Omtn; so that the integrals become regular functions of t, which are known to be solutions of the equations when a = O. DIFFERENTIAL EQUATIONS. 73 13. The main theorems as to the equations t - t + at+fi (t, t2, t) t dt2 Kt2 + bt +tf2 (tl, t2, t)) dt so far as concerns the non-regular solutions, are:When a is not zero, so that the equations do not possess any regular solutions that vanish with t, they possess non-regular solutions that vanish with t. If Kc have its real part positive, not itself being a positive integer, there is a double infinitude of such solutions; they are regular functions of t, tK and t log t. If K have its real part negative, there is only a single infinitude of such solutions, they are regular functions of t and t log t. When a is zero, so that the equations possess a single infinitude of regular solutions vanishing with t, then if Kc have its real part positive, not itself being a positive integer, there is a single infinitude of non-regular solutions vanishing with t which are regular functions of t and tK; but if K have its real part negative, the equations possess no nonregular solutions vanishing with t. These theorems can be established by analysis and a course of argument similar to those which have been adopted, wholly or partially, in preceding cases. The actual expressions for the integrals, when a is not zero, are t, = aO + At + Sgln,,n lOmt' t2 = - t +B+ s t+ mn y Omt tn where the summation is for values of 1, qn, n such that 1 + m + n > 2, the coefficients A and B are arbitrary, [ denotes tK and 0 denotes t log t. When a is zero, all the coefficients gmn, hlmn for values of m > 0 vanish; so that 0 disappears from the expressions for t1 and t2. The resulting expressions then can be resolved each into the sum of two functions: one a regular function of t which involves A, the other a regular function of t and ' which involves B, and vanishes when B = 0. It may be noted that a slight degeneration occurs in the solutions when Kc is the reciprocal of a positive integer; a regular function of t and tK is then merely a regular function of tK. When the equations in their first transformed expression are du t d = mu + at + 0 (u, v. t) dt t = v + /3t + (u, V, t)) the general results are the same as above; the value of KC is -m+ 1, and the critical condition, which is represented by a =0, is stated at the end of ~ 2. VOL. XVIII. 10 74 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF CASE I (c): the roots of the critical quadratic are unequal, and both are positive integers. 14. Denoting the roots by m and n, of which m may be taken as the smaller integer, the equations can be transformed so as to become du t -=u + at + 0 (u, v, t) dt dv t = nv +/3t + 0(u, v, t) They can be modified by substitutions sirnilar to those adopted in the preceding case; such substitutions can be applied m - 1 times in succession, leading to the forms dt1 t = ti +at +f, (tL, t2 ~~~~~dt2~~ t-t = Kt2bt+f(t t t)t where K, =n-m+1, is a positive integer greater than 1, the integrals t, and t2 are to vanish with t, and the functions fi, f2 are regular functions which vanish with their arguments and contain no terms of dimensions lower than 2 in ti, t2, t combined. It has already been proved (~ 3) that the equations possess no regular integrals vanishing with t, unless two relations among the constants be satisfied; one of them is represented by a =, the other by (say) C= 0, where C is a definite combination of a, b, and the constant coefficients in f1 and f2. The theorem as regards the nonregular integrals is: The equations in general possess a double infinitude of non-regular integrals which vanish with t; they are regular fjinctions of t, and t log t. If both of the conditions represented by a = 0, C= 0 are satisfied, the equations possess no non-regular solutions vanishing with t: they are known to possess a double infinitude of regular integrals which vanish with t. The method of establishing this theorem is similar to that for the case when K is unity so that the critical quadratic has a repeated root. As that case will be discussed later in full detail, we shall not here reproduce the analysis and the argument, which follow closely the corresponding analysis and argument in that later discussion. It may be added that the conditions for the equations t d = ~mu + at + 0 (u, v, t) dv teesented for the modified forms by a (,, have already ( 3) been given. represented for the modified forms by a = O, C=O, have already (~ 3) been given. DIFFERENTIAL EQUATIONS. 75 CASE II (a): the critical quadratic has equal roots, not a positive integer. 15. It has been proved that, in this case, the original equations possess regular integrals vanishing with t: and therefore, in order to consider the non-regular integrals (if any) that vanish with t, we transform the equations as in ~ 6, and we study the derived system dt1 = t + 1 (t, t2, t) t dt2 =icti + et2 + +2 (tl, t2, t) where <b1 and 02 are regular functions of their arguments, vanish when tl =0, t2 =O, and contain no terms of dimensions less than 2 in tl, t2, t combined. The integrals t1 and t2 are to be non-regular functions of t, required to vanish with t. The non-regular integrals are given by the theorem: When the repeated root e of the critical quadratic has its real part positive, not itself being a positive integer, there is a double infinitude of non-regular integrals vanishing with t, these integrals being regular functions of t, t*, te log t. When the theorem is established, there is an immediate corollary: If the real part of the repeated root e of the critical quadratic be negative, then the equations do not possess non-regular integrals vanishing with t; the regular integrals possessed by the original system of equations are the only integrals that vanish with t. The forms of the theorem and the corollary are indicated by proceeding nearly to the limit of the theoremrs for the case of I (a) when the roots of the critical quadratic are equal to one another. If = + 3, where 8 is infinitesimal, then tt2 = t (1 + log t +...), so that a function of t, tt1, t2 becomes a function of t, tEl, t' log t; but further investigation is needed in order to shew that, in passing to the limit, the functions under consideration continue to exist. Instead of adopting this method of proof, we proceed independently. It is convenient to take = t, -7 = tt log t. If therefore integrals of the character indicated in the theorem exist, they can be expressed in the forms t2 = a1,,E blmn tm t n 10-2 76 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF and these values must, when substituted, satisfy the differential equations identically. Now t d =, dO - dt dt so that d t (rtt) = (n + l1 + me) lmtn - m_ l~+lm-ltn. Hence equating coefficients of glmtn on the two sides of both equations after substitution of the assumed values of t, and t2, we have {n + (l + m - 1) I} almn - mal_, m+, n = almn {n + (I + m - 1) } bl,, - mb_-1, m+i, n = amn + /'lmnJ where a'ilmn, / lmn, being the coefficients of Ïq1ïr/t" in fib and 02 respectively, are linear functions of the constants in 0b and k2, and are integral functions of the coefficients am'n',, bl,lmf,,, such that l' < 1, mr' n m, n', l' + m' + n' < 1 + m. + n. Assuming that the real part of e is positive but that e is not a positive integer, we see that no one of the quantities n + (I+m-1).I can vanish if l+m+n >2. If l = = n = 0, then a'mn = 0, /'l =0; hence aoo0 = 0, bo000 = Oaooo = O. For values such that +r+n= 1, we have O. aOl0 = 0, that is, ao00 = K, a001 = 0, O. aloo = O, that is, ao10 = L; O. bolo = 0. aolo = OK, boo001 =. a001 =, O. bloo = 0. ao10 = OL. In order therefore to obtain finite values for the coefficients a and b, we must have K=O, L=O, and then boo0, b10, are arbitrary; that is, we have aoi0 = O, aool = O, a1oo = 0; bo010 B, bool = O, bloo = C. To obtain the terms of dimension two in ', rV, t in t, and t2, we require the explicit expressions of S6 and b2: let them be qb = attl + btt2 + Ct12 + ett2 + kt2 +..., <2 = att1 + /tt2 + 3/t12 + etlt2 + tct22 +.... DIFFERENTIAL EQUATIONS. 77 The terms in t, and t2 of dimension one, obtained as above, are tl = O, t2 = C+ Bq, so that, as far as terms of dimension two in ~b and b2 after substitution, we have,b = bt (By + Ct) + k (By + C7)2, 02 = t (By + Ce) + K (Br + CO)2. Accordingly, for I + m + n = 2, we have a020 = kB2, a0l = bB, (2 -) ao2 = 0, a0,, = bC, (1 + ) a110- a020 = 2kBC,:aoo = kC2; b020 = KB2 + a020, bo11 = B + a011, (2 - ) boo2= Oao2, bl,, = /3C + Oa1,, (1 +:) b110 - bo02 = Oa110 + 2/BC, b200 = KCC2 + Oa200; and therefore the terms in t, and t2, of dimensions two in the arguments, V, t, are 2kBC + B2 J C22 + 1 + B-^ + bB2 + + bBt in t,: and (K+ -) - + + +) 2- (+ + Ob) Ct + ( + b) Brt (K + -- + 2KBC 0 (2kBd + B l+_ _ + o (1 +: B, in t2. And so on. The equations, when solved in groups for the same value of I + m + n beginning with a zero value of 1, and solved in successive groups for increasing values of + m + n, give values of amn, blmn which are sums of integral functions of the literal coefficients of fj and 0, and of the arbitrary coefficients B and C, each such integral function being divided by a product of factors of the form n + ( + m - 1). Let the values thus obtained be aImn = Clmn, bimn = iP8mn. As in ~ 9 for the former case, it can be proved that aoop = O, boop, = for all positive integer values of p, so that there are no terms in t1 or in t2 involving t alone; every term involves either ' or r7 or both g and i7. 78 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF To establish the convergence of the series thus obtained, we proceed in two stages as in the corresponding question (~~ 10, 11) when the roots of the critical quadratic are unequal. Extract from t1 and t2 all the terms which are free from Y]; as each of them involves ', their aggregate can be taken in the respective forms >p, r-; and the remaining terms then have qj for a factor, so that we may write t2 = T + 72. t2 =T+17e2. It will be proved, first, that solutions of the form tl = gp, t2 =?T exist, where p and r are regular functions of t and f, p vanishing at t= O and r having an arbitrary value there: so that the functions involve one arbitrary constant, and there consequently is a simple infinitude of such solutions. Then substituting t1= p+ î1, t2=UT+ 2, it will be proved that functions ~1 and 0, exist, so that they are regular in their arguments g, r, t, they involve an arbitrary constant C, ~1 vanishes at t = O and 20 acquires the value C there. Thus for an assigned value of B, these will represent another (and an independent) simple infinitude of integrals. In each stage, the details of the analysis follow the detailed analysis of the former case somewhat closely: it therefore will be abbreviated for the present purpose. 16. Substituting ti= p, t = =r in the equations for t1 and t2, we find p and T determined by t dt = (P, r, ', t) td= op++2(p, T,?, t) dt where the general character of -I and q2 is as before. If these are satisfied by regular functions of t and r, their expressions p = ikq ~t~, T = Zj~mn mtn, must, when substituted in the above equations, satisfy them identically. Accordingly, comparing coefficients of gmtn on the two sides of both equations, we have (n + mo) Cmn, = K',mn (n + m~)j.n = J'mn + 6Omn, DIFFERENTIAL EQUATIONS. 79 where K'm, J'mn are linear in the literal coefficients of p and r, and are integral functions of km,,,' jm'n, such that m' Q m, n' ~ n, m' + n'< m + n. Also, from the form of 1i and 2,, K0'o = 0, J00 = 0; hence we have koo = O. But joo is undetermined, and it can therefore be taken arbitrarily: let its value be B, where B is any arbitrary constant. When the equations for kmn and j,,r are solved, in groups for the same value of m + n and in succeeding groups for increasing values of m + n, they lead to results of the form kmn = Kmnn, Jmn = tmn, where /Cm, tmn are sums of integral functions of the coefficients in -1 and 42, divided by products of factors of the form n + mn. The dominant functions are constructed as before. Let e denote the least value of n +mel for integer values of m and n, so that e is a finite (non-vanishing) quantity; and let 10l=, I C=C'. Also, let a common region of existence for the functions 4k and.2 be given by the ranges \t ]r, \ r r., IPl h, rp T k; and within this region let M be the greatest value of 1 11 and!%1. Then consider functions P and T, defined by the equations _______ P - — E - IMn —Mr- 1-3r E~-T cC + OP M- M i1/ (%M. <Y \ - hr) t hr, k Clearly (e + ()P= e (T- C'), that is, p=- (T -C') - - The value of P is a root of a cubic equation which, when t = O and = 0, has no term independent of P and has a non-vanishing term involving the first power of P: so that it has one and only one root vanishing with t and u, and this root is a regular function. To obtain its expression without actually solving the cubic, we take P =:Kmnm,, 'M tn, where K,, = 0: we expand the right-hand side of the dominant equations as a regular function of t. ', P, T, and compare coefficients. The analysis that leads to the values of Cmn, Lmn can be used to obtain the value of Kmn, by making appropriate changes similar to those in the earlier corresponding case. These changes are now, as was the case before, such as to make | Kmn <K rmn, tmnnl <Krmn; 80 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF and therefore as the series.KmnUqmtn converges, the series kmn C +m2tn, C + 2jmnn,t also converge. The existence of the integrals, connected with the first stage, is therefore established. 17. Now writing ti = 'p + 1O7, t2= + 102, where p and r are the regular functions of t and Ç as just determined, the equations for,1 and 0, are dt t —t = (o~, o2, ~', v, t) ) tde2/ = 0oe + f2(o1, o2, ~ v, t) ) where fi and f, are regular functions of their arguments, vanishing when ~1 = 0 and 02=0; the coefficients of the first powers of,1 and 0, vanish when t =0; and any term, involving e1 and 0, in the form OeXe,, contains <x+i-1 as a factor. The method of proof and the general course of it are the same as before (~ 11). The regular functions of ', q, t, which are the formal solution of the equations, are proved to converge, by being compared with the functions which satisfy the dominant equations êe`@ =i_ __ IMM M _ _ M 1 | _V -)i- _2) ri —) l - I a \ r)\ p)\ pa1 \ \ r \ P M r o'( )1 )(- PI(1 _ o-~2 / and are such that, when t=O, =0,?7=0, then <P is zero and 2 = C l. There exists a single quantity (I, satisfying these equations and vanishing with t, which is expansible as a regular function of t,,; q in a non-infinitesimal region round t, the power-series which is its expression being consequently a converging series within that region. And therefore <>2, being given by is also expressible as a regular function of t,, r which, when t=O, acquires the value | CI. DIFFERENTIAL EQUATIONS. 81 A comparison of the coefficients of l4vmtn in,1 and 20 with those of the same combination of the variables in (I and V2 is easily seen to lead to the inference that the moduli of the former are less than the modulus of the latter; consequently the former series converge and therefore integrals of the equations, defined by the specified conditions, are proved to exist. Their explicit expressions, as power-series, are obtained as in ~ 11. CASE II (b): the critical quadratic has a repeated root which is a positive integer. 18. Denoting the repeated root by m, the equations are du t =u + + 0 (u, v, t) t, = KU + mv + ft + (u,, t, ) where the functions 0, 4 are regular, vanish with u, v, t, and contain no terms of dimensions lower than 2 in their arguments. The equations can be transformed as before (~ 5) by the appropriate substitutions; and this transformation can be effected - 1 times, leading to new equations of the form tdt + = ti + at + 01(t1, t2, t) t t2 = ct+ t++ 0+ bt+ (t,, t2, t) where t, and t2 are to vanish with t; and 01, 02 are of the same type and properties as 0, > in the first form. There are two sub-cases according as K is zero, or K is not zero. 19. First sub-case: Kc=0. The equations can be taken in the form t d= x + at + 01 (x, y, t) tdt =y+bt +0(x, y, t) j the integrals are to vanish with t; and the functions 01, 0, are regular functions of their arguments, which vanish when x = 0, y = 0, t = O and contain no terns of order lower than 2 in x, y, t combined. The integrals vanishing with t are defined by the theorein: The equations possess, in general, a double infinitude of non-regular integrals vanishing with t, which are regular functions of t a.ud t log t; and it is known that there are no VOL. XVIII. 11 82 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF regular integrals vanishing with t. If, however, both a = O and b= O, the equations do not possess non-regular integrals vanishing with t; the only integrals vanishing with t are the double infinitude of regular integrals which the equations are known to possess. This theorem can be established, as in other cases, by the construction of dominant equations and comparison with their integrals which actually are obtained in explicit expression. For this purpose, consider the equations t - oX + At = S 2 Aijp Xi Y tp dt t d Y_ Y + Bt -- ~2 Bijp Xi YJtp where i +j+ p 2 in the two triple summations. The quantities - and p are real, positive, and less than unity: ultimately they will be made equal to unity. It follows, from the theorem of ~ 8, that there is a double infinitude of integrals vanishing with t, these integrals being regular functions of t, tG, tp. Let two new variables 0 and q be introduced such that t = t - (- - 1) 0 + (o- - 1)2 < tP = t - (p - 1) + (p - 1)2; we easily find t j+ t-O=(1-p)(1 —) 0=/3 )! ~~~~~~~dt~ t d + ( = (o + p - + 1) = a I where a and f3 are constants which, when p = 1, o = 1, are equal to 1 and O respectively. The regular functions of t, ta, tP are expressible in the form of absolutely converging power-series; when t' and tP are replaced by their values in terms of 0 and 0, the new functions are regular functions of t, 0, b. To obtain their expressions in this last form directly from the differential equations, we substitute X = X hImn tl S gbn Y= YCY kImn tl Om on in the equations which are to be satisfied identically. Now dX a( dOa d a ~) tdX (t a + t + t X dt a t+t dta dt a95 = 25 {(I + m + an ) hlin tl Am pn - mh.mn tl+l 0m~-1 n - nhm^ t1 0?n+l n-1 + f3mhmn t1 Om-1 ~+l} }; DIFFERENTIAL EQUATIONS. 83 hence, comparing coefficients of t10m O n on the two sides, we have (I + m + an - a) hl - (m + 1) hil,,. l,- (n + 1) h1,m_,n+l- + (m + 1) fhl, m+l,n- = xa'm, Similarly (I + m + an - p) klm, - (m + 1) ki_ m+ n - (n + 1) Ci, m-, n+ + (m + 1) kI, m+,n_ = 3imn Here amn, /31mn are expressions which are linear in the coefficients A,j Bj respectively, being an aggregate of terms of the form Nl Aijphlm,n,... hlim, 'n,'. l,, 'm./n/' 1 l Bijphllnlnl,.. hliminim kllln,'... ki'm'/n/ respectively; the subscripts are subject to the relations mi +... + mi. m1 +... + mf' = mr -n, -... + ni + nl: '... + nj' = n î+ 1+...+ li+ 1,'+...+ Ij/= and the numerical factor N1 is the number of integer-solutions of these equations. In particular, we have hooo = 0, ooo = 0. When + r + n= 1, the equations for the coefficients in X are (1 -o-)hoo h0o = — A, (1 - o) holo- hool = 0, (a - -) hoo, + holo = 0, which are satisfied by h010 = (1 - ) h100oo + A! hoo = (1 - o) holo J and hloo is arbitrary. Similarly, k010 = (1 - p) koo + 8 ^001-( L -p) Polo J and kjoo is arbitrary. When l+ m + n = 2, the equations for the coefficients in X are (2 - a) hoo - ho,, = h o 2 (1 + a - o-) h011 + 2h0o2o0 - h002 = a'o01 (2a - o-) hoo2 + /lholl = a'oo0 (2 - r) h10 - 2ho20 - h101 = a'nlo (1 + a - o) hïoi - ho011 + 3h1o = a'o0. (2 - o-) h0 oo- h110 = 2'oo 11-2 84 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF The first three equations, when solved, determine ho2o, ho, hoo2; when the values of h020 and h,0o are substituted in the next two equations, they determine h1lo, h,,,; the last equation then determines the form of h2oo. Similarly for the coefficients in Y. For values of l + m + n 2, the equations can be solved in a similar way. They are solved in groups for the successively increasing values of + m + n. In each group, say that for which l+m + n= N (so that the coefficients hi'm'n', kl'm'n', such that l'+ ' + n'; Ni- 1, are supposed known), the convenient method is to arrange the equations in sets, determined by the values of i and in sequence according to increasing values of I beginning with 0: in each set, the equations are arranged in sequence according to increasing values of n beginning with 0. In each set, we use the equations in succession to express hIlm in terms of hl,N-l,o and previously known coefficients and constants; when the first N-I equations in the set have thus been used, the remaining equation, on substitution of the values of h1,0,N-, hl,i,Nl-l-î, then determines hv-l,o and so also the values of all the coefficients hl,,,,, such that m + n= V-1. Likewise for the coefficients kmn. And then, as the solutions are known to be regular functions of t, 0, f, the series y,:, h,,.mn t O n~,: k^l, t l, with the values of hln, kimn which have been obtained, converge absolutely. As regards the forms of the coefficients hlmn, klmn, they are the aggregates of positive terms T. The numerator of each term T is the sum of a number of positive quantities: it is an integral algebraical function of the coefficients Aijp, Bijp: it is also an integral algebraical function of h 4+m+n, ki+m+n such that + + n = 1. The denominator of the term T is of the form P + Q/, where P is the product of factors of the types + m + an- o-, + m + an - p, and where Q is an aggregate of quantities, each positive and similar to P but containing two factors fewer than P. As regards the number of factors in P, being a part of a denominator in a term T in hnn or klmn, it can be proved, by an amplification of Jordan's argument quoted in ~ 11, that this number I 31 + 2m + n. It is known that, so long as a and p are different from unity, the convergence of the power-series is absolute: hence this will be the case when =l-, p = 1 -, DIFFERENTIAL EQUATIONS. 85 where e is a real positive quantity that can be taken as small as we please. Proceed therefore to the limit in which o and p acquire the value unity, so that e passes from small values to zero. The effect is to give to 0 and 0 the values 0=-tlogt, = -t(logt)2; to change the differential equations to the forms t -X + At = 2 Ai Ytp dY '/; dt -Y+Bt= Bip XiYyt and to change the integrals to the forms X = s h'lmn t (- log t) {t (10g t)} n whr = 2 C ^ ^nt1 (-t 10g t)m t (log t)2d ' where h'im and k'irn, are the respective values of hlm, and klmn when o-=1, p = 1. It is necessary to compare the coefficients h'rn, and hmn' and likewise the coefficients k'imn and Imn. Let T be one of the terms in hl,,n as explained above: and let T' be its value when- = 1, p=l. The effect of the change on the numerator is to replace (1 - )hlo,+A by A, hooi by 0, (1 -p) kcoo+B by B, k0oo by 0, in every case a decrease: and therefore, as the numerator is a sum of positive terms, the whole effect on the numerator is to decrease it, that is, numerator of T'< numerator of T. As regards the denominator of T, in the form P+Q, the quantity /8 is of the second order of small quantities; Q is an aggregate of a limited number of products, each containing a limited number of factors; hence Q/8 is of the second order of small quantities. Let P' be the changed form of P, obtained from P by changing l+m+an- - into +m+n-1, and l+ m+an -p into l+m+n-1. Now l +m+ n-o — ( +m n- 1)=-(2n -1)e, a small quantity of the first order unless n= O; so that, unless n=O, I + m + cn - I+m+ n-1= 86 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF where y is a positive small quantity of the first order. When n =O, I + m - o - (I + m - 1)(2 - o-) = (2 - I -m), so that as + m > 2, we have I+m-a I 1= 2 - 0 - y', l+m-l where ry' is a positive small quantity of the first order, unless + m=2, and then = O. Hence P' I 1 P 1-7 2- - -y > Il 1 >II ~ > (2 - o)31+2mn ' the difference between the two sides being a small quantity of the first order. Also Qe p' is a small quantity of the second order, that is, a quantity of an order less than the foregoing difference; consequently P' 1 P + Q3 > (2 - a)312m+ The changes depreciated the numerator of T into that of T': hence T' P+Ql/ <T Pt < (2 - o-)3+2nz+n < (2 - o)31+3m+3n This result holds for every terin in hlmn; hence h'lmn < (2 - o-)31+3m+3n himnn Similarly, ki mn <(2 - -)31+3m+3s kImn Let the region of convergence of the power-series EEhImntlOmn, EE kImntlOmonn be defined by the ranges t <r, Ir, 0 r, ~) r2; DIFFERENTIAL EQUATIONS. 87 and let M1, M2 be the maximum values of the moduli of the series respectively within this region; then himn < M rlrlmr2 7n M2 kimn < l. rn' r in rîl consequently h'lmni < " 2 r rir ) { r2} kzmn < M2 r r2 (2 - 7)3j (2 - O') (2 - a)3^ Hence the series converge absolutely for values of t such that t I < r. The existence of integrals of t d= x + at + S a;, cxyx' t dty= + bt + sbbijpxiyitp can be deduced from the preceding result, by choosing al =A, b (=B, aijl= A, Ibij=Bip, as the quantities A, B, Aip, Bijp for those dominant equations. The expression for the integrals is X = 2;S$lmn tlûmpny y = — Z Kimn tlOmon' where HImn is derived from h'lm,, and Kin from 'lmn, by changing A into -a, B into - b, Ajp into ap, and Bij into bi. The effect of these changes is to give H-mn < h'imn, Kimn < k'lmn; and therefore the series for x and y converge absolutely. The actual values are x = at log t + Cit + IIS Himn tlOm"i y= bt logt 2t + C + E Km tlOmt n' where O=-tlogt, b= ~t(logt)2, the summation is for values of 1, m, n such that + m+ n 2, and the coefficients Ci, C2 are arbitrary constants. 88 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF But the formal expression is more general than the actual value. The equations determining the coefficients are (1 + rn + n - 1) Hmn - (2n + 1) H1,, n - (n + ) H, m-i, n+1 = Eljnn (I + m + n - 1) Kmn - (m + 1) i, m+i, n - (n + 1) Kl, n-i, n+li = F~ n with 1lHoo = CI, Holo = - a, H0oo = O, Kl0oo = C Koo = - Koo1 = 0. It is clear that, when + m + n=2, Emn=O, Fin=0, if n=1, 2; hence Hmn, Kimn both vanish for + m + n = 2 if n = 1, 2. Thus for l+m+n=3, Eimn=O, Fin=0, if n=l, 2, 3; hence also Hlm, Kimn, both vanish for I + n+n=3 if n=1, 2, 3. And so on: all the coefficients Himn, Klm vanish if n >0; that is, the quantity ~b does not actually occur in the expressions for x and y which accordingly are regular functions of t and tlogt. The theorem is therefore established. Note 1. Any term in x and y is of the form Kttm (t log t)'n, that is, Ktm+n (log t)n; and therefore the index of log t is never greater than the index of t. If, however, the equations were t d- = x + at + ct log t + aijpq xiyjt (t log t) dy t = y + bt + c't log t + lSlzbijpq xyJtP (t log t)q where i+j+p+ q>2 for the summations, then the values of x and y satisfying the equations are x = - ct (log t)2 + at log t + Clt + SEZH,n tO y =- lc't (log t)2+ bt log t+ Ct + ZsKimn tlOîBnJ) where t, O, q have the same values as above, and the summations are for values of DIFFERENTIAL EQUATIONS. 89 l, m, n such that l + m + n 2: and the coefficients Hifn,, /Kimn are determinable as before. Any term in x is Htl+'m+n (log t)m+2-, that is, the index of log t is not greater than twice the index of t. Note 2. If a vanishes but not b, the solutions are still non-regular functions of t; likewise if b vanishes but not a. In these cases, it is known that no regular integrals vanishing with t are possessed by the equation. If a = 0, b = O, then Hl,= 0, KiM =0, if m > 1: that is, t log t disappears from the expressions for x and y, which then become regular functions and are the double infinitude of regular integrals that vanish with t. In this case, the regular integrals are the only integrals vanishing with t that are possessed by the equation. 20. Second sub-case: Kc not zero. The theorem is: The equations possess in general a double infinitude of non-regular integrals vanishing with t which are regular fuicctions of t, t log t, t (log t); and it is known that there are no regular integrals which vanish with t. If however a= 0, then the integrals can be arranged in two sets; one is a simple infinitude of non-regular integrals vanishing with t which are regular functions of t and t log t; the other is the simple infinitude of regular integrals vanishing with t which the equation is known to possess. (It is necessary that the constant K be different from zero: otherwise some of the coefficients in the second set are infinite unless b also is zero, in which form we revert to the first sub-case already considered.) The method of establishment is similar to those which precede: it need therefore not be repeated after the many instances of it which already have been given. The initial terms in the integrals of the equations as taken in ~ 15 are ti = aO + At+..., t, = Kca + (KA + b) 0 + Bt +..., the unexpressed terms being of higher order in t, 0, he: here A and B are arbitrary, 0O=tlogt, and = t (log t)2. Any term in the expansion of t, or t2 which involves < contains K in its coefficient; the disappearance of the terms in () from the integrals in the first sub-case is thus explained, for Kc then is zero. Concluding Note. 21. Some sub-cases still remain over from Case I(a), when the roots el and:2 of the critical quadratic do not satisfy the conditions that (~ 8) prevent some one (or more) of the quantities (X- 1)VO+L. +I, X1+(~-l)2,+ 2, VOL. XVIII. 12 90 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF EQUATIONS. from vanishing for integer values of X, /a, v such that X + / + v 2. The real parts of l>, 2 are supposed to be positive. The instances that can occur are obviously for X = O in the first set and j =0 in the second set; both are included in the form e = im + ^, where e and q are the roots of the quadratic, and + v > 2. The cases J = O, = 1, have already been discussed. For the remaining cases, we have the theorem: The double infinitude of non-regular integrals vanishing with t are then regular functions of t, t1, t'+v" log t, where, and v are integers. It can be established in the same manner as the similar theorems in the preceding sections. IV. Ueber die Beceutung der Constante b des van der Waals'schen Gesetzes. Von PROF. BOLTZMANN und DR MACHE, in Wien. [Received 1899 August 14.] IN dem Buche von Professor Boltzmann "Vorlesungen fiber Gastheorie, II. Theil" wurde die van der Waals'sche Formel aus der Vorstellung abgeleitet, dass die Gasmolektile Anziehungskrafte auf einander ausûben, deren Wirkungssphare gross ist gegen den Abstand zweier Nachbarmolekile. Der Fall, wo diese Annahme nicht mehr zutrifft, wurde in demselben Buche auf Seite 213 kurz behandelt. Es zeigt sich, dass dann Erscheinungen, wie sie bei der Dissociation zweiatomiger Gase vorkommen, nicht eintreten kônnen, falls die Anziehungskraft gleichmassig nach allen vom Atomcentrum ausgehenden Richtungen wirkt. Die an jener Stelle abgeleiteten Formeln konnen aber benUtzt werden, um die Zustandsgleichung zu entwickeln. Es wurde dort die Annahme gemacht, dass die daselbst mit X bezeichnete Grosse constant ist. Lassen wir diese Annahme fallen, so tritt an Stelle der Formel 233 allgemein der Ausdruck 2n = 1 2 dr e2hf (r). n1 V 2 f( Es wird also jetzt angenommen, dass die Trennungsarbeit von der Tiefe abhàngig ist, bis zu welcher das Centrum eines zweiten Moleküls in den kritischen Raum des ersten eingedrungen.ist. Dagegen soll zunachst der Fall dahin vereinfacht werden, dass die Anziehungskraft innerhalb dieses kritischen Raumes constant bleibt. Dann wird f (r)= C (( + 8- r). Schreibt man zur Abkirzung 2hC= c und fûhrt die Integration durch, so hat man n2r = -{ecs [(co + 1)2 +1] - [(c +, + 1)2 + 1]} = ni2. Es gilt aber allgemein fur ein Gasgemisch aus n1 und n2 Molekulen verschiedener Art die Beziehung p V= -- (1 + n) = MRT (n, + n,). 12-2 92 PROF. BOLTZMANN UND DR MACHE, UEBER DIE BEDEUTUNG Nennen wir a die Zahl der Molekile bei vollkommener Dissociation, so ist a = 1i + 2n2 = n1 + Kn?2. Hingegen ist die Zahl der freien Molektile im betrachteten Zustand a+ ni n = ni +2-.22 Durch Elimination von ni und Entwickeln der Wurzel findet man hieraus den NaherungsC2K wert n = a- und folglich auch weiters a2-MRT pV=aMRT — K. Ist aber m die Masse eines Molekiils, / das Atomgewicht, v das specifische Volumen, endlich r die Gasconstante des betrachteten Gases, so ist M=- am =1 /Jb V V endlich - = r und es wird auch rT art P v 2v oder wenn man auf den Ausdruck fur /c zurickgeht rT 1 2rrT rT A p 2 3 {ec [(co + 1)2 + 1] - [(co + + 1) + 1]} = "' 7j2 7jm L / \ Hiebei ist aber in v noch der von den Deckungspharen der Molekûle ausgefillte Raum 1 p= -. 37r-3 abzuziehen. Wir erhalten also als Zustandsgleichung rT A p=v- p (v- p)p p = _ p)2' Zur Discussion dieser Formel finde noch folgende Betrachtung Raum. Es ist, wie man sich leicht durch Rechnung ûberzeugt, ec [(c- + 1)2+ ] -[(c+ + c 1) + 1] = C2 (C + + 1 2?~= 1 $- I! ~n + 2 + Jf O 2 -8 2 ( - N 2 Ferner ist A=. 27rrSrT 2 (c8)-_ i + + t C i Es gilt weiters die Beziehung-c ---2hC=-. T 1 C8 oSetzt man endlich -. 2rr a = a3, =, 7 ' n9Z7 ~ -- so ist auch p = -. 47r = _ -ae DER CONSTANTEh b DES VAN DER WAALS'SCHEN GESETZES. 93 und es lasst sich die obige Zustandsgleichung in der Form schreiben: rT arT n=lo fl_ l\l 2 2 p v- e (v-a2 1 ) n!+ e + n + 2! Die Constanten dieser Gleichung haben folgende Bedeutung: Es ist a gleich dem halben im Volumen der Masseneinheit vorhandenen kritischen Raume, C3 /3r= - gleich dem Potential der Anziehungskraft auf der Oberflâche der Deckungssphare, endlich e=, gleich dem Verhaltnis aus dem Durchmesser des Molektils und der Distanz, auf welche die Anziehungskraft wirkt. Da die Gleichung 233, von welcher wir ausgegangen sind, voraussetzt, dass die Anzahl der Tripelmolekule gegen die Anzahl der Doppelmolekile verschwindet, so ist auch die obige Gleichung an die Voraussetzung gebunden, dass die Abweichungen des Gases vom Boyle-Charles'schen Gesetze noch klein sind. Es darf also auch das letzte Glied unserer Gleichung, welches ja den Innendruck darstellt, nicht iiber einen gewissen Wert hinaus wachsen. Dies wird um so weniger der Fall sein, je grosser e ist. Aus den Versuchen von Amagat und Andrews fiber die Compressibilitat des Kohlendioxyds berechnet sich e fiir dieses Gas zu ungefahr 100. Nach dieser Vorstellung scheint also der Anziehungsbereich sogar noch relativ klein zu sein gegen den Durchmesser des Molekils. Wir haben bisher unsere Zustandsgleichung abgeleitet, indem wir fur f(r) ein bestimmtes einfaches Abhangigkeitsverhaltnis einfthrten. Lasst man f(r) ganz willkiirlich, so ergibt sich leicht, dass dies den Typus der Zustandsgleichung, auf welche man kommt, in keiner Weise verandert. rT A Es wird stets p = _ - und es ist nur noch A von f(r) abhàngig. v-p (v- p)2 Dies gilt freilich nur solange man die Anzahl der Tripelmolekule und der noch hoheren Congregationen vernachlassigen darf. Ist dies nicht mehr der Fall, so werden noch weitere Glieder hinzutreten, welche in ihren Nennern das v - p in der dritten, vierten und hôheren Potenzen enthalten. Es ergibt sich dann fir p eine Potenzreihe, wie sie ahnlich auch schon Herr Professor Jager von anderen Betrachtungen ausgehend aufgestellt hat. Leider begegnet die Auswertung ihrer weiteren Coeffcienten kaum zu iberwindenden Schwierigkeiten. V. On the Solution of a Pair of Sinmultcneous Linear Differential Equations, which occur in the Lunar Theory. By ERNEST W. BROWN, Sc.D., F.R.S. [Received 1899 July 14.] IN the calculation of the inequalities in the Moon's motion by means of rectangular coordinates a certain pair of differential equations is continually requiirig solution. The left-hand melnbers are linear and always the same; the right-hand members are known functions of the independent variable-the time-and vary with each class of inequalities considered. It has been the practice to obtain the required particular integral by assuming the solution (the form of which is known) and then to determine the coefficients by continued approximation. This method is troublesome to put into a form which a computer can use easily and is besides peculiarly liable to chance errors; a large number of processes would have to be learnt before the computer could proceed quickly and securely. The main object of this paper is to put the solution into a form which will avoid these difficulties, but I believe that some of the results may be found to be of a more general interest. Further, the question of the convergence of the series used to represent the coordinates in the Lunar Theory may be somewhat narrowed. In fact it being granted that the series forming the 'Variation' inequalities and the elliptic inequalities depending on the first power of the Moon's eccentricity are convergent, it is not difficult to demonstrate, by means of equation (14) below, that all the terms multiplied by a given combination of powers of the eccentricities, inclination and ratio of the parallaxes, that is, all the terms with a given characteristic, form a convergent series. The equations to be considered are d2x dy dt 2 dt+ +L'y= R, 2n' + L'x + L"y = R', MR BROWN, ON THE SOLUTION OF A PAIR, OF EQUATIONS, ETC. 95 where L, L" cos L~, are of the forms iqi sn (2i + 1)(n - n)(t - t), R, of the forms iqi cos {i (t- to) + 7 (t - t)} (n- n'), to, tl, 7, n, n,, qi being known constants, and i taking all positive and negative integral values; r is either an integer, in which case t = to, or is incommensurable with an integer. The corresponding particular integral required is, in general, yX = i Pi COS {i (t - to) + r (t - tl)} (n - n'). y -p'i sin If we substitute this solution in the differential equations and equate to zero the coefficients of like periodic terms, we obtain an infinite series of linear equations with an infinite number of unknowns. The series are assumed to be convergent and in most cases the coefficients diminish rapidly as i increases. Nevertheless, it is frequently found necessary to proceed as far as i = + 5, demanding the determination of about 20 unknowns from the same number of equations. In the determination of the latitude the equation d2z dt2 + L1 z = R", occurs; L1, R" are of similar forms to L, R', respectively. If zi, z, be two particular integrals of d2z + L1 z = O, it is known that the particular integral required is z. C= Z2 l R dt - z fz R" dt, where C is a constant given by C = z dz dz, è1t -Z2dt I shall show in what follows how we may obtain a similar expression for the solution of the simultaneous equations above, having a sufficiently simple form to be of use in computations. Later the significance of the solutions is explained and certain exceptional cases occurring in the Lunar Theory are treated. The results obtained have in fact been used in the calculation of the terms of the third* and fourth orders in relation to the eccentricities, the inclination and the ratio of the parallaxes. * I.emon. 1R. A. S., Vol. LIII. pp. 163-202. 96 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR I. In order that the series which occur may be all algebraical instead of trigonometrical, we use the conjugate complexes ut, s, where u = x+yt, s= x-yt. We also put W = exp. (n - n') (t - to), d 1 d dC L (tn-n')dt' ni m = n — to = O t. n - The generality of the results is not affected by the last supposition. The simultaneous equations then take the form (D + m)2 +Mt + Ns = A ` -..........................o..........(1), (D - m)2 + Ms + Nu = A where M, N are of the form pii,, A is of the form pij-2i+1+T + p'/ 2i+1-T, J M=M. The bar placed over a letter or expression denotes here and elsewhere that t has been changed to -, that is, ~-l put for. To obtain the particular integrals of equations (1), it will first be necessary to obtain four independent particular integrals of (D + m)2 u + Mu + Ns = 0 ) (D- m)2s + Ms + =.................... Denote these integrals by ' = j, s=Sj, j=l, 2, 3, 4, so that if Qj denote an arbitrary constant, the general solution of (2) is =.j =jQj sj, j= 1, 2, 3, 4. By supposing the Qj to have variable instead of constant values we can then proceed to find a particular integral of (1) and thence their general solution. In order to make certain of the later arguments clear it is necessary to indicate the manner in which the equations (1) arise. The equations 3 Ku D2u + 2m Du + - m2 ( + s) - - = 0, 2 (us): D2s - 2m Ds + m2 (u + s) ( = 0, 2(us) DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 97 with their first integral, F-Du.Ds+ m2(u+s)2+ =0C, admit a particular solution, U =_ u = O ai 2i+l, 8= s0 = a_i '2i-1, containing two arbitrary constants; these constants are the quantities denoted by n, t0 above. The coefficients ai are functions of n and the known constants present in the differential equations. Put U=Uo+U1, S = o+S1, and, after expansion in powers of 1i, sl, neglect squares and products of these quantities. Omnitting the suffix, and giving proper meanings to M, N, the resulting equations become those denoted by (2) above. The first integral F= C becomes aF aF = u21 + s=O0. auo0 as If, however, we had deduced this first integral directly from (2), it would have been = C', where C' is an arbitrary constant. When the equations (2) are considered independently the constant C' must be retained. Three independent solutions of (2) are known. In finding the principal part of the motion of the lunar perigee Dr Hill* gave one of them, namely, U =Du,, s = Ds, and obtained the forms of the other two; the coefficients of the latter have been obtained by myself. It is therefore only necessary to find a fourth solution, linearly independent of the other three, in order to obtain the generai solution. II. The Fourth Integral of the Equations. (D + m) u + Mu + Ns=.............................. (3), (D - m) s + Ms + u = 0.................................(3'). The known integrals may be denoted by -1 = E2 e++C S1= o? E-.2i -1+C u 2 =:ii 'i2ii+-c S =:i' _ 2i-l-C+ - ii2i+l —, 3-_ i= — 2i —1-C. )....................(4).,3 = Ei (2i + 1) a2i+l, S3 =,i (2i - 1) a_2i-l * Acta Math. Vol. viII. pp. 1-36. + Mem. R. A. S. Vol. LIII. p. 94. VOL. XVIII. 13 98 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR If Q1, Q2, Q3 be three arbitrary constants, then u= jQjuj, s= jQysj, j= l, 2, 3...........................(5) is a solution of the equations. Owing to the introduction of Q,, Q2, Q3, we can consider ui,... 83 completely known; c is a constant which is supposed incommensurable with unity. To discover the fourth integral, the method of the Variation of Arbitrary Constants is used in the usual way, by assuming that 'uiDQ, + u2DQ2 + u3DQ, = 0. By substituting (4) in the differential equations we find Du,. DQ1 + Du,. DQ2 + Du3. DQ3 = O, j (sj D2Qj + 2Dsj. DQj - 2mnsDQj)= 0............................ (6). Put u2Du3-u3Du2 = a1, etc. Then DQ1 DQ2 DQ3 _ Then = L, suppose. ~a, 02 a3 Substituting in (6), the equation for L may be written, (a=s) DL + 2LD (sas) - L (SsDa + 2mcas) = 0........................(6'), where Eas = as, + a2s2 + a3s3, etc. The last term of this equation can be shown to be zero. Substitute ul, si and u2, s, successively in (3): multiply the resulting equations by s2, si respectively and subtract. We thus obtain (D + 2m) (s2Du, - i Du2) + (m2 + M) (s2ul - u2s) = 0. Also, treating (3') in a similar manner, (D - 2m) (u2Ds, - uDDs2) + (m2 + M1) (u2s - sul) = 0. The sum of these two equations is integrable and gives s2Du, - tl1Ds2 + u2Ds1 - s1Dz2 + 2m (s2U1 - u'2si) = C12, where C12 is a constant. It should be noticed that this constant is not arbitrary since the values of u1, si, u2, s. were definitely fixed, so that CI, may be treated as a known constant. Denote the last equation by f 2= C............................................. We find in an exactly similar manner f23 = C23î f31= C31...................................... (7 ). Multiply these three equations by ui, u2, us and add. Noticing the meanings attached to aC, a2, a3, we obtain u1 23 + 23 C31 + U3 12 = $_s. Similarly 0 = u1Df23 + u2Df1 + u2Df2i =,sDa + 2m$as. DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 99 Substituting the last result in (6'), we find DL 2) (es) L+2 -s which, on integrating, gives L Lo Lo (:~S)- () 2 ~ U2 C3 + U 31 + t3 C12)2 where Lo is a new arbitrary constant. Thence =(Q)+L -, etc., Thenee QI = (QI) + L~D-1 (u C23 + u2C,31 + u3C&2)2 in which (Qi) is a new arbitrary constant and D-1 denotes an integration, i.e. the operation inverse to D. If, finally, we now let Q1, Q2, Q3, Q4 represent four arbitrary constants, the general solution of (2) is u = Q121, + Q2u2 + Q3U3 + Q4U4, S = QlSl + + Q Q33 + Q4S4, where 4 = jUjD-'( 3 + u + uC2)2' j = 1, 2, 3. (u1 C03 + 22 C31 + US:)2 This result is true whatever particular solutions are represented by U1, 8 ~12, 82 2 3, 83 as long as they are linearly independent. As, however, the expression for u4 can be very much simplified by using the values given earlier, I shall immediately proceed to the special case under consideration. It is easy to show that C3 0 = C23. For, looking at the forms assumed, we see that u1, s, contain the factor tc, us, s the factor ~-' and u3, 3s have no such factor. Hence f23 has the factor r', f3l the factor g'-". As c is supposed incommensurable with unity, the equations (7') are only possible if C03 =O and C03 = 0. Hence we have u2 = uD u2Du3 - u Du + 2-u3Du - 1Du + u U, D, - u-, Du,,4CI22 = UlD-1 U27)83-U3D~2 + UsD-1 U3 Un U32 The first two terms of the right-hand side are integrable and become U2 U1 1i - - n2 - that is, zero. Whence considering C122 as absorbed in the arbitrary Q4, we have 114 = n3D- (uiDz-2Dui).(8). U US -D~ -1 (U U2~-' ' t'l'0 )................................. (8). We may similarly show that s4 = s3 D- (s Ds2 - sDs,) 13-2 100 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR III. Although this is probably the simplest form obtainable for u4, it is unsuitable for calculation. The values of u1,... are all of the form sum of cosines+ L (sum of sines). To adapt u4 to calculation it is best to express it in the form 3 (P + Qt) where P, Q are real. I shall show that * s1U~ 1s2 0 U- U)8 U,3s + S Us U3 _ -u3+ (U2 8S U2 - (2m+..3....(9). Since 23 = 0 =3 and f- = 12, we have 1 C12 - u2f3 - u1f23.i2 UIDu2 - u2Dul + u2Ds - ulDs2 + s1Duz - s^Du1 U3S3 83 2 U3s3 i32 2u3S3 S1Ut - 21S2 Du3s s - U2U u1S2 DU1 - U2- Du1 S+ D ( - U1s2) U3S -m S3. 32 383 U2-U12 fDît3 -DS3 i 3 U 3 L3 82 -SU,-Us2,(Du3 s) 8?31 -s, U'3S3 U3 S3 S U23 Submitting this to the operation D-1 and transposing we obtain the required expression. It is easy to see that (9) is of the required form. For when we put - for t, that is, ~-1 for g, the expressions U1, U2, S1, s 3, u, s, D-1, D respectively become S2, S, U 2, U S3 -, -3, - u — 1 -D; the first term of (9) is therefore unchanged, while the second term simply changes sign. Hence the first terrn is real and the second a pure imaginary. IV. It is necessary to examine the four solutions and especially the one last found a little more closely. Write u4 = u3 (P + D-1P,). The expressions (4) show that P and P1, being both real, will be expressible as sums of cosines of multiples of the angle 2(n -n')t. As P1 contains a constant term B, D-'P, contains a term of the form tBt (n-n'), and therefore u4 is of the form ut3 [Bt (n -n') + a power series in r2. It is therefore of the same form as u3, except for the part tBtu3 (n - n'). DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 101 We saw earlier that the equations (2) admit of a first integral 5= C', and that this should be derivable from the integral F=C, of the non-linear equations when the former are considered as derived from the latter. The constant C' should therefore in this case be zero. It is easy to see that the constant is zero when we substitute in b the solutions us, s. or u2, s2 or u3, s. For the solution U4, S4, the constant takes the value C12 which is not zero. Hence though (u4, s4) belongs to the linear equations (2) it plays no part in the non-linear equations from which these were derived. The solutions u,, s, and u2, s2 are those used in developing the Lunar Theory; they contain the terms dependent on the first power of the lunar eccentricity. It is necessary to see why the solutions u3, s3 and U4, S4 are not used in the development. The particular solution of the original equations of which use was made was,=Uo), S= So where no = Eiai 2i+1 = iai exp. (2i + 1) (n - n') (t - t). If we add a small quantity Sto to to (which is an arbitrary constant of this solution) the resulting expression will still be a solution. Expand in powers of St0 neglecting squares and higher powers. The additions to u0, s, will be u = Sto = -Duo. Sto, Ss = o to = - DSo. Sto. ato ato These values when substituted for u, s in (2) must satisfy them independently of the value of Sto. Hence t =kDu, s= kDs is a solution obtained merely by altering the arbitrary to and is therefore unnecessary for the development of the Lunar Theory. The other arbitrary constant in Uq is n, and the coefficients ai are functions of n. If we make a small addition Sn to n and proceed as before we see that au, 0 as u=k, s=n o an an is a solution of the linear equations (2). It is only necessary to identify this with U4, s4. The forms for both are evidently the same. For we have a — i {m a + (2i + 1) (t-to) ai exp. (2i + 1) (n - ) (t - to) aai = Si a exp. (2i + 1)(n ')( - t) + (t — to ) Duo. The terms with t as factor agree (to was put zero in the expression for u4) when the proper constant factor is introduced, and the remaining parts are of the same form. As no linear relation can exist between the first three solutions and either of the forms 102 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR for the fourth solution, these two forms must be the same except as to a constant factor. Hence kauo D- (u1D2 -u2DuI) K 213 U~ DU- - u-DLU - ) ln U 32 This relation is a somewhat remarkable one. In investigations where the arbitrary x a~y constants are varied-and there are many such-we have a means of obtaining ', aY an' an (which are the most troublesome to find) when the numerical value of the ratio n'/n has been used in finding x, y. A direct proof of this relation is desirable. This and the theorems which I have given elsewhere* are probably particular cases of some much more general theorem. Thus, of the four integrals of the linear equations two only are required for the development of the lunar theory, the other two arising from additions to the arbitrary constants in the particular solution of the original equations. V. Having obtained the solution of (D + m)2u + Mu + Ns = 0, (D- m)2s + Ms + Nu = 0, in the form u=2Qjuj, s= Qjsj, j=1, 2, 3, 4, the next problem is to find the solution of (D + nm)2t + Mu + Ns = A, (D- m)2s + Ms + u = A, where A, A are functions of the time. Following the usual method of varying the arbitraries we have,Du^j. DQ = A, YDsj.DQj = A (1. ujDQj = 0,:sjDQj = 0 These must be solved in order to find the variable values of the arbitraries. The only difficulty is to find these values in forms sufficiently simple to be of use. The expressions at the end of II. show that we can derive s,4/s, from u,/u, by putting -~1 for Ç and changing the sign. For ul, s2 interchange as do u2, sl, while D changes sign: u3 becomes -S3. Since t4 = u3 (P +'QO), we have 4= s3 (- P + Q). Hence U4S3 - s4 3 = 2,3s3P = S2 s - S2................................(11) by the result obtained in III. * P?'oc. London Math. Soc. Vol. xxvIII. pp. 143-155. DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 103 Again, as the first integral obtained in II. is equally applicable to u4, s4, we have 34 =f34 = s4Du3 + 4Ds3 - u3Ds4 - s3Dt4 + 2m (s4u, - u4S3), which, by inserting the expressions for U4, 84 just given, becomes C34 = - 2 (S3Du3 - u3Ds3) P - 2tu3s3DQ + 2m (s2ul - us), or, using the values of P, Q obtained in III., 34 = - (S3DU3 - 3DS3) SU - 0C + (s1u2 - us2) (2m+ 3 -— Ds3) + 2m (s2i- $- S whence c34 = - C..........................................(12). We can show as in II. that C4=O0= 24. Solving equations (10) we obtain DQj= a where A= Du1, Du2, Du3, Du, Ds,, Ds2, Ds3, Ds4 1 U1, U, U3, ut4 S1, 82, S3, S4 A1 = A, Du2, Du3, Du4 A, Ds, D Ds Ds4, etc. 0, U23, U3, U4 0, 82, S3, S4 In the determinant A the first minor of Du, is DS2 (U3s4 - S3u4) + Ds3 (u4s2 - s4u2) + Ds4 (U2Is - S2u3), = S2f34 + 83f42 + 84f23, = 82 C34 + S3 C42 + St C23 Also, the first minor of Ds, is similarly -(u2 C34 + t3 C42 + 24 C23). The other minors of the elements in the first two rows of A are similar, the suffixes following a cyclical order. We have thus all the minors of the elements A, A in the determinants Aj. Remembering that C34 =-C and that all the other constants Cij are zero, we obtain A = - (sA + u2A) C12, A2 = (s A + u1A) 0C2, A3= (s4A + 4A) 0,, A4 =- (s3A + usA) C12, and A = - (s2Du1 - s1Du2 - s4Du3 + s3Du4) C0. 104 MR BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR But the effect of putting Ç-1 for ' in à is only to interchange an even number of rows and columns and therefore to leave A unaltered. Making this change in the last equation we find A = - (-u, Ds2 + u2Ds, - zu4Ds + s4Du3) C2. Whence, by addition, 2A = - {/i2 - 2m (s2U1 - ZS1) -f34 + 2m (4U3 - u4Ss3)} C12 = (C - (2- 34),12 =- 222, in virtue of (7) and (12). Hence A=-(7~2. A I s2A + ~2À Finally, AQ. = 2A, etc. and Q 1=- D-' (s2A + 2uA), etc. C12 And the particular integral corresponding to the right-hand members, A, A, is u = - {Z)1D-l (s2A + ut2A) - u2D-l (slA + îtnA) 12 - u.D- (s4A + 4A) + uD-1 (3A + A )}......(13), s = C {lD-~ (s2A + Z2A)- s2D-' (sA + t1A) ^12 -- s3D- (s4A + u4A) + s4D-1 (sA + 23A)}. It is easy to see that s is derivable from u (as it should be) by putting g-~ for g. In fact, the coefficient of u1 in the first term is conjugate to that of u2 in the second term, that of i3 in the third term is a pure imaginary and that of u4 in the last term is real. VI. In the applications of this result to the Lunar Theory A is always an expression of the form Sqii'+ i +S i-., i =0, 1, + 2,... where r, qi, qi' are known constants; A is derived from A by putting ~-1 for ~. Thus A, A are conjugate complexes whose real and imaginary parts are respectively sums of cosines and sines. The corresponding particular integral should in general be of the same form. Hence a difficulty arises owing to the fact that t4, 84 contain t in a non-periodic form. I shall now show that in general all the non-periodic parts disappear from the particular integral. Put u = u4' + LBu.t (n - n'), s4 = s4' + tBs3t (n - n'). DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 105 Then,u4, s4' are periodic. The sumi of the third and fourth terms of (13) becomes - t3D-1 (4'A + u1A) + u4D-1 (s3A + JA) - [utD-1 {(s3A + uA) t} + u3tD-1 (sA + U3A)] tB (n - n'). The first line of this expression is in general periodic. The second line becomes, on integrating its first term by parts, u3BD-2 (sA + f3A). The non-periodic part thus disappears. When we perform the double integration involved in this last expression, we obtain u3 [fo + C'i (n - n') t + periodic part} where CO, C7 are arbitraries. The terms containing Co, C, are simply parts of the complementary function and may be considered as contained in Q3u3 + Q4u4. The particular integral may therefore be written 1 = C(sA + 9A) - [uDD (lSA - lUA) + u1D-' (sA + UA) - I3D-1 {s4A + u4'A - BD-1 (sA + t3A)}].....(14), which is its final forni. VII. In general this particular integral consists only of periodic terms. There are, however, two cases in which non-periodic terms may arise. If =an odd integer, that is, if A is of the form q^, '2i+1, the integrals multiplied by 2t4' and t4n might give rise to terms of the form at where a is a constant. In this case, sA + u3A is of the form,/3i (2i -~2i) and therefore its integral will be periodic. The last term of (14) is of the form - uD- costs. + power series in 2) - -U3 (tk + k' + power series in 2), k, k' being constants, the former definite and the latter arbitrary. The terms - u3(tk +k') may be written Ic kI- 4' - k'u - {utBt (n - n) + U4} Bt (n - n') + Bt (n - n') The first two terms of this may be considered as included in the part Qu + Q4ut4 of the complementary function; the last part is definite and periodic. Hence no non-periodic part remains. The second case of non-periodicity occurs when A = iqi C2i+1+C + {iqi' 2i+1-C. Here the first two terms of (14) may give rise to the non-periodic part {uot (n - n') [sA +?uA], - utt (n - n) [s A + ulA]o} - C2, VOL. XVTIII. 14 106 MR BROWN, THE SOLUTION OF A PAIR OF EQUATIONS, ETC. where [lr] denotes the constant term in the expansion of J as a sum of cosines. Now s2A + u2A and s1A +,1A are conjugate. Hence [s2A + z2A]o = [s1A + 1A]o = [(si + S2) A]o Thus the non-periodic part is (1, - 2) [(Si + s2) A] (n - n')t. C2............................. (15). In the applications to the Lunar Theory, the part of the complementary function used is obtained by putting Q3=0=Q4, and the constants in uz, u2 are so adjusted that we can put Q=1= =Q. I shall show that (15) is equivalent to a small addition &c to c in the index of ' in 11 +?t2 = ie+ir2i1C +.is/2i+1-c squares and higher powers of &c being neglected. Put c + c for c in the last expression.* It becomes u1 é8c + 2t2 -8C Remembering that = exp. (n- n') t and expanding in powers of 8c we obtain 21i + h2 + (1i - u2) ctL (n - n') t. Comparing with (15) it is evident that we can put 3c = [(s1 +,s) A]o + C2. This is nothing else than the general form of the expression which I obtained in a paper, "Investigations in the Lunar Theory *." For 12 =f/2 = [f,] = j (2j + 1 + m + c) ej2+ Sj (2j - 1 - m + c) e_2, on substitution of the values (4) in fi. Also s, + s8 is the same as the expression there denoted by Se. The comparison of A with the remainder of the equation of the paper just referred to will follow from what precedes that equation. The general case is given in my memoir on "The Theory of the Motion of the Moon, etc.t." No useful purpose will be served by giving further details of the comparison of the two forms for c. The final conclusion is that the non-periodic terms either disappear of their own accord *or belong to a part of the complementary function which is not to be included in the general development. The last part of this investigation-concerning ec-is of course only applicable to cases similar to those which occur in the Lunar Theory where we proceed by continued approximation and where we require to have only periodic terns. In the general problem the non-periodic terms will remain. * American Jour. Math. Vol. xvII. p. 336, equation (16). + lMe. R. A. S. Vol. LIII. p. 75. VI. The Periodogram of Mlagnetic Declination as obtained from the records of the Greenwich Observatory during the years 1871-1895. By ARTHUR SCHUSTER, F.R.S., Professoer of Physics at the Owens College, Manchester. [Received 1899, Aug. 1.] I. INTRODUCTION. THE science of Meteorology deals with variable quantities which are subject to continuous and apparently irregular changes. Irregularities in the strict sense of the word do not however exist in nature; there is never absence of law, though often an appearance of lawlessness caused by the effects of several interacting causes. Our efforts rnust be directed to disentangle these causes, and to discover for that purpose the hidden regularities of the phenomena. If we look for instance at the curve which represents the barometric changes, we see at once that though irregular, there is a tendency towards an average position, large deviations from that position being less frequent than small ones. Prof. Karl Pearson has investigated statistically the laws of deviation from the mean, and obtained valuable and interesting results. But enquiries of this kind necessarily leave out of account one of the most essential points in the phenomena they deal with, which is the regularity which may exist in the succession of events. In taking the average daily values of barometric pressure and studying their deviations from the mean, the same importance is attached to an exceptionally high barometer when it follows another day of high barometer, as when it follows one of low pressure. But a high pressure is more likely to be followed by a high pressure than by a low one, and the regularity which this succession implies seems to me to be of greater importance than the laws of distribution based on the assumption that successive days are quite independent of each other. I intend in this paper to describe a method, applying it to a particular case, which seems to me to yield some valuable information concerning the hidden regularities of fluctuating changes, though it does not pretend to give a complete representation of ail that it is important to know. 14-2 108 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION The method has been suggested by the analogy between the variable quantities we are here concerned with, and the disturbance in the luminous vibrations. If we could follow the displacements in a ray of light, we should find them to present characteristic properties not unlike those of meteorological variables. There is the same irregular fluctuation combined with a certain regularity of succession, which becomes revealed to us by prismatic analysis, and shews itself in the distribution of energy in the spectrum. Absolute irregularity would shew itself by an energy-curve which is independent of the wave-length, i.e. a straight line when the energy and wave-length or period are taken as rectangular coordinates, while the perfect regularity of homogeneous vibrations would shew itself as a discontinuity in the energy-curve. Fourier's analysis gives us a means of doing by calculation for any variation what the spectroscope does experimentally for the luminous vibrations, and if we construct a curve which represents the relation between the coefficient of Fourier's series for a given period and that period, we have a simple way of representing the regularities of the quantities to be investigated. We shall also incidentally gain the great advantage of separating in a clear and definite way the fluctuations which take place in definite periods, such as the lunar and solar variations, froin the more complex changes on which they are superposed. II. THE PERIODOGRAM. Let f(t) be any function of t, and consider the quantity R determined by the equations,r+nT 7+nT -nTA f(t)costd, = nTB f(t) sin sctdt... (1), R =A + B2.................................................(2), where K = 27r/T and n is an integer. In these equations T represents a certain interval, and, a time which can be varied. In the class of functions f(t) to which this paper refers, a change in T with a constant value of n and T will cause R to fluctuate round some mean value. Let S2 be the mean value of R2 which, still keeping n constant, will in general depend on T. With T as abscissa and S2 as ordinate, draw a curve, which may be called the "Periodograph." I define the "Periodogramn" as the surface included between this curve and the axis of T. It will be seen that the "Periodograph" corresponds exactly to the curve which represents the distribution of energy in the spectrum. The treatment of a few special cases will render this clear, and lead gradually up to the complex phenomena which form the chief subject of this investigation. CASE 1. Let f(t) be a simply-periodic function, so that we may put f(t) = cos (gt + S). FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 109 The integrals A and B are easily calculated and expressed in the form TA cos (a b)+ cos (a-b)sn gnT, nTA= 2 L + sin ln. T y__ c g- +s( lTB = 2 Lsin (a + b) s sin izgT, where 2a = ci + Ca, 2b = /3 + 3f2, and al = T, 1 = -g + s, 2,= (r + fT), /2 = g (r + 2T) +. Hence nTR = 2 sin gnT[2 (g + K2) + 2 (2 - K2) cos 2b]i. 12 - KC 2 ' If the average of R2 is formed for different values of T, the term containing cos 2b will disappear, and therefore writing ry = (g - ),T = T=7r - it follows that S = /2 (2+ tK2) sin y g+K 7r If?n is large, S will only have appreciable values when g and Kf are very nearly equal, and in that case we may put with sufficient accuracy S siny This is the well-known expression, giving the distribution of amplitude in the focal plane of the telescope, when a homogeneous vibration is examined by means of a prism or grating. If we wish to plot down the curve of intensities of vibrations as analysed by a grating-spectroscope, we may define any direction by the period 27r/Kll which has its principal maximum in that direction. If the incident light has a period 27r/g the expression for the distribution of amplitude is sin [7r Nr (g - K)/K] * 73r (g-K)/K which is identical with S if V, the number of lines on the grating, is equal to n,; the number of periods included in the integration. In obtaining the "Periodogram," we have done by calculation precisely what the spectroscope does mechanically. The analogy is complete, and just as a ray of homogeneous light does not appear homogeneous in a spectroscope, there being secondary maxima owing to the finite resolving power, so does a purely periodic function when analysed by Fourier's series shew apparent periodicities * This expression may be obtained either from the in my paper "On Interference Phenomena," Phil. Mag. original papers by Lord Rayleigh on the resolving powers Vol. xxxvnI. p. 509 (1894). of spectroscopes, or more directly from an expression given 110 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION having secondary maxima near the principal one. These secondary maxima I have termed " spurious " periods. Their intensity remains the same when the "resolving power" n is increased, but they approach nearer and nearer to the principal maximum. They are therefore distinguished from the true periodicity by the fact that their position changes with n. CASE 2. The function to be analysed consists of two overlapping simple periodicities. The integrals A and B will split up into two parts which we may call AI, A2, B1, B2 respectively. Hence R2= (A, + A2)2 + (B1 + B2)2. The products A1A2 and B1B2 will vanish in the expression for S2 when the average is formed for varying values of r. Hence S2 = A 2 + B12 + A,'2 + B,22 = R12 + R22, or the Periodogram of two simple periodicities may be formed by the superposition of the separate periodograms *. CASE 3. The function varies uniformly with the time. Putting f(t) = t, and performing the integrations, it is found that 2c 2c A = sinr c; B = - - cos cr, K Kf R2 = S2 = 4c2/K2 = 2T2/r2. Hence the Periodograph is a Parabola. The consideration of this case, which has no analogy in the analysis of luminous disturbances, is of importance in the treatment of secular variations, such as that of the magnetic elements. CASE 4. So far the function f(t) has been taken to be continuous; but cases arise, where f(t) is given numerically for a number of values of t, which we may for the sake of simplicity assume to be equidistant. As Fourier's analysis applies also to discontinuous functions, we may include cases of this kind. Let the different detached values of f(t) follow the law of errors so that, NT being the total number of ordinates, the 2hN number having a value intermediate between / and /3 + d is e-hI82 d8. I have 27r shewn t that in this case S h 2 Nh2 ' * In my paper " On hidden periodicities " (Terrestrial change to the latter form is apparent from the above. Mag'netisnm, Vol. III. p. 13) I defined the ordinate of the t On the investigation of hidden periodicities, loc. cit. Periodogram to be S instead of S2. The advantage of the FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 11 so that the periodograph is a straight line, parallel to the axis of T, the distance between the two lines being inversely proportional to the nurnber of ordinates. CASE 5. The function is given in the form of an irregular curve which satisfies the condition that there is a definite law of probability that the quantity R should lie within assigned limits; this probability being independent of the initial time r. If we consider for instance the curve representing the height of the barometer, excluding lunar and solar periodicities, the changes in the curve will apparently be quite irregular but will satisfy the above conditions. Let A1 and B1 be taken to be components of a vector defined by the equations r~+inT T+ÏrnT InTA f = f(t) cos ictdt, ~ nTBî = f(t) sin /ctdt. 7'+2mT T+2nmT Similarly nTA2 = f(t) cos Ktdt, rnTB, = f(t) sin Ktdt, T+îmT T+mT T+smT rT+snmT and so on until nTA, = f(t) cos ctdt, ~, nTBs = f f(t) sin ctdt, r+ (s-1) mT r+(s-1)mT with the condition that smn=n, m not being necessarily an integer number. We may choose mT sufficiently large to secure complete independence of successive vectors, all directions of the vectors being equally probable. In that case the vector R which is the resultant of the separate vectors A, B, etc., will, as shewn by Lord Rayleigh*, have a value such that the expectancy of R2 is proportional to the number S of vectors; hence keeping mn constant and increasing S, the ordinates of the periodograph will vary inversely with nT. This is the only general conclusion we can draw in this case. CASE 6. The function f(t) is formed by the superposition of one or more simple periodicities superposed on the irregular curve of case (5). This includes the important cases of barometric, thermometric or magnetic changes. The Periodogram may in all these instances be used to separate the real from the accidental periodicities. For the value of the ordinates of the Periodogram has been shewn to be independent of the range of time over which the integration is performed when the periodicities are real (Case 1), but to vary inversely with the time when they are accidental (Case 5). Hence we may obtain a conclusive criterion to distinguish between the two cases. The fundamental proposition on which the separation depends may be stated thus: T The value of f (t) cos Ktdt fluctuates for the functions under consideration about soie value which is proportional to T when f(t) = cos /t and proportional to i/T when f(t) contains no real periodicity of periodic time 27r/1c. * PF1il. lllag., Vol. x. p. 73 (1880). 112 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION The separation of regular and irregular oscillations, by an increase of the time interval, is identical with the spectroscopic separation of bright lines and continuous spectra (e.g. in observing the solar chromosphere) by an increase of resolving power. III. CALCULATION OF THE PERIODOGRAM OF MAGNETIC DECLINATION. I chose as an example of the treatment indicated in the previous pages the record of magnetic declination at Greenwich. The subject interested mne chiefly on account of an alleged magnetic effect connected with solar rotation, and special attention was therefore paid to the periods in the neighbourhood of 26 and 27 days. It will appear that the magnetic declination is not at all a favourable quantity to fix upon for the discovery of possible outside magnetic effects; but as the only real pieces of evidence, so far produced, in favour of a period approximately coincident with that of solar rotation, were derived from magnetic declination and the occurrence of thunderstorms, and as the latter does not lend itself easily to accurate treatment, I had no choice but to attack in the first instance the records of declination. The publication of the Greenwich Observatory contains the average daily values of declination to 'I1 minutes of arc. There are occasional gaps of a few days duration. The way of dealing with these gaps was quite immaterial on account of the large quantity of material used, and a rough process of interpolation was adopted. Thus if there were no records during three days, and if the values given for the days preceding and following the gap were 17'1 and 15'8, the intermediate values were put down as 16'8, 16'4, 16'1. In the few instances in which the records extending over a considerable portion of an adopted period were missing, the whole period was excluded. The first object of the calculation was to find the Fourier coefficients corresponding to a sufficiently large number of periods, so that the curve representing the periodograph might be drawn continuously through the points obtained. The original series of figures were for this purpose arranged according to the usual procedure, in rows corresponding to the selected period. In order to obtain, for instance, the Fourier coefficient for the 24 day period, the first row would begin with the magnetic declination of Jan. 1, 1869, and end with that of Jan. 24, the second row including the values from Jan. 25 to Feb. 17 being written underneath the first. Subsequent rows were added until a date was reached as near as possible to Jan. 1, 1870. This meant 15 rows, the last number being that corresponding to Dec. 26, 1869. The arithmetical sum of the 15 rows was taken as basis for the treatment of the 24 day period during 1869. A similar group of rows was written down for 1870, beginning, in order to secure continuity, with Dec. 27, 1869; but the third group, beginning with Dec. 22, 1869, and ending with Jan. 9, 1872, included 16 rows. I thus obtained a new set of 25 rows (there being 25 years), each of which consisted of a sum of 15 or 16 of the original rows. The subdivision into years was chosen so as to divide the whole material into convenient portions. It will be understood from what has been said that a row corresponding to a particular year has been obtained by making use of observations, the great bulk of which fell FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 113 within that year, but some of which may have belonged to December of the preceding or January of the following year. Table I. gives the figures for the 24 day period, the last three columns indicating the date of the first and the last observation made use of in the corresponding row and the number of rows included in the year, 356' 70 Ineaning the 356th day of 1870. The unit in the first three Tables is 0'1 of a minute of arc; in the remaining Tables, unless otherwise stated, it is the minute of arc. The columns of Table I. and of the corresponding ones for other periods were added up, and the results, after subtracting a constant for each row, are given in Table II. Table II. clearly shews the effects of secular variation, and we must consider in how far it is necessary to take any notice of this variation. If our observations extended over an indefinite time, Fourier's analysis would itself perform all that is required, and each period would be totally independent of all others. But our investigations have been limited to a range of time of 25 years, and the secular variation involves a period much longer than this. The progressive change of declination will add terms to the periodic series which it is easy to evaluate with sufficient accuracy. If we take the change to be uniform and equal to -ct, Fourier's theorem applied to the interval 0 to T gives us cT cT (. 27rt 1. 47rt 1. 67r ) - ct= - s-+- T sin sin T+ +n +...............(3) The effect of such a uniform progressive change would be to leave the cosine cT terms unaffected, and to add T to all sine terms of period T. 7r As it is our object to separate all real from accidental periodicities, we are justified in eliminating all known effects either totally or partially according to convenience. The average magnetic declination at Greenwich during the year 1893 was 2052.7 less than during 1869, giving during 25 years a change of almost exactly 3~. Throughout this investigation the magnetic declination has therefore been assumed to be made up of a uniform progressive diminution of 7'2 per year added on to more or less irregular changes, the latter only being subjected to Fourier's analysis. No assumption is made as to the secular variation being either uniform in character or having exactly the above magnitude. We have eliminated from our results a large portion of the secular variation, but it is immaterial whether it is entirely eliminated or not. Should it be found desirable to return to the uncorrected figures and to calculate the Fourier coefficients, including the effects of secular variation, it will be easy to do so with the help of equation (3). As the unit in Table II. is 01 of a minute, the correction is made by adding to successive columns, successive multiples of 1800/n, where n is the number of days in the period. For example, in the 24 day period, 75 is added to the second number, 150 to the third, and so on. VOL. XVIII. 15 TABLE I. 24 DAY PERIOD. Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Begins Ends Rows 1869 9668 9673 9649 9640 9646 9664 9591 9679 9653 9601 9636 9616 9617 9603 9595 9601 9579 96229609 9605 9595 9600 9599 9565 1' 69 15 1870 8007 8046 7999 8016 7987 7983 8023 8009 7990 7986 7965 7971 8009 8020 7989 7978 7922 7867 7957 8004 7994 7954 7899 8004 361' 69 355' 70 15 1871 6781 6753 6748 6811 6826 6754 6754 6781 6688 6696 6757 6752 6693 6686 6738 6711 6644 6708 6708 6709 6710 6723 6732 6709 356' 70 9' 72 16 1872 5539 5598 5568 5513 5525 5530 5498 5434 5492 5476 5388 5431 5451 5529 5512 5536 5507 5491 5576 5498 5461 5494 5535 5481 10' 72 3' 73 15 1873 5033 5049 5054 5021 5010 4959 4982 4979 4913 49192 4909 4930 4923 4960 4974 4958 4985 4921 4980 4965 4946 4944 4953 5035 4' 73363' 73 15 1874 4405 4371 4367 4346 4371 4394 4347 4357 4375 4431 4366 4328 4303 4314 4315 4329 4288 43284359 4351 4395 4400 4311 4300364' 73358' 74 15 1875 3241 3195 3238 3178 3238 3246 3237 3220 3194 32323244 3255 3235 3202 3228 3170 3192 317413169 3188 3204 3203 3172 3132'359' 74353' 75 15 1876 2638 2590 2609 2577 2582 2612 2627 2600 2579 2605 2548 2544 2580 2519 2537 2503 2523 2485 2471 2478 2471 2481 2496 2479 354' 75 6' 77 16 1877 8601 8592 8593 8619 8602 8587 8574 8539 8531 8516 8561 8539 8545 8587 8562 8590 8571 8581 8572 8530 8531 8585 8551 8545 7 77 1' 78 15 1878 7547 7536 7541 7531 7487 7499 7493 7501 7488 7476 7452 7467 7431 7426 7460 7441 7419 7444 7498 7475 7497 7457 7454 7460 2' 78 361' 78 15 1879 6107 6115 6129 6122 6137 6116 6155 6181 6178 6147 6149 6123 6122 6124 6111 6197 6039 6072 6050 6032 6070 6034 6040 6031 362' 78356' 79 15 1880 4987 4942 4925 4952 4941 4925 4924 4949 4952 4916 4891 4899 4914 4893 4919 4905 4981 4874 4921 4888 4887 4886 4886 4892 357' 79 351' 80 15 1881 4379 4398 4423 4405 4363 4379 4381 4372 4362 4373 4363 4393 4366 4333 4327 4299 4306 4328 4343 4290 4318 4318 4317 4344 352' 79 4' 82 16 1882 3362 3390 3373 3410 3414 3404 3378 3389 3328 3307 3358 3325 3314 3321 3326 3330 3333 3309 3300 3295 3282 3325 3320 3324 5' 82 364' 82 15 1883 2311 2274 2294 2287 2269 2267 2307 2304 2238 2278 2241 2278 2259 2250 2155 2227 2263 2233 2152 2289 2278 2291 2248 2213 365' 82359' 83 15 1884 1193 1230 1232 1175 1198 1204 1186 1155 1189 1168 1171 1178 11551144 1141 1166 1137 1141 1116 1143 1148 1103 1138 1130 360' 1885 3325 3341 3347 3315 3347 3321 3317 3344 3348 3304 3321 3350 3353 3342 3321 3313 3278 3239 3257 3212 3225 3293 3270 3244 355' 84 348' 85 15 1886 8763 8764 8803 8770 8768 8720 8756 8767 8759 8815 8740 8785 8745 8743 8720 8757 8683 8724 8734 8733 8707 S730 8670 8704 349' 85 2' 87 16 1887 7337 7413 7373 7349 7387 7369 7411 7401 7412 7421 7408 7395 7390 7355 7352 7331 7352 7292 7342 7275 7318 7290 7338 7300 3' 87362' 87 15 1888 6100 6072 6090 6060 6067 6038 6065 6071 6104 6108 6075 6067 6085 6062 6088 6079 6056 6053 6036 6039 6040 6052 6065 6049 363' 87 357' 88 15 1889 5272 5264 52805239 5251 5337 5273 5258 5254 5241 5260 5263 5246 5239 5265 5238 5280 5216 5242 5233 5239 5233 5185 5209 358' 88351' 89 15 1890 4614 4647 4616 4651 4602 4594 4595 4613 4510 4598 4579 4631 4604 4595 4583 4565 4584 4576 4553 4565 4550 4550 4579 4536 352' 89 5' 91 16 1891 3537 3536 3611 3523 3516 3460 3479 3466 348913503 3518:3501 3519 3533 3522 3531 35003494 3464 3478 3485 3441 3456 3455 6' 91 365' 91 15 1892 2696 2637 2649 2647 2674 2612 2627 2607 2688 2624 2613 2594 2549 2595 2599 2624 2593 2582 2559 2567 2556 2580 2619 2616 1' 92 360' 92 15 1893 1771 1759 1731 1759 1753 17831759 1753 1783 1723 1741 1755 1748 1751 1745 1719 1689 1684 1684 1701 1744 1715 1673 1698 361' 92354' 93 15 Sums 7214 7185 7238 6916 6961 6757 6739 6729 6497!6457 6254 6370 6153 6126 6084 6098&5704 5138 5652 5543 5651 5682 5506 5455 TABLE Il. No. of daysin 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 2122 23 24 25 2627282930 Period 24 2214218522381916196117571739172914971457125413701153112610841098 704 438 652 543 651 682 506 455 25 2387210321001890173118491809178815981431153114921258139614331245 887 851 873 601 986 795 582 601 388 26 1736174819481533157214371393160013481214128812761040 881 883 9461026 898 969 903 735 605 543 503 488 252 27 19621747167216551366129710781086 99012081098116111541005 936 604 771 798 718 583 610 372 351 339 381 129 229 28 1776 1699 1771 19331732 1645 1717 1474149514571315 1086 1285|1015 902 820 688 648 840 802 527 608 329 362 175 180 232 185 29 21212121214518591980189919411716158715491522143613791032122212271176101411431063 850 817 772 857 763 671 539 394 513 30 230222972247211920902071201419762149 1901 1904 1774168316131915168515571522,18861563149913471136122910101008 748 878 709 585 TABLE III. No. of daysin 1 2 3 4 5 6 7 8 9 10 1i1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Period 24 2214 2260 2388 2141 2261 213221892254 2097 21322004219520532101 2134 2223 1904 1713 2002 1968 2151 225721562175 25 2387 2175 2244 2106 2019 2209 2241 2292 2174 2079 2251 2284 2122 2332 2441 2325 2039 2075 2169 1969 2426 2307 2166 2257 2116 26 736 8171086 741 849 783 8081084 902 837 9801037 870 781 852 984113310741215121811191058106510951149 982 27 962 814 805 855 633 630 478 553 523 808 765 894 954 872 869 604 838 931 918 850 943 772 818 872 981 796 962 28 776 763 9001126 989 967 1103 924 1009 1036 958 793 1057 851 802 878 717 741 997 1024 813 958 744 841 718 788 904 921 C.-i 929 1183 1245 13311107 1291 12721376 1413 1146 1170 1205 1181 1186 90111541221 1232113213231305115411831200134713161286121611331314 30 302 357 367 299 330 371 374 396 629 441 504 434 403 393 755 585 517 542 966 703 699 607 456 609 450 508 308 498 389 325 116 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION Table III. gives the figures so corrected, and these were plotted down on a suitable scale, and curves drawn, joining the ordinates by straight lines. The Fourier coefficients were obtained by means of Coradi's Harmonic Analyser, belonging to the City Guilds of London Institute, which Prof. 0. Henrici kindly placed at the disposal of his assistant Mr H. Klugh for the purpose. Table IV. gives the values of the coefficients of the series a1 cos Kt + a2 cos 2ct +...... + bi sin Kt + b2 sin 2ct +...... TABLE IV, Days in No. of Period ai b1 a, b, a( b3 a4 b4 b5 Periods 24 + 752 +8-92 +6-48 + 008 - 232 - 448 - 332 + 544 + 124 + 308 380 25 -080 - 096 + +444 - -276 - 304 -4-00 + 368 + 648 + 404 +1-76 365 26 -0-68 -13-08 - 6-60 -- 316 + 3-92 + 0-24 - 3-44 -2-16 - 3-36 + 1-92 351 27 + 044 -11-20 +8-76 - 288 + 344 + 628 - 308 - 372 - 116 + 1-16 338 28 + 004 + 8-36 - 596 + 2-56 + 2-16 + 1-96 - 1-52 - 528 - 308 - 020 326 29 + 4-68 - 052 - 456 + 296 - 128 - 3-40 - 056 + 188 + 396 + 108 314 30 -13-64 - 800 - 2-36 + 4-64 + 2-16 - 4-12 - 020 + 0-80 - 0-96 + 3-52 304 To obtain comparable figures a further reduction is necessary. The number of rows included in Table III. and indicated in the last column of Table IV. differs according to the period, being larger for the shorter periods. If Fourier's analysis had been applied to the original series of numbers made up of the actual observed values of declination, the factors obtained would have been smaller than those given in Table III. in proportion of the number of periods included. It is not necessary to perform the division for each coefficient separately, as the ordinates of the Periodograph depend only on the square of the amplitude, viz. r2 =a,2+b,2; r22=a22 + b22; etc. Table V. gives the reduced squares R12, R22, which correspond to R2 in (2), r being the Ist of January, 1869, T the number given in the first column of Table V., and n the number given in the last column of Table III. It is seen that the values of R2 are subject to considerable variations, R12 being for instance more than 100 times larger for the 26 day period than for the 25 day period. According to the reasoning uniformly employed by previous investigators, this would prove a real existence of the 26 day period, but the theory of probability shews FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 117 that such variations are not more than we should expect. Assuming the ordinates of the Periodograph to vary uniformly between the periods of 24 and 30 days, we obtain, by taking the mean of the columns of Table V., the ordinate S2 of the Periodograph corresponding to a period of 27 days. The value of S, or the amplitude of mean square, i.e. the square root of the expectancy of R,2, is thus found to be 0' 0317 (see Table V.). This therefore is the order of magnitude we should expect for the amplitude, TABLE V. Days in Period 12 l Rs2 R 3 1/42 R5 24 946-9 x 10-6 290-7 x 10-6 176-5 x 10-6 281-9 x 10-6 76-5 x 10-6 25 11-7 205-1 189-6 416-4 146-0 26 1392-4 434-5 125 3 134-2 121-7 27 1099-7 744'3 448-7 204' 2 23'5 28 657-6 396-2 80-2 284-4 89'8 29 225 0 299'9 133-8 39-0 171-1 30 2705-8 293-3 234-0 7 4 143-9 Mean (S2)_ 1005-6 x 10-6 380-6 x 10-6 198-3 x 10-6 195-4 x 10-6 110-4 x 10-6 S= 0'0317 0'0195 0'0141 0'-0140 0'0105 if Fourier's analysis is applied to a record of 25 years of Greenwich declination, the period being in the neighbourhood of 27 days. As the expectancy of amplitude varies inversely with the square root of the time-interval, the expectancy of amplitude is as great as 0'1585 for a single year's record. The ordinates of the Periodograph may be obtained in another way, agreeing more closely with the theoretical definition given on page 108. If each of the rows of Table I. is separately treated by Fourier's analysis, and the coefficients afterwards are divided by the number of periods included in each row, we obtain the amplitude of the 24 day period for each year; the mean square of this amplitude is the ordinate of the periodograph for the interval of one year. It was considered sufficient to confine this method of treatment to the 26 and 27 day periods. If Fourier's series is put into the form ri cos (/t - g1) + r cos (2ct - b2) +......, Table VI. gives the values of r2, r2...... r2 for the 26 day period, Table VII. the same values for the 27 day period, and Table VIII. the angles 01 and P2 for the same periods. 118 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION TABLE VI. 26 DAY PERIOD. 2 2 No. of Year 2 r2 r2 2 Perio s 1869) 10-574 3-480 1-116 0122 0930 14 1870 1-082 1-478 1-150 0-824 0'580 14 1871 1-992 4-526 1-682 2-497 0-284 14 1872 13-744 4-802 0268 0'128 2-401 14 1873 12-109 0603 0-404 3-803 0716 14 1874 8-488 0'601 1 892 0504 0213 14 1875 3-624 0-284 1 016 0-678 0'692 14 1876 4004 2'511 2-949 0771 1-604 14 1877 1-297 2-339 2-180 0-569 0 008 14 1878 1 758 0-328 0-811 0216 0179 14 1879 5673 0-578 0-433 0-876 0194 14 1880 1 403 0-305 1.151 0265 1 300 15 1881 2-448 1 504 0-392 0-216 0149 14 1882 7-092 2932 0-014 0 532 0'005 14 1883 2-500 2-938 0-758 1 341 0190 14 1884 3-379 0-437 0-464 1-386 0'052 14 1885 0-315 2512 0-041 0-592 1-632 14 1886 3118 1-850 0-503 0116 1 182 14 1887 4180 4640 1 638 0-763 0-058 14 1888 1-946 2-269 1-550 0-847 1-847 14 1889 2-064 2694 0-354 0-184 0-029 14 1890 1 790 0-240 0-573 0-531 0-834 14 1891 0715 0'090 0-303 0132 0 382 14 1892 0-784 1-271 0-784 2726 0754 14 1893 9-143 9-541 0-362 0583 0-471 14 FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 119 TABLE VII. 27 DAY PERIOD. Year r12 r2 r3 r2 r52 No. of Periods 1869 3-667 0-923 1 205 0-457 0 339 13 1870 13-875 4 326 3'968 1-375 1'800 14 1871 18'320 1181 1-978 1-071 0-301 13 1872 22859 1-256 0-447 0-113 0'088 14 1873 3-963 6-470 0-708 2'609 1-871 13 1874 5-044 1-073 1-260 1-330 2-624 14 1875 0-440 0-743 0'008 0-951 1 404 13 1876 5-297 0-473 3-910 2-403 0'092 14 1877 0-673 0-548 0'105 0'509 0-148 14 1878 5-264 0'192 0-142 1'823 0'709 13 1879 2 078 0-016 0-227 0-036 0-142 14 1880 5722 0794 1-456 0-269 0-331 13 1881 2 258 3 589 2-960 2-662 0-305 14 1882 12-857 1-561 0-693 0-163 0-146 13 1883 2-198 2-844 0-155 1'992 0-680 14 1884 4-348 1-619 0'098 0-875 1 1060 13 1885 1'190 0519 0-816 0009 1-657 14 1886 8 555 1-403 0-094 1-567 0-062 13 1887 4 692 0-167 0-560 0-685 0-033 14 1888 1-742 0-004 0-268 1-079 0 427 14 1889 1-270 2347 0804 0270 1-116 13 1890 1-396 1-074 0-373 1-038 0 211 14 1891 6-548 0-710 0-536 0-715 0-591 13 1892 5-177 1-177 3-533 0'307 0-035 14 1893 14-654 3'059 0908 0'252 1'190 13 120 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION TABLE VIII. Period of 26 Days Period of 27 Days Period of 26 Days Period of 27 Days Year <12, <)2 S1 1) r r2 1869 73~ 155~ 354~ 318~ 3-25 1 87 1'92 i -96 1870 271 354 56 280 1-04 1 21 3-72 2 08 1871 293 275 141 51 1 41 2-13 4-28 1-09 1872 257 201 297 303 371 2-19 478 1'12 1873 253 263 289 268 3-48 0-75 1'99 2-54 1874 250 349 280 143 2-91 0-75 2-25 1'03 1875 261 193 135 70 1'90 0-53 66 0-86 1876 135 43 84 338 200 1-58 2-30 0-69 1877 253 158 226 271 1-14 1-53 -82 074 1878 30 133 78 115 1-33 0-57 2'29 0-14 1879 51 89 151 251 2-38 0-76 1 44 0-13 1880 7 344 317 298 1'18 0-55 2-39 0-89 1881 238 55 110 346 1 57 1-23 1 50 1'90 1882 283 294 261 133 2-66 1-71 3-59 125 1883 67 356 267 31 1 58 1-71 148 1 69 1884 163 177 221 267 184 0-66 2'09 1-27 1885 87 110 238 245 0-56 1-58 1-09 0-72 1886 185 290 198 297 1-79 1-36 293 1-18 1887 31 233 126 54 2 04 2-15 217 0-41 1888 290 134 258 45 1-40 1-51 1-32 0-06 1889 51 288 161 342 1.44 1-64 1-13 1-53 1890 287 359 272 49 1-34 0-49 1-18 1-03 1891 208 206 266 153 0-85 0'30 256 0-84 1892 64 249 350 350 0'89 1-13 2-28 | 1'09 1893 233 211 310 65 3'02 3'09 3'83 1 75 FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 121 The number n of periods included in each row of figures is given in these Tables, and if 82 in accordance with the previous notation represents the expectancy of the square of amplitude: r3 2 = -2 25 n' The values of S2 found in this way are entered in Table IX., the last column giving the average of the two values found for the 26 and 27 day periods respectively. TABLE IX. Amplitude of Periodogram for interval of one year. (The unit is the square of one minute of arc.) Period in Period in Average of Days S2 Days S2 two Periods 26- 1 02147 27 -1 -03145 -026460 26 -2 01117 27 2 -00777 -009471 26 3 -00462 27 -3 -00583 -005225 26 4 -00432 27 4 -00537 -004846 26- 5 '00337 27 5 -00384 -003606 According to the theory founded on the laws of probability the values of S2 for the one year interval should be 25 times greater than for the 25 years interval, and we may obtain an important confirmation of the theory by the comparison given in Table X. TABLE X. Period Ordinate of P. G. Ordinate of P. G. in Days for one year for 25 years Ratio Final Mean Secular Variation 27 26460 x 10-6 1006 x 10-6 26-3 1052 x 10-6 28704 x 10-6 13-5 9471 381 24-9 379 7176 9 5225 198 26-4 208 3189 6 75 4846 195 24-8 192 1794 5'4 3606 110 32-7 139 1148 between the values of S2 which have been found from the 25 years curves (Table V.) VOL. XVIII. 16 122 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION and those just deduced for the shorter interval. The latter being the mean of values obtained for the 26 and 27 day periods should, strictly speaking, be put down as belonging to a period of 26'5 days, but for our purpose it is sufficient to neglect the difference of half-a-day. Considering that the value of S2 for the 25 years interval represents the mean of only seven values, the approximation of the ratio of the numbers given for the intervals of 25 years and one year respectively to the theoretical number 25 is very remarkable. Incidentally this agreement shews that the secular variation has been eliminated sufficiently to leave no appreciable effect on the Periodogram. The last column of Table X. gives the ordinates of the P. G. for a uniform progressive change of 7'2 per minute. The original uncorrected figures would have given, according to our previous deductions (Cases 2 and 3), values for the P. G. made up of the sums of Columns VI. and il. or III. respectively, and the ratios of these sums would have been widely different from 25. Further consideration of the figures shews that, while possibly a small change in the assumed value of the secular variation would have brought the numbers of Column iv. into still nearer agreement with the theoretical number, such a change would amount to less than a percent., and would be quite uncertain. The surface of the Periodogram having been determined with sufficient accuracy for periods varying between 5 and 27 days, it seemed desirable to extend the investigation 1 Unit = 0'0001 -12 / 11- / 10- 9- 7 -6 -4 — 3 -12 -0 5 10 15 20 25 Fig. 1. to shorter and longer periods. The calculation for a period of 2 days gave very little trouble. If the alternate numbers in each of the rows of Table I. are added together, and the differences of these sums are taken, we obtain numbers which, after division by FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 123 the proper factor, give the Fourier coefficients. The average square of amplitude for the year was found to be *003460 and this has to be divided by 25 to get the ordinate of the periodograph for the 25 years interval. The number 1384 x l0-6 so obtained is almost identical with that previously found for the 5'4 day period, which tends to shew that for short periods the expectancy of a Fourier coefficient is independent of the period. Fig. 1 gives the shape of the Periodogram for periods up to 30 days. The vertical ordinates give the heights actually determined, while the curve is drawn continuously so as to pass nearly through these points. For longer periods the monthly averages, as published in the Greenwich records, served as basis of calculation. To obtain the coefficient of the annual period, the interval of 25 years was divided into 5 groups of 5 years, and the harmonic analysis was applied to each of these 5 groups. The average square of amplitude then gave the ordinate of the Periodograph for a range of 5 years, which has to be divided by 5 in order to reduce it to our normal interval of 25 years. Periods of 11 and 13 months were treated similarly and the coefficients obtained for 5 groups of 55 months and 4 groups of 65 months. The average squares of amplitude have in these cases to be divided by 60/11 and 60/13 to reduce to the normal interval. The results are given in Table XI., and it will be noticed that the PeriodTABLE XI. Period in Months S12 S22 S2 S42 S52 11 -04591 -00475 -00158 -00079 -00054 12 -08828 -01610 -00842 -00287 '00218 13 -09344 -01082 -00891 -00237 00196 Average -07588 -01055 -00630 -00201 -00156 Period in Months 12 6 4 3 24 ogram continues to increase rapidly with increasing lengths of period. The conclusion we must draw from the curve in Fig. 1 and the figures of Table XI. is, that the causes which produce the variations of declination are on the whole persistent in character, so that the variations of short periods have on the average a much smaller amplitude than those of longer periods. IV. APPLICATION OF THE THEORY OF PROBABILITY. In a previous paper* I have applied the theory of probability to the solution of the question whether the value of any particular coefficient of Fourier's series indicates * Terrestrial Magnetism, Vol. III. p. 13. 16-2 124 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION a true periodicity or may be accounted for by purely accidental causes. The principal results arrived at may be shortly stated here, as far as they concern the present discussion. The average daily value of magnetic declination, leaving the secular variation out of account, oscillates round some average value. If / is the difference between any observed value and its average, there will be some function f (/) such that f(3) d/ will represent the number of cases in which the value lies between / and / +d/; for instance, if the ordinary law of errors holds, the number of cases in which the deviation from the average 2hN lies between f/ and fi + df/ will be -- e-h22 d/, where h is a constant and N the NV7r total number of days considered. In this case it is found that the probability that the Fourier coefficient of any particular period lies between p and p +dp is Nh2e —Nh2p2 pdp. This expression holds on the assumption that the values on successive days are entirely independent of each other. The expectancy (E) of the square of Fourier's coefficient is in that case p2.,hVe —^h pdp = N-2, and the probability that p2 should exceed a value KcE is simply e-K. This latter expression still holds when the law of distribution is not that of errors, and even if the successive daily values are not independent of each other, as is e.g. the case when the causes which produce the deviations froin the average persist for several days. In the last case the expectancy must be obtained by trial, the mean square of the Fourier coefficients being taken. This expectancy, which according to our definition is the ordinate of the periodograph, should serve as the basis of any attempt to discover real periodicities, and Table XII. will give at once the probability that a coefficient of the Fourier series is due to a periodic cause and not to accident. If for instance the square of a coefficient has been found to be equal to about twice the expectancy, we obtain by the Table the value of e-' for c = 2 as 135, which means that in one case out of about seven, accidental circumstances will cause the coefficient to be even greater than this, and therefore no conclusion can be drawn as to a real periodicity. When the square of amplitude which for shortness we may call the "intensity" amounts to about 12 times the expectancy, the probability of mere chance is only one in 200,000 and we may then begin to be fairly certain of a real effect, or if we are satisfied with a probability of one in 1000, we may begin to count effects as probably real when the intensity becomes equal to about 7 times the expectancy. We may follow the theory of probability a little further in another direction; the expectancy has in most cases to be determined by trial, and for trial, and for this purpose the mean of a certain number of calculated intensities is taken. The question arises how many FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 125 such numbers must be combined in order to obtain a sufficiently approximate value for the expectancy. TABLE XII. K I e- 1 K e-K 05 -9512 6 2-48 x 10-3 '10 -9048 8 3-35 x 10-4 *20 8187 10 4-54 x 10-5 '40 1 6703 12 6-14 x 10-6 '60 5488 14 8 32 x 10-7 ~80 -4493 16 113 x 10-7 1 00 -3679 18 1-52 x 10-8 1 50 *2231 20 2-06 x 10-9 2 00 -1353 25 1 39 x 10-1 3 00 -0498 30 9-36 x 10-1 4-00 -0183 40 4-25 x 10-18 5'00 -00674 50 -2 x 10-22 To calculate the probability with which an average of a finite number of cases approaches the expectancy, we take two quantities such that the probability of either exceeding a certain value IE is given by e-K and find the probability that their sum exceeds 2pE. If the first lies between KcE and (K + dKc)E the second must be greater than (2p - c) E as long as K is smaller than 2p, if greater the second may have any value. Hence the required probability becomes 2p e-p + J e- e-(2p-K) dK = e-2 (1 +2p). By a repeated application of the same process it is found that if there are n quantities, the probability that their average exceeds KE is e-nK L1 + n + 2 23 + 3 +... ~ + n-i lcn-l] which is equal to (ni ) fiKKCL- e-6K dK 126 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION so that the probability that the average of n values should lie between icE and (ic + dK) E is n^ - - n-1 e-K d/c. (n- 1)! If n is large, we may simplify the numerical calculation by putting approximately according to Stirling's theorem log (n - 1)! = (n - 1) log n - n + log 27r, from which it follows that nn )!=e / (n -^r6 27r_ In order to illustrate the law according to which a gradually increasing number of intensities tends to approach the value of the expectancy, I have plotted in Fig. 2 the curve y= (n- 1)!n-1 e-n, K I2 A1 1 H A2 0 '5 i 1-5 2 2 5 3 3-5 4 Fig. 2. for the three cases that n equals one, five, or fifty; fifty being the number on which our Periodograph in the neighbourhood of the 26 day period rests. FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 127 The line HK gives the position of the expectancy, and the curve A1A2, which represents the case n= 1, shews how a single value of a Fourier coefficient generally does not give us even approximately the value of the expectancy. For n=5, and still more for n=50, the probability-curve approaches the line HK. In the conclusions which we shall have to draw on the reality of periodicities inuch depends on the law of distribution of accidental Fourier coefficients. According to the theory the probability that the square of any coefficient exceeds K times the expectancy is e-K; and although the theory rests on a sound basis, it is interesting to obtain an experimental verification. The material collected for this investigation includes the Fourier coefficients of five terms for each of 25 years, for the 26 day and the 27 day period. Hence 250 separate values of amplitude have been obtained. For each of the five terms the average value of intensity gives the expectancy, and calculating the ratio of the intensity to the expectancy we find 250 values of K. Table XIII. shews the comparison between the TABLE XIII. Calculated Observed Range of K number of cases number of cases Above 3 12-5 14 Between 2 and 3 21-4 25, 15,, 2 22'0 19,, 1'0, 1'5 36-2 32-5 8,, 10 20-4 20'5,, '6,, 8 24-8 23,, 4,, 6 30-4 27-5,, 2, 4 37-1 48-5 Under '2 45-3 40 Altogether over 1 92'0 90'5,, under 1 1580 159-5 calculated distribution of these values of K and that actually found, the agreement being very satisfactory. The fraction one-half appears in the column of observed values, because if the value of ic agreed to 2 decimal places with a limiting value, it was considered as being half-above and half-below that value. Thus K = 60 was entered as one-half into the compartment including the values of Kc between '6 and '8, and as one-half into the compartment including the values of K lying between '4 and '6. 128 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION V. CALCULATION OF AMPLITUDES IN SPECIAL CASES. The Fourier coefficients having been calculated for the 26 and 27 day periods in each year, we are able to obtain the amplitudes for periods not differing too much from these values. To shew the process of calculation to be adopted for this purpose, let A1, A2, etc.; B1, B2, etc. be defined by the equations 0n T 22nT snT A1 = jf (t) cos gtdt, A = f(t) cos gtdt, Am=J f(t) cos gtdt, ^O ' nT ( s-l) nT çnT -2nT fsnT B, = f(t) sin gtdt, B2= f(t) sin gtdt, B = f (t) sin gtdt, = JJnT (s- 1) nT where g = 27r/T. It is required to find 2 P T' 2 PT a' = j (t) cos Ktdt, ' = (t) sin tdt, where Kc = 2Tr/T'. If K and g do not differ much from each other we may put approximately înnT mnT f (t) co t ft) os (gt + ) =, cos a,, - B sin a,,...............(4). (m-1) nT J (m-1) n The greatest approach to equality is assured when the curves cos Kt and cos (gt + ca) are made to coincide as nearly as possible throughout the interval, and hence the phases should agree in the middle of the interval, so that for t= (m - )nT, Kt=gt+. This gives a = 27r - (-) nT. We may now put "nT îmn=s m=s f (t) cos Ktdt = 4 A, cos amr - S B, sin a,,, 0 mb=l m=l l'snT. m=s m=s f(t) sin Kctdt = A,,, sin.,, + 1 B, cos ar. -m0 il=1 m=l The coefficients which we suppose to have been calculated are a, = A, b= - B,, etc., 2 f sIT 1 so that s nT J /(t) cos,ctdt = - S (a,, cos a,, - b,, sin an).?îZ o l s FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 129 If snTT=pT' and p is an integer, the left-hand side would represent the coefficient of period T' obtained by analysing the record of p successive periods. If p is not an integer we may still take this to be approximately the case if sn is large, for we may always put snT iT' (p+e) T' f (t) cos Ktdt f(t) cos t dt + f (t) cos Ktdt.....................(5), 0 ' O0 pTI p being the nearest integral to sn, and e a fraction. The second integral will be small compared to the first, if the first includes a large number of periods. We have therefore finally for the required coefficients a' and b' a' -m-, (a, cos a m-bm, sin ma=,rmoS(an + m), pT m=1 pT (............(6), nT m^s nT b' = -, 2 (a, sin a,, + bm cos aqr) =, rm sin (cc. + Cm), pT' =1 where a, = r, cos m, bm = rm sin qm. The fraction nT/pT' may generally be taken to be equal to 1/s. The values of a are those given above, so that a = 7rn - -, 2 -= 37r T', a= (2m - 1) rn T...............(7). It remains to be shewn what error has been introduced by the assumed equality (4) and the neglect of the second integral of (5). For this purpose we imagine the function f(t) to be accurately represented by cos/ct, so that mnT 2Ko KT Am =, cos ~cos t os gtdt = - sin -- cos K (rn - ) nT, (m-1)nT K9 - and as a = ( - g) (m - ) nT, Al= ~+ g sin IcnT cos an, where the lower sign is taken when n is odd. Similarly 2g7 Bm = + _ 2 sin -~KiT sin am,. K 2 -2 2-g By substitution it follows that, using equations (6), 2 2 a'= T' 2 _2 sin i-nT (K/ cos2 au, + g sin2 qaQ) 2 1 = T 2 2 sin ~ /crT ~ {(K + g) + ( -/Y) cos 2an}, VOL. XVIII. 17 130 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION or writing = 2 ni ' nT sin 7 {s + K cos 2a ni7 T sin 7 _t g - Similarly b' T= in? 5 - sin 2am. piT' 'y +g The factor sin ry/y only having appreciable values when y is smali, the value of -g K +g will be small compared to unity, hence the sum of s terms containing that factor will be small compared to s. This reduces the coefficients to nsT sin y pT' r p is defined as the nearest integer to nsT/T', and as ris, the total number of periods included, was about 350 in the cases to which the above investigation will be applied, we may with sufficient accuracy write sin y The original function investigated cos Kt, having unit amplitude, it is seen that the approximate method of calculation gives an amplitude which is reduced in the ratio sin y/y or an intensity reduced in the ratio sin2 7/72. A Table of sin2 y/y2 is given in Mascart's Optique, Vol. I., p. 324, from which it appears that as y takes the values 15~, 30~, 45~, 60~, the function becomes -977; -912; -811; '684. If it is simply desired to decide whether a period is real or accidental, the intensity need not be accurately known, and we may allow ourselves considerable latitude therefore in the value of y. If we fix the extreme value of that angle as 45~ which means a reduction of intensity of about 20 ~/, we obtain a relation betxveen T and T', for in that case T -T' 7r y=7rn T < + i:' 4' T- T' i or T' < +4n If T is 26 days, and n=14, there being 14 periods of 26 days in the year, we find that by the method indicated all amplitudes may be calculated which lie between 25-54 and 26-47 days. If the coefficients of the 26 day and 27 day periods are known for each year we shall be able to calculate those of all intermediate periods with sufficient accuracy, for the extreme reduction in amplitude when T - 7' =-day will be *789, and it is only when the intensity comes very near the point at which it is difficult to distinguish between real and accidental periods that this reduction will make a material difference. FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 131 VI. NUMERICAL APPLICATIONS. Some investigators have cone to the conclusion that several meteorological and magnetic phenomena shew a periodicity having a tiine not far different from 26 days and, not uncommonly, this period is supposed to be connected with solar rotation. I proceed to apply the methods of this paper to test the reality of this period. Hornstein%, on the strength of the declination records for Prague, assigns to it an amplitude of '7 minute of arc or an intensity of '5. Such an intensity would be equal to 500 times the expectancy, if an interval of 25 years is submitted to examination; and if real and approaching Hornstein's value in magnitude, it should stand out above the accidental periods to such a degree that every doubt would be removed. Adolph Schmidtt was led by a discussion of Hornstein's results to a duration of 25a87 days as being the most probable periodic time, while von Bezold finds a slightly shorter period for the frequency of thunder-storms. More recently Professors Eckholm and Arrhenius have published a paper in which a periodicity of 25'929 is put forward as probable, or even proved. As opposed to these investigators Professor Frank H. Bigelow gave a considerably longer time (26 68 days) to the periodicity and has endeavoured to shew that it exists in many meteorological phenomena. To shew whether the Greenwich records confirm. or disprove these results, it is necessary to calculate the intensities for each periodic time, and its corresponding half period. This I have done, the results being collected in the first section of Table XIV. TABLE XIV. Square of Square of Period Amplitude K Semi-period Amplitude K 25-87 -001001 '95 12-935 -000316 -83 25 929 -001027 -93 12-965 -000200 -55 26-68 -000242 -23 13-340 -000132 -35 25 809 -006168 5-86 12-905 -001060 2'80 25-825 -004182 4-07 12-913 -001286 3-39 26-181 -001144 1'09 26-255 -001081 1 04 26-814 -005936 5-64 27-061 -002943 2'80 * Wiener Ber. LXIV. p. 62 (1871). + Kongl. Svenska. Akad. Vol. xxxi. No. 3 (1898). t Ibid. xcvi. p. 989 (1887). 17-2 132 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION The column headed K gives the ratio of the intensity (square of amplitude) to the expectancy; and there is a remarkable unanimity in the smallness of this factor, shewing that the amplitudes are even less than the average amplitudes calculated on the theory of chance. This result must definitely disprove Prof. Eckholm and Arrhenius' period of 25-929, as well as that of Bigelow, as far as the Greenwich records of declination are concerned. The interval of 25 years which forms the basis of this investigation is, however, so long that unless the periodic time is very accurately known beforehand, the existence of the periodicity may escape attention. Hornstein's investigations, as treated by Schmidt, do not claim any great accuracy, and a period of say 2584 days might give a large amplitude. In other words, we can only say that there is no periodicity having a length between about 25'86 and 25'88 days, but a further investigation is necessary if the possibility of an error of more than '01 day in Schmidt's value is admitted. Both Bigelow and Eckholm and Arrhenius claim to have fixed their period to three places of decimals and our result must be considered as conclusive against them. In order to be certain that no periodicity of sufficient magnitude has remained unnoticed the investigation was extended in the following way. A diagram was prepared (Plate I.) in which the phases of the 26 day period, as they are given in Table VIII. for each year, are measured off as ordinates in equidistant vertical lines which represent successive years. If there is a period in the neighbourhood of 26 days which has a large amplitude, the points representing the phases should group themselves more or less round a straight line and from the inclination of the straight lines we may calculate the length of the period giving the increased amplitude. In order to include possible periods which may differ as much as ~5 from 26 days, the diagram must be repeated three or four times so as to admit a phase variation of several revolutions of a circle. Thus for the first year the phase was 73~ and a point is marked on the diagram, not only on the horizontal line corresponding to 73~ but also on that of 433~, 793~ and 1153~, all differing by 360~. In order to be able to give more weight to those years in which the amplitude is great, the points are marked differently according as the amplitude is great, intermediate or small. The manner of marking is best seen on the Plate. If the eye is suddenly moved towards the Plate so as to obtain a general view of the grouping of points, I think there will be no doubt that these shew a decided tendency to group round a straight line marked A1A2. To bring the phases of the points which lie along this line into agreement the phase of the 25th year which is 593~ must become equal to that of the 5th year which is 1333~. This gives a shift of phase of 37~ per year. To obtain the period corrected so as to bring the phases into agreement we may use equation (7), putting T - T7 m - a,_ = 27rn T'= 37. If T= 26 and n= 14 the corrected time T' is found to be 25'809. FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 133 The amplitude was next calculated for this corrected period and its square entered into the second section of Table XIV. The intensity now exceeds the expectancy, being 5'86 as great. There appeared also to be a minor tendency of groupings about the lines B1B2 and (GC, and to bring the phases along these lines into agreement the corrected periods were calculated to be 26'255 and 26'181. Table XIV. however shews that the intensities corresponding to these times barely exceed the expectancy. Plate II. gives similarly the distribution of phases for the 27 day period, the straight lines along which there seems a possibility of clustering are marked on the Plate, the corresponding periodic times being 27'061, 26'814, 27-327 days. The intensities of the two first of these periods are entered into Table XIV. It will be noticed that the two periods which shew the greatest amplitudes are those of 26'814 and 25 809 days. As regards the latter, reference to Table XII. or independent calculation shews that it will happen about once in every 350 trials that, owing to accidental circumstances, the square of a Fourier coefficient exceeds 5'86 times the expectancy. It will of course be noticed that the period which gives the high value for the amplitude has been selected with that special object in view, and regard must be had to the fact that it represents the greatest intensity that can be obtained within the range of periods extending from 25.5 to 27'5 days. The question how many independent trial periods that range may be considered to contain may be answered by our previous investigation (p. 130) from which it appears that two periods T and T' may be considered as independent when T-T' 1 T' >4n' n being the total number of periods included in T. For T= 27, n was 338, and hence T - T' is almost exactly '02 day. As our range covered all periods between 25*5 and 27'5 days, we must consider that we have dealt with 100 independent periods and found the two greatest intensities to be respectively 5'64 and 5'86 times the expectancy. What it comes to therefore is this, that 100 trials have given us one intensity 5'86 times the expectancy, while on the average this should only happen once in 350 trials. Or taking the two greatest amplitudes into consideration, it ought according to chance to happen once in every 150 trials that an intensity of 5 times the expectancy is found, while in the actual case this happened twice in 100 trials. It is obvious that no conclusions as to the reality of the periodicity can be drawn from this argument. There are however two considerations which lead me to pause before finally rejecting the 25 809 period; the high amplitude is accompanied also by a considerable amplitude of the half period, and if these half periods are plotted in a manner illustrated in Plates III. and IV., it is found that a somewhat greater value is obtained if the time were altered to 25'825 days. This however gives a decidedly smaller value for the main period (see Table XIV.). The coincidence of two high intensities for a period and its semi-period much increases of course the probability of its reality, but even if this is taken into account, the excess of intensity over the expectancy is insufficient to establish the period. The second consideration lies in the fact that the most definite result so far in the 134 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIO DECLINATION search of periodicities has been that of Prof. v. Bezold whose work had reference to the frequency of thunder-storms. He gives 25'84 days as the length of his period, but it was really only the semi-period which shewed a large amplitude. The numbers 25 84 and 2a5825 lie so near together that it will be wise to keep an open mind as to the possibility of some real periodic time of that length. But it must be understood that the record of Greenwich declination extending over 25 years shews nothing beyond a slight indication of such a period. An intensity of '006 corresponds to an amplitude of '077 minute of arc, and it can be definitely asserted as the result of this enquiry that there is no period between 25'5 and 27'5 days which had a larger amplitude at Greenwich during the years 1871-1895. VII. LUNAR PERIODICITIES. One of the principal objects of this investigation was to prove or disprove the suspected lunar period in the daily average of magnetic declination. The clustering of phases round the line BB', Plate IV., shews that observation gives a somewhat larger amplitude than the average for a period of 27'327 days which lies very near the length of the tropical month. The two periods, that of tropic revolution and that of synodic revolution, were therefore specially treated, the result being exhibited in Table XV. It TABLE XV.. Square of Square of Period Amplitude K Semi-period Amplitude K 27-32 -002352 2-24 13-66 -000819 2-16 29-53 -000026 -25 14-77 -002876 7-56 is seen at once that there is a strong indication of a period having as its time half the period of the synodic month. The value of Ic which is 7-56 is considerably higher than any other given within the whole range of investigated periods. An accidental coincidence is not excluded, for as calculation shews it may happen once in every 2000 trials that such a large value should be found for K. We can only assert therefore that there is a probability of 2000 to 1 that the moon has a true effect on magnetic declination. The amplitude is only '054 minute of arc and the strong evidence afforded of the real existence of a periodicity having such a small amplitude shews, I think, the value of the method which has been adopted in this investigation. As regards the phase of action no certain conclusions can at present be drawn; the maximum westerly declination occurred on the average during the years under examination between 2 and 3 days after new and full moon. Nothing is of course asserted as to the reason why the moon should affect the declination needle, but the action is probably a very indirect one. It would be important to extend the investigation to the other components of magnetic force and to other localities. It is highly improbable that a westerly force FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 135 should act simultaneously all over a circle of latitude, for that would imply considerable currents across the earth's surface. It is more likely that the principal action takes place along a geographical meridian; and if that is the case, the horizontal force should shew stronger evidence of these lunar periodicities than the declination. There is also the possibility that what is observed in the daily average of declination is only a remnant of a variation having the lunar day for its period. In that case the periodicity should disappear when the average position of the needle in a lunar day is subjected to calculation. If this is the correct explanation it should not be difficult to prove it, for it would require a much greater amplitude within the lunar day to account for the 0'06 amplitude found in the daily averages. How much greater may be seen from the following consideration. If from a periodic function cos Kt another is formed by taking averages over a period 27 we obtain 1 ft+,r cos Kt dt =- sin K-r cos Kt, 2r t-r ICT that is a reduction in amplitude of - sin K-r. If 2T is one solar day, 27r/K one lunar fCT K7- 28-53 day, - = 2953 hence T-r equals 174~ and the amplitude of the curve obtained by taking averages is only about the 29th part of that of the original curve. The comparison of averages of successive days will therefore produce an apparent period having the lunar month as periodic time and, if the period found above is due to this cause, the amplitude of the original lunar variation should be 1'"74. Such an amplitude ought to be traceable without much difficulty. A thorough enquiry into the nature of lunar periodicities of inagnetic records seems to me to be of special importance, but requires considerable arithmetical labour; for, to be conclusive it must be complete. I have been assisted in the numerical calculations which were necessary in the present investigation by Mr J. R. Ashworth, to whom I desire to tender my thanks. The expense connected with the numerical work was partially covered by a small contribution from the Government Grant Fund of the Royal Society. VII. Experiments on the Oscillatory Discharge of an Air Condenser, with a Determination of "v." By OLIVER J. LODGE, D.Sc., F.R.S.. and R. T. GLAZEBROOK, M.A., F.R.S. [Received 9 August, 1899.] PART I. GENERAL DESCRIPTION OF THE METHOD. AFTER a considerable number of experiments on the discharge of Leyden jars, and a qualitative study of the electric oscillations accompanying such discharge, it seemed desirable to make an exact determination of the frequency of alternation given by a standard condenser through a circuit of known self-induction, in order to ascertain whether the well-known theory of the case was accurate or only an approximation. The absolute determinations necessary were three, viz.:(1) The capacity of a condenser, which is K times a length; (2) The self-induction of a coil, which is /u times a length; though it would be natural to measure it indirectly by comparison with the already carefully determined standard of electrical resistance; (3) The period of one oscillation of the discharge, under circumstances when the damping influences are not appreciably disturbing. The resistance of the circuit might possibly enter as a correction into the result, and many other minor determinations might have to be made, but these three are the main quantities involved, and the relation between them is T= 27r (1. K.), and the formula would be verified if the resulting value for the product of the as yet entirely unknown constants, t the permeability and K the inductive capacity of the medium, agreed at all closely with the already otherwise determined value, viz. the square of the reciprocal of the velocity of light. It was hoped indeed that the method might turn out sufficiently accurate to give a useful re-determination of this important quantity. It was with this idea in mind EXPERIMENTS ON THE OSCILLATORY DISCHARGE, ETC. 137 that the following research was undertaken, and much care was accordingly bestowed upon it. It may be here noted that Lord Kelvin himself, in one of his popular lectures *, suggests this method of electric oscillation as just conceivably one of the methods by which v could be practically determined; and he puts the matter in a geometrical way, which it may be interesting freely to paraphrase thus: Take a wheel of radius equal to the geometric mean of the following two lengths, the electrostatic measure of the capacity of a condenser, and the electromagnetic measure of the self-induction of its discharge circuit; make this wheel rotate in the time of one complete electric oscillation of the said condenser (as if it were being driven by an electrically oscillating piston and crank), then it will roll itself along a railway -with the velocity v. And indeed (as Maxwell discovered) ethereal waves excited by the discharge are actually transmitted through space at this very speed. GENERAL REQUIREMENTS OF THE METHOD. The first essential is a condenser of capacity directly measurable from its dimensions. Its dielectric must accordingly be air, its plates must be a reasonable distance apart, and they should be either spherical or have a guard-ring. The necessary smallness of capacity of a condenser satisfying these requirements is a difficulty, especially when a quantity so large as the velocity of light is the subject of measurement. A difficulty of the same sort is, however, common to all methods, and is what makes "v" a quantity so much more difficult to determine than for instance "the ohm." To compensate for the smallness of practicable electrostatic capacity a discharge circuit of very great inductance must be employed, or else the time-determination will be difficult from its excessive minuteness. The inductance must be secured in combination with as much conductance as possible, or the discharge will fail in being oscillatory. To this end Messrs W. T. Glover and Co. were requested to supply a regularly wound hank or coil of No. 22 (s. w. G.) high conductivity copper, very thinly india-rubber covered, of shape such as to give maximum self-induction, and of size estimated to give between 5 and 6 secohms, i.e., in magnetic measure, a length of 5 or 6 earth quadrants. This would be afforded by a coil of 4 inches cross-sectional area and mean diameter 15 inches, with three or four thousand turns of wire. But to guard against the danger of sparking or leaking between layers it was decided to reduce the dangerous tension to one-quarter by having the coil in two halves. Accordingly it was made as follows (to quote Messrs Glover's statement): * Sir W. Thomson's Lectures alnd Addresses, Vol. i. p. 119. Lecture on Electrical Units to the Inst. C. E. VOL. XVIII. 18 138 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY " 4,330 yards of No. 22 tinned copper wire covered with 2 coats of pure indiarubber to the diameter of '035 inch. This was the only covering. In two parallel coils, internal diameter 103 inches, 4 inches deep, and 2 inches wide." See Figure 1. This pair of coils were then packed carefully and permanently in a round walnut box or drum, with a thin sheet of glass between them, and the 2^ terminals of each coil were led to the outside and finished off on four -j i ",, ebonite pillars. They could, therefore, be connected up in series, or parallel, or used separately; but in practice they were usually joined in simple 104 series. With this coil many preliminary experiments were made at Liverpool. yI_ _ 1The self-induction of the double coil was estimated as about - - -2 5 secohms or "quadrants," but no attempt was made to ineasure it with any care at this time, because it was better to do it when all the FIG. 1. apparatus was in position in the basement room set aside for the experiments described in Part II. The chief part of the whole business consisted in taking clear images of a spark on a moving sensitive plate, getting every detail of the oscillation clearly recorded on the negatives, so that they could be subsequently analysed under a microscope and the time of an oscillation accordingly determined. The sparks used were extremely feeble, and each was drawn out by motion into a band, so that in order to get every detail clear the plates had to be super-sensitive. For such plates we were indebted to the kindness of Mr J. W. Swan, who sent on several occasions a special packet of Messrs Mawson and Swan's most highly sensitized plates, which answered admirably. The next principal part consisted in the microinetric reading of the records on the photographic plates. The reading is rather a tedious process as a great many numbers have to be recorded for each plate, and care is necessary to disentangle the several sparks, which to economise time and labour at the experimental end were usually taken during a single spin. The details of the method of obtaining the record will now be described. TIME OF ONE OSCILLATION. The long-established method of observing spark oscillation by means of a revolving mirror was at first used; but this plan, though easy for observation, does not readily lend itself to precise measurement. It is desirable to obtain a photographic record which can be studied at leisure, and it seemed therefore best to form an image of the spark on a plate moving so rapidly that its constituent oscillations were clearly visible. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF c"v." 139 For metrical purposes there are many advantages in thus moving only the sensitive plate, though for mere display Mr Boys's more recent plan of spinning a succession of lenses is able to give more striking results. Accordingly an old packing case was made light-tight, and used as the camera. In it were contained: first the spark-gap, a pair of adjustable brass knobs about half-aninch in diameter, clamped to a glass pillar, one vertically over the other and with a clear space, on the average about 2 millimetres, between them; next the lens, an ordinary camera lens on a special stand; and lastly the sensitive plate in its conjugate focus, arranged so that the image was not very much smaller than the object. The photographic plate is supported firmly in a revolving wooden carrier or frame fixed to the horizontal axle of a whirling machine (one of Weinhold's) which was firmly clamped to a stone pillar outside the camera and was driven by a long carefully spliced whipcord belt by means of one of Bailey's "Thirlmere" turbines standing on a distant sink, and having a large grooved pulley to give the necessary "gearing up." One end of the whirling machine axle passed through into the box in a light-tight manner, and it was supplied with a self-oiling syphon wick. The ordinary speed at which it was driven was 64 revolutions per second; occasionally it rose as high as 85, but the water pressure was not often enough for this. The turbine could have been fed from a cistern in the roof, but greater pressure was attainable in the mains, and though liable to fluctuation this was found at certain times in the day or evening regular enough for good observation. MODE OF CONTROLLING AND DETERMINING THE SPEED. Uniformity of rotation was essential, and to secure it the method employed by Lord Rayleigh in his determination of the ohm was imitated. A small cardboard stroboscopic disk was painted with several circles of radial markings, or "teeth," the ones chiefly used being 3, 4, 5, 6, 8 teeth respectively in a circumference, especially the pattern 4. This disk was watched through a pair of slits carried by the prongs of a large electroinagnetically maintained Koenig fork, whose loads were adjusted to give 128 vibrations per second precisely. The slits permitted vision at the middle of each swing, consequently 256 glimpses a second. Hence whenever the 4 pattern on the stroboscopic disk was distinct and stationary as seen through the slits, it meant that the sensitive plate on the same axle was spinning 64 times in a second. Photographs of sparks were taken only when the pattern was stationary and the speed thus known to be regular. To determine the speed absolutely it was necessary to calibrate or specially observe the period of the fork. To this end two methods were employed: one the ordinary method devised by Lord Rayleigh, for comparing an electromagnetically maintained fork 18-2 140 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY with a large free standard fork*; the other by means of a simple four-figure mechanical counter attached to the axle of the stroboscopic disk. This counter recorded mechanically the actual number of revolutions made by the disk, during say five or ten minutes, and all this time the disk could be watched through the jaws of the electromagnetic fork and some definite pattern kept, on the average, absolutely steady. The control over the speed was obtained, as in Lord Rayleigh's case, by passing the driving cord through the fingers of the observer as he watched the disk through the jaws of the fork, thus keeping on the cord a slight frictional pressure, which, whenever necessary, was increased or relaxed, and thereby regulated the speed. With practice this method of personal government is susceptible of surprising accuracy. It is always however much easier to keep a pattern still on the average, that is, to bring a tooth back if it has slipped forward a little, so as not to allow any unknown escape of the steady pattern from control, than it is to keep the pattern constantly steady, as it ought to be when a photograph is being taken. At the same time it may be noticed that at the customary working speed a retardation or acceleration at the rate of one tooth interchange every second (which is conspicuously bad) makes an error of only 1 in 256, or less than one-half per cent.; and as it is not a systematic error it is likely to disappear from an average, even if so great as this. When the water pressure is regular, and the oiling also regular (a superabundance of paraffin is the easiest way of securing this latter condition) the regulation of the cord is easy. But if the water pressure varies much a duster or pad is necessary between the cord and the fingers, to save them getting burnt, and then some of the delicacy of manipulation has departed. It will be observed that in the experiments for determining the rate of the fork there is no need to run the stroboscopic disk very fast. The 8 or the 12 pattern may be the one kept still; corresponding to 32 or 21~ revolutions per second, a moderate speed which is not liable to heat or otherwise overstrain the counter. The multiplication necessary to get the speed for any other steady pattern is of course precise. The fork was not found to vary on different days; it was set very accurately to 128 vibrations per second (viz. close to the mark 256), and this part of the determination, viz. the absolute speed of the revolving plate, was entirely easy and satisfactory. EXAMPLE OF A RATING OF THE FORK. The following may serve as an example of one of the observations for calculating the speed of the fork. There were three observers: one to watch the disk and control the driving string, so as to keep any selected pattern steady; another to watch the counter and make a tap whenever a figure changed on the 100 dial (the units flew past invisibly, and the tens were inconveniently quick); and the third to read a chronometer and record the time of occurrence of every other tap to the nearest half second. * See Phil. Trans., 1883, Part 1, p. 316. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 141 The correctness of absolute time was secured by comparing the chronometer every day with a standard clock which was rated from the Observatory. The error in the rate of the chronometer was thus found negligible, being certainly not more than one or two seconds a day. Although it was possible to keep the speed constant for ten minutes or so, it was rather wearying and was really unnecessary, two or three minutes being quite sufficient, on this method of observing. Table I. gives a set of readings taken on the 23rd July, 1889, the "eight" pattern being kept steady and every other tap, or every 200th revolution, being timed: TABLE I. h. m. s. h. m. s. xi 32 3'5 xi 33 43'0 9.5 49-5,,,, 9'5.9 15'5., 56'0 22-0, 34 2'0.. 285, 8-5 345, 14-5 410, 20-5 47-0 27-0,,),, y? o,,,, 00..535. ". 33'5... 59.5. 9 39.5 33 5'5, 45'5 120, 52-0 Analysing these figures it will be found that the average time for 16 "taps" of 200 revolutions each is 100 seconds; as it happens exactly. And this corresponds to 32 revolutions per second; appropriate to the steadiness of the "eight" pattern. After this the speed could be increased till the "four" pattern was steady, with the certainty that the plate was then revolving 64 times a second with extreme accuracy. Thus the fork was used merely as an intermediary time-keeper to the chronometer, the media of comparison being the counter and the stroboscopic disk. PROCESS OF TAKING A SERIES OF SPARK PHOTOGRAPHS. The room being thoroughly darkened one of the sensitive plates was extracted from its case, and by the light of an exceedingly dim red glimmer fixed into the rotating frame holder. The spark knobs had previously been focussed on a dummy plate so that the spark length would be exactly radial, and near its outer margin. The packing-case cover being well covered, light was admitted to the room so as to make visible the stroboscopic disk which was watched between the jaws of the vibrating fork, and the turbine was turned on. The patterns were seen steadying themselves one after the other until the 4 pattern was reached and just passed; the water was regulated close to the point by past experience; the cord was then gripped by the observer and the escaping pattern brought back steady. 142 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Meanwhile a small Voss machine, attached to the spark knobs, which formed the terminal of a circuit containing the condenser and the coil, had been excited with its knobs in contact. At a signal from the observer watching the disk they were drawn apart, and one, two, three, or four sparks listened for inside the case. The machine was then short-circuited again, and the lens slightly shifted a felt amount (which could be done without opening the "camera") so as to bring the spark image a trifle nearer the centre, and another ring of sparks was then taken; sometimes with the conditions varied, sometimes with them just the same. Then a third, a fourth, and sometimes a fifth circle of sparks were also taken. The number of sparks which without too much fear of unintelligible superposition could be taken in a single circle depended partly on their strength. With a large condenser a single spark might overlap its own record; with a very small condenser 6 or 8 sparks could be safely taken. In practice either 4 or 5 was the commonest number, and though chance frequently caused some overlap it was not usually difficult to disentangle the records when reading the plate. It was customary to get about 2 dozen sparks on a single plate, though sometimes it would have been wiser to try for fewer. But a bad overlap after all is no worse than if neither record had been attempted. Lastly, a needle point was held on the still spinning plate near its middle so as to centre it by a small circular scratch, and then the turbine was stopped, the room darkened, and the plate removed. An assistant, Mr Robinson, to whose careful manipulation we are much indebted, then proceeded to develop the plate, sometimes using an intensifier when the markings were too faint. Meanwhile whatever conditions had to be varied were attended to, other measurements, such as that of the self-induction of the coil, or the timing of fork, were made, and things were got ready for another spin. This process went on without interruption for some weeks, and a large number of negatives were obtained. The plate at first used was the ordinary half-plate size, but in order to permit larger circles, Mr Swan subsequently sent us square plates, 4 inches square, and on these the final records were taken. The spark-trace exhibited the alternate oscillations very distinctly: one end (probably the cathode) being always brighter than the other, and this brighter end alternated from side to side with every half-period. The beginning and end of each oscillation though clear enough to ordinary vision became furry under magnification, and by far the most definite things to set the crosswire on was a narrow bright radial line or sharp spit, due evidently to the sparking of the knobs into one another: a phenomenon which accompanied the main oscillations of the condenser and marked the beginning of each electrical surge. These spits were so instantaneous that the rotation of the plate had absolutely no effect on their sharpness. They were narrow lines no wider than the crosswires. DISCHARGE 0F AN AIR CONDENSER, WITH A DETERMINATION OF "v." 143 READING OF THE RECORD. The negative when thoroughly finished was subjected to careful rnicrometric examination. To this end the plate was fixed on a horizontal circular graduated plate, part of a spectrometer, reading with verniers at opposite ends of a diameter, and capable of rotation with a slow motion tangent screw. Above the plate was clamped a microscope of moderate power, with crosswires in its eye-piece; and below the plate a scrap of mirror was arranged inclined at 45~ to throw the light up. The centre of the plate was made to coincide with the centre of rotation, and the microscope was placed over one of the spark rings. The plate was turned until the beginning of a spark-trace appeared. Some definite feature of it was then brought under the crosswire, and the verniers were read. Then another feature was sighted, and the verniers read again, and so on, all along the trace of that spark; and similarly with every spark round that circle. Then the microscope was shifted till over another circle, and the process repeated. By far the most distinct features, and the most useful for precise setting, were the sharp spits or radial lines already referred to and visible in the positives or rough copies of some of the preliminary plates. All the readings were done on the negatives, and the best or final series of plates have had no positive copies taken from them as yet. PART II. THE MEASUREMENT OF THE SELF-INDUCTION OF THE COIL. Theory of the Method. The method adopted for the measurement of the self-induction is that devised by Maxwell, in his papers on "A Dynamical Theory of the Electromagnetic Field," Collected Pctpers, Vol. I. p. 549. B F=iG. 2. D FIG. 2. 144 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY The coil whose coefficient of self-induction L is required forms one of the arms of a Wheatstone bridge, Fig. 2. Let P be the resistance of the arm. Two of the other arms R and S are two resistances whose ratio-preferably one of equality-is known, and a balance is obtained by adjusting the fourth arm Q. When this balance is found we have the relation P/Q -R/S. If the connections in the battery circuit be now reversed, a current due to selfinduction in the arm P passes through the galvanometer. Let a the first throw of the galvanometer be observed. Now alter the resistance Q by an amount îQ. In consequence there will be a deflection of the galvanometer needle; let 0 be this deflection; let x, x' be the currents in the arm P before and after the alteration of Q, X the logarithmic decrement, and let T be the time of a complete oscillation. Then, remembering that P and Q are equal, we have (Rayleigh, "On the Value of the British Association Unit in Absolute Measure," Phil. Trans. Part II., 1882) x T ( 2sin 2 L=GQ- -(1X) 2 x 477 tan ' THE RESISTANCE BOXES. In our experiments the coil P, already partly described, was wound in two sections each with a resistance of about 100 ohms, so that when both sections were in use P was approximately 200 ohms. The other resistances were taken from two boxes of coils of platinum silver wire by Messrs Elliott Bros., correct in "Legal Ohms" at 17~ C. The boxes had been calibrated in previous experiments, and the coils agreed closely with each other. R and S were two coils of 100 ohms from one of these boxes; for the arm Q an arrangement of two resistances in multiple arc was used. One of these was 205 ohms, the other was a large resistance of about 8000 ohms, and by varying this a fine adjustment could be easily obtained. DESCRIPTION OF THE GALVANOMETER USED. The galvanometer employed was a ballistic instrument of about 64 ohms resistance. It has two channels of rectangular section. Each channel contains 20 layers of thin copper wire and 16 layers of thick, making about 465 and 202 double turns respectively, so that there are 667 double turns in each channel, and about 2668 single turns on the galvanometer. The two thicknesses of wire were employed in order to fill the channels, and at the same time permit the resistance of the galvanometer to be varied as required. The ends of the wires are connected to binding screws on the bobbin marked A, B, &c., a, b, &c. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 145 A to a is one wire, B to b another. In our experiments the coils were connected up in series, the total resistance being about 64 ohms at a temperature of 13~'2 C. The needle of the galvanometer was suspended from the Weber suspension by three single cocoon fibres of 60 centims. in length. The magnet was a small bar of hardened steel 15 centim. long, '6 centim. broad, and '12 centim. thick; its weight was '708 grm. The magnet was attached by two small screws to a brass stirrup to which the mirror was fixed. A piece of brass wire 6 6 centims. long, with a screw thread cut on it, was fixed to this stirrup at right angles to the plane of the mirror, projecting equally on either side of the mirror. Two small brass cylinders could be screwed along this brass wire, and by means of them the moment of inertia and time of swing of the needle could be adjusted as required. The stirrup and mirror weighed 6'6 grms. The galvanometer has a solid wooden base of about 18 centims. diameter, and this base was supported on three levelling screws. A graduated circle is fixed to the base, and the coils can be turned about a vertical axis, and their position read by means of a vernier. This was found useful in adjusting the coils parallel to the magnetic meridian. The galvanometer rested on a stone bracket built up from the ground. A scale placed approximately north and south at a distance of about 347 centims. from the magnet was reflected in the mirror and viewed through a telescope. The scale rested on a solid stone support on the floor of the room. The mirror, about 1'5 centim. square, was a specially good one, selected by a fortunate chance from among a number in the laboratory. The divisions of the scale were in millimetres, and after practice these could be subdivided by the eye with great accuracy to tenths. The scale itself was of paper; though this material is unsuitable for many purposes because of the changes produced in it by the weather, in our experiments these changes are of small consequence, for we require only the ratio of the throw produced by the induction current to the steady deflection produced by the permanent current; and the time which elapsed between the measurements was only a few minutes. Any shrinking or alteration of the scale will go on very approximately uniformly throughout its length and not alter the ratio of two lengths, which were never very unequal, as measured by the scale. The scale had been carefully compared with the standard metre and the necessary correction applied to the readings. The distance between the mirror and the scale only enters our result in the small correction necessary to reduce the scale readings so as to give the ratio of the sine of half the throw to the tangent of the deflection. It was unnecessary, therefore, to measure it with any great accuracy or to take steps to ensure its remaining the same from day to day; so long as it did not change during the half-hour occupied by each experiment, all the conditions required by us were satisfied. The scale was carefully set so that the line joining its griddle point to the centre VOL. XVIII. 19 146 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY of the mirror was east and west, while the scale itself ran north and south. By taking, however, throws and deflections on both sides of the zero which was at the centre of the scale, the effect of any small error in setting was eliminated from the result. GENERAL THEORY OF THE METHOD. In making the observations the double amplitude, i.e., the distance between an extreme elongation to the right and a corresponding one to the left, was noted. Let a be this double amplitude in scale divisions for the induction throws, c for the deflection due to the alteration îQ, and let d be the distance between mirror and the scale. Then tan 2a =, tan 20=-, and from this we find "d' d' 2sin l a 1 lla2-Sc2) tan = c 1 28d28 ) neglecting higher powers of a/d and c/d. The values of (lla2-8c2)/128d2 varied for the different arrangements from '00173 to '00023. The value of the ratio x'/x was obtained as follows: Let E and E' be the values of the potential difference between the points where the current enters and leaves the bridge, in the two cases when the values of Q are Q and Q +SQ respectively. e the E.M.F. of the battery, which we suppose does not alter*. Let X and X' be the resistances between the points A and D where the current enters and leaves the bridge in the two cases, and Y the battery resistance. Then putting P= Q= 200 in the small terms, and R =S= 100, we find E'='(Q+ Q+ 100 + 'O Q), E =x (Q+ 00), Y E l+ also E = E- Y= 1 +X if a term of the order YSQ/90,000 be neglected. Y is of the order 1 ohm, and îQ of 4 or 5 ohms. x _= Q + 100 Hence x Q+100+37Q' * A combination of large Daniell's cells was used. themselves will afford a test of this. A small change in Except for the correction now discussed, the results are the E.M.F. would only produce a first order change in the independent of changes in the battery E.MI.F., provided value of the correction, and therefore a second order change such (if they occur) go on uniformly, and the experiments in the whole result; it may therefore be omitted. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "." 147 In a second series of observations the approximate value of Q was 100, and in,x' Q+100 this case the formula becomes -= Q23. x Q -100+ Q -+ 8 ' The actual value of the ratio will vary with the value of îQ in the various experiments; in most cases it is from one to two per cent. greater than unity: aQ being negative. Introducing these the formula for L expressed in terms of quantities which can be directly observed is 1 1T x Q+100 a lla2 - 8 4=QQ Q+ ( 1 + )Q c 128d2 } [The coefficient of 8Q in the denominator is in some of the experiments 21.] THEORY OF THE ACTUAL OBSERVATIONS. The above simple theory of the experiment assumes (A) that a perfect resistance balance can be (1) obtained and (2) maintained during the experiment, and (B) that in measuring a throw the galvanometer needle can be brought to absolute rest before a reversal of the current. The coil is made of copper wire; slight changes of temperature therefore affect its resistance, the current itself produces a small heating effect in the wire, and it is practically impossible to maintain an accurate balance. Again to bring the needle accurately to rest before each throw involves time, while to avoid undue heating it is necessary to be rapid in observations; it is better therefore to make a correction for any small swing which may exist at the time of making a reversal. Lord Rayleigh has shewn how actually to make the observations, provided the reversal takes place as the needle passes its equilibrium position (Phil. Trans., 1882, Pt. II., p. 680). The following quotation gives his theory and practice of the method of observation. "In the simple theory of the method the induction throw is supposed to be taken when the needle is at rest, and when the resistance balance is perfect. Instead of waiting to reduce the free swing to insignificance, it was much better to observe its actual amount and to allow for it. The first step is, therefore, to read two successive elongations, and this should be taken as soon as the needle is fairly quiet. The battery current is then reversed, to a signal, as the needle passes the position of equilibrium, and a note made whether the free swing is in the same or in the opposite direction to the induction throw. We have also to bear in mind that the zero about which the vibrations take place is different after reversal from what it was before reversal, in consequence of imperfection in the resistance balance. At the moment after reversal we are therefore to regard the needle as displaced from its position of equilibrium, and as affected with a velocity due jointly to the induction impulse and to the free swing previously existing. If the arc of vibration (i.e. the difference of successive elongations) be a0 before reversal, the arc due to induction be a, and if b be the difference of zeros, the subsequent vibration is expressed by S (a + a0) sin nt + b cos nt, 19-2 148 M5ESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY in which t is measured from the moment of reversal, and the damping is for the present neglected. The actually observed arc of vibration is therefore 2 {(a + ao)2 + b2}, -or with sufficient approximation 2b2 a + ca + -, so that 2b2 a = observed arc + a - - "In most cases the correction depending upon b was very small, if not insensible. The 'observed arc' was the difference of the readings at the two elongations immediately following reversal. As a check against mistakes the two next elongations also were observed, but were not used further in the reduction. The needle was then brought nearly to rest, and two elongations observed in the now reversed position of the key, giving with the former ones the data for determining the imperfection of the resistance balance. As the needle next passed the position of equilibrium, it was acted upon by the induction impulse (in the opposite direction to that observed before) and the four following elongations were read." To find then the correct double throw a, if a, be the observed throw, a0 the throw at the time of reversal, and b the difference between the equilibrium positions before and after reversal, we have 2b2 (a = aC1 + ao -- The sign to be attached to a0 depends on the directions of a, and a0. After two throws right and left respectively have been observed, and the equilibrium position is taken with the battery key in one position-denoted by R, say, in the table-then Q is altered by 3Q and the new equilibrium position is found. This was done by bringing the needle approximately to rest near the new position, by the proper use of the battery key (Maxwell i.) and an auxiliary damping circuit, and reading three elongations in the usual way. From these the position of rest was found. The difference between the two equilibrium positions gives ci the deflexion to the right; the battery key is then reversed and a deflexion to the left found; the resistance 3Q is then removed and a second zero reading taken; from these two, we find the deflexion c2 to the left. The sum of cl and c, gives c the double deflexion required. The values of Q and Q + IQ are calculated from the resistances on the multiple arc in the arms of the bridge. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 149 Thus, on July 18th, for the balance the resistances were 205 and 7750 ohms, for a deflexion 205 and 3950 ohms. Hence 1 1 1 Q 205 + 7750' 1 1 1 Q+ Q~ 20 + 3950' Whence Q=199-713, Q + SQ = 194-884, 3Q =- 4829 "legal ohms." The temperature of the box was 18~'5. Having obtained a value for c as described, a second series of throws were taken, then another series of deflexions, and so on successively. Table II. gives as a specimen the observations for July 20th. Temperature 17~'5 C. Battery 1 Daniell cell Resistances for Balance 205 and 6760 for Deflexion 205 and 3460. Whence Q= 198-9662 legal ohms Q =- 5-4329!1a2 - 8c2 8 = 00107, = 1'0211. x 150 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY TABLE II. Mean Mean Time Zero a, b a throw deflexion a c II. 10 L. 75-65 ' 1 80-3 -05 80-2 80-2 Throw R. 75-6 *2 80 80*2 Deflexion: cI c II. 12 L. 75-65 117 41-35 825 825 Deflexion R. 75-6 34-45 41-15 a60 a b a II. 14 R. 75-6 0 80 -05 80 80 Throw L. 75-65 1 79-9 80 Deflexion cl c2 c II. 17 R. 75-55 34-5 41-05 82-3 82'3 Deflexion L. 75-7 116-95 41 25 a0 ai b a II. 18 L. 75-7 ' 2 79-9 -15 80-1 80-05 Throw R. 75-55 ' 1 80-1 80 Deflexion cl C2 c 1. 22 L. 75-7 117 41-3 82-4 82-4 Deflexion R. 75-65 34-55 41-1 a0 a1 b C6 II. 24 R. 75-65.1 79-7 -05 79-8 79-85 Throw L. 75 7.2 79-7 79-9 11.26 Final Means... 80-025 82-40 Thus the complete set of 4 throws and 3 deflexions took sixteen minutes. We see from the last two columns that there has been a slight change in the value both of the throw and of the permanent deflexion: the current has decreased slightly, but very slightly, during the observations. We can get a series of values of the ratio of a/c by combining an observation of deflexion with the throws on either side, or an observation of throw with the deflexions. The mean value of a/c for this series is -97118. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 151 TIME OF SWING OF NEEDLE. The time of swing was found in the usual way by observing the transits of the zero reading over the cross-wire of the telescope. In this case 12 transits were observed and then after waiting for an interval of 16 transits 12 more were taken. We thus found the mean of sets taken on several occasions, always both before and after the series of throws and deflexions; T=10'713 mean solar seconds. The time was taken on the chronometer already mentioned in Part I. of the paper. The greatest error from the mean in any one of the 12 observations was less than 2 parts in 1000. Thus the time of swing is very accurately known. The value of X was found by reading a series of 42 deflexions. The average value of a large number of observations (which lay between '0134 and '0131) was '01324. From these observations we obtain for the value of L L = 4-6488 Legal Quadrants. The result requires a small correction because $Q was at 17~'5 instead of at 17~ at which the box is right. Introducing this we find as the value 4'6494 Legal Quadrants. Four sets of observations were taken on the two coils arranged in series. Table III. gives the details, from which the results have been calculated. The mean value of '01324 has been employed throughout for the logarithmic decrement X. TABLE III. Tempera- a c T Q P L L Date ture c T QP L t7 ture at 17~ C. July 18th 18-5* 80-71 73-87 10-742 - 4-829 199-684 4-6480 4-6499 July 20th 18-9 77-01 71-46 10-716 - 4903 199-909 4-6460 4-6485 July 29th 17-5 80-025 82-40 10-713 - 54329 198-966 4-6488 4-6494 July 30th 18'9 80-725 77'25 10-723 -5-0553 199-640 4-6471 4-6496 Mean 4'6493 legal quadrants. * There was a slight uncertainty about this temperature. 152 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY It appears that the greatest difference between two results is '0014 in a total of 4-6500, or less than 1 part in 3000. It will be noticed also that the agreement is very decidedly improved by the temperature corrections of the last column. Thus the value of the coefficient of selfinduction has been determined to an accuracy which requires that the temperature of the various coils used should be known to a fraction of a degree. The value given, 4-6493, is in legal quadrants; i.e., the resistance of a column of mercury 106 cm. long has been taken as 109 C.G.S. units. To reduce it to "Henry's" or "International Quadrants" it must be multiplied by the ratio 106/106-3. We then find as the value of the coefficient of self-induction of the coil 4-6362 Quadrants, or 4-6362 x 10lO0u centimetres. SELF-INDUCTION OF EACH HALF OF THE COIL. Since the coil was wound in two parts and one of the parts occasionally used alone, it was thought well to find the coefficients for the two parts separately, and to check the result by observing also the value when they were arranged so that the mutual induction of the two opposed the self-induction. Let L1, L2 be the two coefficients of self-induction of the two parts, M the coefficient of mutual induction between the parts, L' the coefficient of self-induction of the whole with the two parts opposed. Then L = L1 + L2 + 2M, L' = L + L2 - 2M = 2(L + L2)-L. Thus L = 2 (L, + L) - L'. The coefficients are all small and the probable errors of the measures are greater than those in the direct measurement. The following values, however, were obtained: L = 1-405 Quadrants for semi-coil marked A. L, = 1393.. " " B. L' = 0-963 Whence L = 4'633 legal quadrants; agreeing fairly well with the true result. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 153 PART III. CORRECTIONS TO THE SIMPLE THEORY OF THE EXPERIMENT. (i) THE ELECTROSTATIC CAPACITY OF THE COIL. The chief cause of difficulty in comparing the experimental results with theory arises from the fact that the coil has considerable capacity, and further that this is not distributed uniformly along the length of the wire. The coil consists of two similar portions almost identical. Each half is wound with about 60 layers of gutta percha covered wire containing about 30 turns to a layer. The interior diameter is 27'5 cm. and the exterior is 48'7 cm., while the axial depth of the coil is about 5'2 cm. The number of turns of the coil were not counted exactly when it was wound. After the experiments the case was opened and the coil measured as far as practicable. It was found that the number of layers in a radial direction as estimated from those which could be seen and counted was 64, and they occupied 10'35 cm. Thus the average distance between the centres of consecutive layers is 10'35/64 or *164cm. The inner layer contained 28 turns, and of these 25 lie in a space of 3'9 cm.; thus the distance between consecutive turns is '156 cm. The thickness of the uncovered wire was found to be '049 cm.; thus the thickness of two coverings is '107 cm. The two halves are separated by a sheet of glass with a circular hole in its centre; the sheet is about '27 cm. in thickness. The whole coil is enclosed in a wooden box, the ends of the wires being brought to terminals which are well insulated from the wood. Now if we consider any turn of the one coil lying near the glass, it is faced on the opposite side of the glass by a similar turn, which during the experiments will be at a very different potential. Charges will thus accumulate on these turns and their capacity must be considered in the theory. If we consider a turn in the centre of either coil it is surrounded by other turns at nearly the same potential as itself, and does not therefore become much charged. The outer layers of the coil will have some capacity, but if the wood case be treated as an insulator this will be small, and thus we may consider that the chief capacity of the coil lies in the faces in contact with the glass. We may thus represent the two coils in the following diagrammatic manner: Consider a number (n-1) of equal condensers, each of capacity S'; each plate of a condenser represents two adjacent turns of the wire, which lie on the same side of the glass, and face two corresponding turns, representing the second plate of the condenser, on the opposite side of the glass. VOL. XVIII. 20 154 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY In strictness, since the diameters of the turns increase from 27 to 48 cm., the capacities of the condensers ought not to be taken as equal; but unless this is done the solution is very complex, and when the correction is small the error introduced cannot be great. Let the positive plate of each condenser be connected to the positive plate of the two adjacent condensers by wires of self-induction L, and likewise for the negative plates. Each loop of wire represents two adjacent layers in the coil itself. The one set of condensers and loops represents one coil, the other set the second coil. Connect the two plates of the condensers at one end of the series by a loop of wire of inductance 2L, and connect the plates of the condenser at the other end of the series by wires of inductance L to the two plates respectively of a condenser of capacity S. We have thus a representation of the condenser and coil in which the oscillations occur. This is shewn diagrammatically in Fig. 3. V1 V2 X2 V3 i D2 Q Q1 Q4 ç S S' S' s' s' A B FIG. 3. Case i. Let x1, x1' be the currents in the wires connecting the positive and negative plates of the first and second condensers, x, x,' those in the wires connecting the corresponding plates of the second and third condensers, and so on. Let Q1, Q2,... be the charges on the positive plates. Then since the rates of increase of the charges on the opposite plates of any one condenser are equal and opposite, dQt x~ = dt l, X2 -- X ------ t -- X2 X, X = X2', Xs= —, etc. Now let V1, V2, V3 be the potentials of the positive plates, V', V', etc. those of the negative plates, R, L the resistance and inductance of the wires joining consecutive plates, Lxd + iR x = V1- V, Lr1' + Rx-' = V2' - T-. But, ~="',1 = '11, DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 155 V.-V V- -Vz' == Q2 ^-T/=|1, 7,- |,,.L. 21~ + 2Rsx = V,- r1'-(V2- V I') Q2 Now if L', R' denote the inductance and resistance of the two wires joining the plates of any two consecutive condensers, then L'= 2L, R'= 2R. Then if R is the whole resistance, L the whole inductance, R = nR', L = L'. And the equations to find the period are L'd2 + R'lx = - L'd + R'x = Q2 Q -L/ ~Xi2 - ~- - - S 7 1/ dQ2 -1 =i dQ9 ct' d Q xn-l-1 xn= dt Hence we have L'"x + R',X + (- + yS x-S-, 2x, = 0, X i L+ x + R'C' +, x2 - X-, = 0 -S x- + L ''2 + -RR'2, + 2-,= O. Put x - X eiÀt, and let S' XL'-R'X- - 1 = P, S' {x2L'-R't\} -2 = Q, S' {2L'- R'x} - 1 = R. Then the equations become PX, + X2 = o, X, + Qx, + x, = o, X2 + QX2 + X4 = 0, Xn-_ + RX2T = 0. 20-2 156 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Whence to find the periods we have the determinant P 1 0.... =0 1 Q 10... o QI O... 0 1 Q 1. (to n columns). 1 R Now the determinant =P Q 1... +(- 1 i Q Q 1 (n- 1) columns Q (n-) columns i1 l? Q1 0 I1 R 1 0 O =P R 1... +(-1)2-3 R 1 Q.. (n - 1) columns Q (n- 2) columns.. Q Q 1... 1 Q I Q Now R 1 =1R Q 1 0 +(-1),-l 1 0 0... 1 1 i Q 1 (m) columns Q (m - 1) columns Q 0 I Q Q 1 iQ =1R Q 1 0... +(- l)2'-3 Q 1 o 1 Q 1... (m - 1) columns 1 Q I (m - 2) columns Q............ = R\m_i - Am-2, if An stands for Q 1 0 0... to m columns. L Q 1... Q i 0... 0 1 Q 1i... Q Thus the given determinant = P {Ra2 - a_-3} - {RAn_3 -- a 4} = 0. Now if Q=2cos 8, then we know that = sin (m+ 1) O (Rayleigh, Sound.) Hence we have as the equation for the periods P {R in (- ) - s in (n - 2) O - {R sin (n- 2) - sin (n- 3) } = 0. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 157 In the first instance neglect the terms depending on the resistances, then S'2L'- 2 = Q = 2 cos 0, S'2L' = 2 (1 + cos 6)= 4 cos. Now if the whole of the capacity S, were concentrated at one part of the circuit connected by a wire of inductance L we should have 1 I XSL 1n (n - 1) S'L' In the most important of the cases with which we have to deal a large part of the inductance is so concentrated in the capacity S. We shall suppose therefore in solving the equation that X2S'L' is a small quantity of the order 1/n2. So that 2 cos - is of the order 1/n, and 0 is not much different from wr. Put 0 = r- q: ultimately (b will be treated as small, though for the present the solution is general: then S'X2L'= 2(1 + cos 0)= 2 (1 - cos). Whence substituting {S'X2L'- 1} {(S'2L'- 1) sin (n - 1) g + sin (n - 2) b} + (S'X2L'- 1) sin (n - 2) ( + sin (n - 3) - {(S'X2L' - l)sin (n- 1) + sin(n- 2) } =0. Thus 2S'X2L' {sin (n - 2) - sin (n - 1),p} + sin (n - 1) ( + sin (n - 3) q - 2 sin (n- 2) -S sin (n - 2) b - sin (n - 1) + S'X2L' sin (n - 1) b} + S'2X4L'4 sin (n- 1) = 0,.'. 2S'X2L' {sin (n - 2)- sin (n - 1) /} + 2 sin (n-2) ( (cos - 1) S- {sin (n - 2) - sin (n - 1) ~ + S'x2L' sin (n - 1) /} + S'2X4L' sin (n- 1) = 0. But 2(cos -1) = - S' L'....................................(A).. L'{sin (n - 2) b - 2 sin (n- 1) ~} - I {sin (n - 2) q - sin (n - 1) + S';2L' sin (n - 1) b} + S'À4L'2 sin (n - 1) =...................................................(B). 158 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY On eliminating b from (A) and (B) we obtain an equation for X2. Now we have seen that 4 sin2 b/2 is of the order S'L'/EL, where E stands for the whole capacity. In the most important cases S, the external capacity, is large compared with S1, the capacity of the coil; and in this case the whole capacity is large compared with Sl. In the general case on substituting in (B) from (A) we find x2L' {sin (n - 2) q - 2 sin (n - 1) + 2 sin (n - 1) (1 - cos )} = s {sin (n - 2) s - sin (n - 1) s + 2 (1 - cos ) sin ( - 1) 6}. Whence 2L'S = 1 + - (n - 1) sin (n - 2)p - 2 cos 5 sin (n -1)b -= 1sin(n- 1) sin nb = 1 - cos + sin r cot nb, 2L' (S - S') = sin cot n..............................(C), or substituting for X2L' from (A) 2 tan (S - S') = S' cot n....................... (C'), the fundamental equation for the periods. Up to the present no assumption has been made as to the relative values of S and S'. In our case S' is small compared with S1, and then o is small; S may have any value. In the more important cases S is large compared with S1. In this case ne) is of the order (Si/S)L. We may expand cotnas and cot ~/2 and use 1/SL as an approximate value for X2 in the small terns. Now we have 2 (1 -cos s) = XL'S'. Thus s2 = 2L'S' (1 + -2X2L'S' +...). Hence 2 = X2S {l + + 4+... =' 1 n+ + +" where a, b, etc. can be found approximately. Hence expanding in Bernouilli's numbers, 28 2 / +4 * 1 P- L4...* 22B,02 B244 1 Now, as a first approximation, 'n2L'' n S, S ( 1 LS n -1S S \ n) DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 159 Hence n2o2 is of the order Si/S, n2s4 is of order (S1 00)2 r since n = 30 approximately in the experiments. Again, the value of S is in the final experiments 5853 cm. while Sl, when the two coils (A) and (B) are used, is about 1600 cm. (See Part IV.) Thus the important terms are those in n2b2, n4b4..., while a term such as one in n2o4 iS, when S, has its largest value, of the order (S1/S)8; and when S, is only 100 cm. it is of order (S1/S)4. Hence retaining the most important terms, 2LS -i {S1 1 } 22B1n2 2 24B2n4c 4 2 (n - L2 and B1 2 n2 1+) 3 x 22' = n1+~ x?~S {11 S }1 - Thtus 2L8{1 —2} (=-. 1) + P Hence =LS = 1 1 - 2 - 3 l8 { 2 (n - 1) 1 -~ { 2(n-1)}' omitting terns of the order 1/n2 in the coefficient of 81/S. Substituting in the terms in n4 0 and introducing the value of n in the last term, we have approximately L - )........................... (D ). In obtaining the coefficients of (S8/S) the terms in no have been retained when compared with terms in b. Hence for SI/S = 2733, X2LS = 1 - -0896 + -0066 = -9169. In some of the preliminary experiments however the value of Si/S was greater than unity and the series method of solution will not apply. The following graphical method however will apply to all the cases. 160 MEssRs GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY The equations to be solved are X 2 (1- cos ) L'S' tan (2S - 1)= cot no; S'= 1 n, -1=30, =-2733, n-' - for S= 5853 metres, 1= 16 metres. Hence (i- 1)= 21 Hence 218 x tan = cot 310. 2 An inspection of the Tables and a trial shews that s is nearly 56'. By plotting on a large scale the values of cot 31o and 218 tan sb/2 at about 56', we find the curves intersect at 56' 30". Hence = 56' 30". LS Hence 2LS = 2 (1 - cos s) L-S 2(1S = 2 (1 - cos s) n (n - 1) = 919, substituting for s, n and S/Si. Thus practically the same value is found as by the series. If we take S= 1 metre, as in the experiments with coil (A) or (B) singly, then S/S = 58, 2 (n-1) = 60, and tan s/2 {58 x 60 - 1} = cot 310. Thus 3479 x tan ( = cot.310. A similar procedure gives for s the value 14' 45", and substituting in the equation for X2LS we obtain X2LS = 9924. The solution by series already obtained was 9943. The case in which there is no outside condenser is given by putting S = S' in the DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 161 original equation; thus, supposing the coil to consist of n parts, so that nS' = S = 16 metres, we have tan = cotnb, n = + 2n2 X2LSX = 2n2 (1 - cos ) = (2+ 1)2 4 I- n eglecting l/n2 = 238 if n= 31. The case of a continuous coil of uniform capacity s and inductance I per unit of length may be treated as follows. We assume the frequency to be such that the current across each section of the wire is the same at any given moment. Let V be the potential at one end, that at the other being zero, a the length of the wire, v the potential at a distance x from the end at which the potential is zero. Vx Then v=-. a The charge on an elelnent dx at x is s Vxdx q = svdx = /a V2x" Energy -= qv = s a2 dx. The total electrostatic energy of the coil is thus 1 sV2 fa and thus = - x - V sa = 1 VIS, if Si is the capacity of the whole coil. Hence the total electrostatic energy of the coil and condenser is i1 (S+ 1Si). The electrokinetic energy is ~Lu2 if u is the current. Hence 2 = L (S + Si)' This agrees with the result already found for a large number n of condensers connected by wires. (See equation (D), p. 159.) In some of the earlier experiments described in Part IV. in which the whole coil was VOL. XVIII. 21 162 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY used and the value of S1 therefore was 16 metres, the values of S were approximately 2, 5, 5'5 and 10'5 metres. The values of 9b and X2LS can be found for these cases in the same manner and we thus get the following Table. TABLE IV. S S1 S/S1 0 LS 2 16 -125 2~ 37' 48" -2449 5 16 1 312 2 16 40 *4582 5'5 16 -344 2 12 20 -4741 10-5 16 -656 1 52 12 '6498 58-53 16 3-659 56 30 -9169 0 16 238* The experiments in which the external capacity is small are of no value as a means of finding "v." They serve however to test the truth of the formula and of the corrections which we have applied. We may put the correction another way, and say that instead of employing the whole capacity S to calculate the frequency from the formula 2LS = 1 we have to use a capacity S/k, where k has the values given in the last column of Table IV. Throughout the above we have taken L', the effective coefficient of self-induction of 1/n of the number of turns, as 1/n of the whole coefficient, and neglected the mutual induction between the turns; we proceed to justify this. Now the effect of inductance in any wire is made up of the self-induction of that wire, and the mutual induction of the other wires; moreover the currents in the various turns are, owing to the capacity, not the same. Let il be the coefficient of induction of a wire in which the current is &1 due to itself, mn2, nz13, etc. the mutual coefficients. Then the strict equations for any wire joining two of the condensers each of capacity S' will be 1 Ila + %2,2 +... + (XI - x2) = 0; put L'X1 = l1 + n+122 +. and let x2 = xc + y2, X3 = 2i + y3, etc., L'5 = (il + m2 +...) 55i + mm.yo + m,) +.... * For this case S, takes the place of S. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 163 Now y//l depends on S'/S, and m,,, etc. are all finite and less than 1,; (- - 1) S L = (li + m2, +...)+ (- - m, where m is of the order of the arithmetical mean of rnm,, mnl, etc.,.. L'= li 4-:I,2 +... + SThus in the case of a large number of turns, if S,/S is small the equations already used are correct if L' be 1/nth of the whole self-induction, for we may neglect the term 81-n/S compared with the sum lI + mn + etc. There is now the correction for resistance to be considered. In the case of a simple circuit X2=T{J 1 R)2} X, being the uncorrected value; thus we may put X = 1 (1 - k'). Now so far as the inductance of the circuit is concerned, X2= (1 - 1k). And in the more important cases both k" and k' are small. Therefore = -1 (1 - k" - k), where k" has the value already found from Table IV., In some of tbe exyerinments X, is about 27r x 103, = 200 x 109, L = 5 x 109, R 1 X0L - 50 approximately, and the correction is negligible. if the period be 1/120 second as in other experiments,;. — 27r x 12 x 102= 7-2 x 102 approximately, R 1 X0L =.0 approximately, and k' = 400 1600 which is also negligible. 21 —2 164 MEssRs GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY In some of the experiments only half the coil was used. We may represent this case diagrammatically thus (Fig. 4). The upper set of plates and loops represents the coil connected to the main condenser, the lower set represents the insulated coil, the ends of which are insulated. vll vl _ ___ _ y V'2 Q v,2 (?QeQ/ L 22' 2/72PLPJ LP Y'~i2.~ v:, FIG. 4. Case ii. As the main condenser is discharged the electrostatic action of the upper plates causes the charges on the lower plates to vary and oscillating currents are produced in the lower coil. Let x be the current leaving the main condenser, y,, y3, Y4 the currents between the lower plates of the coil condensers, then the currents between the upper plates of the same are x-y2, x- y3, etc., and the equations are L' () = V,- V2 L' (a - y2) = V2 - V3.....o..................... L'3 = Vn- V1I L' y 'L'y = V'n- - vj Hence nL'x = V1 - V' + V2' -V =^,-^/+^,- -7~+j DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 165 =2(- ( -)-2L'') -+ S' (n + 2)L'x'= 2Q Q+ Q8 (n + 2) L'x =- Q- + Q2 Q3 ~S' S" 2y2 (-2y) - Y Hence | YL(à_ 2-3)= 2 y3-, rL' ( 2y -) Y n-2 + 2y nNow we mnay shew that Y2=y y.-i For superpose everywhere on the system a potential v; this will not affect the currents. And now choose v so that the potentials of the plates of the main condenser are equal and opposite, i.e. so that V1 + v=-(V' + ), V =2(L+ L7); the distribution is a symmetrical one and obviously in this case Y2 = yn-1 Y3 - Yn-2, etc. Hence if n = 2m dQm+l _ Ym = Ym-' dt - and we have m - 1 equations L' (-22)=-Y2 + - Y3 - Y L (Y-2m)=- -J +; s=(1+V'> 166 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY put x1 =, x2 =x 1-IY2, x = x-2y3, Y2= -i (x1- 2), y3= (x - ), etc., Y3 - Y2 (x2 - X3), etc. i.. 1 1 (x 1- 2) (m + 1) L'=-S _ 2 xHence < L'2 = _X1 - 2 2 - x3 2 2S' 2S ' \ L'$ = i S' (xn_ - xn);. putting x = Xe-t, i2S' {(m + 1)L' - 1 -1 X1+ X2= 0, X, + {2S'X2L' - 2} X2 + X3 = O, Xn,_ + {2S'x2L'- 1} Xm = 0. Solving these equations as previously, putting S'L'X2 l-cos, we find as the equation for the periods Sx,2L'..n...=1...................(E). + -cos + sin cot mb) Expanding as far as b2 and assuming that 02 may be neglected, compared with;,"2)2, we have S2L' +m + 1, 2S2NowLL, {1 =, 22 = 1'. N{Now 2mL' = L, 52= 2S'X2L', S = (2m-1))S'. Heiice LxS {1+ (2N -L) = 1' or neglecting the terms in (8/8)2, 1- (1...................... DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 167 Thus so far as the term in SI/S is concerned the correction is one-fourth of that to be applied in the first case, and this, when the values of S1 and S are 16 and 58'53 respectively, comes to - 022. If we assume the coefficient of the term in (81/S)2 to be also divided by 4 we have to add to this + 001. Thus to about one part in 1000 we have 2LS = l — 021 = '978. If however in the case represented in Fig. 4, we suppose that the lower coil is uninsulated, the equations can readily be shewn to be those of the first section of this Part, and the formula for the frequency will be X2S = 9169..........................................(G ). There is however another possible arrangement to notice. -VI D VI V, 2__2 ---'v \Q~JJ2 V3 û_Qj2L2^ jLOJ^ V.-1 2^2J^'lFIG. 5. Case iii. If in Fig. 5, AB, CD represent the two coils, we have supposed above in Fig. 3 that B was connected to C, while A and D are connected to the main condenser. In some of the experiments however it appears possible that B was connected to D, and A and C to the external condenser. The distribution of potential would then be as shewn in the figure, and the solution differs from that of the first case. We can write down the equations and solve this case, but it can be shewn thus that it reduces to the second case. For compare Fig. 5 with Fig. 6, which is obtained by putting the coil DC alongside AB, and placing above the two a second similar double coil A'B'D'C' with its ends insulated; the distance between this and the first coil being the same as for the two coils in Fig. 5. The distribution of currents is clearly the same. Now if S~ be the capacity of the two coils AB, DC, or A'B', D'C' in Fig. 6, the correcting term is --- s/12S. 168 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY But Si = 2Sl. Thus the correcting terni in this case is - S or one half of its value 65 in the first case. -V /,' A c i-Vz -~V +V9 __ - VI A V, Vn-i 0 0i -V FIG. 6. We shall assume that if the higher powers of SI/S are included the correction is still one half of its value in the first case. The solution for the case in which there is no condenser attached is best obtained from the original equation (E), p. 166, by putting S = S', i.e. assuming the last section of the coil to be the external condenser. Thus we have 1S'L - c + sin - c o cot n} 2S'L'= 1 - cos b. Also nS'= S, 2nL' = L. Whence n (1 - cos ) cos ( - = cos - cos n. Now assume, guided by the solution on p. 161, that n4 = 2 - where f is small. ' 22 Then n (1 - cos s) sin 2 (S + ) = cos sn and = - approximately; neglecting 212 Thus n.~2 5 = 2 2 )2' {1in+l + } + nb 4& Thus approximately = 16 n2' '7rT2 7r Now X2SL = 2n2S'L' X2 = -2,2 (1 - cos b) = 2.~ = 6. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 169 Thus comparing this with the result on p. 161 we see that the frequency is the same whether the connexions be as in Fig. 3 or in Fig. 5. In order to solve when S1 is not small compared with S we have with the same notation from the above equations sin + ) -n (1 -cos = sin cos Now we may suppose q is small. Hence cot né = tan f = 2 (S 2 S' n'2 S 2 Now if X2SL =k = 2nX2SL', since 2X2S'L' = 2, we have k = ns _-..'. cot 7z> =n S{ S ( Now S' =- = approximately, and when the connexions are as in Fig. 4, k= 25 for S= 2. Hence assuming k is not very different in the present arrangement cot 300 = - (1 + 12), 3.36 cot 30 = — 1-68s. The solution of this gives =- 2~ 50' 30" and k = -2873. If S =5, k is '45 in the first arrangement. Thus cot 30, = 6q approximately, whence b = 2 30'30", and k='558. For S = 5 =5, = 574, while if S=10'5, k ='784. If then we write 2LS = k we have the following values for k according as the connexions are made. VOL. XVIII. 22 170 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY TABLE V. k I Case i. Case ii. Case iii. Coil as in Fig. 3 Half-coil used Coil as in Fig. 5 as in Fig. 4 2 ~245 -287 5 -458 -558 5-5 -474.574 10-5 -659 -784 30-5 1 - 152 1 - -076 190 1- 028 1- 007 1- 014 3000 1- '002 1- -0005 1- '001 58'5 1- 081 1 - 021 1- 040 Again it appears from an approximate calculation founded on the measurements of the coil that the capacity of any one layer on those adjoining it is large, it may even be as great as 600 cm. It is therefore desirable to investigate if possible what effect this has on the formula for the frequency. We may perhaps represent this by supposing the coil to consist of a series of 3 Q3, -Q.I ~ xys. v4-^ ---- ^ ^ y _ _v -Q4 Qn-1 FiG. 7. loops without capacity with a series of parallel plates attached at the centre of each loop. L'c= V1- V2, L' (-f'2) = y2- 13, DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 171 L' ( - yn-) -= Vn - Vn, L'= TFn- Y,, y, vQg dQ2 V2 - F_= J "dt Y2, etc., nL'X - L' (y, + Y2 + y,,-l)= V1- FV = L' (- y_) = - yn-i, (Si'L'2 - 1) Y =(S,'L'x - 1) Y3 = SL' x2X1,,X2nL+(-(n 2) S ='L'2xl 1 1* r - i-^ 2. rs' n SLx2 = 1 - &1',2L, XSL = -2 S1' n2 Now it appears possible from the investigation in Part IV. p. 174, that S' may be as great as 600 cm. so that S'/=S. Also, taking the two coils, n = 60 and 1- - 1- approximately, thus the correction is negligible, and we might give S' a much larger value without modifying our final result. If we have no external condenser then S = Si and we have X2S7'L = 1 — approximately. So that in this case the capacity of each layer of the coil on the next may be the effective factor in determining the period. (ii) EFFECT OF RESISTANCE AND THROTTLING ON THE PERIOD. The critical resistance at which the discharge ceases to be oscillatory is 4L V~8 >s22 2 22-2 172 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Now the resistance of our wire circuit is only 200 ohms as ordinarily measured, but it is well known that under rapid oscillations the resistance of a conductor is increased by reason of the extra peripheral distribution of the current. The spark gap has also a certain resistance which it is not easy exactly to estimate. Some observations were made with a condenser discharging through several known circuits and the same air gap, in order to study the damping and make an estimate of what the resistance of the spark was. These indicate that for feeble discharge a spark resistance is high, while for powerfil discharge it may be quite low. With our feeble spark it is undoubtedly large, and quite eclipses the resistance of the wire part of the circuit, though it does not amount to anything like the critical resistance at which the discharge ceases to be oscillatory; but it cannot be considered as constant, and its complete specification will be difficult. With regard to the throttling by reason of rapid alternation, it must be observed, lst, that the alternations were not excessively rapid, always comparable to 1000 per second; and 2nd, that the wire on the coil was copper and very thin. The coil had a mean diameter of 38 centims. and consisted of 3493 turns of copper wire, half a millimetre in diameter. At 1000 alternations per second uniform distribution of current through such a wire would hardly be departed from, and neither the resistance or the self-induction would be greatly different from their ordinary values. It is important to note that no correction to self-induction is necessary, for even with infinite rapidity of oscillation, when all the current flows by the periphery, the value of the self-induction would not be greatly disturbed; though the throttling resistance would then be enormous. The reason why the self-induction is not very dependent on distribution in a thin wire is that it is only the space inside the wire which ceases to be magnetised by a peripheral distribution, and this is small in comparison with all the space outside. (iii) SEI,F-INDUCTION OF LEADING WIRES. The self-induction of the leading wires between condenser, coil and spark-gap, was about 100 metres; but as the self-induction of the whole circuit was considerably more than an earth quadrant this is entirely insignificant. (iv) EFFECT OF LEAKAGE. The insulation resistance between the two halves of the coil was measured and found to be 20 megohms; hence leakage during a discharge was practically non-existent. (v) EFFECT OF WAVE LENGTH. The electric oscillations have not been assumed quick enough to give waves comparable in length with the circuit, else different parts of the circuit would be in different phases, and some complications would result. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 173 The length of the circuit is 4 kilometres. The wave length in the chief cases is either 3 x 1010 3 x 1010 80 or 1600- centms. 880 1600 and is always bigger than 100 kilometres, so no complication from different phases will arise. PART IV. PRELIMINARY EXPERIMENTS. IN the earlier stages of the work a large number of experiments were made with various condensers; some of these had a small capacity. It was thought at the time that it might be possible to use a guard-ring condenser of which the capacity could be accurately determined and that thus a good value for " v" might have been obtained; at this stage the importance of the correction for the capacity of the coil was not fully realized and it was the discrepancy which was observed when the results of these experiments were compared with a simple theory which led to the fuller consideration of this correction which has been given in Part III. The experiments therefore are chiefly of interest as a test of the theory and as enabling us to see the consequences of the correction. Several measurements were made with a small air condenser consisting of 7 concentric brass cylinders each 45'4 cm. high and '75 mm, thick and of internal diameters 13'25, 9'90, 8-26, 6-92, 5-00, 3-40, and 1-60 cms. respectively. The capacity of this condenser making some allowance for the edges and for connecting wires was calculated at 5'5 metres. Another condenser consisted of eleven circular dises of brass of total capacity, as calculated from the dimensions, of 5 metres. A list of these various condensers is given below. (See p. 175.) Two other condensers were used, one consisting of tinfoil plates on glass, the other a paraffin paper condenser. The capacity of the former calculated from its dimensions is 47'5 K metres, K being the specific inductive capacity of glass. Taking K as 5 this comes to 237 metres. An attempt made however to determine by observation the capacity of this condenser gave as the value 190 metres, corresponding to the value 4 for K which is very low. The capacity of the paraffin paper condenser was - microfarad or 3000 metres. The capacities of these condensers were also determined by the ballistic method, but it must be remembered that with such small capacities accuracy cannot be expected and the values found are therefore only approximate. 174 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY The correction to be made to the simple theory involves as we have seen in Part III. the capacity of the two halves of the coil treated as two plates of a condenser. We may obtain a very rough estimate of this by treating the two sets of opposed turns as two dises separated by the glass plate and insulated covering of the wire. Now we have from the dimensions given in Part III. p. 153 the following data: the thickness of the glass is '27 cm. and of the gutta percha '1 cm.: taking the inductive capacity of glass as 7 and of gutta percha as 3, we have for the equivalent air thickness *27 -1 -- + 8 or '072 cm. 7 o Hence since the interior diameter is 27'5 cm. and the exterior 48-7 cm. wr x 76-2 x 21'2 S1 = 47r x 4 x 072 = 1400 cm. approximately. But the value of Si can best be found by the ballistic method. The two halves of the coil were charged like the two coats of a condenser to a potential difference of 60 volts and discharged through a ballistic galvanometer. A standard condenser of -01 microfarad was similarly charged, and the kicks compared. As a second experiment the galvanometer was shunted with the 1/9th shunt and a condenser of capacity '02 microfarad discharged. A number of concordant readings were obtained with the result that the capacity came out as '0018 microfarad or 16'2 metres for rapid charging. If the time of charging is prolonged the capacity rises apparently and could be got as high as 22 metres; this was due in part to the action of the containing box which behaves as a conductor for slow charging. We have taken then the value 16 metres as that to be applied in the corrections in the final experiment. We have seen that we mnay also require to know the capacity of one layer of the coil on the next. This it is difficult to determine with any approach to accuracy. In each layer there are about 30 turns of wire, its thickness being about '05 cm. The least distance apart of the surfaces of these wires is about '1 cm. while the distance between their centres is about '15. We may as a very rough approximation treat the two layers as two concentric cylinders 1'5 cm. (30 x '05) in height and '12 cm. apart. The mean diameter of these cylinders is 38 cm. Hence if K be the inductive capacity of the dielectric the capacity required is 7r x 38 x 1'5K 47r x '12 cm, or about 120K cm. Assuming K =3 for india rubber we get for the capacity of one layer on the next the value 3'6 metres. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 175 List of condensers used in the preliminary experiments with their capacities as directly measured ballistically or estimated from their dimensions. Capacity. Cylinder condenser, already described, 5'5 K0 metres. 11 plate disc condenser,,, 0 0 5 plate disc condenser, part of the above 20 A Leyden jar about 20'5 A flat sheet glass and tinfoil condenser about 1900 or as found by calculation 237, A large paraffin paper condenser by Muirhead, consisting of six 2 microfarad condensers arranged in series about 3000 After several preliminary photographs at various speeds and modes of connexion we took on 22nd July a careful series of spins with the fork adjusted exactly at 128 and with the 4 pattern of the disc extremely steady. The connections were made as in Figure 8. L being the self induction coil. C and C' the condensers arranged close together. S the spark gap in the dark box, and M the electrical machine. The point E was sometimes earthed. Connexion/:~~Fm 8ith t /C M FIG. 8. Connexion w p ith the machine was made through wooden penholders w, in order to always done, and it is this circumstance which was specially varied; the object being to test the influence of wooden connexions, for subsequent use; e.g. with a guard-ring condenser; where wooden connexion might preserve the potential uniform during slow charge but isolate the guard-ring during sudden discharge. The following are the circumstances of the chief plates taken this day. 176 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY PLATE No. I. On its first or outer circle several sparks were taken, with cylinder and disc condenser joined by wood w'. Second circle, several more with the same. Third circle, with the cylinder condenser alone, the dise condenser being disconnected. Fourth circle, both condensers in parallel, joined as in Figure 8, but by wire, not wood. The following is the micrometric analysis of this plate, the numbers being given rather fully as a specimen. It was the first plate carefully read. We do not quote the actual circle readings but the successive differences or lengths of the constituent halfoscillations; the last one was usually faint, and some were better marked than others. It will be seen that there were very few oscillations in each spark, because of the smallness of the condenser and the resistance of the circuit. It would not indeed have been surprising if the damping had affected the period perceptibly; but the only obvious effect is the lengthening out of the last swing by the high resistance of the decaying spark. Successive alternation intervals on fourth or inner circle of Plate No. 1 for different sparks:6~ 48', 6~ 18', 6~ 5.5', 7~ 6' 6~ 36', 6~ 15', 6~ 40', 6~ 54' 6~ 29', 6~ 41', 6~ 32', 7~ 6' (lower power object glass), 6~ 32', 6~ 35', 6~ 25', 7~ 11' (plate recentred), 6~ 32', 6~ 35', 6~ 26', 7~ 7' (repetition), 6~ 23', 6~ 33', 6~ 16', 7~ 2' (plate recentred), 6~ 32', 6~ 38', 6~ 34', 70 28' (apparatus reset). It is clear that the last or decaying half oscillation is unduly lengthened by reason of the high resistance of the dying spark, so, omitting it, we have as the average of a half oscillation for this circle 6~ 31'. Similarly omitting the last reading, which in nearly all cases is longer than the others, the average length of a half oscillation on the third circle is 5~ 19'; on the second circle 5~ 18'; and on the first or outer circle 5~ 19'. Since the plate was making 64 revolutions per second, this gives as the observed frequency: For the cylinder condenser connected by wood to the dise condenser 2170 per second. For the cylinder condenser alone...2170 For the cylinder and dise condenser properly connected... 1770 These numbers shew that the wooden connectors separating the condensers act as expected, at least in preventing combined discharges, and thus act effectively in isolating the machine terminals from the capacity discharged. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 177 It would be tedious to quote at full length the details of all the preliminary spins, and indeed all of the records have not yet been read. But such as seemed likely to be instructive were carefully examined, and a summary of them is given below. July 25. The l1-plate disc condenser arranged so that the machine charges it through a needle point, an inch or two distant, without direct contact. The needle point replaces the wooden connexion previously used. The following are the lengths of various half oscillations as recorded on a plate spinning 64 turns a second: 4~ 42', 4042', 4~47', 4041', 4 49', 4 41', 4~44', 4 38', 4 38', 4~ 41, 4~ 50', 4~ 48', 4~ 37', 4043', 4~ 38', 4~29', 4~19'. Average of these numbers 4~40'. Frequency deduced from the observation, 2470 per second. Same date. Cylinder condenser, similarly arranged. Frequency 2370. Same date. 5-plate disc condenser, similarly arranged, average 4~ 30'. Frequency 2580. July 30. Cylinder condenser arranged in a different part of the circuit, viz. each set of plates connected to one of the terminals of the two halves of the coil as in Figure 11, p. 181. Average reading 4~ 34'. Frequency 2560. On other circles of the same plate, condenser detached and middle terminals of coil left insulated, so that the only capacity was that of the two halves of the coil: Readings 2~45', 2 54', 2~31', 2~4', 2~4', 2~2'. Average 2~ 30'. Frequency 4630. July 31. Spins taken at the 12-pattern speed (i.e. 21~ revs. per sec.) with the large Muirhead condenser in simple circuit with the whole coil. The outer circle was taken with the condenser attached to middle screws, as on July 30; for the others it was connected in the ordinary way. But no correction for coil or other capacity should be needed with this great condenser. Average reading 30~ 30', the speed not perfectly steady. Frequency deduced 126 per second. August 1. Cylinder and disc condenser in parallel. Average of readings for spark alternations on outer circle 6~ 32'... for another set ditto....6~ 38' ",, for spark on second circle.. 6~ 30',,," for another set ditto.... 6~ 29' 5",,,, for spark on third circle...6~ 32'... for another set third circle.. 6~ 29' ". " for spark on fourth circle...6~ 30' General average for this plate...... 6~ 31' 5" Frequency deduced, 1766. The speed for the outer circle was steadiest. VOL. XVIII. 23 178 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Same date. Cylinder and disc again, with coil connexions reversed, otherwise everything the same. Average of readings off all alternations on outer circle 60 13' 5",,"."~, " ". second circle 6~ 15' 5" ",,~,,",,,, ""third circle 6~ 1 ", ~.......""~ fourth circle 6~ 17' Speed for fourth circle was steadiest; weighted average 6~ 15'. Frequency deduced 1830. Same date. Leyden jar added to cylinder and dise condensers. General average of readings 9~ 48'. Frequency deduced 1180. August 2. Took a spin with the large Muirhead condenser connected not to the entire coil, but only one portion of it, the portion called B. Average of readings (one spark on each circle) at 4-pattern speed was 49~ 40', but the speed was not over steady, and with these heavy sparks the setting of the microscope on a leading feature of each alternation is less definite. Frequency deduced 232 or 233. Same date. Same condenser joined to coil A. Average of readings 50~ 10'. Or omitting the last or drawn-out alternation, and taking the most probable average from the steadiest circle: Estimated reading 49~ 42'. Hence frequency deduced, average 230; most probable 232. Same date. Muirhead condenser joined to complete coil, one spark attempted on each circle, but one apparently missed fire. Average of whole set (with 8-pattern speed) 45~ 15'; or frequency 128. Same date. Muirhead with dise and cylinder condenser added. Speed deduced 127 and 124. August 3. Sheet glass condenser (glass as dielectric). Composed of 8 sheets of glass and 9 of tinfoil. Each tinfoil 38'1 x 54'2 centim. Combined thickness of the 8 plates 2'2 centim. Plate running at 6-pattern speed. Average of readings 17~ 8'. Frequency deduced 450. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 179 Same date. Same condenser through B coil only; frequency 820. This is a sufficient account of the preliminary experiments, whose object was partly to gain experience and partly to find out what sort of condenser was best to use. Decided that a large simple air condenser was advisable, without complication of guard-ring or anything, but with edges that could be allowed for by calculation and with plates large enough to make the correction of relatively small amount. In order to compare these preliminary results with theory it seemed best to calculate the theoretical frequencies, using the formula X2LS = k where k bas the proper value for each combination as given in Table V. in Part III. We thus obtain the following results: TABLE VI. Both coils A and B being used. Capacity in Frequency Frequency Date Condenser centimetres calculated observed July 25 Five-plate disc 2 2590* 2580 July 25 Eleven-plate dis 5 2400* 2403 July 25 2470 July 22 ^ T -2060 2160 July 25 Cyl270 2370 July 22 Cylinder and disc 10'5 1770 Aug. 1 in parallel 1766 Augo. 1 Cylinder, disc and 305 Leyden Jar 190 501 Aug. 3 Sheet Glass 230 51 450 L937 447 July 31 126 Aug. 2 Paraffin condenser 3000 128 127,,,, 128 In the observations marked thus * the calculations have been made on the assumption that the connexions were as in Fig. 5. 23-2 180 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY TABLE VII. using only one coil. Frequency Frequency Date Condenser Capacity calculated observed 232 Aug. 3 Paraffin condenser 3000 233 230 Glass condenser 190 926 820 237 827 Considering the nature of the experiments the agreement may be considered satisfactory. The capacity taken for the cylinder condenser is probably too high; if it were assumed as 5'1 instead of 5'5 the agreement in all the experiments in which it was used would be improved. It is also clear that the value 190 taken for the capacity of the glass condenser is too small; only one determination of this was made, and there may have been some leak which reduced the capacity as measured; an earlier attempt at measuring the capacity was a failure from this cause. It will be noticed that the corrections have been applied as though on July 24 and 25 the coil connexions were those shewn in Figure 5, while on the other days they were those of Figure 3. There is no evidence in the note-book that this was the case; at the time these experiments were made the importance of the direction in which the current traversed the coils was not realized. The results of three of the preliminary experiments are not recorded in the table. On August 1, with the cylinder and disc condensers in use, the connexions were as shewn in Figure 9, and the frequency was 1766; this result is given. Spark gap Earth FIG. 9. The connexions were then altered so that the condenser was to the terminals B DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 181 and C as in Fig. 10. A and D being connected together, the frequency rose to 1830; this result we have not been able to explain as satisfactorily as we could have wished. Spark Earth FIG. 10. The following may however have been the cause. In the figure A and D are terminals connected with the outer turns of the coils, B and C those connected with the inner. Now the capacity of the outer turns is greater than that of the inner, while at the same time the portions of the coils which are nearest to the condenser, and in which therefore the potential difference is the greater, will have most effect on the result. We have however taken an average value of S', 16/30, in calculating the correction. It may be that this average is right for Fig. 9, but that for Fig. 10 it ought to be reduced, for the actual value of S' near C is only 3/5 of that near A. If we assumed = 3 x 16/5 =10 say, or S' =10/30, we should obtain as the frequency the value 1860 which agrees closely with that given by experiment. Again on July 30 the cylinder condenser was connected to the coil as in Fig. 11. The observed frequency was 2560. s ' Spark D Earth FIG. 11. The calculated frequency for this case, assuming the corrections already given, is 2060, or if we assume the connexions to have been as in Case ii., 2270; in either case the result is much too low. It will be noticed however that in Fig. 11 the condenser is connected to the terminals B and C, i.e. to the inner terminals of the coil as in Fig. 10, and we have just seen that the assumption that the effective capacity of the coil is 10 metres when this is the case serves to reconcile theory and experiment. It becomes of interest then to evaluate the frequency, assuming S~ equal to 10 and S' to 10/30. 182 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY The resulting value for the frequency is 2470 which is still below that found by experiment, viz. 2560, but it has already appeared that the capacity assumed for the cylinder condenser, viz. 5'5 metres, is too high. The assumption that the value was 51 which (p. 180) is required to reconcile with theory the experiments recorded in Table VI. would also bring the results of this case into greater harmony. On the same date (July 30) and immediately after the above experiment, the condenser was reinoved and oscillations taken with the coil alone. In this case, assuming the theory developed in Part III., we have S1Lx2= - (1- ) = 238, or if we suppose the capacity uniformly distributed along the coil, SiLx2 =:3. On substituting for S1 the value 16 metres, and for L 4-63 secohms, we find for the frequency the values 3830 and 4300 respectively; the experimental result is 4630. In this case theory and experiment would be reconciled by the assumption that the capacity of the coil was 10 metres instead of 16, and this value fits, as we have seen, the experiments just discussed in which the condenser was used. If we adopt the first of the two formulae and take S, = 10 we find the theoretical frequency is 4820,, while the second formula based on the assumption of a uniform distribution of capacity leads to the value 5360. The observed value was 4630 which agrees best with the first of these two theoretical values, being rather below it. It will be observed however from the record on p. 177 that the experimental results are very variable. Thus these three sets of experiments in which the condenser was connected to the terminals B, C of the coil will be reconciled with theory by the assumption that when the experiment is so conducted that there is a large potential difference between the inner windings of the coil for each of which the electrostatic capacity is smaller than for windings near the outer edge, the effective capacity of the coil S1 of the formula is about 10 metres, possibly rather over 10 metres. These results are given in Table VII. (a). TABLE VII. (a). Both coils A and B being used, but the coil capacity taken as 10 metres instead of 16. Date'] Condenser CapaFrequency Frequency Date Condenser Capacity calculated observed Aug. 1 Cylinder and dise 10-5 1860 1830 July 30 Cylinder 5-5 2470 2560 1,, i Coil only 4820 4630 DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 183 The general concordance of the experimental results with theory appears to shew that the capacity of the coil, layer upon layer, has no rnarked effect; if it be taken into I S/' account a correcting factor of the form 1 - 6, must be introduced, where S,' may possibly be 3 or 4 metres. This would reduce the frequency in the case of the 2-metre condenser by about 1/30th, bringing it to 2510. For S=5 the correction would be 1/75. A sufficient account has now been given of these preliminary experiments; as a resuilt we were led to construct an air condenser of considerable capacity which we could calculate with some degree of accuracy. PART V. THE AIn CONDENSER. We proceeded to make an air condenser of eleven flat plate glass slabs very carefully covered with tinfoil so as to offer a perfectly smooth metallic plane on both sides; folding the tinfoil round the edges so that they were practically slabs of metal. The plates were nearly square, and their size was measured individually, giving as the average result 59716 cm. long by 59'614 cm. broad. The boxwood scale which had been used was then compared with a brass standard metre, which we know to be accurate at 0~; and 60 centims. on it was found to be 21 millimetre longer than 60 centims. of the standard at 18~. The expansion of the brass would make the length of the standard too long by *2 millim., so the total correction is *025 centim. Hence the corrected size of the condenser plates is 59'74 x 59'64 square centim. The thickness of the eleven plates clean and finished and lying close together was measured in eight different places and found to average 3157 inches, or when clamped together tightly 3,116 inches, so the thickness of each plate was -284 inch or *721 centim. We then cut a number of plate glass distance pieces, measured them carefully, and arranged them in the 10 spaces between the plates, 5 in each space, like the pips on a' card. Set the plates on end on a pair of ebonite wedges and clamped them in special wooden frames, making careful contact with each plate by a thin wire lying along the middle of an edge. Connected alternate plates together and proceeded to charge. But found that the glass distance pieces leaked in the most surprising manner. Four of them were sufficient to prevent the machine from charging anything. Tested them separately and found they leaked like wood, giving a distinct brush discharge from their corners. 184 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Hence replaced them by pieces of ebonite all cut out of the same sheet; each piece 7 millimetres square; 52 pieces end to end, measured in vernier callipers, occupied 10'24 inches. So the thickness of each distance piece was *1989 inch or -5001 centim. With 5 of these between each plate, one at each corner and one in the middle, the plates were once more clamped up, connexions carefully made, and an experiment begun. The plates stood vertically on a pair of sharp-edged ebonite wedges, at a height of 1 inch above the floor of the frame, which was tinfoiled to make it definite. The sides and top were at first open, so that the edges of the plates were then free; but afterwards in order to keep the inside air dry for all the best experiments, the box was panelled in. The distances of the wood panelling from the plates were as follows: From edge of plates to wall of case......... 5'15 centim. ". ".. roof.........8 8.".. ". floor..........25 A simple wooden X formed the front and back at a distance from the outer flat of the plates, 4'0 on one face and 3'2 on the other. ESTIMATE OF CAPACITY OF CONDENSER. The method of correcting for the edges of a thin plate is given in Maxwell, vol. I. ~ 293. A term has to be added to the linear dimensions as if an extra strip of a certain breadth were put on all round a uniformly charged plate. This extra breadth, on account of the extra density at terininations of thin parallel planes, is -oge 2, 'n. where "b" is the distance between the plates. But the plates are thick and square edged, and a further correction has to be made for their thickness; Maxwell's further 7r/3 correction for thickness /, cos b, assumes the edges to be rounded and is therefore inapplicable, but acoustic analogies suggest the addition to the dimensions of each plate of a quarter of its thickness, to represent the effect of the edges themselves. (Cf. Rayleigh, Sound, vol. iI. ~~ 307 and 314.) The total correction is thus 22b + '25/B = '11 + 18 = '29 centim. to be added all round. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 185 Thus the capacity of the condenser proper is 10 (59-74 + -58) x (59-64 + -58) 47r x '5001 and this reduces to 5779 centimetres. To this has to be added a term for the proximity of the case to the edges, especially for the proximity of the floor; the floor correction is 5 x 60 x 72 -4 x 2 — = 6'9 centires. 4 x 2'5 The walls and roof together amount to 8'7, or altogether 15 6 centims. Next, the ebonite distance pieces must be allowed for. They are each ~ square centim. in area, and there are 50 of them; their specific inductive capacity may be taken as 3, so the extra capacity due to them is 8 centims. Adding all these we get for the condenser capacity, 5803 centims. Then there is a correction for the charged portion of the wires leading from the condenser and coil to the spark knobs. This was approximately 7'3 metres long and one millimetre thick, with a span of 1~ metres between it and the walls. Its capacity 730 was therefore = 50 centims. 2 loge 150 Thus the whole electrostatic capacity under charge was 5853 centimetres. Connexions were as in Figure 12. The machine was usually connected only across an air space by needle points so as to take no part in the discharge. '". I FIG. 12, VOL. XVIII. 24 186 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY PART VI. FINAL EXPERIMENTS. With the air condenser described in Part V., a number of spark photographs on Mr Swan's 4-inch square plates were taken with the plate revolving 64 times a second. From seven to nine circles were attempted on these plates with three or four sparks on each circle. Tin plates which are lettered from A to Z were afterwards read with great care by Mr J. W. Capstick who writes: "The measurements will be found to be within a very few minutes of the correct reading. In one case I accidentally went over a spark twice, and though I was then at the end of six hours' almost continuous work at them, and the spark was an exceptionally indefinite one, the greatest divergence in the readings was only 3 or 4 minutes. "The plates are very much better than any I had done previously, and the setting of the microscope was generally a simple matter. The sparks were in general so definite and regular that I did not think it necessary to make drawings of them." [This had been done with some of the earlier plates.] Mr Capstick remarks-as will be seen from the Tables-that there is some irregularity in the sparks, and that, unless it is desired to study this, greater accuracy of reading is hardly necessary. The analysis of this long series of plates has been a work of time; we give below the results of a study of all the plates from G to U. In the earlier plates, marked A to F, the work was in some respects of a preliminary character; there was no plate marked Q. In the spin for plate P the coils were in multiple arc, and the coefficient of self-induction for this arrangement was not determined. We give as an example the actual record for two of the circles on plate U*. This illustrates the method of dealing with the results. SPARK RECORD ON PLATE U. Coil B only used. Outer circle. Actual readings. Differences. Averages. Spark (1) 194~ 0' 186 49 140 36' 179 24 (14 45) (172 4) 14 25 1 4 164 59 (14 24) 157 40 * The record for this plate happened to cone first in one of the note-books in which results were recorded. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 187 Actual readings. Differences. Averages. Spark (2) 155~ 14' 14~ 21' / (147 46) 14 14o0 53 140 53 ^ ^ 14 12 14o 22' 133 34 14 33 126 20 Spark (3) 112~ 40' 14 140 11' 106 4 or 105 5 (14 43) 97 57 14 17 14~ 18' 97 57 4 14 9 90 48 14 12 83 48 76 36 General mean for this outer circle 14~ 24'. Second circle. Spark (1) 136~ 23' 129 4 14~ 12' 122 11 14 11 140 14' Spark (2) 118 29 14 14' 114 53 14 19 111 6 107 52 14~ 1S8' 104 11 14 10 14~ 18' 96 56 14 28 89 4 3 Spark (3) 175~ 24' 168 1 14~ 18' (161 6) 14 0 14~ 13' 154 2 14 21 146 25 Spark (4) blurred. Here there was some simple overlapping, giving no difficulty in sorting out. The general average for this circle is 14~ 14'. 24-2 188 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY TABLE VIII. PLATE U. Coil B. Circle... I. II. III. IV. V. VI. VII. VIII. IX. Spark 1 14~ 36' 14~ 12' 14~ 33' 14~ 17' 14~ 21' 14~ 34' 14~ 39' 14~ 46' 14~ 56' 14 45 14 11 14 25 14 6 14 12 14 26 14 40 14 18 14 41 14 25 14 19 14 19 14 25 14 24 14 43 14 24 14 37 14 24 14 56 2 14 21 14 18 14 31 14 9 14 12 14 26 14 45 14 11 14 16 14 12 14 10 14 18 14 5 14 10 14 14 14 23 14 14 14 12 14 33 14 28 14 22 14 15 14 11 14 24 14 36 14 11 14 9 14 49 15 5 3 14 11 14 18 14 29 14 21 14 14 14 14 15 3 14 17 14 39 14 43 14 0 14 19 14 20 14 20 14 5 14 29 14 20 14 10 14 17 14 21 14 16 14 24 14 16 14 23 14 16 14 9 14 14 14 21 14 12 14 7 4 14 33 14 19 14 43 14 23 14 26 14 9 14 38 14 20 14 36 14 25 14 39 14 34 -Mean for circle 14 24' 140 14' 140 27 14~ 19' 140 1 8' 14~ 261'14~ 31 14~ 22 14~ 29' circle Mean of central ientragl 14~ 16' 14~ 8' 14~ 21' 1414' 14 12' 14~ 18' 14 28' 1.4~ 19' 14~17' swings for each circle General mean from plate 14~ 23'. Mean from central swings 14~ 17'. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 189 Thus in Table VIII. will be found the actual length in degrees and minutes of all the oscillations on the plate. The Roman numerals at the head of the columns indicate the circles on which the sparks are to be found; the record for each spark is shewn separately. The mean length of oscillation from the 99 sparks here recorded is 14~ 23'; the range of the readings is rather over 1~; the means for the various circles are given in the Table; they range over 17'. It is clear however that the oscillations in any one spark are not of equal length. As a rule the first oscillation is a long one. This is followed by one or more of shorter period while, as the spark dies away, the oscillations again lengthen; the cause of this has been discussed in Part IV. The lengthening of the latter oscillations is more plainly shewn on some of the other plates. If we omit the longer oscillations, and take only the more regular central swings on plate U, we get the following series of numbers, in which the 14~ is omitted for brevity. TABLE IX. Circle... I. II. III. IV. V. VI. VII. VIII. IX. 25 11 25 6 12 26 40 18 41 12 10 18 10 14 23 24 12 17 O 19 5 11 5 36 14 9 9 20 16 38 29 11 10 26 9 14 20 16 23 20 Averages 16' 8 21' 14' 12' 18' 28' 19' 17' These lead to an average length of oscillation of 14 17'. In taking the average in this manner we have given equal weight to each observation. Now the record of this plate taken at the time of the observation is PLATE U. Air condenser in circuit with B coil. Machine not in circuit but arranged to charge it through a pair of needle points from a distance so that its capacity should not interfere. 190 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY On the outer circle were taken 4 sparks, speed steady. second,,,, 4,, fair. third,. " 3,, steady. fourth,,, 4,, fair. fifth. ", 4,, fair. sixth ",, 4,, quite steady. seventh,,, 4, " steady.,, eighth,,,, 4,,, steady on average. ninth - -, (The number of sparks taken is not with perfect certainty correct, because there was sometimes a difficulty in hearing them.) The remarks as to the speed were noted at the time according as the stroboscopic pattern had successfully been held still or not while the circle was being taken. If attention is paid to these speed remarks it would seem that circles I., III., VI., VII. and VIII. should have most weight attached to them. The averages for these circles are 16', 21', 18', 28', 19', for the middle swings, and their mean is 14~ 20'. The complete averages for these steadiest circles are 24', 27', 26', 31', 22', and the mean of these is 14~ 26'. It would thus appear that the best value for the wave length for this plate is 14 20'; while if all the sparks be included which lie on the circles retained, the number is increased by 6'; if all the circles are included, each of these numbers is reduced by 3'. We may claim then to know the length of the oscillation on this plate to about 5', i.e. to about 6 ~/,. The frequency corresponding to 14~ 20' is 64 x 360/14-33 or 1608. PLATE S. Another series of wave lengths as recorded on plate S, in which coil A only was used, is given in Table X. The notes relating to this plate are as follows. On this plate the sparks photographed were taken from the air condenser through the A coil only. Machine charging via needle points. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF lv." 191 TABLE X. PLATE S. SUMMARY OF READINGS. Coil A. Circle... I. II. III. IV. V. VI. VII. VIII. 14~ 31' 14~ 36' 14~ 37' 14~ 28' 14~ 31' 14~ 53' (blurred) (all overlap) 14 11 14 20 14 29 14 3 14 2 15~ 6' 14 24 14 28 14 28 14 16 14 43 14 33 14 26 14 49 14 39 14 11 14 52 14 32 14 15 14 19 14 41 14 45 14 35 14 39 14 56 13 57 14 21 14 43 13 58 14 4 14 19 14 22 14 11 14 22 14 17 14 9 14 11 14 2 14 14 14 25 14 28 14 51 14 48 14 2 14 32 ( ) 14 35 14 40 14 13 14 15 14 33 14 18 14 19 14 34 14 44 14 26 14 47 14 47 14 38 14 45 14 30 14 19 14 17 14 50 14 42 General 140 22' 140 28' 140 28' 14~ 26' 140 26' 140 38' 140 27' Mean Mean of central 14~1 11' 4 22 14~ 23' 14~ 15' 14~ 6' 14~ 26' 14~ 24' waves General mean 14~ 28'. Mean for centre swings 14~ 19'. 192 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Four sparks were taken on each circle. Circle I. speed inoderately steady. Circle V. speed fair. II.,, fair. VI.,, steady on average. III., quite steady for 3 sparks. VII., slightly backing. IV.,, steady. VIII.,, fair. To save space only the differences are quoted. All the differences read are included. Sometimes overlapping prevented any reading being attempted. The general mean from Table X. is 14~ 28', while the central swings give 14 19'. These means include all the circles. The range of the mean readings is about the same as for plate U, and the frequency calculated for the central swings works out to 1610 oscillations per second. TABLE XI. PLATE R. Complete Coil A + B. Circle... I. I. III. IV. V. VI., VII. Spark 1 26~ 52' 26~ 53' 26~ 34' 26~ 43' 26~ 47' 26 16 26 22 26 11 26 19 26 23 26 18 26 46 26 44 26 55 26 51 26 52 27 1 26 4 26 18 26 15 26 20 26 23 26 51 27 7 3 26 59 26 51 26 55 26 46 26 4 26 12 26 19 I1 26 50 26 37 O 4 i 1 26 38 26 4 General 260 33' 26~ 30' 26~ 32' 26" 37' 260 37' Mean Central 26~ 12' 26~ 15' 26~ 13' 26~ 14' 26~ 21' Mean General mean of plate 26~34'. Mean from central series 26~ 15'. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 193 As an example of a plate in which the whole coil was used the record for plate R is given in Table XI. It will be seen that the means for the separate circles differ by 7' in the case in which all the sparks are considered, and by 9' when only the central swings are dealt with; the difference between the two means is 19'. If we take as the length of wave 26 15', the frequency is 64 x 360/26'25 or 878 oscillations per second. It is not necessary to give the results of the other plates in such full detail. The following Table summarizes them sufficiently. In each case the central sparks only are included. TABLE XII. Complete Coil A + B. Plate K L O R T Number of sparks 11 13 9 15 22 Length of wave 26~ 14' 26~ 9' 26~ 16' 26~ 15' 26~ 5' Mean length of wave 26~ 11'. Coil A. Plate G H M N S Number of sparks 20 28 7 27 33 Length of wave 14~ 14' 14~ 32' 14~ 25' 14~ 18' 14~ 19' Mean length of wave 140 20'. VOL. XVIII. 25 194 MESSRS GLAZEBROOK AND LODGE, EXPERIMENTS ON THE OSCILLATORY Coil B. Plate J U Number of sparks 28 37 Length of wave 14~ 26' 14~ 20' Mean length of wave 14~ 22'. From these we find as the mean length of the wave when the complete coil A+B is used 26 11'. With regard to the observations made with the coils A and B in circuit separately, it will be observed that plates H and J give higher results than the others. Now there is a note in the book that for these two series the outer plates of the condenser were earthed; they were taken therefore under different conditions to the others; if they be omitted we have as the mean wave length for plate A 14~18', and for B 14 20'; if we include plates H and J, the mean for A is 14~ 20' and for B 14 22'. The corresponding frequencies are, excluding plates H and J, for coil (A + B) 880 per second, for coil A 1611 per second, for coil B 1607 per second. If we take the whole series of sparks for A and B we get respectively for A 1607, and for B 1603. While the frequencies given by plates H and J are for A 1583, and for B 1595. It is hardly necessary to work out the frequencies for each plate. For the complete coil A +B the greatest variation from the mean is four parts in one thousand. We may now determine from these spark records the value for " v." We have the formula v= 27r. frequency x k/ k L where k is the constant, the values of which are given in Table V., occurring in the DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF "v." 195 formula LS\ = k. In the case in which the two coils were used there is no difficulty in deciding on the value of k. The formula for X is that given on p. 159 (D), LSX2' i 1 -J-l + 4 _ 1 Xt=l-= 1- 2(3 S - + 40..., and hence k = 916. If only one coil is used two cases may arise; if the lower coil is completely insulated we have the case dealt with in Figure 4; the corresponding formula as far as terms in S1/S are concerned is (F) on p. 166, viz.: LSX2=1- 12 (i ~/ ) 1 and the value of k resulting from this is '978. If on the other hand the lower coil is not insulated the correction necessary will be that indicated in (G), p. 167, and the resulting value of k will be the same as that for the two coils, viz. '916. As far as we know the coil was usually insulated; at any rate it was not intentionally connected to earth except for the two plates H and J. But there is another complication in this case. We assume in this case that the value of L is that for either half the coil; now this assumes that there is no current in the unused coil; but in consequence of the electrostatic induction there is a current in the unused coil. This current will be of the order x S'S if x is the current from the main condenser, and its effect will therefore alter the coefficient of self-induction L of the upper coil by an amount proportional to MS'/S or about 3M/120. Now the value of L1 is about 1'4, and of M about '91. Hence the value of L1 in the experiments with the single coil is uncertain to one part in one hundred and seventy. Omitting however this correction we get the following Table of values. TABLE XIII. Coil used k Frequency L S v Observations A + B '916 880 4'636 58-53 3-009 x 1010 Mean from seventy sparks A ( 978 1611 1 409 58-53 2-939 x 1010 Unused coil assumed insulated (916 1611 1-409 58-53 3 037 x 100,,,, uninsulated ~916 1583 1 409 58-53 2-984 x 1010 Plate H, coil uninsulated B -978 1607 1'393 58-53 2'922 x 1010 Unusedcoilassumedinsulated ~916 1607 1'393 58'53 3'020 x 10~,,,,,, uninsulated ~916 1595 1-393 58'53 2'990 x 1010 Plate J, coil uninsulated 25-2 196 EXPERIMENTS ON THE OSCILLATORY DISCHARGE, ETC. In the fourth and seventh lines of this Table we give the velocity as obtained from plates H and J. We know that in this case the effective coil and one plate of the condenser was earthed originally, and we have therefore used the value of k calculated on the assumption that the free coil was earthed throughout. It will be seen that the resulting values of "v" and that obtained from the experiments with the full coil are in close agreement, being respectively 2-98 x 1010, 2-99 x 1010 and 3-01 x 1010. If we take the other observations for coils A and B, excluding plates H and J, the results are not quite so satisfactory. The assumption that the free coil was insulated leads to the values 2-94 x 1010 and 2-92 x 1010, given in lines 2 and 5 of the Table; on the assumption that it was earthed we find from the same series of experiments the values 3'04 x 1010 and 3-02 x 1010 respectively, given in lines 3 and 6. The truth would appear to lie between the two. If we take the experiments with the complete coil A + B in series, we can determine the corrections with greater accuracy, and we find as the result v = 3'009 x 0101 centimetres per second, while since the corrections can be calculated with more exactness in this case, we attach far greater importance to the result. We do not however look upon the paper as one describing a very exact method of determining " v," but rather as a study in the oscillatory discharge of a condenser which incidentally leads to a determination of "v" by a novel method. VIII. The Geometry of Kepler cnd Newton. By Dr C. TAYLOR, Master of St John's College. [Received 25 August, 1899.] THIS paper consists of two parts (A) and (B), treating respectively of some things in the geometry of Kepler and some in the geometry of Newton, the finisher, in pure mathematics as in physics, of so much of his brilliant predecessor's work. In Fontenelle's Panegyrick of Newton, published in French and English under the title, The Life of Sir Isaac Newton with an Account of his Writings (London, 1728), the third paragraph begins thus, "In studying Mathematicks, he employ'd his Thoughts very little upon Euclid, as judging him too plain and easy to take up any part of his time; he understood him almost before he had read him, and by only casting his eye upon the Subject of a Proposition, was able to give the Demonstration. He launch'd at once into such books as the Geometry of Des Cartes and the Opticks of Kepler. So that we may justly apply to him what Lucan has said of the Nile, whose Springs were unknown to the Antients, That it was not granted to Mankind to see the Nile in a small Stream." (A) KEPLER. Kepler's new and modern doctrine of the Cone and its sections, which historians of mathematics have ascribed to a later generation, was propounded in cap. Iv. 4 of his Ad Vitellionen Paralipomena, quibus Astronomice Pars Optica traditur, a work published originally in 1604, a century before Newton's Opticks (1704), and edited with notes forty years ago by Dr Ch. Frisch in vol. HI. of his Joannis Kepleri Astronomi Opera Omnia in eight volumes. The passage containing the new doctrine is given below line for line, with the addition of numbers for reference, from the edition of 1604: 198 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. PAGE 92. 4. De Coni feéZionibus. Coni varii funt, re&anguli, acutanguli, obtufanguli: item Coni reai feu regulares, & Scaleni feu irregulares aut compreffi: de quibus vide Apollonium & Eutocium in commentariis. Omnium promifcuè fediones in quinq; cadunt fpecies. Etenim 25 linea in fuperficie coni per feéione conftituta aut eft reda, aut circulus, aut Parabole aut Hyperbole aut Ellipfis. Inter has lineas hic eft ordo caufa proprietatis fuae: & analogicè magis quàm Geometricè loquendo: quod à linea redca per hyperbolas infinitas in Parabolen, inde per Ellipfes infinitas in circulum 30 eft tranfitus. Etenim omnium Hyperbolarum obtufiffima eft linea reda, acutiffima Parabole: fic omnium Ellipfium acutiffima eft parabole, obtufiffima Circulus. Parabole igitur habet ex altera parte duas naturâ infinitas, Hyperbolen & Relam, ex altera duas finitas, & in fe redeuntes, Ellipfin & circulum. Ipfa PAGE 93. loco medio media natura fe habet. Infinita enim & ipfa eft, fed finitionem ex altera parte affect, quo magis enim producitur, hoc magis fit fibiipfi parallelos, & brachia, vt ita dicam, non vt Hyperbole, expandit, fed contrahit ab infiniti complexu, fem5 per plus quidem compleétens, at femper minus appetens: cum Hyperbole, quô plus a&u inter brachia comple&itur, hoc plus etiam appetat. Sunt igitur oppofiti termini, circulus & reéla, illic pura eft curuitas, hic pura reditudo. Hyperbole, Parabole, Ellipfis, interieftle, & redo & curuo participant; parabole ex aequo, 1o Hyperbole plus de reditudine, Ellipfis plus de curuitate. Propterea Hyperbole quo longiùs producitur, hoc magis reé&e feu Afymptoto fuæe fit fimilis. Ellipfis quo longiùs vltra medium continuatur, hoc magis circularitatem affeéat, tandemque coit iterum fecum ipfa: Parabole loco medio, femper curuior eft HyI5 perbola, fi equalibus interftitiis producantur, femperque redior Ellipfi. Cumque vt circulus & reéa extrema claudunt, fic Parabole teneat medium: ita etiam vt redca omnes fimiles, itemque & circuli omnes, fic funt & parabole omnes fimiles; folaque quantitate differunt. 20 Sunt autem apud has lineas aliqua pundta praecipue confiderationis, quae definitionem certam habent, nomen nullum, nifi pro nomine definitionem aut proprietatem aliquam vfurpes. Ab iis enim pundis redsæ edu&ée ad contingentes feéionem, punaaq; contaduum, conftituunt æequales angulos iis, qui fiunt; 25 fi punda oppofita cum iisdem punais contaduum connedantur. Nos lucis causâ, & oculis in Mechanicam intentis ea punda Focos appellabimus. Centra dixiffemus, quia funt in axibus fedionum, nifi in Hyperbola & Ellipfi conici authores aliud pundrum centri nomine appellarent. Focus igitur in circulo vnus 30 eft A. isque idem qui & centrum: in Ellipfi foci duo funt BC. æequaliter à centro figure remoti & plus in acutiore. In Parabole DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 199 vnus D eft intra feEtionem, alter vel extra vel intra fetionem in axe fingendus eft infinito interuallo à priore remotus, adeô vt eduaa HG vel IG ex illo caeco foco in quodcunque punurum fetionis G. fit axi DK parallelos. In Hyperbola focus externus 35 f PAGE 94. // dl b F interno E tantô eft propior, quanto eft Hyperbole obtufior. Et qui externus eft alteri feétionum oppofitarum, is alteri eft internus & contra. Sequitur ergo per analogiam, vt in reaa linea vterque focus (ita loquimur de reCca, fine vfu, tantum ad analogiam complen- 5 dam) coincidat in ipfam reEtam: fitque vnus vt in circulo. In circulo igitur focus in ipfo centro eft, longiffimè recedens à circumferentia proxima, in Ellipfi iam minus recedit, & in parabole multo minus, tandem in re5ta focus minimum ab ipfa recedit, hoc eft, in ipfam incidit. Sic itaque in terminis, Circulo & re- Io é5a, coëunt.foci, illic longiffimè diftat, hic plan incidit focus in lineam. In media Parabole infinito interuallo diftant, in Ellipfi & Hyperbole lateralib. bini aau foci, fpatio dimenfo distant; in Ellipfi alter etiam intra eft, in Hyperbole alter extra. Vndique PAGE 95. funt rationes oppofitæ. Linea MN quæe focum in axe metatur, perpendiculariter in axem infiihens, dicatur nobis chorda, & quæe altitudinem oftendit foci à proxima parte feétionis à vertice, pars nempe axis BR. 5 200 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. vel DK. vel E. S. dicatur Sagitta vel axis. Igitur in circulo fagitta aequat femichorda, in Ellipfi maior eft femichorda BF. - fagitta BR. maior etiam fagitta BR. quàm dimidium BP femichordæe feu chord quarta pars. In Parabole, quod Vitellio demonitra10 uit, fagitta DK præcisè æequat quartam chordæe MN. hoc eft D N eit dupla ad DK. In Hyperbole EQ plus eft, quàm dupla ipfius ES. fc. minor eft fagitta ES. s quarta chordæe EQ. & femper minor, atque minor per omnes proportiones, donec euanefcat in reaa, vbi foco in lineam ipfam incumbente, altitudo 15 foci feu fagitta euanefcit, & fimul chorda infinita efficit, coincidens fc. cum arcu fuo, abufiuè fic dito, cùm reta linea fit. Oportet enim nobis feruire voces Geometricas analogiæ: plurimùm namque amo analogias, fideliffimos meos magiftros, omnium nature arcanorum confcios: in Geometria præecipuè fufpicien2o dos, dum infinitos cafus interieélos intra fua extrema, mediumque, quantumuis abfurdis locutionibus concludunt, totamque rei alicuius effentiam luculenter ponunt ob oculos. Quin etiam in defcriptione feaionum analogia plurimùm me iuuit. Etenim ex 5I. & 52. tertii Apollonii defcriptio Hyper25 a boles & Ellipfeos efficitur facilima; potefcque / vel filo perfici. Pofitis enim focis, & inter eos ver/ tice C. figantur acus in focis A. B. anneélatur ad...... acum A filum longitudine AC. ad B. filum longi)-.' \'"'' tudine BC. Prolongetur vtrumque filum æequali30 bus additionibus, vt fi duplex filum digitis comprehendas, iisque à C difcedentibus, bina fila paulatim dimittas, alteraque manu fignes iter anguli, quem vtrumque filum facit apud digitos, ea defignatio erit hyperbole. Facilius Ellipfis defcribitur. Foci fint AB. vertex C. Fige acus firmas in A.B. vtramque filo ample&tere, fimplici amplexu, vt inter AB filum non PAGE 96. a interfit. Fili longitudo fit AC duplicata, & capita fili nodo fint connexa. Infere iam Graphium D in eundem fili complexum cum AB. & tenfo filo, quantum id patitur, circa AB circumduc lineam, hec Ellipfis erit. 5 b Cùm haec tam facilis effet defcriptio, non indigens od.. perofis illis circinis, quibus aliqui cudendis admirationem hominum venantur; diu dolui, non poffe fic etiam Parabolen defcribi. Tandem analogia mofitrauit, (& Geometrica comprobat) non multô O 1' a operofiùs & hanc defignare. Proponatur A focus, C vertex, vt AC fit axis; is continuetur in partes A. in infinitum vfq;, aut quousq; Parabolen placuerit defcribere. Placeat vfq; in E. Acus ergo in A figatur, ab ea fit nexum filù longitudiT5 ne AC. CE. Teneas manu altera caput alter fili E. altera graphium, cu filo extende vfq; in C. g Sit etiam ad CE. erecta perpendiculariter EF. DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 201 igitur graphio C & manu altera E difcede equalibus interuallis à linea AE. fic vt manus altera & fili caput femper in EF maneat, filumque DG femper ipfi AE parallelon; via CD. quam Graphio 2o fignaueris, erit Parabole. Dixi hec de feétionibus conicis tanto libentiùs, quôd non tantùm hic dimenfio refrationum id requirebat, fed etiam infra in Anatome oculi vfus earum apparebit. Tum etiam inter problemata obferuatoria mentio earum erit facienda duobus 25 locis. Denique ad proeftantiffima optica machinamenta, ad penfilem in aëre ftatuendam imaginem, ad imagines proportionaliter augendas, ad ignes incendendos, ad infinite comburen- Machinamléta Optica dum, confideratio earum plan eft neceffaria. Porte. The headlines of the edition quoted are Ioannis Kepleri and Paralipom. in Vitellionemn up to page 221, and afterwards Ioannis Kepleri and Astronomico Pars Optica. PAGE 92. Kepler begins by saying that rays frori the centre of a sphere do not become parallel after reflexion from its inner surface, but converge to the centre. Some other surface then had to be sought which would reflect all rays from some point into parallels. Vitellio in lib. IX. 39-44, in part supplying what was lacking in Apollonius, had shewn that the paraboloid of revolution was of the required form. But the subject of the Conic Sections presented difficulties because it had not been much studied. Kepler therefore-pardon a geometer-proposed to discourse somewhat "mechanically, analogically and popularly" about them. Vitellio or Vitello (Witelo) had proved that at any point of a parabola the tangent makes equal angles with a parallel to the axis and the line from the point to a certain fixed point on the axis. Rays of the sun impinging equidistantly from the axis upon the concavity of a reflecting paraboloid of revolution would therefore all be reflected through a fixed point on the axis, and fire might so be kindled thereat. Of cones right or scalene there are five species of sections (line 24), the right line or line-pair, the circle, parabola, hyperbola and ellipse. From the line-pair we pass through an infinity of hyperbolas to the parabola, and thence through an infinity of ellipses to the circle. Of all hyperbolas the most obtuse is the line-pair, the most acute the parabola. Of all ellipses the most acute is the parabola, the most obtuse the circle. PAGE 93. The parabola is of the nature partly of the infinite sections and partly of the finite, to which it is intermediate. As it is produced it does not spread out its arms in direction like the hyperbola, but contracts them and brings them nearer to parallelism, "semper plus quidein complectens at semper minus appetens" (line 5). The hyperbola VOL. XVIII. 26 202 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. being produced tends more and more to the form of its "Asymptote" (line 12). Parabolas are all similar and differ only in "quantity" (line 19). He then goes on to speak of certain remarkable points related to the sections which had NO NAME (line 21). The lines from them to any point of the section make equal angles with the tangent. He will call them FocI (line 27). He would have called them centres if that term had not been already appropriated. The circle has one focus, at the centre: the ellipse has two, equidistant from the centre, and more remote as the curve is more acute. In the parabola one is within the curve, while the other may be regarded as either without or within it, so that a line hg or ig drawn from that "cecus focus" to any point of the curve is parallel to the axis (line 35). PAGE 94. In the hyperbola the focus external to either branch is the nearer to its internal focus as the hyperbola is more obtuse. In the straight line (or line-pair), to speak in an unusual way merely to complete the analogy, the foci fall upon the line itself. Thus in the extreme limiting cases of the circle and the line-pair, the foci come together at a point, which in the one is as far as possible from the nearest point of the circumference and in the other is on the line itself. In the intermediate case of the parabola the foci are infinitely distant from one another (line 12): in the ellipse and the hyperbola on either side of it they are a finite distance apart. PAGE 95. The line mn through the focus, i.e. the latus rectum, is called the chord, and br or dk or es the sagitta (line 6). In the next line BF is a misprint for BP. The lengths of the sagitta and the chord are compared ill the five sections, and it is said that in the line-pair the one vanishes and the other becomes infinite (line 15), whereas, if e be the eccentricity, they are in the finite ratio 1/2 (1 + e), and vanish together. Kepler commends the principle of analogy in glowing terms, saying that he dearly loves analogies, his most trusty teachers and conversant with all the secrets of nature (line 19). Analogy leads us to comprise in one definition extreme limiting forms, from the one of which we pass to the other by continuous variation through an infinity of intermediate cases. In the next paragraph Kepler shews how to describe an arc of a hyperbola by means of threads fixed at the foci, the difference of the focal distances of a point on the curve being constant. An ellipse is described more easily (line 33), with one thread. PAGE 96. In line 1 "AC duplicata" is inaccurate, the length of the thread being ac+cb. He is shewing how to describe an ellipse by means of a thread fixed at the foci a and b, the point c being a vertex. Having given his construction for this curve without the troublesome compasses (line 6), he goes on to the parabola. To his grief he was long unable to describe this analogously. At length he thought of the construction in the text, in which adg represents a string of constant length ec +ca fixed at the focus a. DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 203 The horizontal line is a fixed ordinate, c is the vertex and d any point of the locus. His construction assumes a case of the theorem that the sum or difference of the distances of a point on the parabola from the focus and a fixed perpendicular to the axis is constant. In conclusion he refers to later passages for applications of his theory of the conic sections. See cap. v. De modo visionis, and cap. XI. prob. 22-23 (p. 375 sq.). THE CONVERGENCE OF PARALLELS. Vitellio, as we have seen, had proved that rays of the sun impinging equidistantly from (i.e. parallel to) the axis upon a concave reflector of the form of a paraboloid of revolution would all be reflected to a certain point on the axis, whereat consequently "ignem est possibile accendi." Hence in different languages the name "burning point" for what Kepler called Focus, in a parabola or other conic. It would appear that the idea of the meeting of parallels at infinity came from the observed fact that solar rays received upon a reflector may practically be regarded as parallel. Moreover it was obvious that the distance, estimated on an infinitely remote transversal, between "equidistant" lines would subtend a vanishing angle at an assumed point of observation. Kepler does not say that his doctrine of parallels is altogether new and strange, when he writes at the end of page 93, "adeo ut...", so that lines from the point h (or i) are parallel,-as if that would be allowed to follow from its being infinitely distant. But it was perhaps a new and original suggestion that h and i at infinity were the same point. Kepler states expressly that he gave the name FocI to certain points related to the conic sections which had previously "no name." With their new name he associated his new views about the points themselves, and bis doctrines of Continuity (under the name Analogy) and Parallelism, which would soon have become known, and would after a time have been taken up by competent mathematicians. An abstract of the passage now quoted at length from Kepler's Paralipomena ad Vitellionem was given by the writer in The Ancient and Modern Geometry of Conics*, published early in 1881, and previously in a note read in 1880 to the Cambridge Philosophical Society (Proceedings, vol. IV. 14-17, 1883), both of which have been referred to by Professor Gino Loria in his writings on the history of geometry. HENRY BRIGGS. Frisch (II. 405 sq.) quotes a letter of Henry Briggs to Kepler dated, Merton College, Oxford, "10 Cal. Martiis 1625," which suggests improvements in the Paralipomena ad Vitellionem. In this letter Briggs gives the following construction. Draw a line CBADC, and suppose an ellipse, a parabola and a hyperbola to have B for focus and A for their nearer vertex. Let CC be the other foci of the ellipse and the hyperbola. Make AD equal to AB, and with centres CC and radius in each case equal to CD describe circles. Then any point of the ellipse is equidistant from B and one * Th1e Ancient and Modern Geometry of Conics is hereinafter referred to as AMGC. 26i-2 204 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. circle, and any point of the hyperbola from B and the other circle. When C is at infinity on either side of D the circle about it becomes rectilinear. Hence any point P of the parabola is equidistant from B and the perpendicular DF to DA. This is expressed by Briggs as follows: A c "Si A sit vertex sectionis, et B, C foci, et AB, AD aequales, et centro C, radio CD describatur peripheria: quodlibet punctum sectionis eandem servabit distantiam a foco B et dicta peripheria. Eruntque...in Parabola (cui focus alter deest, vel distat infinite, et idcirco recta DF vicem obtinet peripheriae) PB, FP aequales." The writer comprehended and accepted Kepler's way of looking at parallels as lines to or from a point at infinity in one direction or its opposite. DESARGUES. The famous geometer Desargues worked on the lines of Kepler, and is now commonly credited with the authorship of some of the ideas of his predecessor. Poncelet in the first edition of his Traité des Propriétés Projectives des Figures (1822) writes with reference to a letter of Descartes, "On voit aussi, dans cette lettre, que Desargues avait coutume de considérer les systèmes de droites parallèles comme concourant à l'infini, et qu'il leur appliquait le même raisonnement" (p. xxxix.). Chasles on the Porisms of Euclid refers to this remark of Poncelet. In his Aperçu Historique (p. 56, 1875) he writes that Kepler " introduisit, le premier, l'usage de l'infini dans la Géométrie," but really with reference only "aux méthodes infinitésimales." The saying that Kepler introduced the use of the infinite into geometry has been repeated by other writers unacquainted with his doctrine of the infinitely great. Dr Moritz Cantor in his Vorlesungen iber Geschichte der Mathematik writes under the head of Girard Desargues (1593-1662), "Wir ntissen einige wesentliche Dinge hervorheben und darunter zunachst die Anwendung des Unendlichen in der Geometrie...Auch Kepler hat 1615, Cavalieri 1635 in Druckwerken, deren Besprechung uns obliegen wird, wenn wir von den Anfangen der Infinitesimalrechnung reden, den gleichen Gedanken zu nie geahnten Folgerungen ausgebeutet, aber bei Desargues waren es ganz andere Unendlichkeitsbetrachtungen als bei diesen Vorgangern" (II. 619, 1892). He goes on to say that Desargues regarded parallels as meeting at infinity, and thus in effect that Kepler did not so regard them. Cantor (p. 620 n.), referring to Poudra's DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 205 ~uvres de Desargues I. 103, states confidently that Desargues could not have held that "es gebe nur einen Unendlichkeitspunkt einer Gerade." "Auch in I. 105...darf man jenen modernen Sinn nicht hineinlesen." But the oneness of opposite infinities followed simply and logically from a first principle of Desargues, that every two straight lines, including parallels, have or are to be regarded as having one common point and one only. A writer of his insight must have come to this conclusion, even if the paradox had not been held by Kepler, Briggs, and we know not how many others, before Desargues wrote. In Poudra's QEuvres de Desargues, I. 210, under the head Traité des Coniques, we read, "Nombrils, point brulans, foyers.-C'est à dire que les deux points comme Q et P sont les points nommés nombrils, brulans, ou foyers de la figure, au suiet desquel il y a beaucoup à dire." Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him the Foci (foyers), including the modern doctrine of real points at infinity. (B) NEWTON. In the fifth section of the first book of the Principia, entitled Inventio orbium ubi umbilicus neuter datur, the determination of conic orbits from data not including a focus, Newton proves the property of the Locus ad quatuor lineas of which no geometrical demonstration was extant, shews how to describe conics by rotating angles and otherwise, and solves the six cases of the problem to determine a conic of which n points and 5 - n tangents are given. Two more problems, each with its Lemma prefixed, complete the section, which ends with the words, " Hactenus de orbibus inveniendis. Superest ut motus corporum in orbibus inventis determinemus." The following pages contain a summary of the greater part of the section, with suggestions for the simplification of some of its contents and a few additional constructions and propositions. The Lemmas and Propositions of the Principia are quoted by their Roman numerals. 1. THE CONIC THROUGII FIVE POINTS. PROP. A. Given five points of a conic to find a sixth. Let A, B, C, D, P be given points of a conic. Through P draw PTSO parallel to BA across BD, AC, CD. It is required to find the point K in which it meets the conic again. 206 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. By a property of conics and by similar triangles, if AB, CD meet in I, OK. OP/OC. OD = IA. IB/I. ID= OS. OT/OC. OD. H Therefore OK. OP = OS. OT, which determines K when the other points are given. Inflect PR to CD parallel to A C. Then the point K is found by drawing CK pacrallel to RT, and PRT, SCK are similar triangles. Cor. 1. To determine the conic through five given points A, B, C, D, P. Having found K, we find H where PR meets the conic again in like manner, nainely by drawing BH parallel to TR. Having two pairs of parallel chords, we can draw their diameters and find the centre. This with either pair of the parallel chords determines the conic, if the pair be unequal. If they be equal, we can use the parallel chord through D in lieu of one of them. Giveln five points A, B, C, D, E, two pairs of parallel chords can also be determined as in Prob. LV. of the Arithmetica Universalis. Let AC, BE cross in H. Inflect DI to A C parallel to BE, and EK to DI parallel to A C. Then, in order that ID, EK may meet the conic again in F, G, we must have wTith Newton's notation for rectangles and proportions, AHC. BHE:: AIC.FID:: EKG. FKD. Cor. 2. To determine the conic touching lines IB, ID at B, D and passing through P. Supposing AC, BD in the figure to coalesce, find K as in the general case, and DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 207 draw the diameter of PK. Then draw the diameter through I, and find its vertices, and those of the conjugate diameter. Cor. 3. Hexagrammum Mysticum. The construction in Cor. 1 for two pairs of parallel chords gives three pairs, AB and KP, AC and PH, BH and KC. Hence Pascal's theorem for the case of parallels. Cor. 4. Given parallel chords AB, KP and a fifth point G of a conic, a sixth point D on the curve can be found as follows. Draw any parallel to CK meeting PK in T and meeting the parallel through P to A G in R. Then BT, CR meet at D on the conic. Cor. 5. In this construction we may say that PR, PT are to be taken in a given ratio equal to SC/SK. See below on Newton's Lemma xx. Cor. 6. The locus of the point (BT, CR) in Cor. 4 is a conic through A, K, C, P, B. Hence the following construction. Take fixed lines PR, PT; fixed points B, C; and a fixed point Z at infinity. Then as the line ZRT turns about Z the point (BT, CR) traces a conic through B and C. Obviously it will likewise trace a conic in the general case when Z is not at infinity. Cor. 7. In other words, the locus of the vertex D of a varying triangle RDT whose base slides between fixed lines PR, PT, while its three sides pass through fixed points B, C, Z respectively, is a conic. This may be shewn independently as follows. Draw CD in any assumed direction, and find R, and then T, and then D. Thus one point D is found on the line through C, and it is a single point of the locus. By drawing the line BC we find that each of the points B, C is a single point of the locus. Thus CD cuts it in two such points, and the locus is therefore of the second degree. Cor. 8. The anharrmonic point-property of conics. In Cor. 4, as D varies, the parallel RT to CK divides PR, PT proportionally, so that the cross ratios of R and T in any four positions are equal to one another. Hence B {D} = {T} = R} =C{D}, or any four points 1D of the conic are equi-cross with respect to B and C, which may be any assumed fifth and sixth. Cor. 9. Hence we can deduce the general case of Cor. 6. Cor. 10. Locus ad quatuor lineas. By similar triangles, PR/PT and SC/SK are equal ratios. Compounding with them other equal ratios we get PR. PQ/PS. PT = SO. SA/SK. SP =f/g, if f, g be the focal chords parallel to A, AB. See also below on Newton's Lemma xvII. Cor. 11. The extension at the end of Cor. 6 follows frorn a simple transformation of the figure by which the parallels RT are turned into convergents. In the figure as 208 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. it stands suppose DBo drawn to PQ. Then, the points A, B, C, P being fixed and D variable, {0} = {R} = T} = {{o}. But P is the position of 0, and likewise of co, when RT vanishes. Therefore 0w passes through some fixed point F. When D is at A the line Oo becomes QS, and when RT passes through B it becomes CH. Thus F is the point (CH, QS), and as Ow turns about F the point D is found by drawing CO, Bw. Cor. 12. By the construction of Cor. 7, as is well known, we can describe the conic through five given points. For example, in the limiting case in which three points A, B, C and the tangents at B, C are given, we can take AB, A C for the fixed lines, and for the fixed points B, C and the intersection Z of the two tangents. LEMMA A. To find the centre of an involution of four points. To find the centre of the involution in which P, K and S, T are conjugate points, through P and S (or T) draw parallels, and through T (or S) and K draw parallels meeting them in R and C respectively. Then RC passes through the centre of the involution (AMGC, p. 258). The converse has in effect been used in Prop. A, where the conic and AC, BD cut a parallel to AB in points of an involution having O for centre. The six joins of any four points cross any transversal in three pairs of points in involution. In the above construction two of the four points are at infinity. 9. Locus AD TRES ET QUATUOR LINEAS. APOLLONIUS OF PERGA. We shall see that Newton mentions Apollonius of Perga in connexion with the problem of the quadrilinear locus. What Apollonius says of the 707roÇ e7r TpEtS Kat reéo-apas 7pax/aaç is translated as follows by Dr T. L. Heath in his edition of the Conics of Apollonius in modern notation (p. lxx. sq., 1896), "Now of the eîght books the first four form an elementary introduction;...The third book contains many remarkable theorems useful for the synthesis of solid loci and determinations of limits; the most and prettiest of these theorems are new, and, when I had discovered them, I observed that Euclid had not worked out the synthesis of the locus with respect to three and four lines, but only a chance portion of it and that not successfully; for it was not possible that the synthesis could have been completed without my additional discoveries." This prepares us to find in the third book of the Conics of Apollonius, if not the synthesis of the locus, the elementary theorems on which it depends. Turning to lib. III. 54, 56 we see the property of the locus proved incidentally for the case of three lines in the proposition thus enunciated by Dr Heath (Prop. 75, p. 120), DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 209 TQ, TQ' being two tangents to a conic, and R any other point on it, if Qr, Q'r' be drawn parallel respectively to TQ', TQ, and if Qr, Q'R meet in r and Q'r', QR in r', then Qr. Q'r': QQ'2 = (P V2: PT2) x (TQ. TQ': QV2), where P is the point of contact of a tangent parallel to QQ'. Dr Heath shews (p. 122 sq.) that this proposition and his next (lib. III. 55), for tangents to one branch and two branches of a conic respectively, "give the property of the three-line locus." The constan.cy of Qr. Q'r' being a corollary from the property of the trilinear locus, we can of course work back from the latter to the former. But more briefly, leaving out r, r', draw the tangents TQ, TQ' crossing any chord RR' parallel to QQ' in K, K'. Then, because the diameter through T bisects both KK' and RR', the intercepts KR, K'R are equal, and likewise KR', K'R. LI D, / T L Q K Therefore RK.RK' (or KR.KR') varies as KQ2. This is the trilinear theorem as proved by Apollonius. Inflect RD to QQ' parallel to QT. Then RK.RK' varies as RD2, and the theorem may be stated thus, The distance of any point on a conic from a given chord varies as a mean proportional to its distances from the tangents at the ends of the chord, each distance being parallel to any given Une. Apollonius does not enunciate the theorem, but he proves and uses it in the course of his propositions mentioned above. VOL. XVIII. 27 210 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. The distances of any point on a conic from the tangents at fixed points A, B, C, D being denoted by a, b, c, d respectively, its distances from AB, BC, CD, DA vary as mean proportionals to ab, bc, cd, da. Hence obviously the four-line theorem, AB. CD =k. BC. DA. Apollonius, who claims to have solved the Locus ad tres et quatuor lineas completely, may very well have deduced the four-line theorem from the three-line theorem in this way. The Lemmas and Propositions quoted below by number are Newton's, whose proofs and diagrams in lib. I. sect. v. of the Principia should be referred to. LEMMA XVII. Case 1. A C, BD being given parallel chords of a conic, through any point P of the curve draw the chord PK parallel to AC and crossing AB, CD in Q, R; and a parallel to AB meeting AC, BD in S, T. Then PQ. QK/AQ. QB is a constant ratio. But, the intercepts PR, QK being equal, the rectangle PQ. PR is equal to PQ. QK, and therefore varies as AQ. QB or PS. PT. Thus Newton's proof for this case is the same as that of Apollonius for the threeline theorem, which it includes, since the parallels AC, BD may be supposed to coalesce. In Case 2, with the help of Case 1, the theorem is shewn to hold when AC, BD are not parallel. In this general case Newton does not use the point K, which might have been found by drawing the parallel to RT through B. This construction leads to the proof of his Lemma xvII. in Prop. A, Cor. 10. The proof in question is given by Messrs J. J. Milne and R. F. Davis in their Geometrical Conics, followed by a corollary in which Lemma xx. is deduced from Lemma xvII., as by Newton. LEMMA XVIII. Conversely, the locus of a point P such that PQ. PR/PS.PT is constant is a conic section. Corol. The trilinear theorem is deduced as a limiting case. Scholium. The term conic section includes the line-pair and the circle. For a trapezium may be substituted a re-entrant quadrilateral; and one or two of the points A, B, C, D may be at infinity. LEMMA XIX. Any line being drawn through A, the point P in which it meets the locus again is determined. Corol. 1. The tangent at a given point is drawn. Corol. 2. It is then shewn how to find a pair of conjugate diameters, and the different species of conics belonging to the locus are discriminated. At the end it is said, with tacit allusion to the algebraic proof of the quadrilinear theorem by Descartes, "Atque ita problematis veterum de quatuor lineis ab Euclide incoepti & ab Appollonio continuati non calculus, sed compositio geometrica, qualem veteres quserebant, in hoc corollario exhibetur." DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 211 3. CURVARUM DESCRIPTIO ORGANICA. LEMMA XX. AB, AC are given chords and P a given point of a conic. Through P draw parallels to AC, AB forming with them a parallelogram PQAS; and across PQ, PS draw CRD, BDT to any sixth point D of the conic. Then will PR/PT be a constant ratio, and conversely. Case 1. The constancy of PR/PT is deduced from the four-line theorem proved in Lemma xvII. Interchanging P, D in this paragraph only, let A, B, C, D in the figure of Prop. A be fixed and P variable. Through D draw a parallel to AC meeting CP in r, and a parallel to AB meeting BP in t. Then, CD being given, Dr varies as PR/RC, and therefore as PR/PS; and, BD being given, Dt varies as PT/TB, and therefore as PT/PQ. Therefore Dr/Dt is a constant ratio. In like manner, with P fixed and D variable as in Lemma xx., PR/PT (p. 206) is a constant ratio. Hence the lne RT is given in direction. See Prop. xxII. and Prop. xxIII., where Pt/Pr is made equal to PT/PR by drawing tr parallel to TR, "actâ recta tr ipsi TR parallelâ" Hence, K being the position of D found by drawing RT through C (p. 206), it follows that RT is parallel to CK. Thus Prop. A is in fact Lemma xx. LEMMA XXI. Take a triangle BPC, and let angles equal to its angles at B and G turn about those points as poles, one pair of the ines or bars containing the angles p intersecting at M on a fixed line or director which cuts BC in N. Then the other pair will cross at a point D lying on a conic through B and C. 27-2 212 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. For inflect PR to CD, making the angle CPR equal to the constant angle CNJM. Then PCR, N/CMI are similar triangles, and PR/NM = PC/NC. Inflect PT to BD, making the angle BPT equal to the constant angle BNM. Then PBT, NBM are similar triangles, and PT/NM= PB/NB. Therefore PR varies as PT, and by Lemma xx., PR and PT being on fixed lines, the locus of (CR, BT) is a conic through B and C, and conversely. The lengths PR, PT in the figure, which differs somewhat from Newton's, are as the perpendiculars from NY to PB, PC. Given four points B, C, D, P, an infinity of conics can thus be drawn through them, for the given point D determines only one point M of the director. Given a fifth point of the conic, the director is determined, and one conic only can be described. To draw the tangent BT at B, make D coincide with B. See Prop. xxII., Corol. 1. In other words, make the angle NC]M equal to the angle PCB, and then the angle MBT equal to the angle PBC. To find the directions of the axes. If the arms BM, CM be made constantly parallel, the intersection D of the others will trace a circle through B and C. This will cut the conic again at the two points found by making the parallel arms successively coincident with BC and parallel to the director. Four points common to the circle and the conic having been found, the axes must be parallel to the bisectors of the angles between a pair of chords joining them. For Newton's construction see Prop. xxvii. Scholium (p. 216). PROP. B. If two angles AOB, AoB of given magnitudes turn about poles O, w, and if the intersection A traces a curve of the nth order, the intersection B will in general trace a curve of the 2nth order. For a given position of the arm OB there are n positions of A and therefore n of B. When OB is in the position Ow all the B's coincide with w, which is therefore an n-fold point on the locus of B, as is also the point 0; and since any line through O (or co) meets the locus of B in n other points, the locus is of the order 2?n. 4. INVENTIO ORBIUM. PROP. XXII. PROB. XIV. To describe the conic through five points. This is done by Lemma xx., and again by Lemma xxI. PROP. XXIII. PROB. XV. To describe a conic through four points and touching a given line. Case 1. When one of the points is the point of contact the construction is effected as in Prop. xxII. DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 213 Case 2. In the general case, HI being the given tangent and BCDP the given points, draw HAI, ICPG, GBDH, and make the ratio compounded of HA2/HD. HB; GB. GD/GP. GC; IC. IP/IA2; a ratio of equality. Thus HA/IA is determined and the point of contact A is found within or without HI. This is Newton's solution briefly stated, and it is identical with the modern solution by what is called Carnot's theorem. When A is found the two conics can be described by the methods used in Case 1. PROP. XXIV. PROB. XVI. To describe a conic through three given points and touching two given lines. Given two points and two tangents, Newton proves that the chord of contact must pass through one of two fixed points. This may be shewn as follows. Let B, D be the given points and GH, GK the given tangents. Take H and K in line with BD, and suppose BD and the chord of contact to cross at R. Then by the trilinear theorem, all the distances being measured along BD, we have BR2/DR2 =- BH. BK/DH. DK. Divide BD within and without at R in the ratio thus determined, and we have two points through one of which the chord of contact must pass. A third given point C taken with B or D determines two points S through one of which the chord of contact must pass. Thus there are four possible positions of RS, giving four solutions. When RS is found the conic can be described as in the first case of Prop. xxIIm. Imaginary Points. In the second case of Prop. xxIIm. and in Prop. xxiv. Newton uses an auxiliary line which is supposed to cut the conic in points X and Y. At- the end of Prop. xxiv. he remarks that the constructions given will be the same whether the line XY cuts the trajectory or not. For the sake of brevity he gives no special proofs for the case in which, as we should say, the points X and Y are imaginary. LEMMA XXII. Figuras in alias ejusdem generis figuras rutare. Here Newton gives a method of homographic transformation, in which the loci of points G, g correspond so that the coordinates X, Y of G and x, y of g are connected by relations of the form, OA. AB y OA.Ly X X0 214 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. By this method, it is remarked, convergent lines can be transformed into parallels; and when a problem has been solved in the simplified figure, this can be retransformed into the original figure. In the solution of "solid problems" one of two conics can be changed into a' circle. In the solution of "plane problems" a line and a conic can be made a line and a circle. PROP. XXV. PROB. XVII. To describe a conic through two given points and touching three given lines. Transform the given tangents and the line through the given points into the sides of a parallelogram. Let these sides be hci, idk, kcl, Ibah, where a, b correspond to the given points and c, d, e are the points of contact. Take m, n mean proportionals to ha, hb and la, lb. Then hc/m = ic/id = ke/kd = le/n, and each of these ratios is equal to the given ratio of hi + kl, the sum of the antecedents, to z + n + ki, the sum of the consequents. Thus the points of contact are determined. It may be remarked that this case is the reciprocal of Prob. xvI. Given two points B, D and two tangents GH, GK, the pole of BD must lie on one of two fixed lines. A third tangent being given, we can thus find four positions of the pole of BD. Having then five tangents and the points of contact of two of them, we can trace the four conics in various ways. PROP. XXVI. PROB. XVIII. To describe a conic through a given point and touching four given lines. Newton's solution is in effect as follows. Let P be the given point, and let two diagonals of the quadrilateral formed by the four tangents meet in O. Draw OPo to the third diagonal, and take Q a harmonic conjugate to P with respect to O, e. Then Q is on the conic, and the case is reduced to that of Prop. xxv. He transforms the given tangents into the sides of a tangent parallelogram; finds the centre O; and finds Q the other end of the diameter PO. In the retransformed figure Q would therefore be found by the previous construction. PROP. XXVII. PROB. XIX. To describe the conic touching five given lines. This is led up to by three Lernmmas, one of which, with a transformation as in Prop. xxv. or Prop. xxvi., would have sufficed for the solution of the problem. LEMMA XXIV. Corol. 2. Using the figure of Lemma xxv., let AMF, BQI be parallel tangents to a conic; A, B their points of contact; FQ, IM any third and fourth tangents. DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 215 Then AM: AF=BQ: BI, and FI, MQ meet on the diameter AB. We can now solve the problem as follows. F M A L D H Complete the parallelogram IKLM by drawing the tangent KL parallel to IM. Then IL, KM cross at the centre of the conic. Conversely, from five given tangents we can determine the conic. Case 1. Let four of the tangents be the sides of a parallelogram, as in the figure, Its diagonals by their intersection give the centre, and FI, MQ also intersect on the chord of contact AB. The diameter AB being known, the conjugate radius is a mean proportional to AM, BI. Case 2. Let the tangents at A, B only be parallel. These with FQ, MI determine a point (FI, MQ) on the chord of contact AB; and with IM, KL they determine a point (IL, KM) on AB. Case 3. When none of the tangents are parallel, the same construction determines AB; for one pair of them, or two pairs, can be transformed into parallels by Lemma xxII. All the points of contact can be found in this way, and the conic can then be traced by various methods. LEMMA XXV. Corol. 1. If IEM, IQK be fixed tangents to a conic and MK rectangle KQ.ME, or (IK - IQ) (IM-IE) is constant. This leads to a tangential equation of the form, a. IE. IQ + b. IE + c. IQ + d = O. 216 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. Corol. 2. The anharmonic tangent-property of conics. A sixth tangent eq is drawn, and it is shewn that KQ: Qq =Me:Ee. Thus the four tangents LK, EQ, eq, LM determine equal cross ratios on the tangents IK and IM. Corol. 3. A tangent quadrilateral being given, the locus of the centre of the conic is the straight line which bisects its diagonals. PROP. XXVII. Hence, five tangents being given, two tetrads of them give two lines through the centre. The parallel tangents can then be drawn, their points of contact found by Lemma xxiv., and the conic described by Prop. xxII. Scholium. The preceding problems include cases in which the centre or an asymptote is given. For an asymptote is a tangent at infinity, and the centre with one point or tangent determines another point or tangent. To find the axes and foci of a conic described by Lemma xxi. Set the arms BP, CP (which by their intersection described the conic) parallel and let them so rotate. The intersection X of the other arms of the two angles will then describe a circle through B, C. Draw its diameter KL crossing the director at right angles in H. When X is at K, then CP is parallel to the major or minor axis according as KH is less or greater than LH; and when X is at L, then CP is parallel to the other axis. Hence when the centre is given the axes are given, and the foci can be found. Newton does not explain his construction for the directions of the axes, which has the appearance of having been first made for the hyperbola, and then stated for the ellipse also as having imaginary points at infinity. Le Seur and Jacquier, in their annotated edition of the Principia, having explained the construction for the case of the hyperbola by means of its asymptotes, or tangents "ad distantiam infinitam," merely remark in conclusion that it applies also to the parabola into which the hyperbola is changed when the intersections of the director with the circle coalesce, and to the ellipse into which the parabola is turned when the director passes outside the circle*. The squares of the axes are as KH to LH. Hence a trajectory of given species or eccentricity can easily be described through four given points. Conversely a trapezium of given species, "si casus quidam impossibiles excipiantur," can be inscribed in a given conic. There are also other lemmas by the help of which trajectories of given species can be described when points and tangents are given. For example, the middle point of a chord drawn through a fixed point to a conic traces a similar and similarly situated conic. "Sed propero ad magis utilia." * Their words are, "Superior autem constructio non Ellipsi in quam vertitur parabola, dum recta MN extra solum hyperbole convenit, sed & parabol in quam hyper- circulum transit," the points M and mn being the interbola mutatur, dum puncta m, M coeunt; atque etiam sections of the director MN and the circle. DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 217 5. PERSPECTIVE AND CONTINUITY. In Lemma XXII. (p. 213) Newton gives a construction made to illustrate his algebraical transformation of an equation of any degree into another of the same degree. After the proof that tangents remain tangents, he remarks that his demonstrations might have been put together "more magis geometrico," but he aims at brevity. With this Lemma should be read his Enumeratio Linearum Tertii Ordinis, where again he has something to say about curves in general. At the end of the preface to his Opticks Newton writes, And I have joined with it another small Tract concerning the Curvilinear Figures of the Second Kind, which was also written many Years ago, and made known to some Friends, who have solicited the making it publick. He is referring to the Enumeratio above mentioned, in which curves of the nth order are called curves of the (n - )th genus or kind, the straight line in this way of speaking not being counted among curves. In this tract he gives the theory of Perspective in space under the name Genesis Curvarum per Umbras, rays from a luminous point being supposed to cast shadows of geometrical figures on to an infinite plane. Thus, he says, the "Parabolie quinq; divergentes" generate by their shadows all other cubic curves, and so from " Curvae qusedam simpliciores" of any genus can be produced all the other curves of that genus. Such genesis of curves by shadows may have been suggested to Newton by some of Kepler's problemata obseruatoria (pp. 201, 203), in which he lets the sun shine through a small aperture into a darkened room, and observes the diurnal course of its projection on the floor. This varies with the latitude of the place, according to which the apparent path of the sun itself in any day cuts or touches or does not meet the plane of the horizon. Thus Perspective as a modern method may be said to have originated with Kepler. The idea of it was not altogether unknown to the ancients, but they were scarcely in a position to put it to effective use, for this could not be done without a more or less complete doctrine of Continuity, including especially the quasi-concurrence of parallels at infinity. See AMGC, p. iv., and the writer's note on Perspective in vol. x. of the Messenger of Mathenmatics (1881). Newton's Lemma xxII. may have arisen from his genesis of curves by shadows. Having seen how to connect varieties of the same order of curve graphically, he would naturally seek to connect such curves algebraically; and this could obviously be done by his transformation of coordinates from X, Y to x, y, with Xx and Yx/y constant. Page 200. 21 quantumuis absurdis locutionibus] Poncelet used "ce qu'il appelle le principe de continuity," which is Kepler's principle of Analogy under a new name. This principle Kepler formulated in terms suitable to its later applications. Including normal VOL. XVIII. 28 218 DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. and limiting forms of a figure under one definition, we are led to paradoxical ways of speaking, "sine vsu, tantum ad analogian complendam" (p. 199. 5-6); as when we think of a hyperbola as a sort of ellipse, and postulate imaginary elements in the one analogous to what we see in the other. Newton in some of his constructions virtually uses imaginary points (pp. 213, 216), whether or not, like Boscovich, he thought definitely of geometrical figures as having imaginary elements. To say that equations in x and y, which represent coordinates, may have imaginary roots (Opticks, p. 151) is to say in effect that there are what may be called imaginary points. Newton doubtless used equations for his own satisfaction in some places where he does not fully explain his geometry. An equation representing the locus described in Lemma XXI. (p. 211), is given in Prob. LIII. of the Arithmetica Universalis (1707). By the method of Fluxions he discovered things which he gave to the world proved "more magis geometrico." Thus he writes: "At length in the winter between the years 1676 and 1677 I found the Proposition that by a centrifugal force reciprocally as the square of the distance a Planet mu st revolve in an Ellipsis about the center of the force placed in the lower umbilicus of the Ellipsis and with a radius drawn to that center describe areas proportional to the times......And this is the first instance upon record of any Proposition in the higher Geometry found out by the method in dispute." Two imaginary points the FocoIDS (AfGC, p. 281), or " Circular Points at Infinity," play a great part in modern geometry. Their existence may be proved in geometrical form as follows. Draw any circle in a given plane, and let sb and b' be the two points in which it cuts the line Infinity. These will be the same for all circles in the plane. For take points A, B on the circle subtending any angle a at the circumference; and take any other two points a, b in the plane. Then the angle AfB is equal to a, because b is on the circle; and the lines fA, Ca are parallel, and likewise OB, Ob, because b is at infinity. Therefore Z acb= z ABB=a, or any two lines through b may be regarded as intersecting at any angle. Hence every circle in the plane passes through b, and similarly through 4'. Conversely, a conic through O and b' is a circle. The orthoptie locus of a curve of the nth class is of the degree n(n - 1-), since its intersection with the line Infinity consists of b and J' taken ~n (n- 1) times. From the equation x2 + y2 - (x + iy) ( - iy) = 0 in rectangular coordinates it seems at first that ~c and 4' are indeterminate, because x (or y) may have any direction. But the angles tan-l + i are indeterminate. DR TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 219 The equation tan (0 + a) = tan 0 reduces to tan a (1 + tan2 0) = 0, and when tan =- 1, then 0 is of the form a + if with 3 infinite. Page 210 ab Euclide incoepti, etc.] Newton has in mind the words of Descartes in La Géométrie, "commencée à resoudre par Euclide et poursuivie par Apollonius, sans avoir été achevée par personne." Apollonius has indeed nothing to say about a locus related to more than four lines, but there is no reason to question his statement that he had solved the problem of the four-line locus. Its complete working out would have supplied ample materials for a book on the scale of his lib. v. on Normals*. Newton assumes Lemma xvII. in Lemma xx., on which his Lemma xxI. depends, thus making the "Organic Description" of conics seem less simple than it is. Having proved Prop. A, make A, B, K, P, C fixed points and D variable, and we have at once RT parallel to the fixed line CK (p. 206) as in Lemma xxI. Page 216 Sed propero ad magis utilia] The Principia, all but some ten or twelve propositions composed previously, having been written in less than a year and a half (Dec. 1684-May 1686), Newton could not have had much time to spare for the two sections (lib. I. 4-5) on Inventio Orbiztn. Maclaurin's constructions of a conic by means of three (p. 207, Cor. 6-7) or more lines through fixed points grew out of a lenma NVeutonianum, as we learn from the preface to Simson's Sectiones Conicce. Newton himself, vith leisure, could have developed the said two sections into a comprehensive and essentially modern treatise. * Of this lib. v. Chasles tells us that it treats of "les maximis et minimis, sur les sections coniques. Dans le questions de maxima et de minima," and that, "On y cinquième Apollonius examine particulièrement quelles sont retrouve tout ce que les méthodes analytiques d'aujourd'hui les plus grandes et les moindres lignes qu'on puisse tirer de nous apprennent sur ce sujet." This astonishing statement chaque point donné à leur circonférence. On y retrouve is a too brief summary of the words of Montucla on lib. v. tout ce que nos méthodes analytiques d'aujourd'hui nous and lib. vI., "Ils traitent l'un et l'autre un des sujets les apprennent sur ce sujet." Chasles goes on to speak of plus difficiles de la géométrie, savoir les questions de normals as the subject of lib. v. 28-2 IX. Sur les Groupes Continus. Par H. POINCARÉ. [Received 25 September, 1899.] I. INTRODUCTION. LA théorie des groupes continus, ce titre immortel de gloire du regretté Sophus Lie, repose sur trois théorèmes fondamentaux. Le premier théorème de Lie nous apprend comment dans tout groupe continu il y a des substitutions infinitésimales et comment ce groupe peut être formé à l'aide des opérateurs (/)= (Xi) df dxi Considérons r opérateurs de cette forme (1) X (f X (f),......, X (); et convenons de poser: XiXk- XkXi = (XiXk). D'après le second théorème de Lie si les symboles (XiXk) sont liés aux opérateurs Xi par des relations linéaires de la forme: (2) (XiXk)= =CiksXs, où les c sont des constantes, les r opérateurs (1) donneront naissance à un groupe. Les relations linéaires (2) pourront s'appeler relations de structure puisqu'elles définissent la "structure" du groupe qui dépend uniquement des constantes c. C'est le troisième théorème de Lie qui attirera surtout notre attention. Quelles sont les conditions pour qu'on puisse former un groupe de structure donnée, c'est-àdire pour trouver r opérateurs X1, X,......, X. satisfaisant à des relations de la forme (2) dont les coefficients c sont donnés? On voit tout de suite que les coefficients c ne peuvent être choisis arbitrairement. On doit d'abord avoir (3) Ckis = - Ciks M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 221 Ensuite d'après la définition même du symbole (XiXk) on a identiquement (4) ((XXb) X,) + ((XbXc) X) + ((XcX) Xb) = 0, d'où résultent entre les c certaines relations connues sous le nom d'identités de Jacobi. Une condition nécessaire pour que l'on puisse former un groupe de structure donnée, c'est donc que les coëfficients c satisfassent à ces identités de Jacobi auxquelles il convient d'adjoindre les relations (3). Le troisième théorème de Lie nous enseigne que cette condition est suffisante. Pour la démonstration de ce théorème, nous devons distinguer deux familles de groupes. Les groupes de la 1ère famille sont ceux qui ne contiennent aucune substitution permutable à toutes les substitutions du groupe. Les groupes de la 2e famille sont ceux qui contiennent des substitutions permutables à toutes les substitutions du groupe. En ce qui concerne les groupes de la 1ère famille, la démonstration de Lie, fondée sur la considération du groupe adjoint, ne laisse rien à désirer par sa simplicité. En ce qui concerne les groupes de la 2e famille, Lie a donné une démonstration entièrement différente, beaucoup moins simple, mais qui permet cependant de former les opérateurs Xi(b) par l'intégration d'équations différentielles ordinaires. Dans une note récemment insérée dans les Comptes-Rendus de l'Académie des Sciences de Paris, j'ai donné une démonstration nouvelle du 3e théorème de Lie. Les résultats contenus dans cette note étaient moins nouveaux que je ne le croyais quand je l'ai publiée. D'une part en effet, Schur avait dans les Berichte der k. sÉchsischen Gesellschaft der Wissenschaften 1891 et dans le tome 41 des Mathematische Annalen donné du théorème en question une démonstration entièrement différente de celle de Lie. Cette démonstration présente la plus grande analogie avec celle que je propose; mais elle n'a pour ainsi dire pas été poussée jusqu'au bout. Comme le fait remarquer Engel, le résultat dépend de séries que Schur forme et dont il démontre la convergence; au contraire Lie ramène le problème à l'intégration d'équations différentielles ordinaires. Je suis arrivé comme Lie lui-même à des équations différentielles ordinaires qui même sont susceptibles d'être complètement intégrées. D'autre part Campbell a donné sous une autre forme quelques-unes des formules auxiliaires qui m'ont servi de point de départ (Proceedings of the London Mathematical Society, tome 28 page 381 et tome 29 page 612). Il m'a semblé néanmoins que cette note contenait encore assez de résultats nouveaux pour qu'il y eût quelque intérêt à la développer. 222 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. Je ramène en effet la formation d'un groupe de structure donnée, à l'intégration d'équations différentielles simples, intégration qui peut se faire en termes finis. Ces équations sont moins simples que celles que Lie a formées pour les groupes de la 1re famille; mais même dans ce cas, il peut y avoir intérêt à les connaître, car elles sont d'une forme différente et me s'en déduisent pas immédiatement. De plus elles sont applicables aux groupes de la 2e famille et dans ce cas elles nous fournissent une solution du problème plus simple que celle de Lie. II. DÉFINITION DES OPÉRATEURS. Soit f une fonction quelconque de n variables xl, x2,..., xn. Soit X un opérateur qui change f en dx 2 dx2,ct' où les (Xi) sont n fonctions données des n variables xl, x2,..., Xn, de sorte que: df X (f)= ((XA) dxj Soient Y, Z, etc. d'autres opérateurs analogues de telle façon que: Y (f )= (Y) d (f= (Zi dxj' clxi. les (Yi), les (Zi),... étant d'autres fonctions de xl, x2,..., ox. Dans ces conditions: X2 (f) = X[X (/)], XY(f) = X [Y(f)], X2Y(f) =X[XY (f)], XYZ(f) = X [YZ(f)],..., seront des combinaisons linéaires des dérivées partielles des divers ordres de la fonction f, multipliées par des fonctions données des xi. Ainsi se trouveront définis de nouveaux opérateurs X2, XY, X2Y, XYZ,..., qui sont des combinaisons des opérateurs simples X, Y, Z,.... On voit que ces produits symboliques obéissent à la loi associative mais n'obéissent pas en général à la loi commutative de sorte que XY ne doit pas être confondu avec YX. Ces opérateurs sont ainsi symboliquement représentés par des monômes; mais on peut définir des opérateurs qui seront symboliquement représentés par des polynômes tels que: 1+aX, aX+bY, aX25 + 2bXY+cY2,...... en convenant d'écrire par exemple: (1 +aX)(f)=f+ aX (f); (aX+bY)(f)=aX(f)+ bY(f)............. M. H. POINCARE, SUR LES GROUPES CONTINUS. 223 On voit que les polynômes opérateurs ainsi définis obéissent à la fois à la loi associative et à la loi distributive; de sorte qu'on aura: (aX + bY) (cX + dY) = acX2 + adXY+ bcYX + bdY2, et en particulier: (X + Y)2S X + XY+ YX + Y2 expression qu'il ne faut pas confondre avec X2 + 2XY + Y2. On peut aussi introduire des opérateurs qui seront représentés symboliquement par des séries infinies. Je citerai par exemple l'opérateur: f+a(X+ Y)(f)+ 2(X+ Y)2(f) + 3(X+ )3 (f )+...... que je représenterai symboliquement par: - ca (X + Y)] (/)' ou plus simplement par 1 1 - c (X + Y)' et l'opérateur: f + aX(f) )+ X3f(f )+ X(+...... que je représenterai par eax(f) ou simplement par eaX. On peut se demander si l'emploi de ces opérateurs représentés par des séries est légitime et si la convergence des opérations est assurée. Il y a des cas où cette convergence est certaine. C'est ainsi que Lie a démontré que etX (f)=f('1, x'2,..., x',,) où les x' sont définis par les équations différentielles: dxi - = (Xi) (x,', x2,... *, I) et par les conditions initiales: x'i = xi pour t = O. Les opérateurs définis par des séries symboliques obéissent évidemment aux lois distributive et associative, ce qui permet par exemple d'écrire des égalités telles que celle-ci: (eYeaxeZ) (e-ZebxeT) = eYe(a+b)XeT. Il y a aussi un cas où ils obéissent à la loi commutative. Soient (X) = a.X, er (X) = ~bnX, deux séries symboliques dépendant d'un seul opérateur élémentaire X. On a alors (x) [4 (X) (f)]= (x ) [(x (f)]. Les deux products symboliqus deux profits symboliques (X) (X) et (X) (X) sont en effet des sommes de monômes dont tous les facteurs sont égaux à X. Si tous les facteurs sont 224 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. identiques, il est clair que l'ordre de ces facteurs est indifférent et que les opérations sont commutatives. Mais cela ne sera plus vrai si les séries symboliques dépendent de plusieurs opérateurs élémentaires différents; il ne faudrait pas par exemple confondre Xn yn eXeY = n mt! ni avec eYeX =, m n n v ni avec eY= E (X + Y)P p III. CALCUL DES POLYNÔMES SYMBOLIQUES. Soient X, Y, Z, T, U,....., n opérateurs élémentaires. Par leurs combinaisons on pourra former d'autres opérateurs représentés symboliquement par des monômes ou des polynômes. Deux monômes seront dits equipollents lors qu'ils ne différeront que par l'ordre de leurs facteurs; il en sera de même de deux polynômes qui seront des sommes de monômes équipollents chacun à chacun. Nous appellerons polynôme régulier tout polynôme qui peut être regardé comme une somme de puissances de la forme: (aX +/Y+ +...)P. Il résulte de cette définition: 1~. Que si un polynôme régulier contient parmi ses termes un certain monôme, tous les monômes équipollents figureront dans ce polynôme avec le même coefficient. Cette condition est d'ailleurs suffisante pour que le polynôme soit régulier. 2~. Que parmi les polynômes équipollents à un polynôme donné il y a un polynôme régulier et un seul. Le polynôme XY- YX jouit de la même propriété que les opérateurs élémentaires, c'est-à-dire que (XY- YX) (f) est comme X(f), Y(f) etc. une combinaison linéaire des dérivées du premier ordre seulement de la fonction f multipliées par des fonctions données des xi. Nous supposerons que les opérateurs élémentaires et leurs combinaisons linéaires sont seuls à jouir de cette propriété. (Si cela n'avait pas lieu, nous introduirions parmi les opérateurs élémentaires tous ceux qui en jouiraient.) Nous devrons donc avoir des relations de la forme: (1) XY- YX = (XY), M. H. POINCARE, SUR LES GROUPES CONTINUS. 225 où (XY) est une combinaison linéaire des opérateurs élémentaires; nous reconnaissons là la relation de Lie dite relation de structure: XXk - XkXi = iks.Xs. Cela posé, deux polynômes seront équivalents lorsqu'on pourra les réduire l'un à l'autre en tenant compte des relations (1). Par exemple le produit (2) P[X T- YX - (X)] Q (où le premier et le dernier facteurs P et Q sont deux monômes quelconques) est équivalent à zero; et il en est de même des produits analogues et de leurs combinaisons linéaires. Les produits de la forme (2) sont ce que j'appellerai des produits trinômrnes. La différence de deux monômes qui ne diffèrent que par l'ordre de deux facteurs consécutifs est équivalente à un polynôme de degré moindre. Soient en effet X et Y ces deux facteurs consécutifs. Nos deux monômes s'écriront PXYQ, PYXQ, P et Q étant deux monômes quelconques, et leur différence P[XY- YX]Q sera équivalente à P(XY) Q, dont le degré est d'une unité plus petit, puisque (XY) est du 1er degré, XY-YX du 2d degré. Soient maintenant M et M' deux monômes équipollents quelconques, c'est-a-dire ne différant que par l'ordre des termes. On pourra trouver une suite de monômes M, Mi, M2,..., Ip, M', dont le premier et le dernier sont les deux monômes donnés et qui seront tels que chacun d'eux ne diffère du précédent que par l'ordre de deux facteurs consécutifs. La différence M- M' qui est la somme des différences M- M1, M - M2,..., M- M' sera donc encore équivalente à un polynôme de degré moindre. Plus généralement, la différence de deux polynômes équipollents est équivalente à un polynôme de degré moindre. Je dis maintenant qu'un polynôme quelconque est toujours équivalent à un polynôme régulier. Soit en effet Pn un polynôme quelconque de degré n; il sera équipollent à un polynôme régulier P',; on aura alors l'équivalence: Pn = P'n + Pn- 1, où Pn-_ est un polynôme de degré n - 1 qui sera à son tour équipollent à un polynôme régulier P'n_,, d'où l'équivalence: Pn-1 -= pn-1 J- Pn-2, VOL. XVIII. 29 226 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. et ainsi de suite; on finira par arriver à un polynôme de degré zéro, de sorte que nous pouvons écrire l'équivalence: P1n = P'n + P'-i + 'Ln-2 +......* dont le second membre est un polynôme régulier. On a donc un moyen de réduire un polynôme quelconque à un polynôme régulier en se servant des relations (1). Il reste à rechercher si cette réduction ne peut se faire que d'une seule manière. Le problème peut encore se présenter sous la forme suivante; un polynôme régulier peut-il être équivalent à zéro? Ou bien encore peut-on trouver une somme de produits trinômes de la forme (2) P [X Y - YX - (XY)] Q, qui soit un polynôme régulier non identiquement nul? Toutes les sommnes de pareils produits sont en effet équivalentes à zéro. Le degré d'un produit trinôme sera égal à 2 plus la somme des degrés des polynômes P et Q. Si je considère ensuite une somme S de produits (2), ce que j'appellerai le degré de cette somme S, ce sera le plus élevé des degrés des produits qui y figurent, quand même les termes du degré le plus élevé de ces différents produits se détruiraient mutuellement. Le produit trinôme (2) peut être considéré comme la somme de deux produits, le produit binôme (2 bis) P[XY- YX]Q, où je distinguerai le monôme positif PXYQ et le monome négatif - PYXQ; et le produit -P(XY)Q, que j'appellerai le produit complémentaire. Soit donc S une somme quelconque de produits trinômes de degré p ou de degré inférieur; je pourrai écrire: S= Sp- Tp + Sp_ - Tp_- +...... + S2 - T2, où Sk est une somme de produits binômes de degré k. (2 ter) P[XY- YX] Q, tandis que - Tk est la somme des produits complémentaires correspondants: -P (XY)Q. Il s'agit de savoir si la somme S peut être un polynôme régulier sans être identiquement nulle. J'observe d'abord que si S est un polynôme régulier, il doit en être de même de S,; car Sp représente l'ensemble des termes de degré p dans S; tandis que (S~_ - Tp), (Sp_- - Tp_),..., (S2 - T3), - T1 représentent respectivement l'ensemble des termes de degré p - 1, p - 2,..., 2, 1. M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 227 On voit immédiatement que Sp est équipollent à zéro; comme zéro est un polynôme régulier, et que deux polynômes réguliers ne peuvent être équipollents sans être identiques, il faut que Sp soit identiquement nul. Soit en particulier p = 3, S3= [XY - YX] Z- Z [X Y- YX], le signe X signifie que l'on fait la somme du terme qui est explicitement exprimé sous ce signe et des deux termes qu'on en peut déduire en permutant circulairement les trois lettres X, Y, Z. On aura: T3= (XY) Z- z (XY), puis = [(X Y)Z- Z (XY)], 2 = [(XY) Z], S= S3- T3+S2-T2= E [XY- YX-(XY)] Z-EZ[XY- YX-(XY)] + ` [(XY) Z- Z(XY) - ((XY) Z)]. Il est aisé de vérifier que S3 et S -T3 sont identiquement nuls, de sorte que S se réduit à - T. Or T, = [(XY)Z] + [(YZ) X] + [(ZX) Y] est un polynôme du ier degré, car [(XY)Z] comme (XY) lui-même est un polynôme du 1er degré. Or dans un polynôme du 1er degré, chaque terme ne contenant qu'un seul facteur, on n'a pas à se préoccuper de l'ordre des facteurs. Tout polynôme du 1er degré est donc un polynôme régulier. Si donc le polynôme T, n'est pas identiquement nul, la somme S sera égale à un polynôme régulier qui ne sera pas identiquement nul. Donc pour qu'un polynôme puisse être réduit d'une seule manière à un polynôme régulier il faut qu'on ait les identités suivantes: (3) [(XY) Z] + [(YZ) X] + [(ZX) Y] = 0. On reconnaît là les identités de Jacobi qui jouent un si grand rôle dans la théorie de Lie. (Si d'ailleurs ces identités n'avaient pas lieu, les opérateurs élémentaires seraient liés par les équations (3) qui ne seraient plus des identités; ils ne seraient plus linéairement indépendants; on pourrait donc en réduire le nombre.) Les identités (3) sont donc la condition nécessaire pour que la réduction d'un polynôme à un polynôme régulier ne puisse se faire que d'une seule manière. Il me reste à montrer que cette condition est suffisante. 29-2 228 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. Je dirai pour abréger une somme régulière pour désigner une somme de produits trinômes qui est un polynôme régulier. Soit alors S = Sp- Tp + Sp_ - Tp +.. une somme de produits trinômes; les deux premiers termes Sp - Tp représentent la somme des produits trinômes du degré le plus élevé, c'est ce que j'appellerai la tête de la somme S. J'ai distingué plus haut dans un produit trinôme trois parties que j'ai appelées le monôme positif, le monôme négatif et le produit complémentaire. Je dirai qu'une somme de produits trinômes forme une chaîne si le monôme négatif de chaque produit est égal et de signe contraire au monôme positif du produit suivant. Le monôme positif du premier produit et le monôme négatif du dernier seront alors les monômes extrêmes de la chaîne. Il résulte de cette définition que tous les monômes positifs d'une même chaîne ne diffèrent que par l'ordre de leurs facteurs. Une chaîne sera fermée si les deux monomes extrêmes sont égaux et de signe contraire. Si Sp- Tp est une chaîne fermée de produits trinômes (Sp représentant la some des produits binômes et -Tp celle des produits complémentaires), il est clair que Sp est identiquement nul puisque les monômes positifs et négatifs se détruisent deux à deux. Nous avons vu que si S est une somme régulière, Sp est identiquement nul, d'où il résulte que la tête d'une somme régulière S se compose toujours d'une ou plusieurs chaînes fermées. Si deux chaînes ont mêmes monômes extrêmes, leur différence est une chaîne fermée. Nous nous servirons de cette remarque pour montrer qu'une chaîne fermée peut toujours de plusieurs manières se décomposer en deux ou plusieurs chaînes fermées. Une chaîne fermée quelconque peut de plusieurs manières être regardée comme la différence de deux chaînes C et C' ayant mêmes monômes extrêmes; soit alors C" une troisième chaîne ayant mêmes monômes extrêmes. La chaîne fermée C-C' se trouve ainsi décomposée en deux autres chaînes fermées - C" et C" - C'. Il s'agit de montrer que toute sonmme régulière est identiquement nulle et en effet quand cela aura été démontré, il sera évident qu'un polynôme régulier dont tous les coefficients ne seront pas nuls ne pourra être équivalente à zéro, puisque tout polynôme régulier équivalent à zéro est par définition une somme régulière. Supposons que le théorème ait été établi pour les sommes de degré 1, 2,..., p- 1; je me propose de l'étendre aux sommes de degré p. Je remarque d'abord que si une somme régulière de degré p est identiquement nulle, il en sera de même de toutes les sommes régulières de degré p qui ont même tête. La différence de ces deux sommes serait en effet une somme régulière de degré p- 1 qui serait identiquement nulle d'après notre hypothèse. M. H. POINCARÉE, SUR LES GROUPES CONTINUS. 229 Il me suffira donc de former toutes les chaînes fermées de degré p et de montrer que chacune d'elles peut être regardée comme la tête d'une somme régulière identiquement nulle. Toute somme régulière S d'ordre p a en effet pour tête une de ces chaînes fermées, par exemple S'; si donc je montre que l'une des sommes régulières dont la tête est S' est identiquement nulle, il en sera de même de toutes les autres et en particulier de S. Pour établir ce point, je vais décomposer la chaîne fermée envisagée en plusieurs chaînes fermées composantes. Il me suffira de démontrer la proposition pour chacune des composantes. J'appellerai chaîne simple de la 1è'e sorte toute chaîne où le premier facteur de tous les monômes soit positifs soit négatifs sera partout le même. J'appellerai chaîne simple de la 2e sorte toute chaîne où le dernier facteur de tous les monômes sera partout le même. Une chaîne simple peut d'ailleurs être ouverte ou fermée. Il est évident que toute chaîne fermée peut être regardée comme la somme d'un certain nombre de chaînes simples, alternativement de la 1ère et de la 2e sorte. Soit donc S une chaîne fermée, Cl, C2,..., Cn, des chaînes simples de la 1e'" sorte, C/', C',..., C', des chaînes simples de la 2e sorte, on aura: S = G C', + + '+ +... + Oî, + c',, le monôme négatif extrême de chaque chaîne étant bien entendu égal et de signe contraire au monôme positif extrême de la chaîne suivante, et le monôme négatif extrême de C'n égal et de signe contraire au monôme positif extreme de CI. Soit X le premier facteur de tous les monômes de CI, Z le dernier facteur de tous les monômes de C'0, Y le premier facteur de tous les monômes de C,, T le dernier facteur de tous les monômes de C', (je n'exclus pas le cas où deux des opérateurs X, Y, Z, T seraient identiques). Soit alors C" une chaîne simple de la 2e sorte ayant son monôme positif extrême égal et de signe contraire au monôme négatif extrême de C',; dont tous les monômes ont pour dernier facteur T; et dont le monôme négatif extrême a pour premier facteur X. Soit C'" une chaîne simple de la 1è'" sorte dont tous les monômes ont pour premier facteur X et dont les monômes extrêmes sont respectivement égaux et de signe contraire au monôme négatif extrême de G" et au monôme positif extrême de C,. La chaîne fermée S se trouvera décomposée en deux chaînes fermées composantes, à savoir: S' = (C" +,) C + CG + (CG 2 + C"), S"=- G" +,C', +... + C '+ - C '". 230 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. S' ne contient que quatre chaînes simples; car (C"'+ C1) et (C' + C") sont des chaînes simples; S" contient deux chaînes simples de moins que S. En poursuivant on finira par décomposer S eni chaînes fermées composantes, formées seulement de quatre chaînes simples. Il nous suffit donc d'envisager les chaînes fermées formées de quatre chaînes simples comme par exemple S'. Les monômes positifs extrêmes des quatre chaînes simples qui forment S' ont respectivement pour premier et dernier facteurs: pour C"'+ C1, X et T,,, C', X et Z,, C, Y et Z, C', + C", Y et T. Soient M1, M'1, M2, M'2 ces quatre monômes. Tous ces monômes sont équipollents entre eux et équipollents à un certain monôme que j'appellerai XYPZT. Nous allons alors construire une série de chaînes simples, comprises dans le tableau suivant, où dans la première colonne se trouve la lettre qui désigne la chaîne, dans le seconde le monôme extrême positif, dans la troisième le monôme extrême négatif; je fais figurer dans le même tableau les quatre chaînes simples qui forment S' et je pose pour abréger: Q = XYPZT; Q'1= XYPTZ; Q, = YXPTZ; Q' = YXPZT: Nom de la chaîne Monôme positif Monôme négatif Nom de la chaîne Monôme positif Monôme négatif C"'+ Cl M, -MD M2 -Q2 GC'"M M' -M, D M' -Q2 C2 M -M' E, Q1 -Q'I C' M', -M, E', Q1 -Q D M, -Q E2 Q2 -Q 2 1D)_ M'1 - Q' E'2 Q'2 - Q1 On peut supposer que tous les monômes de la chaîne -D ont pour premier et dernier facteurs X et T; de sorte que D1 est à la fois une chaîne simple de 1-èr sorte et une chaîne simple de 2e sorte. Il en est de même des autres chaînes D. On peut supposer de plus que les chaînes E se réduisent à un seul produit trinôme de manière que par exemple: E, = X YP [ZT- TZ- (ZT)]. La chaîne fermée: S' = ("' + C)+ C' + C+ C' M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 231 peut être décomposée en cinq chaînes fermées composantes, à savoir: U =C"' + C + D'-El-Di, U'1 = C' + D2 - E' - D'1, U2 = C2 + D,2 - E2-D2, U', = C'2 + C" + D1 - E' - D', V= E, + E' - + E,+ ',. I1 s'agit donc de montrer que chacune de ces cinq chaînes fermées est la tête d'une somme régulière identiquement nulle. Pour les quatre premières, qui sont des chaînes simples fermées, le théorème est évident. On l'a supposé démontré, en effet, pour les chaînes fermées d'ordre inférieur à p. Or il est clair que l'on a par exemple: U,= XH, H étant une chaîne fermée d'ordre p- 1. Quant à V, ce sera la tête de la somme régulière [XY- YX - (XY)] PZT+ YXP [ZT- TZ- (ZT)] - [XY - YX - (XY)] PTZ - XYP [ZT- TZ - (ZT)] + (XY) P [ZT- TZ- (ZT)] - [X Y- YX - (XY)] P (ZT), qui est identiquement nulle. Il reste à envisager ce qui se passe quand deux des opérateurs X, Y, Z, T sont identiques, par exemple X = Y, ou X = Z. Nous devons alors distinguer le cas où les divers monômes positifs ou négatifs de notre chaîne contiennent deux facteurs identiques, l'un jouant le rôle de X et l'autre le rôle de Y (ou l'un le rôle de X et l'autre celui de Z); il n'y a alors rien à changer à l'analyse qui précède. Et d'autre part le cas où ces monômes ne contiennent qu'un seul facteur X. Le premier cas pourra seul se présenter si l'on suppose X = Z, ou X= T, et s'il y a plus de trois facteurs en tout. Le second cas pourra au contraire se présenter si l'on suppose par exemple X = Y; mais on posera alors: Q1= Q'2 = XPZT; Q', = Q2 = XPTZ. La définition des diverses chaînes demeurera d'ailleurs la même et on constatera immédiatement que la chaîne V est identiquement nulle. Le théorème est donc démontré pour les sommes d'ordre p, s'il l'est pour les sommes d'ordre moindre. La demonstration précédente n'est toutefois pas applicable au cas de p=3; car la 232 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. chaîne V n'existe que s'il y a au moins quatre facteurs. Mais la seule chaîne fermée du 3e ordre est la chaîne S3 - T envisagée plus haut et nous avons vu qu'elle est la tête d'une somme régulière qui est identiquement nulle si les identités (3) ont lieu. Le théorème est donc établi dans toute sa généralité. Toute somme régulière est identiquement nulle. Donc un polynôme régulier qui n'est pas identiquement nul ne peut pas s'annuler en vertu des relations (1). Donc en résumé, Si les identités (3) ont lieu, les relations (1) permettent d'une manière, et d'une seule, de réduire un polynôme quelconque àc un polynôme régulier. IV. PROBLÈME DE CAMPBELL. Soient Xl X2i * X Xr r opérateurs élémentaires; supposons qu'ils soient liés par les relations (1) XaXb - XbXa = (XaXb), (XaXb) étant une combinaison linéaire des Xk; supposons de plus qu'on ait les identités (3) ((XAXb) Xc) + ((XbX) Xa) + ((Xc) Xb) = O. D'après le deuxième théorème de Lie, ces opérateurs donnent naissance à un "groupe continu," qui admet r transformations infinitésimales indépendantes. Ces transformations infinitésimales changent f en f + ^x (f), e étant une constante infiniment petite. Soit T = tX~ + t2X9 +... + trXr, une combinaison linéaire de ces opérateurs. Les tk sont des coefficients constants quelconques. La transformation finie la plus générale du groupe s'exprimera par le symbole: e (f). Soient maintenant T = tlX + t2X2 +... + tXr, V = v 4X + v2X2 +. + VrXr, deux combinaisons linéaires des X. Comme les transformations eT forment un groupe, le produit eVeT devra également faire partie du groupe, de sorte que nous devrons avoir: (4) eVeT=ew, M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 233 où W = WlXl + W2X2 +... + wrXr est une autre combinaison linéaire des X. Les coefficients w sont évidemment des fonctions des v et des t. Développons le produit: VmTn eveT= -- mz! n! VmTn Le terme général - est un polynôme d'ordre m + n. Par les relations (1) on m! n! peut le réduire à un polynôme régulier, et cette réduction ne peut se faire que d'une seule manière. Nous pouvons donc écrire: VmTn = Y, W (p, m, n), m!n! où W(p, m, n) est un polynôme régulier et homogène d'ordre p (p m + n); on a donc: eve% = p.. n W (p, m, n). Si nous réunissons les termes de même degré et que nous posions Wp =,%.nr(p, a, ), il viendra: eVeT= p wp. Le second théorème de Lie, que je viens de rappeler, nous apprend que le second membre doit être de la forme ew, et par conséquent que: WP (5) Wp= W1. C'est là une proposition dont la simplicité serait inattendue, si l'on ne connaissait pas la théorie des groupes. Si on pouvait la démontrer directement on aurait, comme l'a remarqué Campbell, une nouvelle démonstration du second théorème de Lie. Mais il y a plus; on aurait aussi une nouvelle démonstration du troisième théorème de Lie. Les égalités (1) nous font connaître des relations entre les opérateurs élémentaires et les combinaisons XY- YX; ce sont ces relations qui constituent la structure du groupe. Cette structure est donc entièrement définie quand on connaît les r3 coefficients c des r2 fonctions linéaires (XY). Mais ces r3 constantes c ne sont pas toutes indépendantes; tous les coëfficients de (XX) doivent être nuls; les coëfficients de (YX) sont égaux et de signe contraire à VOL. XVIII. 30 234 M. H. POINCARE, SUR LES GROUPES CONTINUS. ceux de (XY). Enfin les constantes c doivent être choisies de telle façon que les identités (3) soient satisfaites. J'adjoins donc aux identités (3) les identités suivantes qui sont evidentes: (3 bis) (XX)=O, (X Y)= - (YX). Le 3e théorème de Lie nous apprend qu'on peut toujours trouver un groupe de structure donnée; pourvu que les coefficients c qui définissent cette structure satisfassent aux identités (3) et (3 bis), c'est-à-dire aux identités de Jacobi. Mais supposons inversement qu'on ait démontré directement l'identité (5) et par conséquent la formule (4). Les coëfficients w seront donnés en fonctions de v et de t; et je puis écrire: (6) Wkl= Ok (Vi, ti)Pour former les fonctions dk, il suffit de savoir former le polynôme TW, par conséquent de savoir former les polynômes W (p, m, n); c'est-à-dire de savoir réduire un polynôme quelconque en polynôme régulier; pour cela il suffit de connaître les coefficients c. Soit eVeT'= eW; eweu=eZ; eTeu= eY; où U=C-UkXk, Z= ZkX`Xk, Y = ykXkLe caractère associatif de nos opérateurs nous montre que l'on a: eveY= eZ d'où les relations suivantes: Wk = k (Vi, ti); yk = (ti, Ui). (7) Zk = k (Wi, i) = k (Vi, yi). Regardons dans les équations (6) les t comme des constantes; ces équations (6) définiront une transformation qui transforme vi, v2,..., v,. en w,, w2,..., w,.. Les relations (7) nous enseignent que l'ensemble de ces transformations constitue un groupe. (C'est ce que Lie appelle la Parametergruppe.) Les substitutions infinitésimales de ce groupe sont: Xi (f)= L cl d- P 'dvk dti' où dans ok(vi, ti) on annule les t après la différentiation. Les r substitutions infinitésimales Xi(f) sont linéairement indépendantes. Et en effet, pour qu'elles ne le fussent pas, il faudrait que le déterminant fonctionnel des bk par rapport aux t fût nul, quels que soient les v quand les t s'annulent. Or cela n'a pas lieu car ce déterminant devient égal à 1 quand les v s'annulent. Ayant ainsi défini les opérateurs élémentaires X? (f), leurs combinaisons T -= StX (f), eT, etc. se trouvent définis eux-mêmes. M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 235 Ces opérateurs étant associatifs, on aura eY(f) = eTe (f), c'est-à-dire, en négligeant les quantités du 3e ordre par rapport aux t et aux u: YTU- UT Y=T+U+ 2 D'autre part, d'après la manière dont ont été formées les fonctions bk, on vérifie que Y= T + U + 2 (TU) = Yti Xi + 2UkXk +4 1 2 (tfk - tkui) (Xi Xk), et la comparaison de ces deux identités donne: XiXk - XkXi = (XiXk), où les coefficients des fonctions linéaires (XiXk) sont bien les r3 coefficients c donnés. Le groupe ainsi formé a donc bien la structure donnée et le troisième théorème de Lie est démontré. C'est au fond la démonstration de Schur. Ce que j'appellerai le problème de Campbell consiste donc à démontrer directement la formule (5), ce qui démontre à la fois le second et le troisième théorème de Lie. V. LE SYMBOLE 5 (0). Considérons r opérateurs élémentaires X1, X2,..., X., et une de leurs combinaisons linéaires: T = tIX + t2X2 +...+ tXr. Soit ensuite V un autre opérateur élémentaire qui pourra être ou ne pas être une combinaison linéaire des opérateurs X. Supposons que les opérateurs V et X soient liés par des relations de la forme: VXY- XV = bi. X, + bi.,X2 +... + bi. Xr (ti=l, 2,.... r), on aura alors: VT - TV= ZukXk, où U]k = bi.kti. Je poserai VT- TV= 0(T). Donc O (T) est comme T une combinaison linéaire des X; et les coefficients de O(T) se déduisent de ceux de T par une substitution linéaire. 30-2 236 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. Je poserai 0 [0 (T)] = 02 (T), 0 [Om ()] = 0m+1 (T) de sorte que Om (T) sera comme T une combinaison linéaire des X, les coefficients de dm (T) se déduisant de ceux de T en répétant qm fois cette même substitution linéaire. Si maintenant (0)= 2gkOk.est un polynôme ou une série ordonnée suivant les puissances croissantes de 0, j'écrirai: (0) (T) au lieu de -gkOk (T). Considérons l'équation, dite caractéristique: bl - 0, b2,..., blr (1) b22 - 0,.., =. ( 0..................................... br,..,..., br- 0 Si cette équation a toutes ses racines distinctes et si ces racines sont 01, 02,.., il existe r combinaisons linéaires des Xi, à savoir: (2) Yk = aik Xi, telles que: VYk - YkV= Ok Yk. Si alors on a: T= tXtiX = Etk Yk, on aura: > (0) (T) = b0 (0k) tk Yk. Si nous posons: (O) (T)- = hiXi, nous voyons d'abord que les coefficients hi sont des fonctions linéaires des t; ce sont d'autre part des fonctions des b; étudions ces fonctions. Si b (0) est un polynôme entier d'ordre p en 0, les hi seront des polynômes entiers d'ordre p par rapport aux b. Si donc b (0) est une série ordonnée suivant les puissances de 0, les hi se présenteront sous la forme de séries ordonnées suivant les puissances des b. Nous allons voir bientôt quelles sont les conditions de convergence de ces séries. Des équations (2) on tire en effet: Xi = /3Sik Yk, d'où: tk = - ikti, t) (0) (T) =: (ok) t'kaikXi, M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 237 d'où enfin: hi = Etjo (0k) * aik ' 3jk. Pour déterminer les produits aik3jk faisons t étant une constante quelconque. On a alors: - o (T)= lhXi = H, où hi tjOtia, jk h- -k On tire de là ( - (H) = T, ce qui peut s'écrire: -hi- Zbkihk= ti. De ces équations on peut tirer les h en fonctions des t; on trouve: (3) hi= (tP, où Pj est un polynôme entier par rapport aux b et à t; quant à F(e) c'est le premier membre de l'équation (1) où 0 a été remplacé par e. Le second membre de l'équation (3) est une fraction rationnelle en À; décomposons la en éléments simples; il viendra: hi = Y, J F (k) (k)- ) où Pk est ce que devient Pi, quand on y remplace t par 0k. On a donc: aik -jk = F ) d'où enfin pour une fonction b (0) quelconque: (4) P(0) 5(T tj = jk p (0k) X F'(0k) On voit que les hi s'expriment rationnellement en fonctions des b, des 8k et des (b(0k). La formule (4) peut se mettre sous une autre forme; nous pouvons écrire: 1 (ofS) () ~CtjPoiX (4 bis) (0) (T) = 2i1 d (t) tJ l'intégrale étant prise dans le plan des t le long d'un cercle de rayon assez petit pour que la fonction J (O) soit holomorphe à l'intérieur; nous le supposerons de plus assez grand pour que les points 01, 02,..., 0r soient à l'intérieur du cercle. Cela nous amène 238 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. à supposer en même temps que le rayon de convergence de la série (:) est plus grand que le plus grand module des quantités 01, 02,..., r. On a alors pour tous les points du contour d'intégration: | |> e tl, > |> 0,..1 >., | >...r, d'où il résulte que la fonction rationnelle P7j F(t) est développable suivant les puissances croissantes des b. Il en est donc de même des hl. Nous avons dit plus haut que les hi sont développables en séries procédant suivant les puissances des b; et d'après ce qui précède, il suffit, pour que ces séries convergent, que le rayon de convergence de la série s (f) soit plus grand que la plus grande des quantités 1 | 0 [, | |,......., | 1r Si donc b (e) est une fonction entière, les hi seront des fonctions entières des b. Qu'arrive-t-il maintenant si l'équation caractéristique F (0)= O a des racines multiples? Il est aisé de s'en rendre compte en partant du cas général et en passant à la limite. Je suppose par exemple que 01 soit une racine triple. Alors F (t) contient le facteur (t-0)3. Si je décompose le second membre de (3) en éléments simples, trois de ces éléments deviendront infinis pour y= 01. Soient Al(i) A2(i) A3(i) + ces trois éléments simples. Alors il faudra dans la formule (4) remplacer le terme: F/ ('1) (qui n'aurait plus de sens dans le cas d'une racine multiple) par les trois termes suivants: 2A1(i> Xix (0) - (1!) SA,2() X ' (0,) + (2!) 2A3() Xib" (01). On opérerait de même pour les autres racines multiples. Donc les hi, dans le cas des racines multiples, sont des fonctions rationnelles des b, des Ok, des (0Ok) et de leurs dérivés '(0k),./"(0k),.....; on pousse jusqu'à 4(P>(0k) si 0k est une racine multiple d'ordre p+ 1. Remarquons que je n'aurais pu faire ce raisonnement par passage à la limite, si je m'étais restreint dès le début en supposant que V est une combinaison linéaire des M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 239 X, et que les X sont liés par les relations (1) et (3) du N~ IV. (relations de structure et identités de Jacobi). Alors en effet les cas où l'équation caractéristique a des racines multiples ne pourraient plus être regardés comme des cas particuliers de ceux où toutes les racines sont distinctes. On aurait pu, il est vrai, démontrer directement la formule (4 bis) et se servir de cette formule; mais j'ai préféré ne pas m'imposer au début cette hypothèse restrictive, quitte à l'introduire dans la suite du calcul, de façon à avoir le droit de raisonner par passage à la limite. Quoi qu'il en soit, le cas le plus intéressant au point de vue des applications à la théorie des groupes, c'est celui où cette hypothèse restrictive est satisfaite. Supposons donc que V soit une combinaison linéaire des X: V = V1X1 + V2X9 +... + VrX.. Supposons de plus que les X soient liées par les relations (1) du N~ précédent XiXj - XjXi = =CijsXs) et que les constantes c satisfont à des relations telles que les identités (3) du N~ précédent aient lieu. On aura alors: 0 (T)= CiisVitjX, d'où: bti.î =CiL.,V + C2.-i-kV +. + C -ik,.v Les résultats, démontrés dans le cas général, seront évidemment encore vrais dans ce cas particulier; si donc on pose: ( (0) (T)_= hiXi, les hi seront des fonctions linéaires des t, et des fonctions rationnelles des v, des 0k, des ( (0k) et de quelques unes de leurs dérivées. Les 0k sont les racines d'une équation algébrique dont le premier membre est un polynôme entier homogène de degré r par rapport aux v et à l'inconnue 0. De plus les hi ne dépendent que linéairement des p (0k) et de leurs dérivées. Si ~ () est une fonction entière de:, les hi sont des fonctions entières des v. Dans tous les cas, le symbole p (0)(T) se trouve entièrement défini. Je terminerai par deux remarques: 1~. Si, (:) est le produit des deux fonctions <p () et r(tr), on aura: < (0) () [ (T)] = (0) [< (0) (T) = ( ) (T). 2~. Si on a: (o) (T)= U, on aura: Cette dernière égalité n'a de sens que si <p (:) ne s'annule pas pour = 0, de telle facon que <p7 () soit développable suivant les puissances de 0. 240 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. VI. FORMULES FONDAMENTALES. Considérons l'expression (1) e-aVeT eaV, V et T ayant même signification que dans le ~ précédent, tandis que a et 3 sont des constantes très petites. Développons cette expression en négligeant les termes du 3e ordre par rapport à a et à f; il viendra: (1- iav2~ ) (î+fT i+ 2 )(~V+a 4+ 2 2) OU fiT2 1 + -3 c- -af3 (VT- TV), ou avec la même approximation: eg T-a (VT- TV). On aura donc, toujours avec cette approximation: (2) e-'vePTeVeuer, où U= T-aO(T), ou encore avec la même approximation: (2 bis) e-eV egTeoV= e u, où U=e-ao (T). Je me propose maintenant de démontrer que la formule (2 bis) est vraie quelque loin que l'on pousse l'approximation; et d'abord qu'elle est vraie quand on néglige le carré de f et qu'on pousse l'approximation par rapport à a aussi loin que l'on veut. Supposons donc qu'on pousse l'approximation jusqu'aux termes en / et jusqu'aux termes en am inclusivement. Dans l'expression (1) nous remplacerons ePT par I + fT, eav et e-~v par les m + premiers termes de leurs développements; en effectuant le produit (et négligeant dans ce produit arm+l) nous obtiendrons un polynôme symbolique que nous pourrons rendre régulier par les procédés du N~ III. Soit ^>(a, 3)= SAin, le polynôme régulier ainsi obtenu; II est un monôme symbolique, et A son coefficient qui est un polynôme entier en a et 3. Nous avons alors: (3) 0 (a + da, f) = e-(+d)> V ePTe(+~a) v= e-d v (a, f) ed. v. En effectuant le produit du 3e membre de cette double égalité, et négligeant le carré de la différentielle da, on obtiendra un polynôme régulier de même forme dont les coefficients sont eux-mêmes des polynômes du 1er degré par rapport à da d'une part, par rapport aux coSfficients A d'autre part. Telle est la forme du polynôme b (a + da, f). M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 241 D'autre part on a: dA (3 bis) b (a + da, /)- 0 (a,/) = d E -. da/ dA Cette égalité, rapprochée de la remarque que nous venons de faire, montre que dest une combinaison linéaire des coefficients A. Donc ces coefficients A, considérés comme fonctions de a, satisfont à des équations linéaires à coefficients constants. De plus pour a=O ils doivent se réduire aux coefficients de eT. Ces conditions suffisent pour les déterminer. Or je dis que l'on peut y satisfaire en faisant (conformément à la formule 2 bis): f (a, f) = el'; U = e-a (T). En effet cette formule nous donne: ( (a + da, f/) = ePu', U' = e-(a+da (T), et il s'agit de vérifier que: e- da. V elU eda. V e '. Or la formule (2 bis) démontrée quand on néglige d'une part le carré de fS, d'autre part le carré de a, peut s'appliquer ici puisque nous négligeons le carré de f8 et celui de da. Nous avons donc e-da. V eu eda. V=e", U"=e-da. (U), d'où: U" = e-da. 0e -aO (T)]= - a -0(T) = U'. On a donc bien: b (a + da, /) = e-' V ed~ eeda v= ee". C. Q. F. D. La formule (2 bis) satisfait donc à nos équations différentielles et comme ces équations ne comportent qu'une solution, cette formule se trouve vérifiée. Poussons maintenant l'approximation aussi loin que nous voulons tant par rapport aà f que par rapport à a. Nous avons: (a, )= e- aveTeav; d'où: f (a, f/ + dSi) = e-Ve(+dp)Tea = (e-aeeave) (e-aVedp. TeaV) OU (a, 3+d 3)= (a, /) (a, d3). Comme nous négligeons le carré de dfi, je puis écrire: O(a, d/.)= eda; U= e- (T); VOL. XVIII. 31 242 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. d'où: (4) c (a,+ ( /a )= (+, d) e=. E. Cette formule (4) représente sous forme condensée des équations différentielles de même forme que les équations (3 bis), auxquelles doivent satisfaire les coefficients A de À (a, B)= SA.1T. C'est ainsi que la formule (4) représentait sous forme condensée les équations (3 bis). On peut satisfaire à ces équations par la formule (2 bis); cette formule donne en effet: ( (a, 3 + d/) = e(+dP)U=eU eds e u=d (a, 3) ed. U. Les équations différentielles ne comportant comme les équations (3 bis) qu'une seule solution, la formule (2 bis) se trouve vérifiée dans tous les cas. Cette formule (2 bis) n'est d'ailleurs que la traduction symbolique d'une formule bien connue et, si j'ai développé la démonstration, c'est uniquement pour mieux faire comprendre les symboles employés et pour faire connaître un mode de raisonnement applicable à des questions analogues; je veux parler de celui où s'introduisent les équations différentielles (3 bis) ou les équations analogues. Il importe avant d'aller plus loin de préciser la portée de la démonstration que nous venons de donner. Pour qu'elle soit valable, il faut que tout polynôme puisse être réduit d'une manière et d'une seule à être régulier. Or, d'après le N~ III., cela a lieu dans deux cas. 1.~. Si V et T sont des combinaisons linéaires des opérateurs X, V = v Xi, T = XtiXi, et si ces opérateurs sont liés par des relations XiXk XkXi = $CikSXS, les constantes c satisfaisant aux identités (Xa (XbX,)) + (Xb (Xca)) + (X, (XXb)) = O; si en d'autres termes les opérateurs X définissent un groupe de Lie et si eaI, eT7 sont deux transformations quelconques de ce groupe: Dans ce premier cas la formule (2 bis) est toujours vraie. 2~. Elle sera donc vraie en particulier si on suppose que V; XI, X2,..., Xr sont r + 1 opérateurs liés par les relations (5) VX -XiV= _b.kX. M. H. POINCARE, SUR LES GROUPES CONTINUS. 243 et (6) XiXk-XkX = 0. Ces relations entraînent en effet l'identité (V (XiXk)) + (Xi (XkV)) + (Xk (VXt)) =, en désignant suivant la coutume par (VXi) et (XiXk) les seconds membres des relations (5) et (6). On aura donc dans cette hypothèse: (2 bis) e-aVesTer= eeU; U =e- (T). On aura de même en permutant V et T: (2 ter) e-TeaVeT= esW; W= e-P' (V), e-~ étant un symbole analogue à e-"o et défini de la manière suivante: le symbole v est formé avec T comme le symbole 0 avec T; on a donc, si Y est un opérateur quelconque: (Y)= TY- YT. On aura donc: V(V)= TV-VT=- (T), et en vertu des relations (6) (X)= 0; (V) =; ( V; ()= O, e-P (V)= V - r (V) = V+ O (T). La formule (2 ter) devient ainsi: (2 quater) e-OTeaVeT = eaV+as0(T). Si l'on suppose maintenant que les relations (5) subsistent, mais que les relations (6) n'aient plus lieu, les formules (2 bis) et (2 quater) cesseront d'être vraies quels que soient a et R. Cependant supposons que l'on regarde les opérateurs X comme très petits et qu'on en néglige les carrés; à ce degré d'approximation, les relations (6) dont les premiers membres sont du 2d ordre par rapport aux X se trouvent satisfaites d'elles-mêmes. Les relations (2 bis) et (2 quater) sont donc vraies, si l'on néglige les carrés des X, ou, ce qui revient au même, si l'on néglige le carré de T, ou encore si on néglige le carré de / (puisque T ne figure qu'affecté du facteur 8). Si donc V et les X sont r + opérateurs liés par les relations (5), les relations (2 bis) et (2 quater) ont lieu aux quantités près de l'ordre de f2. Au même degré d'approximation la formule (2 quater) peut s'écrire: eaV+aSo(T) = eV_ /aTeaV + eav/T, ou encore: eav+afîo(,)_ ev e TeaV + e~VePT, 31-2 244 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. ou en vertu de la relation (2 bis): eaV+apos(T)= eaV eaveu+ eaVeT; U=e-aO(T); ou, toujours en négligeant le carré de 8: eaV+ap (T) = eV(1 - 3U + /T) = e&Ve (T- U) Si nous posons: +aor(T)=W; T-U=Y; il vient: 1 -e- ~ (7) eaV+P = eaVeY; Y= e (W). Soit W= CwiXi une combinaison linéaire quelconque des Xi; peut-on déterminer les coefficients t de la combinaison T = -tiX de telle façon que l'on ait + a (T)= W? Cela est évidemment toujours possible si le déterminant des coefficients bik n'est pas nul. Dans ce cas la formule (7) est vraie quel que soit W. Si maintenant ce déterminant est nul, il suffit de partir du cas où ce déterminant n'est pas nul, de faire varier les coefficients b d'une manière continue de façon que ce déterminant devienne de plus en plus petit et de passer à la limite, pour démontrer que la formule (7) est encore vraie quel que soit W. Si enfin V, au lieu d'être un opérateur indépendant des X, n'est qu'une combinaison linéaire des X, la formule (7) est évidemment encore vraie, puisqu'elle ne peut cesser de l'être par suite de l'introduction de nouvelles relations entre nos opérateurs. Remarquons que ce raisonnement par passage à la limite n'aurait pas été possible, si nous nous étions restreints dès le début en supposant que V et T sont des combinaisons des opérateurs X, que les X définissent un groupe de Lie, que eaV et ePT sont deux substitutions finies de ce groupe de Lie. Dans ce cas en effet le déterminant des bik aurait été constamment nul. La formule (7) peut s'établir directement: En effet en négligeant le carré de 3 on a: (aV~C i + W)n a__-1 eav+Pw= = e +n ) e_ (V~ -VnW+ (Vn_-2WWV+ Vn-3WV2 n! ni +... + VWVn-2 + WVn-1) Or on trouve aisément Vn-i W + Vn-2 WVY... + WVn-i = Vn- W- v- " (W)\ Vn-3 02 ( n 1) 1 T V-(W- nO! (W), + * T-n-30 / -\ _. Yn-î W\T Q M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 245 d'où: eav+~W = ea + /: L (-p)!p! V- ( O)P- () ' n! ^ (n - p)p\! ou ou eav+PW = ea"(1 +/3Y) = e'Vey; Y= -- (W). C. Q. F. D. VII. FORMATION DES SUBSTITUTIONS INFINITÉSIMALES D'UN GROUPE DE STRUCTURE DONNÉE. Soient donc Xl, X2,..., X, r opérateurs élémentaires liés par les relations (1) XiXk - XkXi = (XiXk) = iksXs, les c étant des constantes telles que les identités de Jacobi du N~ III. aient lieu. Soient T.= -tiXi, U = uiXi, V = `viXi, W = WiX diverses combinaisons linéaires de ces opérateurs. Considérons le produit effectuons le produit qui sera une série de polynômes symboliques; réduisons chacun de ces polynômes à des polynômes réguliers en nous servant des relations (1); je me propose d'étudier la nouvelle série ainsi obtenue que j'appelle b (a, 3); le raisonnement sera le même que dans le N~ précédent, mais je le développerai un peu plus. Tous les termes de cette série (a, /3) sont des polynômes réguliers; et les coefficients de ces polynômes se présentent eux-mêmes sous la forme de séries développées suivant les puissances de a et de 3. Je puis ordonner (a, /3) suivant les puissances croissantes de /3, en groupant tous les termes qui contiennent en facteur une même puissance de /3. J'obtiens ainsi: (a, 3) = 0 ~+/3i +22 +... D'autre part j'ai: c (a, /3 + d/3) = eaVeTed.T= (a, (C ) e/)d.T = (a, /3)(1 + d/3. T), ou ou: OU: (3) mm-= m-_,.T; ces conditions jointes à (4) bo = eV suffisent pour déterminer <p. 246 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. Or on y satisfait de la manière suivante. Faisons: ) (a, /) = eW, (a-, /3 + d3) = eW+d; soit 7 un symbole qui soit à W ce que 0 est à V. Il s'agit de satisfaire à l'équation (2) ou ce qui revient au même à - (a, +d/3)= (a,/3)edP.T on doit donc avoir: eW+adW = eWed. T Or en vertu de la formule (7) du N~ précédent, on satisfera à cette condition si l'on a: (5) d3. T= (dW). Cette formule (5) représente symboliquement un système d'équations différentielles auxquelles doivent satisfaire les coefficients wi. En vertu de la formule (4 bis) du N~ V., ces équations peuvent s'écrire: (5 bis) tide = I f. le- ~=P, (i=l, 2,...,r). Si l'on a: WXi- XiW = $Ck. i.WkXs, F(J) est le déterminant dont l'élément est (pour la ie ligne et la se colonne) - (Ci.s.w8i + C2.i.sW2 +.. i. 'sw.), sauf les éléments de la diagonale principale (i=s) qui sont égaux à -(Cl.i.iZ.iw + C2.i.iW2 f... + Cr.i.iWi.) + t; les P/j sont les mineurs de ce déterminant. L'intégrale du second membre de (5 bis) est prise dans le plan des Ô, le long d'un contour fermé enveloppant toutes les racines de l'équation F()= 0. La condition (2) sera donc satisfaite, si les w satisfont aux équations (5 bis); la condition (4) le sera également si les valeurs initiales des w pour / =0 sont Wi = Vi.* Les équations (5 bis) admettant toujours une solution telle que pour / = O on ait wi=vi, et d'autre part les conditions (2) et (4) suffisant pour déterminer >, on aura: + (a, l) = ew, w = wiXi, les w étant des fonctions de /3 définies par les équations (5 bis) et les conditions initiales Wi = Vi. La série f (a, /3) n'est donc autre chose qu'une exponentielle dont l'exposant est M. H. POINCARE, SUR LES GROUPES CONTINUS. 247 une combinaison linéaire des Xi; c'est le théorème que j'ai annoncé au N~ IV.; et comme d'autre part ce théorème a été établi en s'appuyant simplement sur les relations (1) et en en faisant des combinaisons purement formelles, le problème de Campbell est résolu et le troisième théorème de Lie, en vertu de la remarque faite dans ce N~ IV., se trouve démontré: Il est aisé de se rendre compte de la forme relativement simple de ces équations (5 bis). Soient 1, l2,......, p les p racines distinctes de l'équation F(e))=0; ce sont des fonctions algébriques des w, puisque F () est un polynôme entier par rapport à t dwj et aux w. Les d- seront donnés par des équations linéaires dont les seconds membres dp/ seront des constantes; tandis que les coefficients des premiers membres seront des fonctions rationnelles des w, des k et des e-; ces coefficients ne dépendront d'ailleurs que linéairement des exponentielles e-t; ce seront des fonctions symétriques des racines. Résolvons ces équations par rapport aux d,' nous trouverons: dwj (6),= iAl.jti+ A2.jt2 +...... + A.jt,., les coefficients A étant rationnels par rapport aux w, aux ek et aux e-tk. Le problème qui se pose à propos du troisième théorème de Lie est ainsi complètement résolu. Il s'agit de trouver X opérateurs X (f,), X2 (f)....... X^ f), satisfaisant aux relations (1); on y satisfait en faisant Xdf+ df' df Xi (f ) = + Ai....... + A X * dw' d~ + dw<. Les équations (5 bis) peuvent se mettre sous plusieurs autres formes. Soit ZCk. i.sWk bi.s On aura (puisque les Pij sont les mineurs du -déterminant F): t:Pij- =bkiPkj = 0 pour i j et Pii - bkiPki = F pour i=j. Nos équations (5 bis) tid/3= =,__ d-P - j. F 248 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. donnent: I2 f 1-e- d/3itibik 2= -1 jd-. F Zjdwi4bikP, ' d'où 2w / 1 J F 2- / rJ La deuxième intégrale étant nulle, nous pouvons écrire tout simplement: (5 ter) Eitibik = 2w — I dl (1 -e- ) (k=1, 2,..., r). D'autre part l'équation (5) peut s'écrire: (7) -d 2 - (T), dp 1 - e-n d'où dwi= 1 f td ytjpi d,3 27r V.l - e-e F() ' ce qui donne: Xi (f- = f - p..- d Xt (/) 2 -V _ J (1-e-t)F(e) J ' dwj' Cette dernière intégrale doit être prise le long d'un contour enveloppant toutes les racines de F() = 0, mais n'enveloppant pas les points -=2k7r/ —l (k=+l, ~+2,... ad inf.). VIII. FORMULES DE VÉRIFICATION. Soit eV+V - eVeY V= viXx, SV= ESviX,, Y= SyiX; on aura en vertu de la formule (7) du N~ VI. 1 e- eT = o (yV) (posant: (T) = VT- TV comme dans le N~ V.). Soit maintenant e-vereV= eu, on aura par la formule (2 bis) du N~ VI. U = e0- (T). M. 1. POINCARÉ, SUR LES GROUPES CONTINUS. 249 Soit e-(V+sV) eTeV+V= eU', on aura: U-= e-(O+~) (T), où 6 + s0 est un symbole qui est à V+ V ce que 0 est à V. On aura d'autre part: eU' = e-Ye-eTeeY = e-YeeY. d'où en négligeant le carré de Y qui est infiniment petit: eU' =e - Yeu + eY= eU+UY- Y D'où U'- U= UY- YU. Si je conviens de poser: e-(O+ 0)- e- = (e-0), il viendra: T'- = 8(e-)(T). Nous arrivons ainsi à la formule symbolique suivante: (1) S(e-) T=-[e-(T)] L (ov1 - k' eV)1] [e-0(T)]. Pour mieux expliquer le sens de cette formule rappelons que nous avons trouvé plus haut: (2) < (O) (T)= 2w Jde (e) YhXi, où les hi sont des fonctions rationnelles des t, des v et des e données par les équations: (3) ehi - ibkihk = ti; bki = Ci.k. iVi + C,.k i,. +... + Cr.k. iVr. Alors on aura: Se-h (T) -, où les îhi sont les accroissements que subissent les fonctions hi quand les variables vk subissent les accroissements Îvk. Si alors les h'i sont ce que deviennent les hi quand on y remplace les tk par les Îvk, la formule (1) pourra prendre la forme (1 bis) 27rw-1: Xifde-ethi =Y(XiX-XkXi) X)fd^ e h'f de-e7hi. Dans le 1er membre le signe 2 se rapporte aux r valeurs de l'indice i; dans le 21 membre aux r(r-l) arrangements des deux indices i et k (l'arrangement i, k étant regardé comme différent de l'arrangement k, i). Cette formule nous fait connaître un certain nombre de relations auxquelles doivent satisfaire les expressions XiX k-XkXi ou (XiXk). Ces relations sont curieuses; mais VOL. XVIII. 32 250 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. la plupart ont déjà été démontrées par Killing et il semble que les autres pourraient se démontrer facilement par les procédés de Killing. Je n'y insiste donc que comme sur un procédé de vérification. Les deux membres de cette équation sont d'une forme particulière. Le premier nmembre est linéaire à la fois par rapport aux symboles Xi, par rapport aux ti, aux Svk, aux exponentielles e-0i (les Oi étant les racines de l'équation F=0). Les coefficients de cette fonction linéaire sont eux-mêmes des fonctions rationnelles des v et des Oi. Le second membre est linéaire à la fois par rapport aux symboles (XiXk), par rapport aux ti, aux Îvk, aux exponentielles e-0 et e-i-k (Oi et 0k étant deux racines de F=0). Les coëfficients de cette fonction linéaire sont encore rationnels par rapport aux v et aux Oi. Les Oi étant les racines de l'équation F = sont des fonctions algébriques des v. Dans les deux membres de l'équation (1 bis) entrent en outre linéairement un certain nombre de fonctions transcendantes; il y a d'abord les exponentielles e- i et il y en a autant que l'équation F= O a de racines distinctes. Il y a ensuite les exponentielles e-(Oi+k) qui peuvent être distinctes des précédentes, niais qui peuvent également ne pas en être toutes distinctes si l'une des racines de l'équation F= O est constamment égale a la somme de deux autres racines. Supposons qu'il y ait q exponentielles et soient el, e02,..., ei ces exponentielles. Les deux membres de l'équation (1 bis) seront alors des fonctions linéaires des produits de la forme (4) tonS îevh i, où rn et h peuvent prendre les valeurs 1, 2,..., r, et où /L peut prendre les valeurs 1, 2,..., q. Les coefficients de ces produits sont des fonctions algébriques des v, ne dépendant ni des t, ni des Av. Pour que l'identité puisse avoir lieu, il faut que l'on puisse égaler dans les deux membres de (1 bis) les coefficients d'un même produit (4). Nous aurons ainsi un certain nombre de relations linéaires entre les symboles Xi d'une part, les symboles (XiXk) d'autre part; les coefficients de ces relations linéaires sont des fonctions algébriques des v. Ces relations linéaires doivent être identiques aux relations de structure ou en être des conséquences. J'examinerai seulement le cas particulier où F (t) = O a toutes ses racines distinctes. Je puis alors supposer que les opérateurs élémentaires Xi ont été choisis de telle sorte que: VXi - Xi V = iXi, Oi étant l'une de ces racines. M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 251 Egalons alors dans l'équation (1 bis) les coefficients de t~,,,v; il vient: -1 r X fc-1 didtr = (Xm ) -XXh) e-0m. dvhdt — oh eLe premier membre ne dépend que des exponentielles e-0i, mais le second membre outre l'exponentielle e-~m contient encore e-"-^"m. Égalons les coefficients de e-0hm. Si Oh + Om n'est pas égal à une racine de F= 0, cette exponentielle ne figurera pas dans le 1er membre; nous aurons donc (XmXh) = o. On reconnaît là l'un des théorèmes de Killing. Si au contraire Oh + 0m est racine de F=O, l'exponentielle pourra figurer dans le 1er membre et (XmXh) pourra ne pas être nul. Je n'insisterai pas sur les autres vérifications, ni sur le cas où les racines ne sont pas distinctes et où on retrouverait les autres théorèmes de Killing. Je me bornerai à faire remarquer que la vérification de la formule (1 bis) n'est pas immédiate, et qu'il faut pour la faire avoir recours aux identités de Jacobi et aux théorèmes que Killing en a déduits. IX. INTÉGRATION DES ÉQUATIONS DIFFÉRENTIELLES ET FORMATION DES SUBSTITUTIONS FINIES DES GROUPES. Soit (1) eV+dV = eed4A où: V= viXi; dV= Zdvj. Xi; dA = Zda. X,. On aura en vertu de la formule (7) du No VI.: (2) dA = (cV). Cette formule, identique sauf les notations à la formule (5) du N~ VII., comprend, sous la forme symbolique, r systèmes d'équations différentielles; ainsi que je l'ai déjà fait remarquer au N~ VII. Annulons tous les da, sauf dck; égalons ensuite les coefficients de X1, X2,..., Xr dans la formule (2). Nous aurons r équations différentielles qui définiront dvl dv, dv,. dak' d 'a ' dCk en fonctions des v. Ce sont là comme nous l'avons vu au N~ VII., les équations différentielles qui définissent une des substitutions infinitésimales du groupe, si l'on prend les v comme variables indépendantes. 32 —2 252 M. H. POINCARÉ, SUR LES GROUPES CONTINUS. En donnant à l'indice k les valeurs 1, 2,..., r, on obtiendra r systèmes d'équations différentielles correspondant aux r substitutions infinitésimales du groupe. Nous devons prévoir que ces équations peuvent se ramener, au moins dans le cas des groupes de la P1re famille (vide supra NO I.), à des équations linéaires, puisque c'est là un résultat bien connu obtenu par Lie. Voici le changement de variables qu'il faudrait faire pour retrouver ces équations; soit: U=>u2iXi; e-VeUev=eL; L=2liX; on aura: (3) L = e- (U). Cette équation symbolique (3) nous apprend que les li sont des fonctions des v et des u, linéaires par rapport aux u, et nous permet de former ces fonctions. Si alors on pose: e- V-dVe UeV+dV = eL+dL on aura: eL+dL = e-dAeLedA ou, puisque A est infiniment petit: (4) dL = dA- dA. L. Cette formule (4) représente symboliquement r systèmes d'équations différentielles qui ne sont autre chose que ce que deviennent les r systèmes d'équations différentielles représentées symboliquement par la formule (2) quand on prend les li pour variables nouvelles. Celui de ces systèmes que l'on obtient en annulant tous les dc sauf dak s'écrit: dL (4 bis) dL = LXk- XkL. Ces équations sont linéaires et à coefficients constants et s'intégrent immédiatement; ce sont celles auxquelles Lie arrive par la considération du groupe adjoint. Il importe de remarquer que la réduction des équations différentielles (2) aux équations (4) par le changement de variables (3) n'est pas immédiate et qu'on ne peut la faire qu'en tenant compte des identités de Jacobi. Considérons de plus près le cas des groupes de la 2e famille. Nous pourrons alors choisir les opérateurs élémentaires Xi de telle manière qu'on en puisse distinguer de deux classes. Ceux de la 2de classe seront permutables à tous les opérateurs, ce seront les X"i; quant à ceux de la 1ère classe que j'appellerai les X'i, ils seront caracterisés par la propriété suivante: aucune combinaison linéaire des X'i ne sera permutable à tous les opérateurs. Pour mettre en évidence cette distinction, j'écrirai quand il y aura lieu: viXi = v' X'i + v"iX"; V' = v',X'; V" = v"X"; V = V' + V". M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 253 Les v'i seront ainsi les coefficients des X'i et les v"i ceux des X"i. Les lettres Ui, u.i; Vi, i;, U"; L, L"; etc. auront une signification analogue. Il est clair qu'on aura: V"T - TV" = V'T" - T" V' = 0, d'où 0(T)= VT-TV= V'T'-T'V'. J'introduis alors un symbole nouveau; soit: 'T -T'V' V= -X',X' + XX"; je poserai: 6' (T) = YXAX'i; 0" (T)= X"X', et je définis (0') à l'aide de 0' comme j'ai défini <p(0) à l'aide de 0. On a alors: (x",= o; O[O"(T)]=o; ~[(O)(T")=O; et on trouve aisément: p (0) (T) = < (0) (T') = < (0') (T') + 0" () ( 0) (T')] + (0) T". Remarquons que les expressions: 0 (T), O'(T) (T), dépendent des v' et des t' mais sont indépendantes des v" et des t"; et il en est de même de < (0).(T) si < (0) est nul. Les li étant linéaires par rapport aux iu, je puis écrire: ~,dli i = dUl ^d=li-Uk Les dl- sont des fonctions des v. Voyons combien de ces fonctions sont indépendutk dantes les unes des autres. Je dis d'abord que ces fonctions ne dépendent que des v'. Nous avons en effet (eZ étant une substitution quelconque du groupe): eV'+V" = eV'eV", e-V"eZeV" = ez, d'où eL = e-V-eUeV'- V' e"e -V''" - 'e-'euee = e v'eUeV ' ce qui montre que L ne depend que de V', mais pas de V". dl Je dis maintenant que le nombre des fonctions ~- indépendantes les unes des autres est précisément celui des variables v'. En d'autres termes, si l'on pose: eL = e-VeueV, eL1 = e-VeUev, 254 M. H. POINCARE, SUR LES GROUPES CONTINUS. l'identité L =L1 si elle a lieu quel que soit U entraîne l'identité V' = V1. Si en effet L=LI, on aura quel que soit U: eV e-veUeve —v = eu, ce qui montre que eve-vl est permutable à toutes les substitutions du groupe. C'est donc une substitution qui ne dépend que des X"i de sorte que je puis écrire: eVe-V = e"', W" étant une combinaison linéaire des X"i; on en tire: eV= eTr'eV = el+ W", d'où V = V1 + W", V' = V '1; V"= V"' + T". Donc V'= V'. C. Q. F. D. dl Nous pourrons prendre comme variables les cd et les v", au lieu des v' et des v". Les d- sont définis par les équations (4 bis), qui étant par rapport à ces variables des équations linéaires à coefficients constants s'intégrent immédiatement. Les équations (4 bis) nous font donc connaître les d — et par conséquent les v' en fonctions de la variable ak. Pour obtenir les v", revenons aux équations (2); si nous posons: 1 - e- =0+ 02+ (0), elles peuvent s'écrire: dA' =dV' +0' r (0")(dV'), dA" = dcV" + 0"// (0') (dV'). On a dA' = da'k. X'k; dA" = Eda"kX"k. Si on annule tous les dc' et tous les da" sauf da"k, nos équations donnent simplement: v'i= const.; v"i= const. (i k); V"k = ak+ const. Si on annule tous les da' et tous les da" sauf d2'k les équations deviennent X' da' -=dV' +O 0'i ( ')(dV'), O = dV" + O"// (o')(dV'). M. H. POINCARÉ, SUR LES GROUPES CONTINUS. 255 La première de ces équations, équivalente aux équations 4 bis, est susceptible comme nous l'avons vu d'être ramenée à la forme d'un système d'équations linéaires à coëfficients constants. L'intégration est immédiate et nous donne les v' en fonctions de la variable ak. La seconde équation est équivalente à un système d'équations de la forme: dv"i + dv',Fi, + dv',2F( +... + dv'9F'9 = O, les F étant des fonctions données des v'. En remplaçant les v' par leurs valeurs en fonctions de a'k, elle prend la forme: dv"i + c (ak) da'k = 0 et s'intègre immédiatement par quadrature. X. Contact Transformations and Optics. By Professor E. O. LOVETT. [Received 15 September 1899.] "Ayant vu combien les idées de Galois se sont peu à peu montrées fécondes dans tant de branches de l'analyse, de la géométrie et même de la mécanique, il est bien permis d'espérer que leur puissance se manifestera également en physique mathématique. Que nous représentent en effet les phénomènes naturels, si ce n'est une succession de transformations infinitésimales, dont les lois de l'univers sont les invariants?"-SoPHUS LIE*. IT is the object of this note to elaborate, and in fact in n dimensions, certain ideas which the lamented Sophus Lie sketched for ordinary space in a short papert presented to the Leipzig Scientific Society in 1896 and which were more or less developed for the plane in the first volume of the geometry of contact transformations + which appeared with the cooperation of Scheffers in the same year. 1. Attending to a few preliminary details, consider a family of ool transformations in n variables x,, x2,..., xn' xi= 1X (1,..., xn, a), f2= X2 (1,..., xn, a),... x, Xn (i,..,, a)...... (1) where the functions X,,..., X, are regular analytic functions of x,,..., x, and an arbitrary constant a; suppose in particular that the family contains the identical transformation, that is, that for some value of a, say a= 0, the equations (1) reduce to the form 1 = Xl, X2 X2 2,.., Xfn = -n. Then for a value of a, say St, infinitesimally different from zero, the equations (1) will yield an infinitely small transformation. With the assumptions made relative to the functions X,,..., X, the transformation (1) for a = 8t has the form xfi = XI + el (1,..., xn) ît +..., '2 = x2 + 2 (w,,..., x,,) t +...,............................................ ().......................................... xii = Xn + n (Xi,..., Xn) St +.... Under this infinitely small transformation (2) x,,..., x, receive the infinitesimal increments 8x1 = êtlt + *,= t......, 82, 8 = +t +...................... (3). * Le Centenaire de l'École Normale, p. 489.-Paris, Math.-Phys. Classe, Bd. 48, 1896, pp. 131-133. Hachette et Cie, 1895. î Geometrie der Beriihrungstransformationen, dargestellt + " Infinitesimale Beruhrungstransformationen der Op- von Sophus Lie und Georg Scheffers, Bd. i, Leipzig, tik," Ber. ii. d. Verh. d. k. sêichs. Ges. d. Wiss. zie Leipzig, Teubner, 1896. See in particular, pp. 97, 100-103. PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 257 Such a transformation is called an infinitesimal transformation. The expression.f 3f af Uf- el, + + I....( + f.(4) - ax, ax ax................................. is adopted as its symbol, since Uf. t is the increment assigned to any function f (x,..., xn) by the infinitesimal transformation. 2. If the transformations of the continuous ensemble (1) are so related that the successive application of any two of them is equivalent to a transformation belonging to the same family, (1) is called a continuous group of oo transformations. Let the family (1) be a continuous group; suppose further that the group contains the inverse transformation of every transformation in it: that is, that the resolution of the equations (1) with regard to x,,..., x,, gives a system of the form X1=Xi (1,..., Xni, b), X2 = X2 (X',..., n b),...xn = Xn (x,../,... x,,', b)... (), where b is a constant depending only on a. Under these conditions it is easy to see that the group contains an infinitesimal transformation; for, if T, is the transformation of the group corresponding to the parameter value a, the inverse Ta~- of T, is also found in the group. Further the transformation Ta+sa corresponding to the parameter value a + Sa, is the transformation of the group differing infinitesimally from Ta. The product Ta+s Ta-' which, by the assumed group property, belongs to the group, differs then infinitesimally from the transformation T7a Ta-'; but the latter is the identical transformation; thus the group contains a transformation possessed of the properties attributed to an infinitesimal transformation in the preceding paragraph. 3. Conversely, every infinitesimal transformation is contained in a determinate continuous group. This may be made clear in the following manner. The given infinitesimal transformation assigns the infinitesimal increments 1= 1(1,... )t,., îx=*.(, Zn) t,.................... (6) to the variables x,,..., x,n, on neglecting infinitely small quantities of a higher order; if t be interpreted as the time, x,..., x, as point-coordinates in a space of n dinmensions, ît as a time increment, and îx1,..., Sx, as the corresponding increments of i,..., xn, then the equations (6) determine a stationary flow in space of n dimensions. After an interval of time t the point (x,,..., x,) will have assuined the new position (X/,..., xn); the latter position will be obtained by integrating the simultaneous system dx - dx,' dx,' ddn (L, *, x,., X/) "' - n (1/..3..,X d) VOL. XVIII. 33 258 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. with the initial conditions that x,',..., x' shall reduce respectively to x1,..., xn for t= O. The n integral equations may be taken in the form Ui(X )..., x )= U1 (æ1) *.. XPI A U2 (X/, )..... )= tU (xi,. ),...................................................................... (8 ); U(x-1/,...(, n/)= Un- (, *.,.)+ tx), ITRl (xl X * * ' 7 ) = Un (XI,..., w-n) + t; ) the form of these equations shows that when resolved with respect to x1',..., x,' they represent a group with the parameter t; accordingly the given infinitesimal transformation is said to generate a one-parameter continuous group. 4. Consider in particular the case where the preceding transformations are contact transformations. The equations X.=X1(x,.... (, n Z pl..., pn),...,, x*n = X, (..., *, p. n p), z =Z(P,..., Z,,, Z, p1, * *, P pn) pi =Pi(x,..., zp,, i,p), J (i = 1, 2,...,)............................................. (9).are said to define a contact transformation when they give rise to a differential relation of the form i=n i==n dz' - pi dxi' = p (x,..., x,, z, p,..., p,) (dz - Ypidxi).............(10); i=1 i=1 the corresponding geometric characterization is that the property of tangency is an invariant property under contact transformations. Point transformations are then a particular category of contact transformations. The explicit formulation of this notion, contact transformation, is due to Lie; implicitly it is to be found in particular form in many directions and may be traced to Apollonius. Lie has determined all infinitesimal contact transformations in a space of n+ 1 dimensions, in the following manner. By definition the equations z = Z + (i,..., Xn Z Z, pl,..., pn) t +..., * = Xi + ei(Xi,..., *,n Z, Pi,..* * n) ît +..., pi'=pi+7ri (x,,..,, xI, z, pi,..., p) t +, (=, 2,.., n).......(11) can represent an infinitesimal contact transformation only in the case where the relation,(10) is a consequence of these defining equations (11). On substituting (11) in (10) we have dz + dt +... - (i + it +...)(d dt...)= p dz - pid........(12). i=l i=I PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 259 The left-hand member of this equation is a series of ascending integral positive powers of St; thus the function p must be an ascending series in integral positive powers of ît; as the term of zero degree in the left-hand series is dz - pidxj, p must therefore have the form p = 1+ + t........................................... (13). Inserting this value of p and equating the coefficients of corresponding powers of St we have d - p idi - rxidxi = -(dz - Cpidxi)...........................(14), or d ( - piti) + i dp - ri dxii = o (dZ - p.idxi)..................... (15). This linear and homogeneous condition in dz, dxi, dpi must be true for all values of these differentials; hence, writing Ç-spi p =- (, X,... æ P,.. p......................(16) for convenience, we have n i - 7ri = pi, z —, p- i =..........................(17). Eliminating a and solving (16) for Ç we find i= P, r=2jifpi-f, 7rri= —fl-pAi..................... (18). The infinitesimal transformation is therefore completely determined, g, ei, 7ri being given by an arbitrary function f2. 5. Let the preceding results be now applied to the infinitesimal contact transformation defined by the characteristic function fi = v/1 + Pl2 + p 22 +... + pn2 The formulae (18) show that the coordinates of a surface element, by which we mean the ensemble of a point and a plane through it, receive the infinitesimal increments i =' aEit, 8z = - 2 St, p~ = 0.(19). 8x^ pi î-t, îZ ît, îPi- 2~0.....................(19). Vl+ Sp.i2 Vl+2/1 + -Pi This infinitesimal transformation generates a one-parameter group of contact transformations, namely the group of dilatations, whose finite equations are found by integrating the simultaneous system _ _ _ _ _ _ _ _ _ _ _ >V 1 ~ 3pi d p1 _ dp n 1 - =... --- P — dxn- _ dz = p=.....(20) p/ p~' - 1 0 0 the integration effects itself, without any difficulty, and yieids the integral equations piti t xi = xi + \ z z-, i'= pi,...... (i = 1,..., n)......(21) v1 + Epr 41 + E pi2 where t is an arbitrary constant. 33-2 260 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. These transformations are obviously characterized geometrically by the property of changing the surface-element (xi,..., xn, z, pl,..., pn) into the surface-element (X/,... Xn,, z', p'., pn') in such a manner that the point of the second lies on the normal to the first and at a constant distance t from its surface. They transform the surface-elements of a point into those of a sphere, and change parallel surfaces into such. 6. As Lie has pointed out for ordinary space the theory of wave-motion in an isotropic elastic medium is intimately related to the one-parameter group of dilatations of the space filled by the medium. Consider a wave-motion originating at a center of disturbance P0 of an isotropic n + 1-diinensional elastic medium; in an interval of tirne the motion will have advanced to all points P of a sphere whose center is at P0 and whose radius is t, say, in precisely the same manner as the dilatation (21) would change the surface-elements of the point P0 into those of the last-named sphere. Every point P of this sphere can now be regarded as the center of new elementary waves which in a second interval of time, say tl, will have advanced to spheres of equal radii t, about the points P as centers. These elementary waves have an outer envelope, which by Huygens' principle is the identical wave that would have been developed from the original center P0 in the total time elapsed. But in exactly the same manner the dilatation pitl t, = l + + I =V - i pf=, '... (i=l,..., n)......(22) carries every point P of the sphere about P0 into a sphere of radius t, about P as center, so that the sphere of center P0 will be changed by the dilatation (22) into the sphere of center P0 and radius t1 +t0, that is into the sphere into which the point P0 is changed by the successive application of the dilatations (21) and (22). Thus the principle of Huygens finds its mathematical expression in the fact that all dilatations form a one-parameter continuous group. The importance of this particular group of contact transformations is further exhibited by observing that reflections and refractions from one isotropic medium to another are contact transformations which leave the infinitesimal dilatation invariant; the reflections have the additional property of being commutative with the latter. To establish these facts it is only necessary to make the ordinary illustrative constructions in a space of n + 1 dimensions and apply the principle that all the surfaces of a complex f that touch a surface b have in general an envelope cP, and hence the passage from q to P is a contact transformation. 7. Let the characteristic function be an arbitrary function of pi,..., pn, say nzi=nt(a, (pi, P,.......................................... (23); the infinitesimal transformation defined by I is represented by the equations Sxi = nT,Ït, Sx = 2pn, - nI, Spi = 0,...... (i= l,..., n)...............(24). PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 261 By integrating the simultaneous system dxl dxI,, dz' dpn- = d. (25), I -Ip/.... -I,- — i-pi, —1 —[ — 0 - '" -- -0.....................(25), H,' Y-pi ï Hp i n- l H O -O we have the corresponding one-parameter group of contact transformations X = x + rnptP, ' = z + (=PiUpi - t, =...... (i = 1, -..., n).........(26). Let t have for the moment a fixed value; the corresponding contact transformation of the group changes the point (x,..., x,,, z) into a surface whose equation in current coordinates (xl',..., x,', z') is obtained by eliminating the pi from the first n + 1 equations; this elimination yields the equation (t ' t ' t " t ' t )......... (27). The form of this equation enables us to find the characteristic property of these transformations as the following considerations will make evident. 1~. In the first place it is clear that contact transformations in n + 1-dimensional space may be determined by a system of r equations 1 (X... -, Zn, Z 1,..., Xn)= 0, 2 = 0,..., O,.= 0...................(28), where r may have all values from 1 to n + 1; in the last case the transformations if existent will be point transformations, since the n + relations will give the n+ 1 quantities x', z', as functions of the n + 1 quantities xi, z alone. In fact the problem of determining all finite contact transformations of a space of n + 1 dimensions is that of resolving the total differential equation n n dz' - pidx' - p (dz - pidxs) = O () 0,...... (,,n)...............(29) i 1 where the z', x', pi' are functions of the 2n +1 variables z, xi, pi to be determined. This equation shows that there ought to exist at least one relation between the variables z', x', z, xi containing z' and z*. Taking the general case of r different relations expressed by (28), the equation (29) ought to be a consequence of do1 = O, do = O,...,. =.................................(30); that is, it ought to be possible to find r coefficients ÀX,..., X,. such that the identity n n r dz' - pi'dxi'- p (dz - lpidxi) = 2Xidci 1 1 1 exists. This demands the following equations: 1 = l...,,z Pi i au),...........(3 ); p = — x a. =i — ii;* See Goursat, Leçons sur les équlations aux dérivées partielles du premier ordre, Paris, Hermann, 1891, p. 258, 262 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. the 2n 2 +r equations (30) and (31) in general determine the 2n 2 + r functions z', Xi', pi', Xj, p as functions of z, xi, pi. Eliminating p we can write the following n + r+ equations for z', x,', Xj, S^^+^2X^ii_.. L-~=0, j=l,..., n, i=l o,....a= Si=i~, cl=0., coi )..=. 0 resolving these for z', xi', Xj, the remaining functions pi', p are found by substituting the values of the former in the remaining equations of the system (30) and (31). 2~. In the second place two transformations S and T are commutative when the symbolic equation ST= TS obtains. Consider the contact transformation S and the point transformation T. That the point P is changed into the point Pl by the transformation T is expressed by the symbolic equation (P) T= (P1). In the same manner, that S transforms P into the surface Z is expressed by the equation (P) S= (). Then if (P) ST= (P) TS, we have also (Pl) S () T. That is, if S transforms the point P into the surface E, and T changes the point P into the point P1, the latter is changed by S into the surface into which the surface 2 is changed by T. 3~. In the third place let S be a contact transformation of an n + 1-dimensional space commutative with all translations T of that space. If S changes a definite point P into the surface 2, the surfaces into which all other points are changed by S may be determined, for there always exists a translation which carries the point P to any other arbitrary position Pi; then by the second paragraph above, the point P1 is changed by S into the surface Y, into which 2 is changed by the last-named translation; hence all points are changed by S into congruent surfaces similarly situated. Accordingly the contact transformations that are commutative with all translations of a space of any number of dimensions are determined by a single function of the form V((x1-x X, X=-X21,..., Xn,-xn, Z - ') =........................(33); it is not to our purpose to construct the explicit forms of these transformations here; the most general one in the plane has been given by Lie in his geometry of contact transformations to which reference has been made. PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 263 Thus the equations (27) and (33) show that all the transformations of the one-parameter group (26) are commutative with all translations. 8. It is evident either from the last-named property or directly from the form of equations (27), that by varying t and this operating on a point (x1,... x,, z) with all the transformations of the group (26), the point is changed successively into similar surfaces and similarly placed. The point P0 is changed by the transformation whose parameter is t, into the surface $. Operating on all the points P of 2 with the transformation whose parameter is t2, these points P will be changed into congruent surfaces that are similar and similarly placed to 2. These latter surfaces have an outer envelope, a surface Il into which the surface l is changed by the second transformation. The successive application or product of the two transformations is equivalent to the transformation whose parameter is tl+ t2; the latter transformation carries the point P0 directly into the new surface Ci, and this surface must then be a similar and similarly placed surface to 2. The preceding geometrical operations and their results suggest the phenomena of wave-motion in an elastic n + -dimensional medium. If such a space is filled with such a medium in which motions originating at a point advance in different directions with velocities depending only on the direction, then a center of disturbance P0 gives rise to a series of waves similar and similarly placed with the common center of similarity P0; accordingly the above geometric operations present a pure mathematical interpretation of Huygens' principle for a non-isotropic elastic medium, and this principle finds its equivalent in the fact that the oc' contact transformations (26) form a group. 9. The group (26) may be generalized and specialized. 1~. Much more general wave-motions nay be designed by using in a similar manner the most general infinitesimal contact transformation defined by the characteristic function S1 (x x z, P... X, pl); a simple geometric construction shows that the normal velocity of the wave is given by the expression.2//1 +;p?2. 2~. The case applying to the optics of a double refracting crystal is given by the particular form 1 1 = a 2 + a pi2, (=,...,n)........................,n)(34). Observing that f o =,2 -1 a.~,-.......................................... (35), we have - = - -a2Q-1...................................... (36); 264 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. hence the finite equations of the group of contact transformations generated by the infinitesimal transformation (34) are xi' = x + a2piD-st, zX - z - ao2p1n-t, pi'= Pi.....................(37); eliminating t by means of the first n + 1 equations, we have the ellipsoid (Xw - X)2 (x; - X2)2 (x'- - ) ) (+t )'l+. + (n) + =...............(38); (alt)2 (at)2 (ant)2 (aot)2 thus the transformations of the one-parameter group (37) change the points of space of any dimensions into ellipsoids of that space; any particular point is changed by all the transformations of the group into similar ellipsoids similarly placed and concentric with the point as common center. 10. Lie might have included in this order of ideas certain other contact transformations *. Thus far the finite contact transformations studied in detail have been defined by a single equation connecting the coordinates of the points of the two spaces. The following however is an interesting example giving a category of such transformations which are determined by two equations in the point variables. Consider the two equations z22 - z2 + (xi" - x_2) = 0, ni - ff _ n 1.(3).....................(39), (zz' + xi'x)2_ - k2 ( + x/) (z2 + i2 ) = 0 1 1 1 ) where k is a constant. By means of the formulae developed in ~ 7, 2~, the finite equations of the transformations can be determined, and the fact that they form a one-parameter group established. If Xlf, X 2f,..., X +.......................................(40) are the infinitesimal rotations of n + 1-dimensional space written in the symbolic form (4), the expression n+1 = 2 (X f)2.......................................(41) may be taken as the characteristic function of the infinitesimal contact transformation which generates the one-parameter group of contact transformations determined by the equations (39). Observing that two infinitesimal contact transformations are commutative only in the case when the relation (Vf)- V(Uf - O.......................................(42) * "Beitrage zur allgemeinen Transformationstheorie," Leipziger Berichte, pp. 495-498. PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 265 exists between their symbols, we can verify by this principle that the transformations of the above group are commutative* 1~ with all dilatations, 2~ with rotations about the origin, 3~ with all spiral transformations starting from the origin, 4~ with all pedal transformations, 5~ with all point and contact transformations commutative with all rotations about the origin, 6~ with all transformations of the infinite group whose characteristic function is _1 _2 X +1 ), 2)(44) ) X <> ) *............................... (44), where 1, X= i $a +.................... (45). 1 axi aZ ) \W axJ The first case of commutation is especially interesting because of reasons given in ~ 6. The second may be shown even more simply by introducing polar coordinates. The aequationes directrices (39) themselves exhibit certain geometrical properties of the transformations. For example they show that every point (z, xi,..., xn) is changed into a circle whose points are at the same distance from the origin as the point (z, xi,..., xn) itself. Further the radii vectores of (z, xL,..., Xn) and (z', x,..., xn') make an angle with each other whose cosine is k. 11. The particular transformation of the above group, namely that corresponding to k = and accordingly defined by the two equations Z2- z2 +- (i/2 - x i2) = O, + Z iXi' 0........................(46), 1 1 was first studied as a contact transformation by Goursat, in three dimensions t. If in equations (46) z', x',..., xn' be regarded as constants and z, xz,..., x, as current coordinates, these equations define a certain circle C in n + 1-dimensional space, the locus of (z,,..., xn). That is the equations make a circle C correspond to every point (z', x',..., Xn'), and similarly, since the equations are symmetrical in both sets of variables, to every point (z, xz,..., xn) there corresponds a circle C' in the current coordinates (z', xz,..., xn'). When the point (z, xz,..., x,) describes a surface S, the circles C' relative to the several points of 2 form a congruence. The focal surface of this congruence is the surface,' into which Z is transformed. -' is also the locus of the points (z', xz',..., X,') such that the corresponding circles C' are tangent to S. The focal surface of the congruence of circles C' is a plane passing through the radius vector OP and the normal PN to the surface at P. Thus to construct the point P' corresponding to P it is only necessary to draw, in the plane passing through OP and the normal PN, the perpendicular OP' to OP, cutting off a distance OP' equal to OP. * In the last loc. cit. Lie shows indirectly that the enumerated commutative properties appertain to these transformations in three dimensions. + See loc. cit. p. 267. VOL. XVIII. 34 266 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. The geometric construction shows that we have here the long known construction by which the apsidal surface of a given surface is derived. Accordingly the above contact transformation is possessed of the very important property of changing ellipsoids into Fresnel wave surfaces. The finite equations of the transformation (46) expressing z', xi', pi' as functions of z, xi, pi may be obtained without difficulty by the method of ~ 7. If this transformation be combined with those of the one-parameter group (37) we shall have o02 contact transformations which change the points of space of any dimensions into the wave surfaces of that space. 12. This suggests the interesting problem of finding all those contact transformations which change every wave surface into a wave surface, that is, those contact transformations which leave the family of all wave surfaces invariant. Analytically the problem may be approached either by determining the finite transformations or the infinitesimal transformations which leave the partial differential equation of the wave surface invariant. From either starting point the difficulties in the way of integrations to be effected are well-nigh insurmountable. This ought not to be surprising since all contact transformations of ordinary space changing plane into plane have not been determined (though Lie has found all those that change surfaces of constant curvature into surfaces of constant curvature irl ordinary space, and lately the most general contact transformation leaving unaltered the family of developable surfaces of n+ 1-dimensional space has been found). An indirect method for finding contact transformations transforming wave surfaces into such may be employed by using the results of a beautiful memoir of M. Maurice Lévy, "Sur les équations les plus générales de la double refraction compatibles avec la surface de l'onde de Fresnel," Conptes Rendus, t. 105, pp. 1044-1050. Without making any assumption whatever relative to the nature of a luminous vector Lévy proposes to find its most general form compatible with the Fresnel wave surface. His problem narrows itself to determining the most general expressions of the second derivatives, with regard to the time, of the three components of the luminous vector as functions of the various second derivatives of these components with regard to the coordinates of the point of the medium which produces the light, by means of the condition of reproducing the equation of velocities and hence the wave surface. The equations to be invariant in this method are more numerous, but simpler in form than the partial differential equation of the surface of waves. For reference Lévy's system of equations is appended here. Letting u, v, w be the components of the luminous vector, t the time, x, y, z the coordinates of the point of the medium which produces the light, a, b, c the reciprocals of the principal indices of refraction, a, /3, 7 three arbitrary constants, and X,,/, z three other arbitrary PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 267 constants entering only by their mutual ratios, Lévy finds the following 4. co5 solutions of the proposed problem: a2 u 2ut a2u a2u a2v v a2w 2 = a ~+c2-b+ b2 2 + 2 (-C2) +-Qy + -b2) ( watg ôc 2 3ay2 + a2 a axay x / x- a a2 a2v a2v a2v v v w X (C) a2u (A ) =c2 + + a2 ( - -2)a2) - C2) at 2 y2 az2 ayaz ayax a2w a2w a2w a2w X a2, a2v = b2 +a2 -+ - ( b+- 2) -a- + - (,2 a2) 3t2 ax2 ay2 a92 ^ v zox v azaôy a2u a2u 2a2u a2u. a2V V a2w d2 =ac-2 +c2y2 +b2) + (cac)aa +x( axa)8z' C 2 - + b2, 2+ ( 2) + (a - b2) ata z ay2 ax h axay X axaz (a2 + — 2 + + (+ - C2) + (/ - C2) (B) = 2 ~j+a + a2)+ +( + - ( c) aya aw a2u a2W aw x' a2 a_2v at= =b22 +ay a2 + azax v azay' 1 ~a 2 u a 2 u + 2 ( a U a u + / ( 2 ) a -V, ~ ( a 2. a 2 w at2 ax2 ay2 azx axay X ac v a2a v 72 a2V\ a a aVv a2./ (C2 = j-2 V + b2 ay2 + (7 - b2) - + - ( - b2) 2c) aX Wzay2-^ ~a^a^)+ a^~ ayax' ura h of line etry. This m r vw a2 tions of ontat transformations to optic as. at2 az2 X2 ay2 axay v azy C(D) oueves to od a sp o a2V e eie X a2p a2w a2w (^ a9-v a2w\? a2î u,1z, a2v However, these half-dozen possibilities or tentatives towards the solution of the problem of finding contact transformations which leave the family of wave surfaces invariant have so far yielded nrio further result than the trivial one formed by the repetitions of the reciprocal apsidal transformation. 13. Assuming the rectilineal propagation of light the theory of optics becomes a branch of line-geometry. This familiar view opens up other possibilities in the applications of contact transformations to optics. Confining ourselves to ordinary space for convenience of expression these applications may be made either by means of the contact transformations which change straight lines into such, or by means of other correspondences set up by contact transformations between two spaces such that straight lines are changed into the elements of some other four-dimensional manifoldness. 34-2 268 PROF. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. The simplest four-dimensional manifoldnesses in three-dimensional space are that of all straight lines and that of all spheres. For this reason those contact transformations between two three-dimensional spaces or which change a three-dimensional space into itself in such a manner that straight lines are changed into spheres, are the first to attract attention and have so far been the most fruitful. Lie constructed such a transformation in his memoir on complexes in the fifth volume of the Iathematische zAnnalen which has led him to a generalized form* of the theorem of Malus. Lately this manner of changing straight lines into spheres by contact transformations has been found not to be unique; in fact infinite groups of infinite numbers of such line-sphere contact transformations have been constructed. The above observations increase the demand for the resolution of the problem of determining all continuous groups in four variables. But such contact transformations need not necessarily be contact transformations of a three-dimensional point space into itself; for example, if the four variables be interpreted as line-coordinates or spherecoordinates, the corresponding invariant Pfaffians by no means provide that the conditions for contact transformations of the three-dimensional space into itself be satisfied. It is precisely because of such a confusion that we find these notions used loosely in a recent memoirt on the employment of infinitesimal transformations in optics. * "Lichtstrahlen, die in Pseudonormalensystem bilden, Pseudonormalensystem auf die Pseudokugel des betreffenden gehen bei jeder Reflexion und Refraction in ein Pseudo- Raumes," Leipziger Berichte, 1896, loc. cit., p. 133. normalensystem ùber. Sind bei einer solchen Refraction t Hausdorff, "Infinitesimale Abbildungen der Optik," die beiden in Betracht kommenden Pseudokugeln (d. h. Leipziger Berichte, 1896, pp. 79-130. Wellenfiichen) weseltlich verschieden, so bezieht sich jedes XI. On a Class of Groups of Finite Order. By Professor W. BURNSIDE. [Received 30 September 1899.] AMONG the groups of finite order that earliest present themselves, from some points of view, to the student are the groups of rotations of the regular solids. An admirable account of these from the purely geometrical stand-point is given in the first chapter of Klein's Vorlesungen itber das Icosaëder. Of the six types included in this set of groups there are three which, though quite unlike in other respects, have a distinctive property in common. These are (i) the dihedral group of order 2n (n odd), (ii) the tetrahedral group of order 12, and (iii) the icosahedral group of order 60. They are defined abstractly by the relations:(i) A2=1, Bn=l, (AB)2 =, n odd; (ii) A2=1, B3=1, (AB)3=1; (iii) A2= 1, B3= 1, (AB)5= 1. The order of each of these groups is even, while the only operations of even order which they contain are operations of order two. While they have this property in common they are otherwise of very distinct types. The first has an Abelian (cyclical) self-conjugate subgroup, order n, which consists of the totality of its operations of odd order. The second contains a self-conjugate subgroup of order four, this being the highest power of two which is a factor of the order of the group. The third is a simple group containing five subgroups of order twelve, each of which has a self-conjugate subgroup of order four. It can be represented as a triply-transitive substitution group of degree five. I propose here to determine the groups of even order, which contain no operations of even order other than operations of order two. The determination is exhaustive; and it will -be seen that the groups in question arrange themselves in three quite different sets of types of which the groups (i), (ii) and (iii), defined above, are representative. 1. Let G be a group of even order N, which contains no operations of even order 270 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. other than those of order two. To deal first with the simplest case that presents itself*, let N= 2m, where m is odd. Since no operation of order two is permutable with any operation of odd order, G must contain m operations of order two which form a single conjugate set. Let these be Al, A2, ~......., Am. If ArAs were an operation of order two, 1, A,, As, and ArAs, would constitute a subgroup of G of order four. No such subgroup can exist, and therefore ArAs is an operation of odd order. The m operations A-rAi, ArA2,......, ArAm, which are necessarily distinct, are therefore the m operations of odd order contained in G. These m operations may similarly be expressed in the form AlAr, A2Ar,......, AmAr; and since Ar. ArAs. A. = AsAr, Ar transforins every operation of G, of odd order, into its inverse. Hence ArAp. AqAr = AqAp = qAr. ArA,; and this shews that every pair of operations of G, of odd order, are permutable. Hence the m operations of G of odd order, including identity, constitute an Abelian group, and this is a self-conjugate subgroup of G. Conversely, if H is any Abelian group of odd order m, generated by the independent operations S, T,..., and if A is an operation of order two such that ASA = S-, ATA = T-1,..., then A and H generate a group G of order 2m, whose only operations of even order are those of order two. When r is given, s can always be taken in just one way so that A,.A is any given operation of G of odd order. Hence every operation of G of odd order can be represented in the form ArAs in just m distinct ways. This property will be useful in the sequel. The groups thus arrived at are obviously analogous to the group (i) above. 2. Next let N= 2nm, where m is odd and n is greater than one. The operations of order two contained in G form one or more conjugate sets. Suppose first that they form more than one such set; and let A, A',..., and B, B..., * This first case is considered in my Theory of Groups of Finite Order, pp. 143 and 230. PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. 271 be two distinct conjugate sets of operations of order two. The operation AB must either be of order two or of odd order. If it were of odd order, /,, the subgroup generated by A and B would be a dihedral subgroup of order 2/u; and in this subgroup A and B would be conjugate operations. Since A and B belong to distinct conjugate sets in G, this is impossible. Hence AB is of order two, or in other words A and B are permutable. Every operation of one of the two conjugate sets is therefore permutable with every operation of the other. The two conjugate sets therefore generate two self-conjugate subgroups (not necessarily distinct) such that every operation of the one is permutable with every operation of the other. The order of each of these is divisible by two, and therefore the order of each must be a power of two; as otherwise G would contain operations of order 2r (r odd). The two together will generate a self-conjugate subgroup H' of order 2n'. If n' is less than n, there must be one or more conjugate sets of operations of order two not contained in H'. Let C, CC..., be such a set. As before every operation of this set must be permutable with every operation of H'. Hence finally G must contain a self-conjugate subgroup H of order 2n. No operation of G is permutable with any operation of H except the operations of H itself; and G is therefore a subgroup of the holomorph* of H. It follows that G can be represented as a transitive group of degree 2n. Moreover, since G contains no operations of even order except those of order two, the substitutions of this transitive group must displace either all the symbols or all the symbols except one. Hence m must be a factor of 2 - 1; and G contains 2n subgroups of order m which have no common operations except identity. With the case at present under consideration may be combined that in which G has a self-conjugate subgroup of order 21, the 2' - 1 operations of order two belonging to which form a single conjugate set. In this case m must be equal to 2 - 1. We thus arrive at a second set of groups with the required property of order 2nm, where m is equal to or is a factor of 2 - 1. They have a self-conjugate subgroup of order 2n, and 2'1 conjugate subgroups of order m; the latter having no common operations except identity. These are clearly analogous to group (ii) above. 3. Lastly there remains to be considered the case in which the operations of G of order two form a single conjugate set, while G contains more than one subgroup of order 2n. If Il and H' are two subgroups of G of order 2n, and if I is the subgroup common to H and H', then since H and H' are Abelian (their operations being all of order two) every operation of I is permutable with every operation of the group generated by H and H'. This group must have operations of odd order, since it contains more than one subgroup of order 22. Hence I must consist of the identical operation only; or in other words, no two subgroups of order 2n have common operations other * Theory of Groups, p. 228. 272 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. than identity. It follows from an extension of Sylow's theorem that the number of subgroups of order 2n contained in G must be of the form 2nk+ 1. If K is the greatest subgroup of G which contains a subgroup H, of order 2n, self-conjugately; then K must be a subgroup of the nature of those considered in the preceding section, and its order must be 2~/, where /u is equal to or is a factor of 2 -1. Also no two operations of H can be conjugate in G unless they are conjugate in K*. The 2n- operations of order two in K therefore form a single conjugate set; and hence /u must be equal to 21 - 1. The order of G is therefore given by T = (2nkc + 1) 2 (2n - 1). That G must be a simple group is almost obvious. A self-conjugate subgroup of even order must contain all the 2k + 1 subgroups of order 2n, since the operations of order two form a single set. In such a subgroup the operations of order two must form a single set, and therefore a subgroup of order 2n must be contained self-conjugately in one of order 2nI(2n-1). Hence a self-conjugate subgroup of even order necessarily coincides with G. If on the other hand G had a self-conjugate subgroup I of odd order r, I would by the first section be Abelian and every operation of G of order two would transform every operation of I into its inverse. This is impossible; for if A and B were two permutable operations of order two in G which satisfy the condition, then AB is an operation of order two which is permutable with every operation of I, contrary to supposition. Hence G must be simple. If A and B are any two non-permutable operations of order two in G, AB must be an operation of odd order w, and A and B generate a dihedral group of order 2/L. Hence G contains subgroups of the type considered in the first section. Let 2m, be the greatest possible order of a subgroup of this type contained in G; and let If be a subgroup of G of order 2ml, and J1 the Abelian subgroup of order ml contained in 1,. Every subgroup K of Ji is contained self-conjugately in I1; and, for the reason just given in proving that G is simple, no two permutable operations of order two can transform K into itself. Hence Il must be the greatest subgroup that contains K self-conjugately; as otherwise 2mn would not be the greatest possible order for the subgroups of this type contained in G. Let pa be the highest power of a prime p which divides ml; and let K be a subgroup of Ji of order pa. If pa is not the highest power of p which divides 1, then K would be contained self-conjugatelyt in some subgroup of G of order pa+l. This has been proved impossible. Hence ml and NV/ml are relatively prime. Again no two subgroups conjugate to J, can contain a common operation other than identity; for if they did Il would not be the greatest subgroup of its type contained in G. If Il and the subgroups conjugate to it do not exhaust all subgroups of G of order 2,u (/i odd), let I2 of order 2m2 (m, odd) be chosen among the remaining subgroups of G of * Theory of Groups, p. 98. t Ibid. p. 65. PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. 273 this type so that m, is as great as possible; and let J2 be the Abelian subgroup of I, of order m2. Then J, has no operation other than identity in common with J, or with any subgroup conjugate to Ji; also no two subgroups conjugate to J2 have a common operation other than identity, and m2 and N/m2 are relatively prime. All these statements may be proved exactly as in the former case. If the subgroups of G of order 2~L (,a odd) are still not exhausted, a subgroup I3 of order 2m3 containing an Abelian subgroup J, of order m3 may be chosen in the same way as before; and the process may be continued till all subgroups of G of the type in question are exhausted. Now JI is one of NV/2m conjugate subgroups and each contains n - 1 operations which enter into no other subgroup conjugate to J1 or to J2 or J3.... Hence the subgroups conjugate to J, J2, J3,... contain 2n1 (ql)2 + 2m ((n2- 1) + (m3- ) +.. distinct operations other than identity. If 13 actually existed, this number would be equal to or greater than N, which is impossible. Hence there can at most be only two sets of conjugate subgroups such as Il and I,. It was shewn in section 1 that each of the m - 1 operations of JI other than identity can be represented in mr distinct ways as the product of two operations of order two. Similarly each of the m, - 1 operations other than identity of J2, if it exists, can be represented as the product of two operations of order two in m2 distinct ways. Moreover these and the operations conjugate to them are the only ones which can be represented as the product of two non-permutable operations of order two. Now G contains (2Jc + 1) (2n-1) operations of order two, and any one of these is permutable with exactly 2n- 1. Hence the number of products of the form AB, where A and B are non-permutable operations of order two and the sequence is essential, is (2nk + 1) (2n - 1) 2nc (2n - 1) = Nk (2 - 1). On the other hand as shewn above this number is 1T N (mi -1)+- (m2- ) N 2or 2( 2-1 or 2 (mi-1) according as Is actually exists or does not. Hence if I2 does not exist Ml = 2+ (2n - 1) + 1; and at the same time m1 is a factor of (2~1 + 1)(2' - 1). VOL. XVIII. 35 274 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. These conditions are obviously inconsistent. Hence I2 does exist, and m + m2=2 {k(2- 1)+ 1l}. It follows that, ml and m2 being positive numbers of which %n is the greater, ml > 2k + 1 - k. On the other hand, since no two operations of order two contained in I, are permutable, while G contains only 2'2k + 1 subgroups of order 21, m n 2nk + 1. Hence there must be an integer 1, less than k, such that mn1 = 24k + 1 - l, and m2 = 2'k + 1 + 1- 2k. Now ml and m2 are relatively prime factors of (2 + 1)(2n- 1). Hence (2nk + 1)2 - 2k (2nk + 1) + 2kl - 12 < (2n2k + 1) (2n - 1), and à fortiori since i is less than k, and 2k + 1 is positive, 2nkc + 1- 2k < 2n- 1, i.e. k 1. The group G can therefore only exist if k is unity, and this necessarily involves that i is zero. Hence NV= (2n + 1) 2n (2- _ 1), m= 2 + 1, m= 2- -1, and these are the only values of iV, ry,, and m, consistent with the existence of a group G having the required property. Since G is simple, it can be represented as a substitution group of degree 2 + 1. The subgroup of degree 2n, which leaves one symbol unchanged, has a self-conjugate Abelian subgroup of order 2n, and 2n conjugate Abelian subgroups of order 2 - 1; the latter having no common substitutions except identity. Hence the subgroup of G which leaves one symbol unchanged is doubly-transitive in the remaining 2" symbols; and therefore G can be represented as a triply-transitive group of degree 21 + 1. The Abelian subgroup of order 2 - 1 which transforms a subgroup of degree 2n is shewn in an appended note to be cyclical. Assuming for the present this result, the subgroups of G of order 2 (2n-1) are doubly-transitive groups of known type. Now G contains just n-l1 operations of order two which transform each operation of a cyclical subgroup of a c l s p o degree 2 - 1 into its inverse. Since each of these leaves on]y one symbol unchanged, each must interchange the two symbols left unaltered PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. 275 by the cyclical subgroup of order 2-l1. But there are only just exactly 2n-.1 substitutions of order two in the 2+ 1 symbols which satisfy these conditions. Hence for a given value of n the group, if it exists, is unique. That such groups exist for all values of n is known*. In fact the system of congruences az+e z'-_z+ (mod. 2), yz + (m where a, f, y, 8 are roots of the congruence X2"-1 1, (mod. 2), such that a S-y O 0, (mod. 2), actually define such a group; and the permutations of the 2 +1 symbols 00,,, X2,...,X2"-1, where X is a primitive root of X2-1-_ 1, (mod. 2), which are effected by the above system of congruences, actually represent it as a triplytransitive group of degree 2 + 1. The set of groups thus arrived at are the analogues of group (iii) above. Finally, every group of even order, which does not belong to one of the three sets thus determined, must contain operations of even order other than operations of order two. NOTE. Let H be an Abelian group of order 2n2 whose operations, except identity, are all of order two; and suppose if possible that H admits two permutable isomorphisms of prime order p one of which is not a power of the other, such that no operation of order two is left unchanged by any isomorphism generated by the two. So far as a set of p2 operations of H are concerned the two isomorphisms, being permutable, must have the form (A11AI..2 ) (A... Ap)) (A2A22... Ap). (AA... A), and (An11A... Ap1) (A2A22... Ap)......(ApA... Ap); A,1, A,2............. App, Moore: "On a doubly-infinite series of simple groups," Chicago Congress Papers (1893); Burnside: "On a class of groups defined by congruences," Proc. L. M. S. Vol. xxv. (1894). 35 - 2 276 PROF. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. being the p2 operations. Moreover any cycle of an isomorphism generated by these two has the form (Ar, -Ar+, s+y...... A+(p-i), s-+(p-i)y), the suffixes being reduced mod. p. Since no operation of H except identity is left unchanged by any one of these isomorphisms, the product of the p operations in any orle of the cycles must give the identical operation. Hence AlA21...... Ap, = 1, AA22...... Ap, p = 1, A11A23 1... Ap, p-l= 1............................. AA2p.... Ap,2 =1; and therefore on multiplication A11 = 1, or AJn =1. The supposition made therefore leads to a contradiction. Hence if H admits a group of isomorphisms of order pm, no one of which leaves any operations of H except identity unchanged, this group has only a single subgroup of order p. It is therefore cyclical*. If then pmn is the highest power of p which divides 27 - 1, the subgroup of order p1n in the Abelian group of order 2n-1, considered above, is cyclical. Hence the Abelian group is itself cyclical. * Theory of Grozups, p. 73. XII. On Green's Function for a Circular Disc, with applications to Electrostatic Problems. By E. W. HoBsoN, Sc.D., F.R.S. [Received 7 October 1899.] THE main object of the present communication is to obtain the Green's function for the circular disc, and for the spherical bowl. The function for these cases does not appear to have been given before in an explicit form, although expressions for the electric density on a conducting disc or bowl under the action of an influencing point have been obtained by Lord Kelvin by means of a series of inversions. The method employed is the powerful one devised by Sommerfeld and explained fully by him in the paper referred to below. The application of this method given in the present paper may serve as an example of the simplicity which the consideration of multiple spaces introduces into the treatment of some potential problems which have hitherto only been attacked by indirect and more ponderous methods. THE SYSTEM OF PERI-POLAR COORDINATES. 1. The system of coordinates which we shall use is that known as peri-polar coordinates, and was introduced by C. Neumann* for the problem of electric distribution in an anchor-ring. A fixed circle of radius a being taken as basis of the coordinate system; in order to measure the position of any point P, let a plane PAB be drawn through P containing the axis of the circle and intersecting the circumference of the PA circle in A and B; the coordinates of P are then taken to be p = log PB, which is the angle APB, and b the angle made by the plane APB with a fixed plane through the axis of the circle. In order that all points in space may be represented uniquely by this system, we agree that 0 shall be restricted to have values between -7r and 7r, a discontinuity in the value of 0 arising as we pass through the circle, so that at points within the circumference of the circle, 0 is equal to 7r, on the upper side of the circle, and to - r on the lower side of the circle, the value of 0 being zero at all points in the plane of the circle which are outside its circumference. As * Theorie der Elektricitdts- und Widrme-Vertheilung in einem Ringe. Halle, 1864. 278 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, P moves from an infinite distance along a line above the plane of the circle up to any point inside the circle, and in its plane, 0 is positive and increases from 0 to 7r, whereas as P moves from an infinite distance along a line below the plane of the FIG. 1. circle up to a point within the circumference, 6 is negative, and changes from O to - 'r. The coordinate qb is restricted to have values between O and 27r, and the coordinate p may have any value from - o to +oo, which correspond to the points A, B respectively. The system of orthogonal surfaces which correspond to these coordinates consists of a system of spherical bowls with the fundamental circle as common rim, a system of anchor-rings with the circle as limiting circle, and a system of planes through the axis of the circle. If we denote by e the distance CN of P from the axis of the circle, and by z the distance PN of P from the plane of the circle, the system Icos 5, sin (, z will be a system of rectangular coordinates, which can of course be expressed in terms of p, 0, b. Let the lengths PA, PB be denoted by r, r' respectively, then r/r' = log p; we have 2rr' cos 0 = r2 + r'2 - 4a2 = 2rr' cosh p - 4a2 2a2 hence rr = 2a cosh p - cos O Again, z. 2a = rr' sin 0, a sin 0 hence z= - cosh p - cos O' also since r2 + r'2 = 2a2 + 2CP2, we have CP2 = rr' cos 0 + a2, WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 279 cosh p + cos 0 whence we find CP'2 a2 p + cos O cosh p - cos O' ~~~~hen~ce ^2 = 2ya sinha p ~henc~~e ~~ = (cosh p - cos 0)2 thus f, z are expressed in terms of p, 0 by means of the formulae a sinh p a sin 0 cosh p - cos 0' cosh p - cos O' 2. To express the reciprocal of the distance D between two points (p, 0, a) and (po, 00, bo), we substitute for e, z and io, z, in the expression 1 - = {(z - zo)2 + 2 + o2 - 2o cos (s - o)}-f, their values in terms of p, 0 and po, 0o; we then find i _ 1 (cosh p - cos 0)3 (cosh po - cos 0o)2 D a V2 {cosh a - cos (O - o0)} where cosh a denotes the expression cosh p cosh po - sinh p sinh po cos (b - bo). If we suppose the expression {cosh a - cos ( - o)}- is expanded in cosines of multiples of 2 cosr os r7-s 0-0o., the coefficient of cos, ( - 0o) is - -h djvr which is equal* to vr o (cosh a - cos r)2 7/' 2 -- Q_ (cosh a) when Qm-o denotes the zonal harmonic of the second kind, of degree i 1 i m -; thus = (os p - os 0) (coshpo- cos 80) E 2Qm, (cosh a) cos m (0 - 0o), where the factor 2 is omitted in the first term, for which vm=0. The series in this expression for 1/D may be summed, by substituting for Qm_ (cosh a) the expression i rf e-mu /12 J (osh - -cosh d' (loc. cit. p. 519); 2a (cosh u - cosh a)'2 we find 1 = 1 _ (cosh p - cos 09) (cosh po - cos 02fc - 1+ 2Ze-nb cos m (0 - 09o)} d, D 'ra 2J (cosh u - cosh a)' and thus we have the formula 1 1 "'O.i/x i\ 0 1 sinh zu 1- - -a (cosh p - cos 0)' (cosh po - cos 0) - o h sinh du, D 7ra v\2 a /cosh u- cosh a cosh u- cos (0 - 0o) where a is given by cosh a = cosh p cosh po - sinh p sinh po cos (b - bo). * See page 521 of my memoir "On a type of spherical harmonics of unrestricted degree, order, and argument," Phil. Trans. Vol. CLXXXVII. (1896) A. 280 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, GREEN'S FUNCTION FOR THE CIRCULAR DISC. 3. In order to obtain Green's function for an indefinitely thin circular disc, which we take to coincide with the fundamental circle of our system of coordinates, we shall apply the idea originated and developed by Sommerfeld*, of extending the method of images by considering two copies of three-dimensional space to be superimposed and to be related to one another in a manner analogous to the relation between the sheets of a Riemann's surface. In our case we must suppose the passage from one space to the other to be made by a point which passes through the disc; the first space is that already considered, in which 0 lies between - r and rr; for the second space we shall suppose that 0 lies between 7r and 3wr, thus as a point P starting from a point in the first space passes from the positive side through the disc, it passes from the first space into the second space, the value of 0 increasing continuously through the value wr, and becoming greater than 7r in the second space. In order that a point P starting from a position PO(po, 00, bo), say on the positive side of the disc, may after passing through the disc get back to the original position P0, it will be necessary for it to pass twice through the disc; the first time of passage the point passes from the first space into the second space, and at the second passage it comes back into the first space. Corresponding to the point p,, 00, <0 where 00 is between -7r and 7r, is the point (po, o + 27r, <Po) in the second space, whereas the point (po, + 47r, 0o) is regarded as identical with the point (po, 0o, <0). The section of our double space by a plane which cuts the rim of the disc is a double-sheeted Riemann's surface, with the line of section as the line of passage from one sheet into the other. Let p,, 0o, 0o, be the coordinates of a point P in the first space, on the positive side of the disc, thus O < 00< wr; taking the expression for the reciprocal of the distance of a point Q (p, 0, 0) from P, given in the last article, we have, since sinhu sinh 1 sinh 1 sinh u _1 _ 2 __ 1 __ 2 _sh coshu-cos (0- 00) 2 eoshI 2- s+ eh + 2 +t2 i c2(eosh u - coso ( - 0 o) co2 sh 1 u cos 0- 0o) sinh u i i.. n\i /, ~~i rOI) 12 pQ = 2 V/2wra (coh P - cos 0)* (,coshl Pi - cos 0j - -osh - cosh -cosh -sinh -u du 2Q 2 - cos 1 (0 - 0,) sinhU + 1- (cosh - cos 0)i (cosh P - cos c du 2 /27ra a /eosh u - cosh a eosh u - os - 27r). we 1~ ~ / aastto we thus see that 1/PQ is expressed as the sum of two functions, the first of which involves the coordinates po, 0o, <po of P, and the second is the same function of the * See his paper "Ueber verzweigte Potentiale im Raume," Proc. Lond. Math. Soc. Vol. xxvIII. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 281 coordinates po, 0+ 27r-, /b of the point P' in the second space, which corresponds to P. If Q moves up to and ultimately coincides with P, we have cosh a=l; it will then be seen that the first function becomes infinite at the lower limit, but that the second one remains finite at that limit. Consider then the function W(po, 8, b0o) given by W(po 0o, 0o,) = 2 - (cosh p - cos 0)Y (cosh po - cos 0,) sinh 2 f Vcosh u - cosh; 1 JaVcosh - cosha cosh U - cosh os - 0o) the above equation may be written - = (p, o0, o0) + W (po, 0 + 27r, 0bo). PQ It is clear that the function W is uniform in our double space as it is unaltered by increasing 0 by 4rr; it will now be shewn that it is a potential function. We nay express W in the form W = (cosh p - cos 0)B (cosh p, - cos 0o) f 1 j + 2/e-4mu cos (0 - _Oo) du, 2 v27ra \a /cosh - cosh a 2 which may be written in the form W= (cosh p - cos 0) (cosh p, - cos 0o,) Q_- (cosh a) + 2 Z Q (co ) cos ( - 0), W7 = 'r5 P - - ' since the formula - e-(n+~) u V\-2 Qn ('cosh cc) e.-, ----, du, 2 Q (cosh a) cosh osh- cosh a) holds for all values of n such that the real part of n + ~ is positive (loc. cit. p. 519). Now (cosh p - cos 0)$ (cosh p, - cos 0,) cos s (0 - 0o) Qs (cosh a) is a potential form whatever s may be, and thus W is a potential function, and is expressible in the form W = - (cosha)2(cosh pa) cos (cosh p cos (cosh a) + 2Qo ( 0 - 0o) + 2Q~ (cosh a) cos (O - o) +.., the value of W (po, o + 27r, o0) being 2vra '{ 2- (cosh p-cos ) (cosh po- cos 0o)\ Q_ (cosh a) - 2Qo (cosh a) cos (0- 0) + 2Q_ (cosh a) cos (- 0) -...}; the two expressions added together give the expansion of l/D obtained in Art. 2. VOL. XVIII. 36 282 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, 4. To evaluate the definite integral in the expression for W write cosh u =S I 1(i-0),th cosh a = -, cos (0 - 00)=, then 2 2 sinh 2 u l- os sl 2 --- --- - du='/-2 dx /c - cosa scosh - cosh (O o ) - 2 ( - rT) =/2 /- 72 5-S, + Sm-n-' / v-' ( '2 ( -) where the inverse circular function has its numerically least value; we thus obtain the expression 1 (cosh p - cos 0) (cosh p, - cos 6) r7. ( 1 1 1 W v = c-oo [j- + sin- cos _ (0- o0) sech a1, 7raV\/ 2 {cosh a - cos(- o)2 L(2- )sech + n which may also be written in the form W=Q + sin-l cos (0-00) sech..................... (1). This expression TW has the following properties:-it is, together with its differential coefficients, finite and continuous for all values of p, 0, 6 in the double space, except at the point P in the first space, and it satisfies Laplace's equation; when Q coincides with P, the inverse circular function approaches, and the function becomes infinite as 1/PQ; when however Q approaches the point in the second space which corresponds to P, the inverse circular function approaches -, and the function does not become 2' infinite. The expression (1) is then the elementary potential function which plays the same part in our double space as the ordinary elementary potential function 1/PQ does in ordinary space. 5. In order to find a potential function which shall vanish over the surface of the disc, and shall throughout the first space be everywhere finite and continuous except at a point P (po, 0o, 0o) in the first space on the positive side of the disc (0< 0o <7r), we take the function W(po, 0o, bo)-W(po, 2wr-00, bo) which is the potential for the double space due to the point P and its image P'(po, 27r-0o, bo), which is situated in the second space at the optical image of P in the disc. This function is equal to 1 (cosh p - cos O) (cosh p, - cos!0 (- +0) sech I af | 7rc /2 {cosh a - cos (0 - 0o)} L2 ( 2 2 ) 1 (cosh p - cs 0)t (cosh p0 + cos 0) 0 r. + I 41 ) 7ra42 {cosh a + cos(0+ 00)} L 2 / 2 WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 283 which is the same thing as =pQ + - sin-l cos (O - 8) secha -p + -sin- cos0 + o)sech a............ (2), where P' is the optical image of P in the disc. On putting in this expression (2), for U, the values == r, 0 =- r, and remembering that over the disc PQ = P'Q, we verify at once that U vanishes on both surfaces of the disc. If Q coincides with the point (po, -00, 00) the function U remains finite. The Green's function GpQ which is a function that is finite and continuous throughout the whole of ordinary (the first) space, everywhere satisfies Laplace's equation, and is equal to 1/PQ over both surfaces of the disc, is given by GpQ= P-U, hence the required value of GPQ is GPQY = PQ - sin-l cos (8 - 0) sech 2 a + P +- sin- {-cos ( + o) sech } a =Q* -cos1 cos- (O(9-0) sech a- +/. - c oscos (O + 0o) sech.........(3), PQ 7 r i 2 7r 2 2 the numerically smallest values, as before, of the inverse circular functions being taken. It will be observed that in interpreting these formulae (2) and (3), the second copy of space, having served its purpose, may be supposed to be removed. THE DISTRIBUTION OF ELECTRICITY ON A CONDUCTING DISC UNDER THE INFLUENCE OF A CHARGED POINT. 6. If we suppose a thin conducting disc to be placed in the position of the fundamental circle of the coordinate system, to be connected to earth, and influenced by a charge q at the point P(p,, 0 b, ) on the positive side, the potential of the system at any point Q is q U where U is given by (2), and the potential of the charge on the disc is - q. GpQ. We shall now throw these potentials into a more geometrical form. We have sin-l {cos 2 (O - 0o) sech a 4} = tan-l'" ( -,) 1Cosh2 -a - cos2 ( - 00) \/2 cos - (0 -,o) =tan-' 2 } =/cosh a - cos (O - 0o) now take an auxiliary point L, of which the coordinates are po, 0 I-, o0, the upper or lower sign being taken according as 0 is positive or negative (-rr < 0 < r). Thus L and Q are always on opposite sides of the dise; using the formulae of Art. 1, we find L- 2 a2 cos0 -2a'cos- 0 CL2 - au = a - _CQ - cosh p, + cos 0 ' cosh p - cos 0 ' PL - f 1 + cos (0-0o) )t Çcosh p - cos 0\' PQL cosh a - cos (0 - 0,) cosh po + cos Qj ' 36-2 284 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, hence sin- {cos (0- 0,) sech 1 = + tan-1 (PL V/az- 1 ); p p/ FIG. 2. in order to determine the sign on the right-hand side, we observe that the inverse sine is positive unless 0 lies between - (r - 00) and - r, that is unless Q lies within the sphere passing through P and the rim of the dise, and is on the negative side of the disc; thus the sign on the right-hand side is to be taken positive unless Q lies within this spherical segment. Similarly we find sin-l -- cos 2 ( + 0,) sech a = t tan-l (Q / 2 - ) where the negative sign is to be taken unless Q is on the positive side of the dise and within the sphere which contains the rim and the point P'. We have thus as the expression for the potential of the system at any point Q (p, 0, b) V= Q L1 + -tan-(' v F tan-2 C.) (4) P _PQ - CL 2-a/2 2P 1 ~Q.2 'tan-PQ /CL-.(2\a when the ambiguous signs are assigned in accordance with the above rules. The auxiliary point L may be found from the following construction: Draw a spherical bowl through the rim of the dise on the opposite side to that on which Q lies, and equal to a sirnilar bowl which passes through Q; draw a plane PA'B' through P and the axis, cutting the rim in A', B'; this plane intersects the bowl in a circle; on this circle L lies, and is found by taking it so as to satisfy the relation LA': LB' = PA': PB'. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 285 In the case in which the influencing point is on the axis of the disc, we have po=O, hence a= p, and the auxiliary point L is on the axis of the disc at the point where this axis is cut by the sphere through the rim and the point Q, on the opposite side of the disc to Q; the formulae for the potential then become V=Q + - sin-' cos (O - O) sech p }] - P + -sin- -cos (0 + 0) sech } q 1ta- __ /a2 - CQ' qP'L a - CQ2 q L1 + an-lP C - + tan- C2PQ - 7. P 017 - a)] 2P' Q 7 P (Q CL - a'] ( 5.; the sign in the first bracket is positive unless Q lies in the segment ApB, and the sign in the second bracket is negative unless Q lies in the segment Ap'B. 7. To find, in the general case, the induced charge on the disc, it is sufficient to examine the limiting value of the potential at a point Q, as Q moves off to an infinite distance from the disc in the direction of the axis. In the expression for - q. GQ given by (3), let 0=0, p= 0, then a =p,, and PQ, P'Q become infinite in a ratio of equality; the expression for the potential of the induced electrification on the disc bas therefore the limiting value - 2q I i o Q s- c os -o 00 sech -2P, therefore the whole charge on the disc is - q. - cos- (cos2 0 sech o) which is equivalent to _ 2 t-,(>/a-CL ) - tan-' ~ q PL;' when L is a point in the plane of the dise which lies on the bisector of the angle APB. This expression may be interpreted thus:p FIG. 3. Let PL be the bisector of the angle APB, draw the chord NLM perpendicular to AB; the total induced charge is L NPM -q......................................(6). q'F 286 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, When the point P is on the axis of the dise, the induced charge is -q. -~, where 0o is the angle subtended at P by a diameter of the disc. When P is in the plane of the dise, the angle NPM becomes the angle between the tangents frorn P to the circular boundary of the disc. 8. The surface density at any point of the disc is given by the formula 47r av when dv is an element of normal and is given by + adO dv = - cosh p - cos ' We thus find for the density p, at the point (p, 7r,,) on the positive side of the dise, =po =-4 * PQ3 {1 + -sin- (sin 2~ sech 2 cos O0 q 1 cosh p + 1 o 47r2 PQ a o1 4rcosh2 -Sa - sin2 this expression can be put into a more geometrical form by introducing the auxiliary point L (po O-rr,,o) of Art. 6. The point L is now in the plane of the disc, and external to the disc; denoting this position of L by Lo, its coordinates are p, O, b0,. We have sin- (sin o 0 sech a) = tan-' (p~ A Q) w, -, T(P. CL L-a2\ which is equal to -tan-' ( ); 2- tn-~ P-Lo a2_ on reducing the second term in the expression for p, remembering that a sin 0o cosh po - cos o' we find that it becomes qz i /CL 2_-a2 -27r92 PQ2.PZo V 2 _ CQ2 ) 2wr'PQ'.PLo a -CQ' and thus the expression for the density at any point Q on the positive side of the disc is given by q PN q PNT PQ L- a' -a 2 (PQ CL'-' -a., po 27r PQ3 2 r2 PQ3 {JPLo a- CQ2 tPLV a- G } (7) where PN is the perpendicular from P to the plane of the dise, and L, is a point on AB produced, such that ALo: BLo= AP: BP. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 287 The value p, of the density at the point (p, - r, b) on the negative side of the dise is found in a similar manner to be q PN PQ CLo/ -a' 1PQ /CL2 -a2- ) 2P 2 P=Q 2 jPLO a' - CQ' ta PL 2 c...... ( 8) Thus the densities at corresponding points on opposite faces of the dise satisfy the relation q PN Po- Pl= - PQ3 When P is on the axis of the dise, L, is at infinity, and the formulae (7), (8) become q PN q PN x PQ tnQ po-2r ' PQ3 2w7.2 PQ3 IAQ BQ tan A.BQ q PN PQ tan- PQ \ P-= 27r {Q /nAQ.BQ tan- (AQ BQ) B(9) The expressions (7), (8), (9) agree with those obtained by another method by Lord Kelvin *. When P is in the plane of the disc it coincides with L,; in this case we find that the density on either side of the disc is given by 1 /CP2 a2 P= 27r2Q2 a2-CQ2............................ (10). 9. If the influencing point P is on the axis at 0o, we find from (5) the following expressions for the potential at points on the axis:-On the positive side of the disc q q (0-0,)- q2 ( +,) when 0>00, PQ 2wr. PQ 2,. P'Q q + 2 Q (0-0)- pQ (O + 0o), when 0 < 0. PQ 2r.PQ > 2r. P'Q On the negative side of the axis -Q+ PQ q( 0) + (2 + 0,), xvhen 0 + 0o is positive, PQ 2w. PQ (-0) 2r. P'Q PQ+- PQ (0 - 0)- q (0+ 0,), when 0+0, is negative. PQ 2w. Q( 2Pr.p Q If we denote by z, the distance of P from the disc, and 'by z the absolute value of the distance from the disc of a point Q on the negative side of the disc, the potential at Q is given by the expression q q q z _ q t o -t-X + Z ct-l (z +- C ot-l - o t-l ~; z+~z See his a)er on "z M-zl Ma" a * See his papers on "Electrostatics ancd Magnetism," p. 190. 288 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC, if z0 be given as a multiple of a, say z0= na, the expression q__ - q _a q (t Z q - cot- z- + cot-1 n) - q (cot - cot- n) z +na r (z + na) c a (na -z) a might be used to tabulate the values of the potential at points on the negative side of the axis. When z= 0, z= oc this expression is zero, and it will have a stationary negative value z for some value of z which may be approximately determined by plotting out the value of the function. Corresponding to this value of z there is a point of equilibrium which is completely screened from the effect of the influencing point P by means of the dise; the lines of induction from P which pass through this point, separate those lines of induction which end on the disc, from those which go to infinity. THE ELECTRIFICATION INDUCED ON A DISC PLACED IN ANY FIELD OF FORCE. 10. The potential of the electricity induced on the disc, which is connected to earth and placed in a field of constant potential, may be deduced from the expression (5) by taking the point P on the axis, and letting it move off to an infinite distance, the strength q of the charge increasing so that the ratio -q remains finite, say equal PQ to -A. We can easily shew that sin- (cos 2 0 sech = - sin-' ( 2a+ where ri, r2 are the greatest and least distances of the point (O, b, p) from the circular rim of the disc. We thus find for the potential of the electricity on the disc, the 2A 2a well-known expression sin-1 2, which is the potential of an insulated disc elec7r ri +r2 trified freely to potential A. 11. To find the potential due to the charge on the disc when placed in a field of force of potential,ux, when x is a coordinate measured from the centre of the disc in a fixed direction in the plane of the disc, suppose charges of strengths q and -q to be placed at the two points P(po, 0, 0), P'(- p, 0, 0) on the axis of x; the potential of the charge induced by these on the disc is at any point (p, 0, b) cos-O __ ( cos O -_pQo ------ 2 -o PQ cos — 2 osh cosh cos a where cosh a = cosh p cosh po - sinh p sinh po cos b, cosh a' = cosh p cosh po + sinh p sinh po cos b; WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 289 now let po become very small, as P, P' move away from the origin, the expression for the potential becomes, when higher powers of po than the first are omitted, cos ) cosh cos sinh pcos 2q P 1 PO s- Pi C 2 - 7' CP+ ~ — cosh 2 cP osh2 2 p-cos2 0 cosh2 ^ /cos^ \ cosh 2 p ^cos sih pCOS 1 1 1 i1 2q p c! p cosh - p cos-0sinh 2 a.CP CpV keosu g p) / h pcOS2 P C osh2 1 now C(7P = a hnco b P = -, hence if q be made indefinitely great so that -p2 =,,we cosh p - 1 po P2 find for the required potential + _ 2 - (, X /O S 1 1 1 s e c h I _\ 2/ S C O S2co s - 0 s i n h p r cos- (cos 0 sech - p - a cos. / s 7r -2osh2p/p </cosu p - cos 0) 2a /cosh - cos 0 a sinh p now - = -- x = cos ri + r2 1 cosh p - cos ' hence we find that the potential due to the induced electricity, in a field of force of potential xLx, is 2 X s 2ai 2a 2a /(r - + r)2 - 4 (11 ri~' + r,cs(r~ r,)c 12. In order to find an expression for the potential of the induced electricity on the dise, when it is placed in a given field of force, we apply the well-known theorem that if a- is the surface density at the element dS of the surface of a conductor when acted on by a unit charge placed at an external point Q, the potential function at Q which has values V given at every point of the conductor is fVa-dS, the integration being taken over the whole surface of the conductor. Suppose V(p, f) to be the given potential function at the element p, <b whose area we denote by dS, on either side of the disc; the potential function at the point (p,,,0, ) external to the dise which on the dise takes the values V(p, ) is then, using the expressions found in Art. 8, whi (1 + cosu p) cos 0, + t |in t2akR o v e h hl tan-s u / coshcf cosd, V(p, /)dS, VOL. XVIII. 37 290 DR HOBSON, ON GREENS FUNCTION FOR A CIRCULAR DISC, the value of the required function, R denoting the distance PQ. We now introduce new coordinates r, v, b instead of p, 0, 0, these being given by x = r2 + a2 sin. cos, y = r2 + a2 sin sin, z= rcos' ç; to express rv in terms of p, 0, we have x2 + y2 = (r2 a2) sin2 ( 27 + a2) sin2 Q2- a2 z2 hence cos4 7+ - a' cos2S= 2, where CQ2 =x2+ y2 + z2; hence we have a a2 /os22 - = (CQ2 - a2)2 + 4a2z COS2 77 \yi a + \_-_ Q 49 ~2a + 2C2a2 2a2 cos 0 2a2 Now CQ2 - a2= cos - and it is easily found that /(CQ2- a2)2 + 4a2z2 = a; cosh p - cos ' cosh p - cos 9 hence we have 1 -cos 0 cos r/ cosh p - cos ' and therefore cos v = / 1 cos 00 cosh po- cos 00 Also as P is on the plane of the disc (r=0), we have CP =a sin q, hence eP= + sin 1- sin vJ from which we find 1 + cosh p = 2 sec2 r. Reimembering that 1 _ 1 l + cosh p Vcosh po - cos 0 R a V/2 /cosh a + cos 0o we have AV/ 1 - cos 0O a cos 7 cos lo cosh a + cos 0o R and also i I (1 + Cshp) os 2 / + coshpp cosh 2 2 2 42, 2, V/eh2 2- 1 i*o R' V/coshpo -cos 0 R' cos ' a /2 cos o aR cos7 cos 'o' cosh2 a - sin2 00 then since dS = a2sin cos dd cdb, we have for the potential function at an external point rt, qo, 0o, which has the value V(ÇY, b) at the point v7, / of the dise, the expression z I a cos T7 cos acos 77 cos d7... (12); V= 7r cos jji 28sin. Vnq, 7) l +a cos 0 tan (o)}d d..(2); here the coordinates of the external point at which the potential is found are the elliptic coordinates given by z = r cos j0o, x = /r2 + a2 sin o0 eos o, y = /r + a2 sin Vo sin bo, the coordinate 770 alone appearing explicitly in the expression. This formula agrees with one obtained by Heine by a different and somewhat complicated procedure*. * See his Kugelfunctionen, Vol. II. p. 132. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 291 THE DISTRIBUTION OF ELECTRICITY ON A CONDUCTING BOWL UNDER THE INFLUENCE OF AN EXTERNAL ELECTRIFIED POINT. 13. In order to adapt the method of this paper to obtain corresponding results for the case of a spherical bowl, we must suppose the surface across which the passage from the first space to the second takes place, to be a spherical bowl with the fundamental circle for its rim. If the angle of the bowl is /3, we must suppose that in the first space 0 has values from /3-27r, on the negative side of the bowl, up to 8/ on the positive side, and that as we then pass through the bowl into the second space, 0 increases from /3 up to 3+ 27r, when the positive side of the bowl has again been reached. If the convexity of the bowl is upwards, 3 is less than 7r; if downwards, 3 is greater than 7t. The image of a point P (po, 0o, s0) in the first space and above the bowl is the point P' (po, 2/3 - o, fo) in the second space, and below the bowl. The expression U =, [+ +-sin- {cos 1 (0e - ) sech } ai V cosh p, - cos (2/3 - 0) 2 ( 2 2 JJ - \Xcosh po-Po cos0~ 02) P'Q + si- {cos 2( + 0o- 2/3) sech a....... (13) corresponds to the expression in (2); it is a potential function which vanishes over the disc, and of which the only infinity in the first space is at P; where it becomes infinite as 1/PQ. The Green's function GpQ is therefore given by the formula GPQ = P - cos-l cos 2 (8 - 0) sech 2 c} PQ?r + \ cosh %p-s cos 0(2)3 ~ f ) POQ s + - si 2( + 9- ) sech 2 a......(14). cosh pL- cos 80- ) P'Q 2 9 By introducing an auxiliary point L whose coordinates are po, 0 F /, 0, this expression may be thrown into a geometrical form corresponding to (4), and the expressions obtained by Lord Kelvin for the density on either side of the disc may be deduced; it is however hardly worth while to give the details of the process, as it is precisely similar to that which has been carried out in the case of the circular dise. 37-2 XIII. Demonstration of Green's Formula for Electric Density near the Vertex of a Right Cone. By H. M. MACDONALD, M.A., Fellow of Clare College. [Received 13 October 1899.] IN a footnote in his Essay on Electricity Green makes the following statement*: "Since this was written, I have obtained formulæe serving to express, generally, the law of the distribution of the electric fluid near the apex O of a cone, which forms part of a conducting surface of revolution having the same axis. From these formule it results that, when the apex of the cone is directed inwards, the density of the electric fluid at any point p, near to it, is proportional to rn-1; r being the distance Op, and the exponent n very nearly such as would satisfy the simple equation (4n+2), =3=7r; where 2/3 is the angle at the summit of the cone. If 2/3 exceeds 7r, this summit is directed outwards, and when the excess is not very considerable, n will be given as above: but 2/3 still increasing, until it becomes 27r-2 2y, the angle 2y at the summit of the cone which is now directed outwards, being very small, n will be given by 2 2n log - = l." The method by which he obtained these results was never published and the problem was not again attempted till 1870 when Mehler+ gave a solution for the electrical distribution on a right cone under the influence of a point charge; but the expression given by him for Green's function is so complicated as to make it difficult to obtain results from it, and the form of the expression does not exhibit the fact that it is discontinuous. In the following analysis a solution for the distribution near the vertex of a right cone forming part of a surface of revolution freely charged (Green's case) is obtained; also solutions for the distributions on a right cone, and on a surface whose form is the spindle formed by the revolution of a segment of a circle about its chord, under the influence of point charges on the axis. Solutions for both these latter problems have also been given by Mehler~. The cases when the point charge is not on the axis can easily be deduced, but present no special interest. The solutions here given are examples of a general method, which depends for its application on the fact that the writer has recently been able to determine the values of n in terms of u for which the harmonie Pn (u) vanishes. * Green, Essay on Electricity and Mlagnetism, 1828; + I have been unable to obtain Mehler's paper conMathematical Papers, p. 67. taining the results for the cone and have had to rely on t Green's statement is quoted and applied by Max- Heine's account of it, Theorie der Kugelfunctionen, Vol. II. well, Cavendish Papers, 1879, p. 385, with the' remark pp. 217-250. that no proof had ever been given. ~ Cavendish Papers, loc. cit. MR MACDONALD, DEMONSTRATION OF GREEN'S FORMULA, ETC. 293 ~ 1. Green's case. With the usual notation, the expression V - Arn P, (t/) is a solution of Laplace's equation in the neighbourhood of the vertex of the cone which is equal to V, on the surface of the cone for which P (cosa) vanishes, where a is the semivertical angle of the cone. That it may be the required solution Pn(,) must not vanish for any value of 0 between a and wr; for if it vanished for a value a', where a'> a, the expression would then be the solution for the space between the two coaxal conducting cones whose semivertical angles are a and a', or for some other space not entirely bounded by the cone whose semivertical angle is a. Hence n must be such that Pn (/u) does not vanish for a value of 0 which is greater than a; now the kth zero of Pn (/) considered as a function of n diminishes as 0 increases, therefore n must be the least zero of P, (cos a). Therefore the potential in the neighbourhood of the vertex of a right cone of semivertical angle a, forming part of a conducting surface which is charged to potential Y0, is V0 - ArnPn (p), where n is the least zero of P,(cosa) and A is a constant depending on the form and size of the surface. Hencet the density of the distribution in the neighbourhood of the vertex of the cone varies as r"-l, where r is the distance from the vertex and n is given by n = o/c/a where x0 is the least zero of JO (x), when a is small, by (4z + 2) a = 37r, when a is 2 nearly 7r/2, and by 2nlog-=1, when a is nearly 7r and 7r- a =y. Thus Green's results 7 are verified. ~ 2. Mehler's cases. (1) The distribution of electricity on a right cone under the influence of a charge on its axis. Let the space to be considered be the space bounded by the two concentric spheres r = b, r = a and the cone 0 = a, where r, 0, > are polar coordinates, and let there be a charge q at the point r=r', 0=0. The conditions to be satisfied by the potential are V= 0, when r= a and a > 0 > 0, V = 0, when r = b and a > 0 > 0, V= 0, when = a and a > r > b, and 2V 28V 1i a ( V and arV + 2a - + 1 a (1 - 2) av- + 47rp = ar" r ar +rL ( + 4 r=O throughout the space. Put r= ae-À, then the equation to be satisfied by V becomes ax2 a*a + 4Maa2e-2pld = 0; Macdonald, " the zeros of the ha ic () considered as a function of Poc. Lon. M. Soc. 1899. t Loc. cit. 294 MR MACDONALD, DEMONSTRATION OF GREEN'S FORMULA and, writing V= Ue2, U has to satisfy the equation 2-4 + (1 - /2) + 4ra2e p =0 with the same boundary conditions as V. Assume U = Z Wm -sin 1 Xo where = log -; this satisfies the first two boundary conditions and will be the solution required if Wm can be determined to satisfy the conditions Wm = 0, when =a and a> r> b, and also a2Wm /m'2 1\ Mi 7rX { T2(1-2) - } -+ )mWm sin + 4wrae 2 p =0, i k_^[ '/3 \A0X 4' J k that is 1 f/-/ a) / m22 1 87-ra2 fo 52 7 m7rX' {1- m) - ( + Wm + — a p 2sin dX = 0. a -) )/aWj ( Xo2r4 Xjo +- mo mo Assuming Wm = SAnmPn (,u), all the conditions are satisfied if this summation extends to all the values of n which make Pn (cos a) vanish and Anm is determined so that / 1`2 m27)r2 87Ta2 o. m7rX Anm n +. + ) + 2) P, ()pe 2 sin - dX, (\AnmAo ( ) o Jo Xo that is, if {'( 1 2-.2) f t 87ra2 i, OÀ m7X An + +- 4 {P, n (L}= dp pe 2 P, (/) sin d dXdr, where uo = cos a. Now f iPn(i/)dIJ 1 "- g$2 {Pn( L) Pn_ therefore (I 1 m2 f22) 1 2 a 8-a2 À f. 7 5, Anm + 1 pe 2 P () sin x dXd/. ++ Xo2.2n +1 l an ap X 2oo /o Making p vanish except at the point r= r', 0 = 0, where q = - 27rpa3 e-~ 'd/d', FOR ELECTRIC DENSITY NEAR THE VERTEX OF A RIGHT CONE. 295 the expression for V becomes I\ X' M7X77-n, 4 e oo' (2n +1) sin m7 sin P (Pn) aXo 1ai - a {n a nJ +m2w2 ' effecting the summation with respect to mn, this becomes 2 A cosh (n + 2) ( - + ) cosh (n + 2) (o - X - ) p 2q =qe 2 _2/ V=- a ----— a --- — - p, ----n — - pn) ~a sinh +n ) X(l (1 - -!~o x ~o~) ~ /02) when X > ', and A+~' cosh (n + ) (Xo -X+ ) - cosh (n + ) (Xo - ') '= ^ 2qe-2 E /p, (/, when X < X'. Making X,\ o the space becomes that bounded by the cone =a and the sphere r= a; and the potential inside an uninsulated hollow conductor of this form under the influence of a charge q at the point r' = ae-' on the axis is given by A+A' 2qe 2 e-(n+)(-À')-e- (n+) (À+À') (1 - jo02) na ' 3n aS/ Pi when >', and by V= 2qe 2 e(n+t) (-X') - e-(n+i) (A+X) a - aP),p, a *Pn an2 al when X< X', that is by ri' rn ny'n\ P (., V=- 2q(r, n+l t n+l ' -P, aPn r a7 c when r'>r, and by ( r/n rar'n\ an ( a, V=-2q\ a-^). -3P 3P when r >r'. To obtain the potential in Mehler's case when the cone extends to infinity put a = o and then 2 ___ P, (/) V=- 2q,- r ~' (- p), dPf, 296 MR MACDONALD, DEMONSTRATION OF GREEN'S FORMULA when r' >r, and In 9 P n( ) rn+l- ap aP ' (1 -/o2) ~ ~ an a~j when r >r', where the summations extend to all the positive values of n which make P, (cos a) vanish. When a =r/2 aP, aP,. 1, an aLP rn and V= 2q 2 rn+ Pn (/), when r'>r, where the summation extends to all the positive odd integers, that is V = q q /r2 + r'2 - 2rr' cos 0 N/r2 + r'2 + 2rr' cos 0 which agrees as it ought to with the expression for the potential due to a charge q at a point distant r' from an infinite conducting plane at potential zero. (2) To find the potential at any point due to the spindle formed by the revolution of a segment of a circle about its chord, when its surface is freely charged. This is immediately obtained by inversion from the above case. Let:0 be the angle in the segment of the circle whose revolution describes the spindle, ~ the angle in any other segment of a circle on the same chord, j = log -, where r1, r2 (r >r2) are r2 the distances of a point on a segment from the extremities of the chord; then putting q = - Vr' and observing that the cone of angle 0 in the dielectric inverts into the spindle the generating segment of which contains an angle eo, the potential at any point due to the spindle when charged to potential Vo is given by e- *</______ P- (cos f) V= Vo + 2 V0 /2 (cosh V - cos:)2 e- P (. ) (lL02) aLI-O where zO =cos o and the summation extends to all the positive values of n which make P,n (cos ef) vanish. The case of the sphere is that when = 7r/2. It may be verified that the density of the distribution on the spindle near one of the conical points agrees with that found ~ 1. For the density at any point on it is given by _ _O {2 (cosh - cos ^)}2 _ _;;; FOR ELECTRIC DENSITY NEAR THE VERTEX OF A RIGHT CONE. 297 and near one of the conical points this becomes V0 rn- 1 27r r ' aPn sin san where r' is the length of the axis of the spindle, r the distance of the point on the surface from the conical point and n is the least zero of P (cose). Now when e0, equal to r - y, is nearly 7, is - r and the values of n which occur are k+n0, where o 'r-, i s an sin nrr 2 k is any positive integer and 2no log -= 1 *; on substitution and summation, the expression for the density at any point becomes -where r >r, 2r7rlno r21+no where > r. * Loc. cit., Proc. Lond. Math. Soc. 1899. VOL. XVIII. 38 XIV. On the Effects of Dilution, Temperature, and other circumstances, on the Absorption Spectra of Solutions of Didynmium and Erbium Salts. By G. 'D. LIVEING, M.A., Professor of Chemistry. [Received 15 October 1899.] IN November 1898 I made a preliminary communication to the Society giving results of observations on the absorption spectra of aqueous solutions of salts of didymium and erbium in various degrees of dilution. Since then most of the observations have been repeated with improved apparatus, whereby several anomalies in the photographs have been removed, and a great many additional observations made, so that it will probably be best to make this communication quite independent of the prelirilinary one, and, at the risk of a little repetition, complete in itself so far as it goes. APPARATUS. The observations were made in part directly by the eye with an ordinary spectroscope, and partly by photography. On the former I rely only for the part of the spectrum below the indigo, on the latter for the more refrangible part. The spectroscope chiefly used for the former had two whole prisms of 60~ and two half-prisms, all of white flint glass, telescopes with achromatic object glasses of 12 inches focal length, and eye-piece of very low magnifying power. It was useless to employ higher dispersion or magnification, because the absorption bands, even the sharpest of them which is that of didymium at about X 427, are all diffuse, and higher dispersion or magnification renders some details invisible. In comparing by eye the spectra produced by two solutions, one was thrown in by reflexion in the usual way, and, after making the comparison, the positions of the solutions were interchanged and the observation repeated, in order to correct any error arising from a difference of intensity between the light entering directly and that coming in by reflexion. For photography the spectrum was formed by one prism of 60~ and two halfprisms, all of calcite, the object glasses of the telescopes were quartz lenses of 18'5 inches focal length for the sodium yellow light. The photographic plate was of course PROF. LIVEING, ON THE EFFECTS OF DILUTION, TEMPERATURE, ETC. 299 inclined to the axis of the telescope so that, as far as the doubly refracting character of the calcite prisms allows, the image might be in tolerably good focus across the whole width of the plate, two and a half inches. To concentrate the light, and make it, for the parts of the spectrum not subject to absorption, nearly uniform whatever the thickness of the absorbent stratum of liquid, a quartz lens of three inches focal length was fixed at that distance in front of the slit, and a similar lens fifteen inches further off, and three inches beyond the second lens was fixed a screen with a circular hole in it about one-eighth of an inch in diameter, and beyond that was of course the source of light. The centres of the hole in the screen and of the two lenses were aligned with the axis of the collimator. The distance between the lenses was fixed so as to allow of the interposition of the longest trough, used as a water bath for maintaining the temperature of the tubes containing the solutions. These troughs were of brass fitted with a plate of quartz at each end, and each had in it two V-shaped septa on which the tube with solution rested, and thereby took up at once its right position in the course of the pencil of light between the lenses. The tubes holding the solutions were of glass, fitted at the ends with quartz plates. These plates were held in position by outer brass plates with central circular perforations, connected by three wires passing along the outside of the tube and furnished with screw nuts by which the plates could be firmly pressed against the ends of the tube. The joint between the quartz plate and the end of the tube was made water-tight by a washer of thin rubber. The washers all had the same sized circular opening which determined the cross section of the pencil of rays falling on the slit. This seemingly complicated arrangement was adopted because it was necessary to have joints which would not be affected by a temperature of 100~, or by dilute acids, or by alcohol, and could be easily taken to pieces for cleaning the tube or plates. Each tube had a branch on its upper side which was left open for the purpose of filling the tube, and to allow of expansion of the liquid when it was heated. Tubes of four lengths in geometrical progression, namely of 38mm., 76 mm., 1525 mm., and 305 mm., and a cell with quartz faces having an interval of 6'7 mm. between then, were used to hold the solutions; and for a few observations a cell of only 5 mm. thickness was used. For observations on the effects of temperature, the trough containing the tube with solution was filled with water and a photograph of the spectrum taken at the temperature of the room; the trough was then heated by one or more gas lamps until the water boiled, the gas lainps were then lowered so as to maintain the bath 3 or 4 degrees below the boiling point, bubbles adhering to the quartz plates swept off with a feather, and when the whole appeared to be in a steady condition another, photograph was taken. Unless the solution in the tube were a very dilute one there was not much trouble with bubbles in the solution, but bubbles in the bath were very troublesome, and had to be removed because they impeded the passage of the light, and thereby 38-2 300 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE affected the photograph. A similar effect is produced by convection currents of unequal density. These were pretty well avoided within the absorbent liquid, but could not be completely avoided in the water of the bath. The difference of temperature, and consequent difference of density, of the currents in the water was, however, small, and the thickness of water between the end of the tube and the quartz window of the trough also small, so that the currents were not of much consequence. Attempts to use temperatures below that of the room were abandoned because of the dew which settled on the quartz windows. Wetting the quartz with glycerol was no remedy, because the glycerol gravitated, destroyed the plane figure of the window, and dispersed some of the light. Very fair observations by eye of the effect of heat on a solution, not too dilute, were made by fixing two similar test tubes containing the solution, one in front of the slit and the other in front of the reflecting prison, and after adjusting their positions until the two spectra, seen simultaneously, were identical, heating up one of the test tubes by placing a lamp under it. For dilute solutions, requiring a greater thickness to give absorption bands of sufficient intensity, two of the tubes used for the photographs were employed, one of them being heated up in its water bath. As a source of light a Welsbach incandescent gas lamp without chimney was chiefly used. This was placed 5 or 6 inches from the screen so that the network of the mantle was quite out of focus at the slit. It gave a good light up to a wave-length of X 370, but beyond this point it would not produce a good photograph without an exposure too prolonged for the less refrangible part of the spectrum. For the region above X 360 a lime-light was used. Inasmuch as the bands observed are all more or less diffuse, and fade away gradually on either hand, any variations of the intensity of the source of light, of the sensitiveness of the photographie plates, or of the development of the image, tend to mask the effects of varying the composition, or the temperature, of the solutions; so that two photographs can be fairly compared, for the sake of determining these effects, only when they have been taken with the saine light, on the same plate, with equal times of exposure, and have been developed together. This has been attended to throughout. The photographs to be compared with each other have always been taken in succession on the same plate, with no other change than the necessary shift of the plate and the substitution of one tube of liquid and its bath for another. The photographs taken thus in succession do very well for comparison of the intensities and other characters of the absorption bands, but cannot be depended on for the detection of a very small shift in the position of a band. That could be done if the two spectra to be compared were in the field at the same time, one of them reflected in, but I have not attempted to photograph two spectra in this way, and have been content to detect alterations of wave-length, in the bands most easily visible, by the eye without photography, ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 301 THE SOLUTIONS EXPERIMENTED ON. These have been chiefly those of salts of didymium and erbium. Most coloured salts have only very wide absorption bands which fade on either hand very gradually, so that it is extremely difficult, or even impossible, to recognise small changes in them. On the other hand, didymium and erbium salts have a great many absorption bands, of various degrees of sharpness and of intensity, and distributed through a wide range of the spectrum. No other salts seem so well adapted for my purpose. However, I made a number of observations on uranous chloride, but found it so prone to chemical change when in solution that I could not with certainty distinguish the effects of dilution, or of elevation of temperature, from those due to chemical change. The absorption spectra of salts of cobalt have already been investigated by Dr Russell, though not exactly from my present point of view, and they are not as good for my purpose as the salts of the two metals to which I now confine myself. Both series of salts had been purified as far as possible, by my assistant Mr Purvis, by a long series of fractional precipitations. The didymium was spectroscopically free from lanthanum, but it had not been found possible to get it, or the erbium, so free from yttrium*. No attempt was made to separate the neodymium from the praseodymium, and there is no method at present known for separating the various metals of which ordinary erbium is supposed to be a mixture. Indeed for my purpose there would be no advantage in doing so; though for a quantitative estimation of the concentration of absorbent material in the solutions it was important to get rid of an admixture of unabsorbent salt. In order to obtain solutions of the salts of different acids in equivalent concentration the metal was precipitated as oxalate, washed, dried, and ignited in air until it was reduced to oxide. Weighed quantities of this oxide were dissolved in the several acids, and, in the case of nitric and hydrochloric acids, the solutions evaporated and excess of acid driven off. The residual salts were then dissolved in measured quantities of water. The most concentrated solutions of didymium employed contained, respectively, of the nitrate, 611'1 grams to the litre, and of the chloride the equivalent quantity, namely 462'9 grams of anhydrous chloridet. These each contain 1862 gram-molecules of the salt * Lanthanum and yttrium cannot be recognised by of three sets in the green and citron of which the brightest any absorption bands, but when induction sparks are taken begin at X5599 and X5380 respectively, and the third at from solutions of their salts, each gives a very character- X5173. There is another weaker set in the orange beginning istic channelled spectrum, by which it is easily recognised at X5865, and two sets in the indigo beginning at X4419 and in a solution containing one per cent., or even less, of the X4370 respectively. My measures were not made with any salt. The yttrium channellings are in the orange, the large dispersion and the last figure of the measured wavebrightest of those of lanthanum in the citron and green, length may not be quite correct, but near enough for recogand both fade towards the red. Thalén in his paper (1874) nition of the channellings which are easily seen with a on the Spectra of Yttrium and Erbium, and of Didymium small spectroscope, especially the two first mentioned. and Lanthanum, gives the wave-lengths of the sharp, t The (crystalline) didymium chloride in this solution more refrangible edges of the yttrium channellings, one was dissolved in just about twice its weight of water; the set beginning at X6131 and the other at X5970'5. He does equivalent solution of nitrate had still less water. not give those due to lanthanum. These I find to consist 302 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE per litre, and as the specific gravity of the solution of chloride is 8-295, it appears to contain one molecule of the chloride to between 27 and 28 molecules of water. Didymium sulphate is rather sparingly soluble in water, so that the most concentrated solution of it employed contained only 58'11 grams of it per litre. For comparison with it, the strongest nitrate, or chloride, had to be diluted to 9'16 times its bulk. Of erbium the most concentrated solutions used contained, respectively, of the nitrate 935 2 grams to the litre, of the chloride 726'6 grams. These each contain 2'67 grammolecules of the salt per litre. The solution of the nitrate was a saturated one at a temperature of about 15~. Less concentrated solutions were also prepared and used, containing, respectively, 566 grams of nitrate of erbium, and 440 grams of the chloride to the litre, or about 1'61 molecules in grams to the litre. The more dilute solutions were obtained from these by taking measured quantities of them and diluting up to the required volume. In fact the most concentrated of these solutions were the stock solutions, and may conveniently be described as of strength No. 1. Half strength will mean such a solution diluted until the bulk was doubled, one-quarter strength will mean No. 1 diluted until its bulk was quadrupled, and so on. Other salts and solvents were employed, and will be described when the experiments upon them are described. The solutions of nitrate and chloride above mentioned were, as a rule, the standards of concentration. THE ABSORPTION BANDS OBSERVED. The didymium absorption bands of which I have taken notice in this investigation, are as follows: A band in the red at about 679. A weak band at about X 623. A rather weak band at about X 596. The strong group extending from about X 590 to X 570, consisting of a number of bands overlapping one another. A rather weak band at about \ 531. A strong group of about four, more or less overlapping, bands, extending from about X528 to X 520. A less strong group of two diffuse bands with the centre about \ 510. ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 303 A well marked triplet at about X 483, 476 and 469, of which that in the middle is decidedly weaker than the other two. A broad weak band, with its centre at about X462, and extending nearly down to the most refrangible band of the triplet above mentioned. A very broad band with its centre about X444. A very weak band,,,,, 433. A strong, narrow, sharply-defined band at about X 427. A very weak diffuse band with its centre about X 418. A still weaker one with its centre about 415. Another weak diffuse band at about X 406. A very broad strong band with its centre about X403. A very weak diffuse band at about X 391. A diffuse band at about X 380. Another, wider, at about 375. A weaker band at about X 364. Four, nearly equally distributed between X 358 and X 350, which in all but the weakest solutions run into one broad band extending beyond the above-mentioned limits. A weak diffuse band at about X 338. And a broad diffuse band with its centre about X 329. These bands appear all to belong to didymium, or to the metals associated under that name, for though they may be modified in character, and even in position, by the solvent and other circumstances, they all disappear in the absence of didymium, and they retain so much the sane general character under all circumstances, that it is reasonable to infer that they have the same primary cause. A reference to plate No. 19 (at the end of the volume) on which are reproduced photographs of the spectra of didymium chloride in solution in water, in alcohol, and in alcohol charged with hydrochloric acid, will make my meaning evident. The erbium absorption bands of which I have taken notice in this investigation are as follows: A group of four bands in the red, of which the most refrangible but one is much the strongest and has a wave-length about X 653. A group of four, of which the more refrangible two are much stronger than the others, lying between À 536 and À 549. 304 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE A weak band at about X527. A very strong one at about X 523. A weaker one at about X520. A rather broad band, strongest on its more refrangible side and fading towards the less refrangible, with its strongest part at about X 491. A strong band at about X 488. A weaker one at about X486. A broad but weak band with its centre about X472. A sharp but weak band at about X 467. A broad, diffuse band with centre about X 454, reaching almost up to a stronger, and narrower, band at about X 449. These two are merged into one with concentrated solutions. A weak band at about X 441. A narrow one at about X 422. A weak one at about X 418. A broad band, fading on its less refrangible side, and extending from about X 415 nearly down to the band at X418. A pair of nearly equal bands, rather strong, at about X 404 and X 407. A very faint but broad band extending from about X:396 to X 402. A well-marked, rather narrow band at about X379, And a weaker one almost touching it on the more refrangible side, which becomes merged with it, and with a still weaker diffuse band at about X 377, in solutions a little stronger. A weak diffuse band with centre about X 367. A strong band at about X365, accompanied by One rather less strong at about X363, which become merged together when the solution is rather stronger. A band rather weaker than the last at about X 357, and A broad weaker band with centre at about X353, which soon merges in the former when the solution is a little increased in strength. All these bands more refrangible than X 404, expand rapidly and become very diffuse at the edges as the solution is more concentrated, so that they may easily be ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 305 confounded with a diffuse continuous absorption which extends from the ultra-violet down the spectrum as the solution becomes more concentrated; but they are common to the nitrate and chloride, and may be seen with a solution of the former when with an equivalent solution of chloride the advancing continuous absorption has obliterated them. The superposition of this continuous absorption, even when it is very weak and scarcely otherwise perceptible, strengthens and widens the bands. EFFECTS OF DILUTION. For observing the effects of dilution equal volumes of the stock solutions were diluted to 2, 4, 8, 45-5, 61 or 91 times their original volumes, and the absorptions produced by thicknesses of these solutions proportional to their dilutions observed and photographed. In the spectra of either didymium or erbium chloride, starting with solutions half the strongest, or less strong, in thickness of 38 mm., I can find no change with dilution, when accompanied by proportional increase of thickness, below X 390: see plate 3, at the end of the volume. With the strongest solution in a thickness of 38 mm. a diffuse absorption creeps down from the most refrangible end of the spectrum, as may be seen in the uppermost spectrum in each of the plates 10 and 11. Above X 375, or thereabouts, it seems to cut off all the light, but the diffuse edge extends with the strongest didymium chloride as low as X 415, making the absorption bands look wider and stronger by its superposition. On comparing with the eye the spectrum of a thickness of 5 mm. of the strongest solution of didymium chloride, with that of 305 mm. of the same solution diluted to 61 times its volume, both spectra being in the field of view at the same time, I could detect no difference between them. Again, photographing the spectrum of a thickness of 6'7 mm. of the strongest didymiium chloride, and that of 305 mm. of the same solution diluted to 45'5 times its original bulk, I can find no difference between the photographs, which take in a range from about X 350 to X600. Plate 7 is a reproduction of these photographs. This identity of the spectra extends to the intensities, even of the weakest bands that I can see, as well as to the positions of the bands, and even to the apparent extinction of the diffuse absorption which is produced by a greater thickness of the strongest solution at the ultra-violet end. Also erbium chloride of half the strongest concentration, in a thickness of 5 mm., gives a spectrum which cannot be distinguished by my eye from that given by 305 mm. of a solution 61 times as dilute. And photographs of the spectrum of the same solution, half the strongest, in a thickness of 6 7 mm., are identical with those of 305 mm. of the same solution diluted to 45'5 times its bulk, below a wave-length of about X 380. Plate 9 is a reproduction of these photographs. The triple band at about X 378 comes out more strongly with the stronger solution, but I am not sure whether this is not an effect due to the superposition of the diffuse absorption creeping down from the more refrangible end. In the region above 355, a thickness of 152 mm. of a very dilute solution of didymium VOL. XVIII. 39 306 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE chloride transmits a sensible amount of light as high as X315 (the highest part of the spectrum included in my photographs) but with a gradually fading intensity frorn about 348 upwards. And this diffuse absorption creeps further down as the solution is stronger until with a solution half the strongest, in the same thickness, it reaches X 360. Didymium bromide produces a similar diffuse absorption which extends lower than in the case of the chloride; and didymium sulphate shews something of the same kind. This diffuse absorption, which creeps far down the spectrum of the most concentrated solutions of the chlorides of both didymium and erbium, seems to belong to a different category from that to which the other bands belong. For not only is it diminished by dilution when the thickness of the stratum is proportioned to the dilution, but it is diminished by diminishing the thickness of the strong solution, without diluting it, at a greater rate than the other bands are diminished, for some of the ultra-violet bands which are quite obscured by it when the liquid is 38 mm. thick are visible in the photographs when the same liquid is only 6'7 mm. thick. The obvious suggestion is that it is due in some way to the common element, the chlorine. Most chlorides, however, produce no such absorption. I have tried solutions of calcium, zinc, and aluminium chloride, respectively, and found them, in a thickness of 305 mm., very nearly as transparent as water for the range of the spectrum included in my photographs, namely below X 355. One chloride I have found, when in a concentrated solution, to behave like the didymium and erbium chlorides, and that is hydrochloric acid, whether it be dissolved in water or in alcohol. Plate 12 is a reproduction of a photograph of the spectra of solutions in alcohol, and in water, of hydrochloric acid, in several thicknesses, and in proportional -degrees of dilution, along with one of distilled water for comparison. The increasing extent of the absorption with increasing concentration of the solution is manifest; and the most probable cause is some action between the molecules of acid during their encounters, for it seems to depend on the number of molecules of acid (or salt) and on their concentration, jointly. We cannot ascribe the absorption to the chlorine ion, because the number of chlorine ions increases with dilution; but the close correspondence of the effects strongly suggests a common cause in all the solutions which give those effects. It should be observed that the percentage of chlorine in the concentrated solution of the acid used in these experiments bore to that in the most concentrated solution of didymium chloride the ratio of about 39 to 14'5. The extent, down the spectrum, of the absorption now in question, is increased, as might be expected, by adding hydrochloric acid to the didymium solution, and also by raising the temperature as described below. In connexion with this it may be remarked that concentrated neutral solutions of didymium, and erbium, chloride lose the clean pink tint, by transmitted light, of their dilute solutions, and take up more of an orange hue, due of course to the diminution of the rays at the blue end of the spectrum. As above stated I have been unable to obtain a solution of didymrnium sulphate so concentrated as my strongest solution of chloride; but using the solution containing ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 307 5811 grams to the litre, and diluting it to twice, four times, and eight times its bulk, I could find no change in the absorption spectrum produced by it when the thickness of the absorbent liquid was proportioned to the dilution, either when directly viewed or when photographed. See plate 4, which however does not include any part of the spectrum below the green. Nor could I detect any difference between the spectrum of the sulphate and that of an equivalent solution of the chloride. Didymium nitrate in four dilutions, beginning with the strongest in thickness of 38 mn., and ending with one-eighth strength in thickness of 305 mm., gave spectra which could not be distinguished from each other, in the range photographed. See plate 11, where the spectra are those of equivalent solutions of the chloride and nitrate alternately, beginning with 38 mm. of the strongest solution of chloride, next the equivalent nitrate, then 76 mm. of the solutions of half strength, 152 mm. of one-quarter strength, and ending with 305 min. of the two solutions of one-eighth strength. This appearance of identity is brought about, however, by the diffuseness and strength of the absorptions by which the details of the groups of bands are obliterated. When the spectra of the same solutions in much less thickness are examined, it is seen that the bands of the stronger solutions of nitrate are more diffuse, or wider, than the bands produced by equivalent solutions of the chloride. The weak bands look washed out, the strong are wider than the corresponding bands of the chloride, and in the strong groups the component bands are merged together. By increasing dilution the several bands contract themselves and become better defined, until, with solutions of -3 strength, I am unable to see any difference between the bands of the nitrate, chloride, and sulphate in equivalent solutions. In the stronger solutions the weak bands look weaker as well as broader with nitrate than with chloride, the strong bands are broader but look no weaker; but I think that when an absorption is very strong the eye does not pereeive, nor a photographic plate always record, a small difference of intensity. There is no indication of an increase of intensity of the bands of the nitrate by dilution with corresponding increase of thickness. There are, on the other hand, indications of a shift of the positions of greatest absorption in the bands in the yellow and green, which remind me of the much greater shift of these bands by the use of alcohol and other solvents instead of water. Comparing small thicknesses (5 mm.) of solutions, the big band in the yellow expands with the nitrate beyond that produced by the equivalent solution of chloride, especially on the less refrangible side. Of the four strong components of this band the least refrangible seems, with the nitrate, to be displaced a little towards the red, and a less strong diffuse band extends still further beyond the corresponding band of the chloride on the red side. The less refrangible of the two strong groups in the green, which for the chloride consists of two nearly equal strong bands separated by a narrow chink of light, and of a fainter very diffuse absorption extending some way down towards the red, has for the nitrate the less refrangible strong band widened out by diffusion, some way beyond its limit for the chloride on the red side, and the more refrangible is weaker with the nitrate. The more refrangible group in the green appears with the 39 —2 308 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE nitrate as a single band narrower than the two given by the chloride, and the middle band of the triplet in the blue is more diffuse with the nitrate. The apparent shift above mentioned may be an effect of the overlapping of the diffuse bands, and though a real shift does not seem to me improbable, it is not in this case sufficiently decided to found an argument upon. Plate 6 reproduces the spectra of 6'7 mm. of the strongest solution of didymium nitrate and of 305 mm. of the same solution diluted to 45'5 times its bulk. The bands of the strong solution are more diffuse and look somewhat washed out, notably the narrow band about X 427, and the middle band of the triplet in the blue; and the strong group in the yellow extends further towards the red and has the appearance of being stronger with the strong solution than with the dilute. Erbium nitrate behaves quite in the same way as didymium nitrate in regard to the greater diffuseness of its bands with strong solutions, and their gradual contraction and growing sharpness as the solution is diluted, until they come to be identical with those of the chloride. This is better seen in the photographs of the erbium spectra than in those of the didymium: see plate No. 5. In plate 8 the spectrum of 6'7 mm. of solution containing 467 grams of erbium nitrate to the litre is contrasted with that of 305 mm. of the same solution diluted to 45'5 times its bulk. The greater diffuseness of the bands of the upper spectrum, which is that of the strong solution, and apparently greater intensity of the ultra-violet band on the left will be noticed. It may be compared with the corresponding plate No. 9 for the chloride, in which however the lower spectrum is that of the stronger solution. Plate 10 contrasts the spectra of equivalent solutions of erbium chloride and nitrate, in four degrees of dilution, the uppermost spectrum being that of the strongest chloride. The greater diffuseness of the bands of the nitrate can be seen, and the gradual approximation to identity in the spectra of the two solutions as they become more dilute. It is the counterpart for erbium of plate 11. The nitrates, as well as the chlorides of both metals, shew a general absorption creeping down from the most refrangible end of the spectrum with increased concentration of the solutions; but though similar in the two salts, that given by the nitrates is not identical with that of the chlorides. Its edge is not so diffuse, but cuts off the spectrum more sharply than that of the chloride; and in the strongest solutions it does not extend so far down the spectrum as that of the chloride. On the other hand with the weak solutions of didymium it extends lower than that of the chloride. With a solution of didymium nitrate of -4 strength in thickness of 152 mm. all light above X333 seems to be absorbed, while with the chloride light gets through beyond X 315; and the strongest solution of the nitrate in a thickness of 38 mm. does not entirely cut off the light below 360, while the equivalent solution of chloride cuts it off much lower. ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 309 There are here four facts to deal with: 1. The identity of the spectra of the different salts of the same metal in the dilute condition. 2. The constancy of this spectrum in the case of chloride and sulphate in different dilutions so long as the thickness of absorbent is proportional to the dilution, a constancy holding good in the chlorides for a great range of concentration. 3. The modification, for I take it to be only a modification, of this spectrum in the case of the nitrate, by some cause which has increasing effect with increasing concentration. 4. The absorptions at the most refrangible end of the spectrum, which are somewhat different for different salts of the same metal, and diminish with increased dilution. The first of these facts is certainly strongly suggestive of the interpretation put on it by Ostwald, that the spectrum common. to all the salts of the same metal is due to the metallic ions. Against this the second fact militates, for the ionization is supposed to increase with dilution, and the absorptions by the ions should increase in intensity by dilution when the total quantity of salt, dissociated and undissociated, through which the light passes remains the same. The third fact points to some cause, affecting the diffuseness of the bands, which is more effective in concentrated solutions. This cause may be encounters between the molecules of the salt, or of its products in solution, which would be more frequent in more concentrated solutions. Ionization should be increased by heating the solutions, and diminished by the addition of acid. I proceed to describe what I have observed of the effects of heating and of acidification on the absorption spectra. EFFECTS OF TEMPERATURE ON THE SPECTRA. The rise of temperature which could be employed was, as described above, only from the temperature of the rooin, about 20~, to a few degrees below the boiling-point of the water bath, or to about 97~. This rise of temperature produced the same kind of effect on all those absorption bands which are common to all the salts of the same metal, whether it be didymium or erbium, and that effect was to render them more diffuse, to spread them out, make their limits less definite, and in the case of weak bands make them appear weaker. The effect of heat was also the same in kind on dilute as on concentrated solutions. Heat also caused the broad diffuse absorption at the most refrangible end to extend itself downwards in a marked degree. Plates 13, 14 and 15 are reproductions of photographs of the spectra of three salts, in various degrees of dilution, cold and hot. It will be noticed that the absorption bands are not increased in intensity by heat, but from the greater diffusion they seem weaker, except the very strong bands which are so intense that they bear diffusion without letting enough light through to affect the plate. The creeping down with the higher temperature of a diffuse absorption from the most 310 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE refrangible end is seen in all, and with the nitrate and sulphate seems to be independent of the concentration, while with the chloride it is barely noticeable with any but the most concentrated solution. In the last exposure with the sulphate the light is a little weaker throughout. The solution was the weakest and in the longest tube, and therefore most likely to be troubled with bubbles on the inner faces of the terminal quartz plates which could not be removed. I have no doubt this general weakening of the light was due to this cause. A general weakening of the light has the effect of making the absorption bands appear stronger. This appearance is deceptive; for the examination of a great many photographs, as well as direct observations of the spectra by eye, have led me to the conclusion that the effect of heat is to diffuse and not to strengthen the absorption bands which are ascribed to the metals. On the other hand it looks as if the diffuse absorption at the most refrangible end, which certainly creeps down lower with hot solutions, were strengthened as well as diffused, for in the region above that included in the plates, the limit of complete extinction of photographie effect is considerably lower with the hot than with the cold solutions. On the whole the effects of heat on the spectrum afford no confirmation to the supposition that the absorptions are due to an increase of the number of ions; but rather suggest that they may be due to the increased energy of the motions of translation of the molecules, causing more frequent encounters. EFFECTS OF ACIDIFYING TIHE SOLUTIONS. The solutions compared with a view to ascertain these effects had in every case equal quantities of the metallic component per litre, but while one was neutral the other had twice as much of its acid component as the first; and they were usually compared in various degrees of dilution and in thicknesses proportional thereto. With didymium salts, chloride and nitrate, the acid made very little difference in the bands, as will be seen by examination of plate 18, which gives the spectra of four solutions of the chloride, two neutral and two acid, The creeping down of the absorption at the most refrangible end is, however, very evident in the most concentrated solution of acidified chloride; and some diffusion of soine of the bands of the nitrate by the addition of the acid is just traceable in photographs of some of the weaker bands of the more concentrated solution. The increased diffusion of the bands of the nitrate by the addition of nitric acid can be easily seen directly by eye, using weak solutions in no great thickness. The addition of acid also produces a slight shift of the places of greatest absorption in the strong groups in the yellow and green. Whether this is due only to the expansion, and consequent overlapping, of the several bands in these groups, or whether there is a real shift, I have not been able to satisfy myself; but the general appearance resembles the changes produced in those bands by the use of different solvents which are described below, and it is very likely that similar causes are at work in the two cases. Nothing of this kind can be seen on the addition of hydrochloric acid to the chloride. ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 311 With erbium nitrate the addition of acid produces more marked effects:` see plate 17. All the bands which are more diffuse with the neutral nitrate than with the equivalent chloride solution, are still more diffuse with the acid nitrate; and the effect regularly diminishes as the solution is made more dilute. There is however no indication that there is any weakening of the intensity of the bands by the presence of acid, but rather a strengthening of them. With the chloride, on the other hand, there seems to be no more difference between the absorptions of the neutral and acid solutions than there is between the corresponding solutions of didymium chloride. Comparing the spectra by eye, I can see no appreciable difference between the acid and neutral solutions of equal thickness and equal erbium concentration. Plate 16 gives a reproduction of photographs of the absorptions of two pairs of equivalent neutral and acid solutions of erbium chloride, the upper pair being those of the strongest solution. The creeping down of the continuous absorption with the acid solution is visible in both pairs of spectra, but more evident with the stronger solution, where it sensibly affects the apparent intensity and breadth of the broad band at about X 451. The second pair of spectra on this plate were taken with solutions made by diluting those used for the first pair of spectra until their volumes were three times as great as before, and they were put into tubes four times as long as those used for the first pair. There is no indication of any weakening of the absorptions by the addition of acid. The absence of any diminution of intensity either of the didymium or erbium bands by the addition of acid, taken in conjunction with the fact that rise of temperature does not increase their intensity, go a long way to negative the supposition that these bands are produced by the metallic ions; and the facts recorded in the preceding pages rather suggest that the metallic bands are the outcome of chemical interactions between molecules of the salt with each other and with those of the solvent, while the general absorption at the most refrangible end, which is evidently of a different class and resembles the absorptions of glass and many other substances which absorb the more rapid vibrations but are transparent to waves of less oscillation-frequency, may perhaps be due to encounters of molecules without chemical change. The effects on the spectrum when different solvents are used may throw some light on this question. Accordingly I made some experiments with didymium salts in various solvents. EFFECTS OF DIFFERENT SOLVENTS. Didymium chloride solution evaporated at 100~ retains some water, and seems to have the composition of the crystalline salt. Dried at a higher temperature it may be had anhydrous, but in that state appears to be quite insoluble in alcohol. Dried at 100~ it dissolves with tolerable facility in absolute ethyl-alcohol, and in glycerol, but will not dissolve in benzene. The alcoholic solution deposits beautiful pink crystals on evaporation. The absorption spectrum of this solution shews the same bands as an aqueous solution, but they are somewhat modified. They are more diffuse so that the weaker bands look as if they were washed out, and the positions of maximum absorption are all moved 312 PROF. LIVEING, EFFEOTS OF DILUTION, TEMPERATURE, ETC. ON THE towards the less refrangible side, and the diffuse absorption at the most refrangible end extends lower down the spectrum than with an aqueous solution of equal concentration. The general relation between the spectra of the two solutions will be seen on comparing photographs (1) and (2) of plate 19, of which the former is given by the aqueous, the latter by the alcoholic solution. The shift of the bands towards the red is visible in the photographs, but as the plate had to be shifted between the exposures, no reliance can be placed on the appearance of a shift in such photographs, when the amount of displacement of the bands is small. This defect is, however, met by direct eye-observations, with the two spectra in the field of view at the same time. In this way it is seen that all the bands that are visible are shifted towards the red, but are by no means all equally shifted. At the same time the strong groups of bands in the yellow and green have, by the action of the alcohol, undergone a modification of their general appearance which simulates the addition of some new bands; but by examining solutions of different concentrations I have satisfied myself that no new bands make their appearance, but the simulation of them is due to the widening and unequal shift of the bands, whereby their overlapping, and the consequent relative positions of the maxima of absorption, are modified. The modifications are such as we may reasonably ascribe to the influence of the bulky colloid molecules of the alcohol, amongst which the vibrating absorbent molecules move and from which they can hardly ever get free, loading them but loading them unequally, and on the whole degrading the rates of their vibratory motions. A very remarkable, and by far the most excessive, modification of the bands that I have observed, is produced by passing dry hydrochloric acid into the alcoholic solution. The third photograph of plate 19 shews the effect. The colour of the solution is changed by the acid from pink to bluish green, and the reason of this is obvious from the photograph. The molecules seem so loaded as to be nearly incapable of taking up the more rapid vibrations corresponding to the bands in the indigo and blue, while they seem to absorb more strongly those of slower rate in the yellow and citron. At the same time these are more degraded than by alcohol alone, and the group in the yellow so spread out that some of the components are distinctly separated. Of course the acid makes the solvent a complicated mixture, including ethyl-chloride and water as well as the unaltered components. The modifications of the spectrum by glycerol are of the same character as those produced by alcohol. The bands are generally shifted towards the red, and are more diffuse, but otherwise not much modified. Plate 20 shews the spectrum of the glycerol solution above and below that of an aqueous solution of didymium nitrate of nearly, but not exactly, equal concentration. Observed directly by eye it is seen that the band in the red at X 679 is not sensibly affected, the group in the yellow and the less refrangible of the two groups in the green, are distinctly shifted towards the red, but otherwise not affected in character; while the more refrangible group in the green is not sensibly shifted, but appears weakened by diffusion. The still more refrangible bands are all rendered more diffuse by glycerol, and are also degraded with the exception ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 313 of the middle band of the triplet in the blue, which does not appear shifted, but of this I am not sure for the photographs shew a trace of a washed-out band about midway between the two extreme bands of the triplet in addition to the stronger band which is more refrangible. With glycerol the continuous diffuse absorption also creeps down the spectrum as with alcohol. In order to observe the effect of a crystallizable solvent other than water, some didymium acetate was prepared and dissolved in glacial acetic acid, and for comparison with it an aqueous solution of didymium nitrate was made of equal concentration. Plate 20 shews the photographs of their spectra. Comparing the absorptions directly by eye, the band in the red appeared stronger in the acetate and sensibly shifted to the less refrangible side, the feeble band in the orange also was shifted in the same direction, the strong group in the yellow considerably extended towards the red but its more refrangible edge not apparently shifted, doubtless because the widening of the bands compensated the shift which was visible in all the other bands of the acetate though they otherwise had the same general appearance as those of the nitrate. The shift and change of character produced by acetic acid was less than was produced by alcohol. Didymium tartrate is very insoluble in water, but the compound produced by potassium hydrogen tartrate acting on didymium hydroxide dissolves in a solution of ammonia. The spectrum given by this solution is contrasted with that of an aqueous solution (not exactly of the same concentration), of didymium chloride in plate 23. With the exception of the group in the yellow, the less refrangible of the groups in the green, and the narrow band in the indigo, the bands seem all a good deal washed out. All the bands are shifted towards the red, and the apparent shift increases as the bands become more refrangible, but probably this appearance is the effect of the greater dispersion of the more refrangible rays. I had no crystals of didymium salts sufficiently large to enable me to see how the diminished freedom of the molecules in the solid would modify the spectrum, but had a rod of fused borax coloured with didymium. This was made by mixing weighed quantities of didymium oxide and dried borax, fusing the mixture, and sucking the fused mass into a hot platinum tube. After cooling the rough ends were cut off and polished, and I was thus able to compare the spectrum given by a thickness of 25 mnm. of this glass with that of an equivalent solution of didymium chloride. Photographs of these spectra are shewn in plate 21. They are somewhat marred by dust on the slit of the spectroscope, but this does not prevent a fair comparison. It will be seen that the modifications produced by the glass are on the whole similar in character to those produced by some of the liquid solvents. The strong group in the yellow is much expanded and the components of the group unequally shifted towards the red, the less refrangible of the groups in the green is shifted and its appearance modified for the same reason. The more refrangible bands are much washed out and their shifts appear very unequal. Nevertheless they appear to be still essentially the same bands modified as to their rates of vibration by the diminished freedom of the molecules producing them. VOL. XVIII. 40 314 PROF. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE On a review of the whole series of observations I conclude that the characteristic absorptions of didymium compounds, namely those which are common to dilute aqueous solutions, and are only modified by concentration, by heat, and by variations of the solvent, are due to molecules which are identical in all cases, though their vibrations are modified by their relations to other molecules surrounding them. The like conclusion holds for erbium compounds. It appears to me quite incredible that the atoms of didymium should retain in chemical combination so much individuality and freedom as to take up their own peculiar vibrations unaffected by the rest of the matter combined with them, as must be the case if we supposed the combined didymium in the nzolecules to give the common spectrum of all the salts in dilute solution. When I speak of atoms of didymium in the salts, I mean of course masses equal to the atoms of didymium metal, but having different energy, which means different internal motions, probably different structure, and different capabilities of vibration. No chemical compounds shew the absorptions which their separate elements exhibit. Sodium vapour, though monatomic, has a very strong absorbent power which is quite lost when it has parted with energy in combining with chlorine. Nevertheless the molecule of a chloride breaks up, in general, into masses equal to those of the atoms of its elements more easily than in any other way, and there is pretty good evidence that in encountering a molecule of water this also is sometimes broken up, and ultimately, if not immediately, new molecules of hydroxide and acid are formed, as well as, by a similar process, new molecules of the salt. In the interval between the rupture of a molecule and the recombination of its parts with each other, or with parts of other molecules, the parts have a certain freedom, and capability of vibrating, which they do not possess in combination. Now if we suppose the number of such parts as have the capability of taking up vibrations of frequency corresponding to the characteristic absorptions of didymium to be directly proportional to the concentration of the didymium salt and to the time of their freedom, the observed facts will be all in agreement with the hypothesis. Increased concentration, and increased temperature, will mean more frequent encounters amongst the molecules, and more frequent ruptures, but at the same time more frequent encounters of the parts and consequent shortening of their times of freedom. These effects will exactly compensate each other and leave the average number of absorbent parts of molecules constant under changes either of concentration or of temperature. The continuous absorption of the more rapid vibrations increasing with concentration and rise of temperature points to an action depending only on the number of encounters of the molecules of the salt with one another. It is not every encounter which is attended with disruption, and the continuous absorption may be due to molecules in encounter without rupture, but at all events it seems due to the condition of the molecules during encounter, but not to occur at the encounters of a molecule of salt with the very much less massive molecules of water. Encounters of a molecule of salt with a molecule of acid will in all probability cause effects very similar to those of encounters between two molecules of salt, and this supposition is quite in agreement with the observed facts. The time of complete freedom of a vibrating part of a molecule must be very ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 315 short, but probably shorter when the complementary part is more massive, as in the case of a nitrate, than it is in the case of a chloride. But between complete freedom and complete incorporation in a chemical compound there is a considerable gradation, and the capacity of the part to vibrate at particular rates will have a corresponding gradation, and the part may moreover be frequently under the influence of molecules, or parts of molecules, with which it does not combine. This influence will probably be greater as the molecule exerting the influence is greater whether more massive, or, as in the case of such colloids as alcohol, more voluminous. These considerations reconcile all the facts as to the spectra I have observed with the hypothesis I have made. There are, however, other facts to be reconciled with that hypothesis. I mean the facts of ionization, of osmotic pressure and the correlative facts of the rise of boiling point, and fall of crystallizing point, of solutions. In regard to all these effects the freedom of the parts is the primary postulate, far more definitely so than in the case of vibrations such as my observations relate to. The laws I have tried to investigate appear to hold good up to the point of saturation of the solutions, which is not the case with the laws of osmotic pressure and of change of boiling and freezing points, which have been established for dilute solutions. Further, ionization implies a certain distribution of energy in the field, the ions are charged with electricity. That is not necessary for the absorption of light, which will depend, primarily at least, on the form of the internal energy of the vibrating mass, that is on its structure. That a redistribution of energy occurs at every rupture of a molecule seems certain, solution is attended with thermal effects and so is dilution, and it is only when equilibrium is reached, and as much change takes place in one direction as in the opposite, that the manifestation of such redistribution ceases. How much of the intrinsic energy of the molecules takes the form of heat and how much is retained in the field at the rupture of the molecules we do not know. It is however quite conceivable that the circumstances under which the rupture takes place may determine whether any, or how much, energy is retained by the field, that is whether any, or how many, of the ruptured parts become ions. The plates, which are all reproductions of photographs, will be found at the end of the volume. 40-2 XV. The Echelon Spectroscope. By Professor A. A. MICHELSON, Sc.D. [Received 19 October 1899.] THE important discovery of Zeeman of the influence of a magnetic field upon the radiations of an approximately homogeneous source shows more clearly than any other fact the great advantage of the highest attainable dispersion and resolving power in the spectroscopes employed in such observations. If we consider that in the great majority of cases the separation of the component lines produced by the magnetic field is of the order of a twentieth to a fiftieth of the distance between the sodium lines, it will be readily admitted that if the structure of the components themselves is more or less complex, such structure would not be revealed by the most powerful spectroscopes of the ordinary type. In the case of the grating spectroscope, besides the difficulty of obtaining sufficient resolving power, the intensity is so feeble that only the brighter spectral lines can be observed, and even these must be augmented by using powerful discharges-which usually have the effect of masking the structure to be investigated. Sorne years ago I published a paper describing a method of analysis of approximately homogeneous radiations which depends upon the observation of the clearness of interference fringes produced by these radiations. A curve was drawn showing the change in clearness with increase in the difference of path of the two interfering pencils of light,-and it was shown that there is a fixed relation between such a "visibility curve" and the distribution of light in the corresponding spectrum-at least in the case of symmetrical lines*. It is precisely in the examination of such minute variations as are observed in the Zeeman effect, that the advantages of this method appear,-for the observations are entirely free from instrumental errors; there is practically no limit to the resolving power; and there is plenty of light. There is however the rather serious inconvenience that the examination of a single line requires a considerable time, often several minutes, and during this time the character of the radiations themselves may be changing. Besides this, nothing can be determined regarding the nature of these radiations until * In the case of asymmetrical lines another relation is necessary, and such is furnished by what may be called the "phase curve." PROF. MICHELSON, THE ECHELON SPECTROSCOPE. 317 the "visibility curve" is complete, and analyzed either by calculation or by an equivalent mechanical operation. Notwithstanding these difficulties, it was possible to obtain a number of rather interesting results, such as the doubling or the tripling of the central line of Zeeman's triplet, and the resolution of the lateral lines into multiple lines; also the resolution of the majority of the spectral lines examined, into more or less complex groups; the observation of the effects of temperature and pressure on the width of the lines, etc. It is none the less evident that the inconveniences of this process are so serious that a return to the spectroscopic methods would be desirable if it were possible (1) to increase the resolving power of our gratings; (2) to concentrate all the light in one spectrum. It is well known that the resolving power of a grating is measured by the product nzn of the number of lines by the order of the spectrum. Attention has hitherto been confined almost exclusively to the first of these factors, and in the large six-inch grating of Prof. Rowland there are about one hundred thousand lines. It is possible that the limit in this direction has already been reached; for it appears that gratings ruled on the same engine, with but half as many lines, have almost the same resolving power as the larger ones. This must be due to the errors in spacing of the lines; and if this error could be overcome the resolving power could be augmented indefinitely. In the hope of accomplishing something in this direction, together with Mr S. W. Stratton, I constructed a ruling engine in which I make use of the principle of the interferometer in order to correct the screw by means of light-waves from a homogeneous source. This instrument (only a small model of a larger one now under construction) has already furnished rather good gratings of two inches ruled surface, and it seems not unreasonable to hope for a twelve-inch grating with almost theoretically accurate rulings. As regards the second factor-the order of the spectrum observed, but little use is made of orders higher than the fourth, chiefly on account of the faintness of the light. It is true that occasionally a grating is ruled which gives exceptionally bright spectra of the second or third order, and such gratings are as valuable as they are rare, for it appears that this quality of throwing an excess of light in a particular spectrum is due to the character of the ruling diamond which cannot be determined except by the unsatisfactory process of trial and error. If it were desirable to proceed otherwise-to attempt to produce rulings which / /,/ / / / / / /! / f / /, / / / / / / / / / / / FIG. 1. should throw the greater part of the incident light in a given spectrum, we should try to give the rulings the form shown in section in Fig. 1. 318 PROF. MICHELSON, THE ECHELON SPECTROSCOPE. I am aware of the difficulties to be encountered in the attempt to put this idea into practical shape, and it may well be that they are in fact insurmountable-but in any case it seems to be well worth the attempt. Meanwhile the idea suggested itself of avoiding the difficulty in the following way: / / / / / / / / / / / / / / / / / / FIG. 2. Plates of glass (Fig. 2), accurately plane-paralleled and of the same thickness, were placed in contact, as shown in Fig. 2. If the thicknesses were exactly the same, and were it not for variations in the thickness of the air-films between the plates, the retardations of the pencils reflected by the successive surfaces would be exactly the same, and the reflected waves would be in the same conditions as in the case of a reflecting grating-except that the retardation is enormously greater. The first condition is not very difficult to fulfil; but in consequence of dust particles which invariably deposit on the glass surfaces, in spite of the greatest possible precautions, it is practically impossible to insure a perfect contact, or even constancy in the distances between surfaces*. If now instead of the retardation by reflection we make use of the retardation by transmission through the glass, the difficulty disappears almost completely. In particular the air-films are compensated by equivalent thicknesses of air outside, so that it is no longer necessary that their thickness should be constant. Besides, the accuracy of parallelism and of thickness of the glass plates necessary to insure good results is now only one-fourth of that required in the reflection arrangement. In Fig. 3 let ab = s, the breadth of each pencil of rays; bd = t, the thickness of each element of the echelon; 0, the angle of diffraction; a, the angle adb; m, the number of waves of length X corresponding to the common difference of path of the successive elements. The difference of path is mX = t-ac. t Now ac = — cos (a + 0), cos a v * Nevertheless I have succeeded with ten such plates, phenomena such as the Zeeman effect, the broadening of silvered on their front surfaces, in obtaining spectra which, lines by pressure, etc.-but evidently the limit has been though somewhat confused, were still pure enough to show nearly reached. PROF. MICHELSON, THE ECHELON SPECTROSCOPE. 319 or, since 0 is always very small, t ac = -- (cos a- sin a)= t (1 -0 tan a), COS a and hence = ( - 1)t + s.........................................(I). a, b I i i Il i / I / / I FIG. 3. To find the angle corresponding to a given value d\, differentiate for X and we find do0 1 d~sL X = - [m -t -. dX s \ t dx) Putting in this expression the approximate value of t m = (-l), we have dx/X = (/ -d....... (II). For the majority of optical glasses b varies between 0'5 and 1'0. The expression (II) measures the dispersion of the echelon. To obtain the resolving power, put e = d/X for the limit. For this limiting value the angle 0 will be \/ns, where n is the number of elements; whence ns = the effective diameter of the observing telescope. Substituting these values we find t...........................................( I II). To obtain the angular distance between the spectra, differentiate (I) for m; we find dmn s ' 320 PROF. MICHELSON, THE ECHELON SPECTROSCOPE. or putting dn= unity, d =................................................(IV ). s The quantity d\/X = E corresponding to this is found by substituting this value of dO in (II), whence E=...................................................(V ). Hence the limit of resolution is the nth part of the distance between the spectra. This fact is evidently a rather serious objection to this form of spectroscope. Thus in observing the effect of increasing density on the breadth of the sodium lines, if the broadening be of the order of \/bt the two contiguous spectra (of the same line) will overlap. As a particular case, let us take t = 7 mm., E= 17000 It will be impossible to examine lines whose breadth is greater than the fourteenth part of the distance between the D lines. It is evidently advantageous to make t as small as possible. Now the resolving power, which may be defined by -, is proportional to the product Wt. Consequently, in order to increase it as much as possible it is necessary to use thick plates, or to increase their number. But in consequence of the losses by the successive reflections, experience shows that this number is limited to from 20 to 35 plates, any excess not contributing in any important degree to the efficiency. I have constructed three echelons, the thickness of the plates being 7 mm., 18 mm. and 30 mm. respectively, each containing the maximum number of elements-that is, 20 to 35, and whose theoretical resolving powers are therefore of the order of 210,000, 540,000 and 900,000 respectively. In other words, they can resolve lines whose distances apart are the two-hundredth, the five-hundredth and the nine-hundredth of the distance between the D lines. Consequently the smallest of these echelons surpasses the resolving power of the best gratings, and what is even more important, it concentrates all the light in a single spectrum. The law of the distribution of intensities in the successive spectra is readily deduced from the integral s/2 A = cos pxdx, J - s/2 27r where p= -. sin2 wr - 0 Hence I = A2 = (4O)2 PROF. MICHELSON, THE ECHELON SPECTROSCOPE. 321 This expression vanishes for = +X/s which is also the value of dco, the distance between the spectra. Hence in general there are two spectra visible as indicated in Fig. 4. 277 -7 O;7 207 FIG. 4. By slightly inclining the echelon one of the spectra is readily brought to the centre of the field, while the adjacent ones are at the minima, and disappear. The remaining spectra are practically invisible, except for very bright lines. As has just been indicated, the proximity of the successive spectra of one and the same line is a serious objection, and as this proximity depends on the thickness of the plates -which for mechanical reasons cannot well be reduced below 5 or 6 mm.-it is desirable to look to other means for obviating the difficulty, among which may be mentioned the use of a liquid instead of air. In this case formula (II) becomes de t 1 C (ICL - ( l)1 t dcx/X s - ( - -, - X) cX j= cSand formula (IV) becomes dO x dm /-1S Repeating the same operations as in the former case, we find nct' and E = -. ct The limit of resolution is still the nth part of the distance between the spectra, but both are increased in the ratio b/c. Suppose for instance the liquid is water. Neglecting dispersion the factor would be 3 55. Hence the distance between the spectra will be increased in this proportion, but the limit of resolution will also be rnultiplied by this factor. But as there is now a surface water-glass which reflects the light, the loss due to this reflection will be VOL. XVIII. 41 322 PROF. MICHELSON, THE ECHELON SPECTROSCOPE. very much less, so that it will be possible to employ a greater number of elements, thus restoring the resolving power. At the same time the degree of accuracy necessary in working the plates is 3a55 times less than before. For many radiations the absorption due to thicknesses of the order of 50 cm. of glass would be a very serious objection to the employment of the transmission echelon. I have attempted therefore to carry out the original idea of a reflecting echelon, and it may be of interest to indicate in a general way how it is hoped the problem may be solved. Among the various processes which have suggested themselves for realising a reflecting echelon, the following appear the most promising: In the first a number of plates, 20 to 30, of equal thickness, are fastened together as in Fig. 5, and the surfaces A and B are ground and polished plane and parallel. They are then separated and placed on an inclined plane surface, as indicated in Fig. 6. A FIG. 5. FIG. 5. FIG. 6. If there are differences in thickness of the air-films the resulting differences in the height of the plates will be less in the ratio tana. An error of X/n may be admitted for each plate-even in the most unfavorable case in which the errors all add; and consequently the admissible errors in the thickness of the air-films may be of the order PROF. MICHELSON, THE ECHELON SPECTROSCOPE. 323 X/nc. For instance, for 20 plates the average error may be a whole wave-length if the inclination a is -1. As there is always a more or less perfect compensation of the errors, the number of plates, or the inclination, may be correspondingly greater. Accordingly it may be possible to make use of 50 elements and the plane may be inclined at an angle of 20~ to 30~. It would be necessary in this case however to use a rather large objective. Possibly this may be avoided by cutting the surface A to a spherical curvature, thus forming a sort of concave echelon. The second process differs from the first only in that each plate is cut independently to the necessary height to give the required retardation. The first approximation being made, the plates are placed on a plane surface as in Fig. 7 (side view) and Fig. 8 (front view). FIG. 7. The projections a and b are then ground and polished until the upper surfaces are all parallel, and the successive retardations equal. The parallelism as well as the height is verified by means of the interferometer. a _ b FIG. 8. These processes are, it is freely conceded, rather delicate, but preliminary experiments have shown that with patience they may be successful. 41-2 XVI. On Minzinmal Su faces. By H. W. RIICHIMOND, M.A., King's College, Cambridge. [Received 10 November 1899.] 1. IN a short paper read before the London Mathematical Society on Feb. 9 last, and since printed in the Proceedings of the Society, Vol. xxx. p. 276, Mr T. J. I'A. Bromwich has noted an interesting form of the tangential equation of a minimal surface, by which the determination of such surfaces is made to depend upon a particular type of solution of Laplace's equation. The idea of thus establishing a connexion between certain of Laplace's functions and minimal surfaces is one that presented itself to me several years ago, and led me then (in 1891-92) to consider at some length to what extent the study of these surfaces given by Darboux in Part I, Book III, of his Théorie générale des Szrfaces might be modified by this connexion. Although the familiar treatment of Laplace's equation led me, (in many instances by simpler paths than Darboux), to a number of the chief known theorems concerning minimal surfaces, yet I never succeeded in reaching untrodden ground, and for this reason laid aside iny work; but the appearance of Mr Brornwich's paper has caused me to look through my notes, and to consider with some fulness a special family of algebraic minimal surfaces to which the method is peculiarly applicable. So thorough a discussion of the history and properties of minimal surfaces is given by Darboux, in Book III. of his Théorie générale des Surfaces, that it will seldom be necessary to refer to other sources of information: references to Darboux will be made simply by the letter D. followed by the number of the paragraph in question;-thus (D. ~ 175). In all that follows it is supposed that a system of real rectangular Cartesian axes is employed. 2. The tangential equation of a surface, (p. 1), 2., n)= 0, (where b is a homoyeneous function of p, 1, n, n but not necessarily algebraic), expresses the condition that the plane x + y + nz = p......................................... (1), should be tangent to the surface. Should / be rational, integral and homogeneous of the kth degree, the surface is algebraic and of the kth class. MR RICHMOND, ON MINIMAL SURFACES. 325 The equation 0b (p, 1, m, n)= 0 will always be regarded as defining a dependent variable p as a function of three independent variables 1, m, n;p =,(,, )............................................(2); but the function r is of necessity homogeneous and of the first degree. The coordinates of the point of contact of the plane (1) with the surface enveloped by it are ap. =3 a1 ap ap; Y=aP'....... so that x, y, z are expressed as homogeneous functions of 1, mz, n, of degree zero, i.e. as functions of the ratios: m:n. It is therefore possible to eliminate 1, m, n from equations (3) and so to obtain a relation in x, y, z, alone, the equation of the surface in point coordinates. The condition that the surface should be minimal is established without difficulty, viz. ap+ p ap (4) - ',+ -. =o..........................................4 Hence:-When p is a function of 1, m, n, homogeneous and of the first degree, which satisfies Laplace's equation, the envelop of the planes (1), or the locus of the point (3), is a minimal surface. When the condition (4) is satisfied, I shall say that p has a minimal value, or is a minimal function of l, m, n. It is of importance to observe that, in what precedes, the condition 12 +,2 + n2 = 1, is not imposed: provided only that p is of the first degree in 1, m, n (which is always to be understood in future), it is absolutely immaterial whether the sum of the squares of these quantities be equal to unity or no. When (4) is satisfied it is easy to establish the theorem of M. Ossian Bonnet (D. ~~ 202, 203), that the horograph of a minimal surface is a conformable map of the surface. 3. I now consider very briefly to what results the common manipulation of Laplace's equation leads. Since p satisfies the equation, so also do its three partial differential coefficients, which, as we have seen, are the coordinates of points of the surface, expressed in terms of the ratios 1: m: n. Now the solutions of Laplace's equation which are of degree zero in the variables are of the form, F (v) + F1 (ut), ~where l~i r+ + it- r + zn - im? r + n where u=- =. u.= = r- - tim' r - n I + in' and r = (12 + m2 + n2). These quantities u and u1 are thus the same as those of Darboux (cf. D. ~~ 193, 195). The formulae of Weierstrass (D. ~ 188, equation 17) are readily deduced; while if we take new variables v and vy, the former a function of u and the latter of u1, we reach the solution of Monge (1D. ~~ 179 and 218). 326 MR RICHMOND, ON MINIMAL SURFACES. Although the integration of Laplace's equation presents no difficulty, it is not easy to say what is the best form of solution of the first degree in the variables which we should take as the value of p. The formulae due to Weierstrass (D. ~ 188, equations 18), may be obtained from the value P = r [f (u) +fi/ (u,)] - (i - im)f(u) - (i + im) fj (u,): but a value which is preferable for the present purpose, in that it is more naturally attained by integration and leads to simpler results, is p = r [X (t) + t ' (u1)] - i [X (t) + X1 (ut)]........................ (5); and this is the value which will be used in the following applications. From it I derive, by differentiation with respect to 1, m, n, the expressions = x' (t) +2 1 (1 - uX2) x () + 1< (u1) + 2 i (1 - u12) x 1 (t1); y = iXy (u) + 2 iu (1 + u2) %" (u) - i1 (ut1)- iut1 (1 + u,12) X1I (1); z = - X (u) + tx' (t) + t2x" (0) - XI (t1) + UX1' (u.) + u1 "1 (u). It will be seen that the two forms are in agreement if f(u) = UX (t); f, (2t) = uXI (ut). 4. As an illustration of the use of these results I consider two methods of solving the problem of determining a minimal surface which has a given plane as a plane of symmetry, and cuts that plane at right angles along a given curve; or, as Darboux (~ 251) expresses it, has a given plane curve as a geodesic. It is clear that if X =X, (which in the case of a real surface implies that X is a real function), the surface has zOx as a plane of symmetry and cuts it orthogonally: moreover, if we fix directions by Euler's two angles, 0 the colatitude and S the longitude, (so that: n: r: sin cos b sin 0 sin b:cos 8: 1, and u- = e. c-ot 0, ul = e-i. cot ), 2 2 the functions X and XI are determined by the equation X (cot o ) = Xi (cot o) = - )osec. pd0, the quantity p being the length of the perpendicular from the origin on any tangent of the given plane curve, laid in the plane zOx, and 0 the inclination of that perpendicular to Oz. 5. But the following solution is of greater interest, in that it is adapted to cases when the given plane curve is irregular, being composed of portions of known curves MR RICHMOND, ON MINIMAL SURFACES. 327 or straight lines, united so as to form a closed contour. Let this contour be enveloped by a straight line which moves round it, turning always in the same direction; let the plane of the contour be xOy; let po denote the perpendicular from O on the enveloping line, and 0b0 the inclination of that perpendicular to Ox. In a complete circuit of ihe contour, the enveloping line will turn through some multiple of two right angles, and return to its original position; po is therefore a periodic function of b0, —the period being a multiple of 7r,-and may be expanded in a Fourier's series even when p, or its differential coefficients have discontinuities: thus po = S (ak sin k0o + bk cos kto). In the case of an oval curve or a closed convex polygon the period of po is 2vr; k will then receive only integer values. In a cardioid the period is 37r, and 3k will always l)e an even integer, etc., etc. The minimal surface sought will be represented by the tangential equation p = E {(k - cos 0) cot 0 + (k + cos 0) tank 2 0 (ak sin kq + bk cos kc0) + 2k. For this typical term may be obtained from the general formulae (5) by making x (u)=-K (tk - u-k), x (y1) =K (tuk - ut-k); K and K1 being constants suitably chosen; and we may deduce z = 2 E( k) cotk -tan (ak sin k + bk cos k); so that, when 0= 7r, z vanishes and p has the correct value. Interesting special cases arise when the given plane curve is an epicycloid or hypocycloid; for the series for p, then reduces to a single term p = A cos kdp, and the required surface is obtained by making in (5) x (Zt) = B (utk - u-), (It1) = B (ulk - I1l-k). It is clear however that special surfaces such as this fall under the cases to which the methods of Darboux are applicable; I therefore pass on to a result which I do not remember to have seen explicitly stated, (although it follows almost immediately from several theorems of Darboux), and to some considerations suggested by it. Enough has been said to shew that integration of Laplace's equation leads rapidly to many of the chief known results concerning minimal surfaces. 6. Since Laplace's differential equation is linear, the sum of any two of its solutions is itself a solution: if then pi and p1 be two minimal functions of 1, rm, n, p1i+p is also a minimal function. Stating this theorem in geometrical language, we enunciate the noteworthy property: 328 MR RICHMOND, ON MINIMAL SURFACES. If any two minimal surfaces be taken, the locus of the middle points of lines which join the points of contact of parallel tangent planes is also a minimal surface. But, conversely, the possibility that a given minimal value of p may be resolved into the sum of two or more simpler values is suggested by the theorem. I propose to carry through this idea in the case of rational algebraic minimal functions;-to prove that every rational algebraic minimal function may be expressed as the sum of a finite number of such functions each belonging to certain standard types, much in the way that every rational fraction may be broken into partial fractions. In other words, I hope to establish that by taking a finite number of minimal surfaces of certain normal types, disposed in space with various orientations, and constructing the locus of the centre of mean position of the points of contact of parallel tangent planes, we may arrive at any minimal surface whatever, for which p is a rational algebraic homogeneous function of 1, m, n, of the first degree. When p is such a function, the surface, whether minimal or not, will have one and only one tangent plane parallel to any given plane: if the surface be of class k + 1 it will have the plane infinity as a k-fold tangent plane, and must therefore be reciprocal to what Cayley called a Monoid surface: (Comptes Rendus, t. 54, 1862, pp. 55, 396, 672). A paraboloid is the simplest instance of the surfaces we are considering. Now the analogous curves in plane geometry presented themselves to Clifford's notice in the course of' that wonderful chain of reasoning, the Synthetic Proof of Miquel's Theorem, (Collected Works, p. 38), and were named by him double, triple,... k-fold, parcabolas. Following his example, 1 call a surface of class k +l, which has the plane infinity as a k-fold plane, a k-fold paraboloid; and the family of such surfaces, (the value of k not being specified), Multiple Paraboloids. 7. The tangential equation of a k-fold paraboloid will be written as p=PV U, U and V being rational integral homogeneous functions of 1, m, n, of degree k and k + 1 respectively. If for the moment partial differentiations with regard to 1, m, n, be indicated by suffixes 1, 2, 3, respectively, the condition (4) that the surface should be minimal gives us the identity V ( ll + U22 + U33)- U(Vll + V22 +V33)+ 2 (U1 U2V+ U2 +U3V3)-2 V(U12 + U22+ U32) U; and so proves that ( U12 + U22 + U32) U is a rational integral function of 1, m, n:-a result possible only if U be the product of factors which are powers either of 12 + q2 + n22, or of linear functions such as al + bm + cn, in which a2 + b2 + c2 = O. MR RICHMOND, ON MINIMAL SURFACES. 329 But it will appear further that 12-m12 + n2 cannot be a factor of U; for if in the above identity we substitute U= ( 1 + m2 + n2)ST= r2T and take account of the fact that V and T are homogeneous functions of degree k+ 1 and k-2s respectively, we find that (2s2 + 3s) VT2. r2 is identically equal to an integral function of 1, m, n: but this is an absurdity and we are compelled to infer that s =0. 8. The denominator U of a rational minimal value of p is thus wholly composed of factors, each an integral power of a linear function of 1, m, n, al + b + n, whose coefficients a, b, c, are such that the sum of their squares vanishes. Any one such factor vanishes for one and only one real systein of values of the ratios 1: rm: a; and, if the corresponding real direction be taken as the z-axis in a new coordinatesystem, is reduced to the form C(l + irm), the quantity C being a complex constant. Proceeding now to the consideration of minimal values of p in which the denominator U is a power of a single linear function of 1, m, n, we may without loss of generality suppose the linear function thus reduced, and confine our attention to values of the form p = V ( (l + imnk. That such values actually exist is shewn by the formulae (5), in which if we make (1) = A (- )-k; X1 (t) = A ( )k; we obtain a value of p of the kind sought, viz. p = - A {(n - kr) (n + r)k + (n4 + kr) (n - r)k (I + in)k. The numerator of this fraction, when the special value 1 -2k(k.- 1) has been assigned to A, will be denoted by ~/k( n); thus 2k (k - 1) k (u) + (n - kr) (n + r)k + (n + kr) (n - -)k 0. The function Lk ('h) is real and may be expanded in powers of n and r2; or, by rearrangement of the terms, in powers of n and (12 + m)2: moreover on account of the value given to A the coefficient of the highest power of n in the latter form is unity: we might in fact write ~/k (n) = nk+i + nZk-1 (12 + m2) + + mk-3 (12 + rn2)2 +..., VOL. XVIII. 42 330 MR RICHMOND, ON MINIMAL SURFACES. ni, m2,... being real numerical constants. The corresponding minimal value of p is of the form p = k (n1) - (I + im)k = k +1- (l + im)k + W - (l + ir)+-1, TW denoting some rational integral homogeneous function of 1, nm, n (with complex numerical coefficients), of degree k. It will be seen that of integer values of k the value k = 1 alone fails to give a function /k (n). It may be easily proved, and will be assumed in what is to come, that no minimal value of p exists whose denominator is + im and whose numerator is a rational integral function of the second degree. 9. In order next to determine the most general rational integral function V of degree k+1 such that the surface p = V (l + i)k is minimal, it will be convenient to write for a time f - l + im, gl-i im, and to use f, g, n, as independent variables in place of l, m, n. The differential equation of a minimal surface is now -L + 4 g = 0, 8n2- a =ag and is to be satisfied by p= V -fk Substituting and multiplying by fk, we find that aV g -f + an integral function = 0, and deduce that the part of V that does not contain f inust consist of a single term, *C. k+l, C being a constant. It follows that by subtracting a numerical multiple of the foregoing particular solution we obtain a new minimal function pi, viz. p = {V- C0 (n)} f k, in which a factor f is common to numerator and denominator, and may be removed. By repetition of the argument and process we continually diminish the class of the surface, and finally establish the theorem:The most general rational mininzmal value of p which has ( +im)k for its denominator is p = al+ /m + r/3 n 7 +:C (n) + (l + im)8; (s = 2, 3, 4,.../ ); the quantities a,,8, r, C2, C3,... Ck being complex constants. MR RICHMOND, ON MINIMAL SURFACES. 331 10. The same method is applicable to the case when the denominator U contains other factors besides f: for if we substitute p= V (fh.S) in the differential equation we find on multiplying by fh that, if S do not contain f as a factor, avr as ag ag must be divisible by f, and infer that the terms in V that are independent of f must be equal to those in S multiplied by nh+1 and a constant. If h be equal to unity we must therefore have V= A. n2. S+ terms divisible by f; but substitution in the differential equation proves that A must vanish: if on the other hand h be greater than unity, we may, by subtracting a properly chosen multiple of h (n) +fh, obtain a new minimal function whose denominator does not contain so high a power of f as fh. It follows that the most general rational minimal function with denominator ( + im)h. S may be obtained by adding to a value with denominator S the terms C,. s (n). (1 + im)s: (s = 2, 3, 4,... h): C2, C,... Ch, being complex constants. The factors of S may now be subjected to the same treatment; that is to say, first reduced to the form 1 + im by a real transformation of axes, and then made to yield a series of fractions of the types already discovered. The most general minimal value of p which is a rational function of 1, m, n, may therefore be resolved into the sum of a number of terms each separately capable of being reduced by a real transformation of axes to one of the types already quoted. 11. The simplest value of p of the kind we are considering is obtained when k= 2, viz. 2p (1 + im)2 = 2n3 + 3n (12 + m2), and leads to a surface, 2 (x + iy)= = 18 (x + iy) + 27 (x- iy), of class and order three: but, as imaginary surfaces such as this are of minor interest, we may pass on to the discussion of the case when the surface is real. In order that the surface should be real, each of the typical complex terms into which p was broken up must be accompanied by the conjugate complex term, the numerical constants multiplying each also being conjugate imaginaries: a rotation of the 42-2 332 MR RICIHMOND, ON MINIMAL SURFACES. coordinate planes about the z-axis will bring both these numerical coefficients to the same real value A. For real values of p the typical real component fractions are therefore A.,k () {(l + im) + (l - i)k} - (1 + ); ( = 2, 3, 4,...) Every real minimal value of p which is a rational function of 1, m, n, may be expressed as the sutm of a finite number of real fractions, each separately reducible by real transformation of axes to one of the forms just quoted. Terns such as al + f3m + yn may also be present, but are ignored since a change of origin will remove them. If we again introduce Euler's angles 0 andcl, as in ~ 4, the surface corresponding to the above value is p = A. k (n). {(l + i)k + ( - im)k} ( + m2); ( = 2, 3, 4,...) = B. cos C. {(k- cos 0) (cot 6 - (k + cos 0) t-tan 2 0} and may be described as the standard minimal multiple paraboloid of the kth type: the origin of coordinates is called its centre and the z-axis its axis. The class of every real multiple paraboloid that is a minimal surface is necessarily odd; thus the above standard surface is a 2k-fold paraboloid and is of class 2k +1. The theorem established now admits of the following statement:By placing a finite numinber of standard surfaces (defined above) with their centres coinciding but zoith various orientations, and taking the locus of the centre of mean position of the points of contact of parallel tangent planes, we can obtain every minimal surface which is a multiple paraboloid. Corresponding to any selected real direction, a multiple paraboloid has, as was pointed out, one and only one tangent plane; there is therefore no ambiguity in the foregoing construction: certain of the planes may however be at an infinite distance. If the surface be minimal, the number of infinitely distant tangent planes must be finite, their directions being normal to the axes of the standard surfaces from which the given surface may be derived. Given a minimal multiple paraboloid, the directions of the axes of the component surfaces are thus plain geometrically. XVII. On Qzartic Sunfaces which admit of Integrals of the first kind of Total Differentials. By ARTHIR BERRY, M.A., Fellow of King's College, Cambridge. [Received 15 November 1899.] CONTENTS. ~ 1. Introduction. ~ 2. Analysis of the fundamental differential equation. ~ 3. Integration of the differential equation, leading to five possible surfaces. ~ 4. Tabular statement of results. ~ 5. Birational transformation of the surfaces into cones. 6. Numerical genus of surfaces which admit of integrals of the first kind.! 7. Geometrical characteristics of the five surfaces. ~ 1. INTRODUCTION. THE theory of the Abelian integrals associated with an algebraic plane curve can be generalised in two distinct ways when we pass from a plane curve to a surface in three dimensions, that is when we are dealing with an algebraic function of two independent variables. Given an algebraic equation, f(x, y, z) = 0, between three non-homogeneous variables, we may study either double integrals of the type f R (x, y, z) dxdy, where R is rational, or single integrals of total differentials of the type (Pdx + Qdy), where P, Q are rational functions of x, y, z, which satisfy in virtue of f= the condition of integrability aP Dq ay a' Such integrals of total differentials were introduced into mathematical science by Picard about fifteen years agoo, and have been the subject of several memoirs by himl. They have also been studied to some extent by Poincaré++, Noether~, Cayley and others. The most important results hitherto obtained are given in the "Théorie des * Comptes Rendus, t. 99 (1 Dec. 1884). drticke," Math. Ann. t. 29 (1887). t The most important appeared in Liouville, ser. iv. il Note sur le mémoire de M. Picard "Sur les intégrales t. 1 (1885), and ser. Iv.. 5 (1889). There have also been de différentielles totales algébriques de première espèce," a series of notes in the Comptes Rendus. B'ull. des Sciences lMath. ser. II. t. x. (1886): Coll. iMath. + Comptes Rendus, t. 99 (29 Dec. 1884). Pcpers, t. xII. no. 852. ~ "Ueber die totalen algebraisehen Differentialaus 334 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS Fonctions Algébriques de deux variables indépendantes" recently (1897) published by Picard and Simart, a book to which it will in general be convenient to refer. Integrals of total differentials, like ordinary Abelian integrals, fall into three classes, of which the first consists of integrals which are always finite. But whereas the number of linearly independent integrals of the first kind associated with a plane curve is at once expressible by a simple formula in terms of the singularities of the curve, and such integrals always exist if the curve has less than its maximum number of singularities, the corresponding problem for integrals of total differentials is far less simple and has only been solved for special classes of surfaces. On a cone, an integral of a total differential is equivalent to an Abelian integral on a plane section of the cone, so that no new problem arises. Moreover, according to Cayley*, any ruled surface niay be birationally transformed into a cone, the genus (deficiency) of a section of which is equal to that of a general plane section of the original surface; hence the number of integrals of the first kind on a ruled surface can at once be determined, but I am not aware that there is any known process whereby the transformation can in general be effected or the integrals actually constructed. For other classes of surfaces the most important results so far obtained are negative in character; thus it is evident that no integrals of the first kind can exist on a rational (unicursal) surface, and the same proposition has been established for surfaces without any singular points or singular lines. The determination of surfaces or classes of surfaces which admit integrals of the first kind of total differentials appears therefore to be a problern of some interest. Since quadrics and cubic surfaces (other than non-singular cones) are rational, they can possess no integrals of the first kind. Two non-conical quartics possessing such integrals were discovered by Poincaré+, and stated to be the only possible ones. Poincaré's results have been adopted by Picard, who has given a proof in outline. The object of this paper is to establish the existence of certain other quartic surfaces which have the property in question, but have apparently been overlooked by the two eminent mathematicians just named. The method which I have adopted appears to shew also that the list given is complete. 2. ANALYSIS OF THE FUNDAMENTAL DIFFERENTIAL EQUATION. It has been shewn by Picardl| that if a surface of order n, of which the equation in homogeneous point coordinates is f(x, y, z, w)=0, admits of an integral of the first kind, then f satisfies the partial differential equation 01 + 0~ f~ + 03 04 o............................(1), '6w Dy az aw * " On the deficiency of certain surfaces," Math. Ann. + Comptes Rendus, t. 99 (29 Dec. 1884). t. II. (1871): Coll. Math. Papers, t. vIII. no. 524. ~ Picard et Simrart, pp. 135, 136. + Picard et Simart, pp. 113, 119, 120. l Ib., Chapter V. OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 335 where 01, 02,,0, 04 are quantics of order n - 3, which satisfy the equation ao, ao2 a,03 4 + + w =..................................... (2). ax ay az azf These equations being satisfied, the differential dx, dy, dz Of, ~y z aw 01 02, 03 satisfies the condition of integrability, and its integral is finite everywhere with the possible exception of certain singular points and lines. When n =4 the quantics 0 are linear and the equation (1) becomes a familiar partial differential equation. If we write 0i - aix + biy + ciz + diw, (i = 1, 2, 3, 4), then, in accordance with the usual elementary theory, the integration of the equation (1) depends upon the roots of the algebraic equation /x — a1-s, b1, c1, d1 =0.(3). 1a2 b2 - s c, c, \ac, b3, C3-s, dS a4, b4, c4, d4-s ] If the roots of this equation are all distinct we can at once obtain three independent integrals of the auxiliary system dx dy dz dw = - Y....................................... (4** 01 02 03 044) and deduce the general integral of the partial differential equation. But if two or more roots of A=0 are equal the integration of the system (4) is less simple, and one or more of the integrals is in general logarithmic, though these integrals may again become algebraic if the coefficients of the 0's satisfy certain further conditions. Although the complete discussion of these cases by quite elementary methods presents no serious difficulty it is rather long and tedious, and the work can be considerably abbreviated by reducing the equations (4) to a standard form by means of the method which was given by Weierstrass as an application of his theory of bilinear forms*. This method, stated in a forin applicable to our particular problem, depends on the resolution of the determinant A into "elementary factors" (Elementartheiler). If s - a occurs p-tuply as a factor of A, p p-tuply as a factor of each first minor of A, p2-tuply as a factor of each * "Bemerkuugen zur Integration eines Systems liîlearer Differentialgleichungen mit constanten Coefficienten," lIathenmatische Werke, I. pp. 75-6. 336 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS second minor, and so on, and if p-pl=a, p1-p2=a..., and 3, ' y...,7,...are the numbers corresponding similarly to the other factors, s - b, s - c,... of A, then (S - a)a, (s - a )... ( - ), (s- )., are defined as the elementary factors of A. These factors are shewn by Weierstrass to be invariant for linear transformation of the variables, and the system of differential equations dx dy dz dw = -1, ' -, — --- 04, dt dlt dt dt is shewn to be reducible by linear transformation of the dependent variables to a standard form, in which there are as many distinct sets of equations as there are elementary divisors of A, the set corresponding to an elementary divisor (x - a)P being of the form dx, dx2 dxp dt a dt = ax2 + x1, dt = a + -.........................(5). Applying this theory to our equation we see that the possible ways in which A can be resolved into elementary factors are as follows: (I) All the roots of A = O equal: (i) (s - a)4, (ii) (s - a), (s - a), (iii) (s - a), (s - a), (iv) (s - a)2, (s - a), (s - a), (v) (s-a), (s-a), (s - a), (s - a). (II) Three roots of A = O equal: (i) (s - a)3, (s - b), (ii) (s - a)2, (s - a), (s - b), (iii) (s - a), (s - a), (s - a), (s - b). (III) Two pairs of roots of A = 0 equal: (i) (s - a)2, (s-b)2, (ii) (- a)2, (s - b), (s - b), (iii) (s - a), (s - a), (s - b), (s -b). (IV) One pair of roots of A = 0 equal: (i) (s - a)2, (s-b), (s-c), (ii) (s - a), (s - a), (s - b), (s- c). (V) All the roots of A= 0 distinct: (s - a), (s - b), (s - c), (s - d). Also the equation (2) shews that the sum of the roots of A =0 vanishes, so that we must have in CASE I. a = O, CASE II. b = - 3a J 0, and we may evidently take a = 1, b = - 3, OF THE FIRST KIND 0F TOTAL DIFFERENTIALS. 337 CASE III. b=-a 0, and we may take a= 1, b =-1, CASE IV. b + c =- 2a, CASE V. a + b+c+ d = O. It follows at once that the Case I. (v) is impossible. For the purposes of our problem we do not want the general integral of the equation (1), but only such integrals as are homogeneous quartics; we may also leave cones out of account, and we must reject solutions giving degenerate (reducible) quartic surfaces; we find also that in one or two other cases we arrive at surfaces which are obviously rational and must therefore be rejected. ~ 3. INTEGRATION OF THE DIFFERENTIAL EQUATION, LEADING TO FIVE POSSIBLE SURFACES. We have in all (after rejecting I. (v)) thirteen cases to consider, which will now be dealt with seriatim. In each case the transformed variables will still be denoted by x, y, z, w, and the auxiliary equations will be expressed in the usual Lagrangean form, the variable t used by Weierstrass being omitted. I. (i). The auxiliary equations are: dx dy dz dw 0 x y z' three integrals of which are: x = const., y2 - 2zx = const., y3 + 3x2w - 3xyz = const., so that the general integral of the equation (1) is f= (w, y2 - 2zx, y3 + 3x2w - 3xy), where ~b is an arbitrary function. The only quartic of this form is a sum of terms X4 (2 (y2- 2z), (y2 - 2zw)2, x (y3 + 3w2w - 3xyz), so that w occurs linearly or not at all, and the surface is therefore a cone or rational. I. (ii). The auxiliary equations are: dx_ dy_ dz dw O O y z three integrals of which are: x = const., y = const., z2 - 2yw = const., so that fVO- L.,, XVII- 2yI. 4). VOL. XVIII. 43 338 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS The general quartic of this type is (z - 2yw)2 + 2 (2 - 2yw) (x, )2 + (, y)4 = 0........................ (6), where (x, y)n denotes an arbitrary quantic of order n. I. (iii). The auxiliary equations are: dx dy_ dz _ dw O x O z leading to x = const., z = const., yz - xw = const., and f- 4 (, z, yz - xw). The general quartic of this type is (yz - xw) + 2 (yz - xw) (x, )2 + (x, )4= O........................... (7). I. (iv). The auxiliary equations are: dx dy dz dw x 0 O' leading to the cone ff,,(x æ, Z )= O. II. (i). The auxiliary equations are: dx dy dz dw x x+y y+z -3w' of which ole integral only, viz. x3w = const., is algebraic, the other two being logarithmic. Thus the only possible form off is p (x3w), leading to a set of planes. II. (ii). The auxiliary equations are: dx dy dz dw x x+y z -3w' three integrals of which are: y/x- log x = const., z/x = const., x3w = const., so that the only algebraic form of f is (z/lx, x3w), and the quartic is the degenerate surface (z, x)3 W = 0. II. (iii). The auxiliary equations are: dx dy dz dw x y z - 3w' three integrals of which are: y/x = const., z/x = const., x3w = const., which lead as before to a degenerate surface (x, y, z)3 w = 0. OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 339 III. (i). The auxiliary equations are: dx dy dz dw x x+y -z z-w' three integrals of which are: y/x - log = const., w/z + log z = const., zx= const., so that the only possible quartic is the degenerate surface z2X2 = 0. III. (ii). The auxiliary equations are: dx dy dz dw x x+y -z -w three integrals of which are: y/x - log x = c onst., /x = c ons t.,w = costt, leading to f=- (zx, z/w), which gives a cone. III. (iii). The auxiliary equations are: dx _dy dz dw x y -z -w' leading to y/x = const.,t., x= costt, const., whence f/_ (xz, xw, y/x), so that the quartic is quadratic in x, y and in z, w, viz. of the form 2 2 (x, y= z, w ).......................................... (8). IV. (i). The auxiliary equations are: dx dy cdz dw ax x + ay bz cw where 2a + b + c = 0. If a 0, three integrals are: ay/ - log x = const., z nt., zw/xb = c on s t., onst., of which the first is essentially logarithmic, so that we have f ()(za/xb, wa/Sx), leading to a cone. If a = 0, so that c =-b 0, three integrals are: x = const., x log z- by = const., zw = const., so that f - (zw, x), leading again to a cone. 43-2 340 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS IV. (ii). The auxiliary equations are: dx dy dz dw ax ay bz cw' where 2a + b + c =0. We may distinguish at once three sub-cases which may arise, viz. (a) a=o, b=-c -0. (/3) b=0, c=-2a0,. (y) a$0, b#O, c O. In SUB-CASE (a), three integrals are: x = const., y = const., zw = const., so that f- ) (x, y, zw), and the quartic is z2w2 + zw (x, )2 + (x, y)4=...................................(9). In SUB-CASE (f), three integrals are: z = const., yx = const., x2w = const., so that f= - (x2W, y/x,, z). The only possible quartic terms are: zw(x, y)2, z4 so that the surface degenerates. In SUB-CASE (7) it is a little simpler to work directly with the corresponding partial differentia] equation ax- +~ay- + bz + cw -f=0, x iay az awI and to consider the possible terms in f. Since the differential operator only alters the coefficient of any term, each term of f must separately satisfy the differential equation. We verify at once that the terms z4, w4, (x, y)4 cannot exist. If a term of the type (x, y)3z exists, then 3a + b =0, whence a= c, contrary to hypothesis; similarly no term of the type (x, y)3 w can exist. Similarly no terms of the types (x, y)2z2, (x, y)2w2 can exist, as their existence would involve b=c. Any term of the type (x, y)2 zw satisfies the equation. If a term of the type (x, y) z exists, then a + 3b = 0, whence c= 5b, so that the equation is - 3 (xf~ + yfy) + zf, +.wfo = 0, OF THE FIRST KIND 0F TOTAL DIFFERENTIALS. 341 of which the general integral is f- f(z3x, x2zw, y/x), leading to 3 (x, y) + zw (, y)2= 0, a degenerate surface. Similar reasoning shews that no term of the type (x, y) w3 can exist. If a term (x, y) z2w, or (x, y) zw2 exists, then a =b or a = c, contrary to hypothesis; and if a term z3W or zw3 exists, then also a = b or a = c. Thus the only possible terms are of the type (x, y)2 zw and the surface consequently degenerates. V. Since no two of a, b, c, d are equal, one of them at most can vanish. We may therefore distinguish two sub-cases: (a) d=O, a+b+c=O, (3) a+0, b+0, c+O, d$O. SUB-CASE (a). Proceeding as in IV. (ii) (y) we see that the terms x4, y4, z4 cannot exist in f, but a term w4 may exist. If a term sxy exists, then b =-3a, c = 2a, so that the differential equation reduces to xf, - 3yfy + 2zfz = 0, whence f -b (xy, Z/X2, w), so that the only possible terms are x3y, xyzw, w4. The quartic is therefore rational, since y only occurs linearly, if at all. If a term x2y2 exists, then a + b= O, and therefore c = 0, contrary to hypothesis. For the same reasons no terms of the types (y, z)4, (z, x)4, (x, y)4 can exist. If a term x2yz exists, then 2a + b + c = O, whence a = 0, contrary to hypothesis. Thus no term of the type (x, y, )4 can exist, so that f contains w as a factor and is degenerate. SUB-CASE (/3). Under the conditions assunied it is evident that no terms such as x4, or x2yz can exist; and there cannot be more than one term belonging to a group of the type (x, y)4. Let the term xc3y exist, then b =-3a, c+ d = 2a, and no other term involving 342 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS x3 can exist. If x2z2 exists also, then a + c = 0, d = 3a. The differential equation is now fxJ - 3yfy - Zfz + 3wfw = 0, whence f- (x3y, z3, xz), so that the most general form of the surface is (X, 3w, o 2z2, y2 2, yzw )l=........................(10). Let the terms xSy, xz3 co-exist, then b=-3a, a=-3c, d =- 7c; and the differential equation is - 3xfx + 9yfy + zfz - 7wf,l = 0, whence f= b (x3y, xz3, X7W3), and the only possible quartic terms are x3y, xz3, xyzw, so that the surface degenerates. If the terms x3y, xyzw co-exist, then we get the surface (10) again. The cases thus considered and those obtained by a mere interchange of variables exhaust all possibilities, if a term such as x3y exists. If no term cubic in any one variable exists, then the possible terms to be considered are of the two types x2y2, xyz'w. If only one or no term of the former type exists the surface degenerates; if terrns such as x2y2, x2z2 co-exist, then b=c; if x2y2, z2W2 co-exist we revert to the case of (10). We have thus considered all possible cases. ~ 4. TABULAR STATEMENT OF RESULTS. The preceding analysis shews that if we exclude conical and degenerate surfaces, there are five and only five types of quartic surfaces, given by equations (6), (7), (8), (9) and (10), which satisfy Picard's differential equation, and are not pricma facie rational surfaces. Surfaces which can be obtained from one another by linear transformation of the coordinates are of course not counted as distinct. After making some slight changes of notation with a view to greater uniformity, arranging the surfaces in the order (9), (8), (7), (6), (10), and adding for convenience the corresponding values of' 0, 02, 02, 0, we get the following table: OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 343 Surface 01 02 603 4 Reference letter 2y2 + 2y (z, w)2 + (z, W)4= x -Y O O A z2 (, y) + - (x, yy)22 w2 (, + )32 =0 y -z -W B (XW - yZ)2 + 2 (xw - yZ) (z, w)2 + (z, w)4 = z w O O C (2xw - y2)2 2 (2xw - y2)(z, )2+ (z, w)4 = O w O O D ax2y2 + bz2w2 + cxyzw + dxw3 + eyz3 = 0 3 -3y z -w E Of these surfaces (A), (B) are the two which Poincaré discovered*; the existence of integrals of the first kind on the other three surfaces was pointed out by the author in a note published in the Conptes Rendus for 2 Sept. 1899. ~ 5. BIRATIONAL TRANSFORMATIONS OF THE SURFACES INTO CONES. Corresponding to each of these surfaces we can at once construct a total differential by the formula already given (~ 1); and since the 0's are unique, there cannot be on each surface more than one such differential, the integral of which remains finite. But it has not been proved that such an integral does actually remain finite everywhere. This could be done by examining its behaviour at each singularity of the surface. But it is of interest to shew that each surface can be birationally transformed into a nonsingular cubic cone, and as such a cone admits necessarily of an integral of the first kind, we are thus incidentally assured of the finiteness of our integrals. The transformations which effect this object are as follows: (A) We write the equation in the form fxy + (Z, W)2}2 = {(Z, W)22 - (Z, w)4, and choose as a new variable w one of the factors of the right-hand side, so that the latter becomes w (z, w)3. The quadric transformation:: y: z: w =x'w'-(z', W'): )y: y'z' y 'w) x ' y': z: w' = y + (, w)2: yw: zw: w2 w * Loc. cit. 344 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS then leads to x ' = ( ', ')........................................ (A (B) Choosing as a ne\ variable y one of the factors of (x, y)32, we can write the equation: z2 (x, y)1 + zw (x, y)22 + w2y (x, y)1 = 0. The quadric transformation: x: y: z: w =x'z': y'z': y'w': z'w' x' y' z'.: w= xz yz:yw:zw then leads to y' (x, y') + z' (', y') + (', y') = 0........................... (B'). (C) The coordinates can evidently be chosen so that the point x = y = z = O lies on the surface; z is then a factor of (z, w)4, which may accordingly be written z (z, w)3. The quadric transformation: x: y: z: w = z' (x' + y'): y 'w': z2w' ': y:: ' = xw-yz: yz: 2 zw then leads to z' + 2x' (z', w')' + (z', w')= 0................................. (C'). (D) Changing the variables as in (C) and employing the quadric transformation: x: y: z: = (x'z + y'2): y'w: z': w 2 ': y': z': w' = 2xw -y: yz: z: zw we get x"2' + 2' (z', W')2 + (', w)=...................... (E) The cubo-quartic transformation: x: y: z: w = y/2z': 'w3: x'y'z'w': y'yw'2 xz: y z: z:w' = z2w: XW2: Xyz: Xyw converts the surface into ay'z'2 + bx'y' + cx'y'z' + dx'y' + ex'2z' = 0..........................(E'). The five surfaces (A')-(E') are cubic cones, which are in general non-singular, so that each possesses an integral of the first kind. The birational transformation of such an integral converts it into an integral of the first kind on the corresponding qiiartic surface. Moreover, if the coefficients which occur in the equations are left arbitrary, the five cones are perfectly general cubic cones, though they occupy special positions relatively to the coordinate planes. Hence we see that a quartic of any of the five types can be birationally transformed-if necessary via a cubic cone-into a quartic belonging OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 345 to any one of the other types. But in order that two such quartics with given coefficients should be transformable into one another it would of course be necessary that there should be a relation between their coefficients equivalent to the condition that the corresponding cubics should have their absolute invariants equal. It should be noted moreover that we have supposed our quartic surfaces to be the most general of their respective types. For special relations between the coefficients one of the quartics might become a cone-a case that we have excluded-or the corresponding cubic cone might become rational or degenerate, in which cases no integrals of the first kind could exist. ~ 6. NUMERICAL GENUS OF SURFACES WHICH ADMIT OF INTEGRALS OF THE FIRST K1ND. It appears from the preceding analysis that the only quartic surfaces which admit of integrals of the first kind are cones or birational transformations of cones; consequently the (numerical) genus* is in each case negative; the numbers being -3 for a non-singular quartic cone, - 2 for a quartic cone with one double line, and otherwise -1. In the course of an investigation dealing with quintic surfaces I have miet with several surfaces which admit of integrals of the first kind, and these surfaces likewise have negative genus. On the other hand Humbert in his well-known memoir on hyperelliptic surfaces has given some octavic surfaces which admit of integrals of the first kind but are of positive genus. Whether such integrals can exist on any surface of order 5, 6, or 7 with positive genus appears to be at present unknown. ~ 7. GEOMETRICAL CHARACTERISTICS OF THE FIVE SURFACES. The surface (A) occurs in Kummer's well-known paper on quartic surfaces which contain families 'of conics+. The surface touches itself at each of the points, y=z=w=O0, x=z=w=O; any plane section through these points consists therefore of a plane quartic curve touching itself twice, that is of a pair of conics having double contact. The two points belong to a class of singular points of surfaces which seem to have been little studied; such a point may be defined as a uniplanar double point, which is further a quadruple point on the section by the tangent plane, and is consequently a tacnode on a general section through the point. Kummer speaks of a " Selbstberihrungspunct"; tacnode or tacnodal point seems a convenient English name~. It can easily be seen that a tacnode diminishes the order of the reciprocal surface by 12, so that for this purpose it is equivalent to six ordinary double points. As Picard and Simart point out, * Genre numérique, deficiency. Cf. Cayley's paper t Liouville, sér. iv. t. 9 (1893). "On the deficiency of certain surfaces," quoted before; $ Crelle, t. 64 (1864). Picard et Sirmart, ch. viII. ~ iv; Castelnuovo & Enriques: ~ According to Picard and Simart this is the name " Sur quelques récents résultats dans la théorie des surfaces given by 'les géomètres anglais,' but I have not been able algébriques," Math. Ann. t. XLVIII. (1897). to find any such authority for it. VOL. XVIII. 44 346 MR BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS the surface can be transformed by linear substitution (x'= x + iy, y'= x- iy) into the general quartic surface of revolution. The birational transformation ernployed in ~ 5 establishes a one-one correspondence between points on the conics and points on the generators of the cubic cone. The surface (B) is the well-known quartic scroll with two non-intersecting double lines, which is Cayley's first* and Cremona's eleventht species of quartic scroll. The surface (C) is Cayley's fourth and Cremona's twelfth species of quartic scroll, and is the limiting form assumed by the preceding surface, when the two double lines coincide without cutting one another, thus giving rise to the higher singularity sometimes called a tacnodal line+. The generators of the surfaces (B) and (C) correspond to the generators of the cones into which the surfaces can be transformed. The surface (D) has a double point at y=z = = 0, which is for some purposes at least equivalent to two tacnodes, as defined above; and the surface can be regarded as a limiting form of the surface (A) when the two tacnodes coincide. A section by a plane through = w =O breaks up into two conics which have contact of the third order at the singular point. This singularity can be defined-in a form applicable to a surface of any order-as a uniplanar double point such that a section by an arbitrary plane through some fixed tangent line at the point has two branches meeting one another in four points at the singular point. This property implies that in the case of a quartic the section breaks up into two conics. As far as I am aware neither this singularity nor the surface has hitherto received any attention. As before the conics correspond to the generators of the cubic cone. It may be observed that though the surfaces (C) and (D) can be regarded, from a geometrical point of view, as limiting cases of Poincaré's surfaces (A) and (B), they are not analytically special cases of them, that is, the equations (C) and (D) cannot be derived from (A) and (B) by giving special values to the coefficients. The remaining surface (E) does not appear to have been studied hitherto. It has two precisely similar uniplanar points of a rather complicated character, which can be stated in a form applicable to a surface of any order somewhat as follows. The section by the plane tangent at the point has a triple point, there, as always happens with a uniplanar or biplanar point; but in addition the three branches at the triple point coincide in direction, and if we call their common tangent the singular tangent line, this line meets the surface not merely in 4 but in 5 coincident points: thus in the quartic case this tangent line lies wholly on the surface. At an ordinary uniplanar point a section by a plane through a singular tangent line has a tacnode (equivalent to two * "A Second Memoir on Skew Surfaces, otherwise + "Sulle superficie gobbe di quarto grado," Mem. di Scrolls," Phil. Trans., t. 154 (1860): Coll. Math1. Papers, Bologna, ser. II. t. vImI. (1868). t. v. no. 340. + Salmon's Geometry of Three Dimensions, ~ 556. OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 347 ordinary double points), but in this case the singularity of the section is of a higher order, one of the branches having an inflexion, so that the singularity is equivalent to three ordinary double points. In the quartic case the section is a cubic and a tangent to it at an inflexion. When the surface is birationally transformed into the cone (E') the generators of the cone correspond to a family of twisted cubics on the quartic. For to the generator '=Xy', y' = uz' (where X, /. are connected by a cubic equation), corresponds in the x, y, z, w space the variable part of the curve of intersection of the quadrics z2= Xxw, w2 = ~yz; but these have in common the fixed straight line z - w = O, so that their residual curve of intersection is a twisted cubic. 44-2 XVIII. An Electromagnetic Illustration of the Theory of Selective Absorption of Light by a Gas. By Professor HORACE LAMB, M.A., F.R.S. [Received 13 December 1899.] THE calculations of this paper, so far as they are new, were undertaken with a view of obtaining a definite mathematical illustration of the theory of selective absorption of light by a gas. The current theories of selective absorption apply mainly to the case of molecules in close order, and it has not been found possible to represent the dissipation of radiant energy except vaguely by means of a frictional coefficient. It seems therefore worth while to study in detail some case where the dissipation can be exactly accounted for; and to consider in the first instance the impact of a system of plane waves on an isolated molecule. If we assume that the molecule has a spherical boundary, then, whether we adopt the electric or the elastic theory of light, the requisite mathematical machinery is all ready to hand. It is necessary, however, for our present purpose to devise a molecule which shall have a free period of vibration, whether mechanical or electrical, of the proper order of magnitude. The inechanical analogy was in the first instance pursued, the aether being represented by an incompressible elastic medium. This enables us to illustrate many special points of interest, but for the purpose of a sustained comparison with optical phenomena the elastic-solid theory proved in the end to be unsuited from the present point of view, as well as on other well-known grounds. As regards the electric theory, the scattering of waves by an insulating sphere has been treated by various writers*, with however the tacit assumption that the dielectric constant (K) of the sphere is not very great. In the present paper attention is specially directed to the case where K is a very large number. On this supposition free oscillations (of two types) are possible, whose wave-lengths (in the surrounding medium) are large compared with the periphery of the sphere, and whose rates of decay (owing to dissipation of energy in the form of divergent waves) are comparatively slow. And when extraneous waves whose period is coincident, or nearly coincident, with that of a free oscillation encounter the sphere, the scattered waves attain an abnormal intensity, and the original wave-system is correspondingly weakened. * Lord Rayleigh, Phil. ag., Feb., 1881, and April, 1899; Prof. Love, Proc. Lond, Math. Soc,, t. xxx., p. 308; G. W. Walker, Quart. Journ. Mlath., June, 1899. PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION, ETC. 349 The conception of a spherical molecule with an enormous specific inductive capacity is adopted here for purposes of illustration only; and is not put forward as a definite physical hypothesis. In order to comply with current numerical estimates of molecular magnitudes, it is necessary to assume that for the substance of the sphere K has some such value as 107. This assumption may be somewhat startling; but it is not necessarily inconsistent with a very moderate value of the specific inductive capacity of a dense medium composed of such molecules arranged in fairly close order. And it may conceivably represent, in a general way, the properties of a molecule, regarded as containing a cluster of positive and negative 'electrons.' In any case the author may perhaps be allowed to state his conviction, that difficulties (such as they are) of the kind here indicated will prove to be by no means confined to the present theory. The main result of the investigation may be briefly stated. For every free period of vibration (with a wave-length sufficiently large in comparison with the diameter of a molecule), there is a corresponding period (almost exactly, but not quite, coincident with it) of maximum dissipation for the incident waves. When the incident waves have precisely this latter period, the rate at which energy is carried outwards by the scattered waves is, in terms of the energy-flux in the primary waves, +................................................ where X is the wave-length, and n is the order of the spherical-harmonic component of the incident waves which is effective. In the particular case of n= 1, this is equal to 477X2. Hence in the case of exact synchronism, each molecule of a gas would, if it acted independently, divert per unit time nearly half as much energy as in the primary waves crosses a square whose side is equal to the wave-length. Since under ordinary atmospheric conditions a cube whose side is equal to the wave-length of sodiumlight would contain something like 5 x 106 molecules, it is evident that a gaseous medium of the constitution here postulated would be practically impenetrable to radiations of the particular wave-length. It is found, moreover, on examination that the region of abnormal absorption in the spectrum is very narrowly defined, and that an exceedingly minute change in the wave-length enormously reduces the scattering. It may be remarked that the law expressed by the formula (1) is of a very general character, and is independent of the special nature of the conditions to be satisfied at the surface of the sphere. It presents itself in the elastic-solid theory; and again in the much simpler acoustical problein where there is synchronism between plane waves of sound and a vibrating sphere on which they impinge. It has unfortunately not seemed possible to render this paper fairly intelligible without the preliminary recital of a number of formulae which have done duty before, notably in Prof. Love's paper. The analysis has however been varied and extended in points of detail, with a view to the requirements of the present topic. In particular, 350 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY the general expression for the dissipation of energy by secondary waves, which is obtained in ~ 5, is found to take a very simple form, and may have other applications. Some notations which are of constant use in the sequel may be set down for reference. We write / d\ sin 1 __ 4 ) sn) ( )1l.3... (2n+1) 2 32 ). (2(2)n+ 3) 2(2) + )a d' cos 1.3..2(2n-1) ) 2 ) n,(~) = _^ ~6 =. 1 - 2(1 -2n) 2.4(1-2n2)(3- 2n) 2n (-.() These may be taken as the two standard solutions of the differential equation dI2F 2 (n + 1) dF + 2( + = = o.................................... (); the solution rn(,) being that which is finite for =0. In the representation of waves divergent from the origin we require the combination / d )~ne-i~ f ( ) =,- - = )- (T).......................... () The functions * (n), 1n (), fn(a) all satisfy formulae of reduction of the types % I (=) —1%+~1 ( ~)....................................... (6), n ' ( ) + (2n + ) k () = n- (r).............................. (7), from which (4) can be verified. We have also the formula % ' (0) % - ())- n () 4 ' (>) = ( 2 +2.............................. 1. The equations to be satisfied in a medium whose electric and magnetic permeabilities are K and /L may be written, as in Prof. Love's paper, dy _d/ K da dy K d_), ca c dy dz' c dz dx' c dx dy..dZ dY p dX dZ /. dY dX - - a= -,- - -7-, — / /3=. =.............. (1 ), c dy dz' c dz dx' c / dx dy where (X, Y, Z) is the electric force, (a, 3, y) the magnetic force, and c denotes the wave-velocity in the aether. Assuming a time-factor eit, we find (V72+h)X=O, (72+ h2) =0, (V2 + h2) Z==O.....................(11), dX dY dZ with d.~ + d- + d - = ~dx dy dz, where h2 = Kir2/c2...(13). * See Hydrodynamics, ~~ 267, 305. t See Lord Rayleigh' Sound, ~ 327. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 351 When values of X, Y, Z satisfying these equations have been found, the corresponding values of a, /3, y are given by (10). Or, we may reverse the procedure, determining the general values of a, f, 7 by means of equations similar to (11) and (12) and thence the values of X, Y, Z by means of (9). The solutions of (11) and (12) subject to the condition of finiteness at the origin are of two types. In the first place we may have X = {hrh,î' n(h + )) +(h (n + 1),nhr (hrr) xr-2 T, Y= {hrqfJ' (hr) + (n + 1) Jr (hr)} rT - lTn~rq' (hr) yrn-2Tn, -...... (14), dy Z = {[hrn (hr) + (n + 1) % (hr)} d rTl nhr i Jr' (hr) zr"-2 T where Tn is a spherical surface-harmonic of order n*. These make io-K (ci /d d m =-iK I (hr) (y -z ), ciK dx d\ =-iE-, (hr) z- dx -- )rl2^T TT. l)j ( dx =-K C h (hr) d-nT It follows that xX + y Y + zZ = n (n + 1) (hr) n...........................(16), xcO + y + zy =0.(;............................................. (17); also that yZ - zY = hr,,' (hr) + (n + 1) (hr)} (y -z r', &c., &c............(18), jo-K __ nxrd~ 2T~ y-z3fi= = r2r, ((hr) r1T (d -nxr' -2T,, &c., &c............................9) In the solutions of the second type we have a = {hrn' (hr) + (n + 1) (hr)} - r U - nhrf,' (hr) xr'2U a = ihrefrn' (hr) + (n + 1) Jn (hr)} fdr' U - nhreJt' (hr) yr2 Un,.........(20), dy y = {hrJr,,' (hr) + (n + 1) ^ (/tr)} - rUn - nhrl4t,' (hr) zrV-2Un * These are equivalent to the forms given iin Hydro- formula relating to spherical solid harmonies, such as dynamics, ~ 305 (6), divided by 2'n+1. The proof of the =r2 (ddp n,+1 \ d n ) equivalence requires the use of (6) and (7), together with '=t 2?+1 \ dx dx r2n~1) * 352 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY where U, is a surface-harmonic. From these we deduce Tr iafj /~ 7n) \ d d = i2/î (h, (hr,)( d -C rd U1),. C.(2). (ix dz c X=- (hr) X y-d lir | T-r Z~icrLL, / 7 x d d '\ i,x X + zz = O................................................... (23); also yy - z,3 = {h ) + (Y + 1) i( (h1, )} y d z r &c., &c..........(24), /-sY=-5 r2+ (hr)-,d,-,l- 2 U ), &c....., &c........(25). It is known that the most general solution of our equations, consistent with finiteness at the origin, can be built up from the preceding types, by giving n the values 1, 2, 3,.... 2. Let us now suppose that a sphere of radius a, having the origin as centre, whose electric and magnetic coefficients are K and,/, is surrounded by an unlimited medium (the aether) for which K= 1 and u = 1. The disturbance in this medium rnay be regarded as made up of two parts. We have, first, the extraneous disturbance due to sources at a distance; this is supposed to be given. Secondly, we have the waves scattered outwards by the sphere. The general expression for the extraneous disturbance is conditioned by the fact that if the medium were uninterrupted the electric and magnetic forces at the origin would be finite. It is therefore made up of solutions of the type already given, provided we put K= 1, /= 1, and replace h by k, where k = -/c.............................................(26 ). As usual, 27r/k is the wave-length of plane waves of the period 2vr/-. In the corresponding expressions for the divergent waves, we must further replace (n(hr) by fn (kr), where fn is the function defined by (5). This is necessary in order that the formulle may represent waves propagated outwuards, the complete exponential factor being then ek (ct-r) It is necessary to have soine notation to distinguish the surface-harmonics used to represent different parts of the disturbance. Those harmonics which occur in the expression for the imposed extraneous disturbance will be denoted by T_?, Un, simply; those relating to the scattered waves by T,\, Un,; and those relating to the inside of the sphere by.T") 0'' OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 353 We have next to consider the conditions to be satisfied at the surface r = a. It appears at once from (16) and (22) that the solenoidal conditions of electric and magnetic induction require that 4an (ka) T, + fn (ka) Tn' = KJn (ha) Tn"...........................(27), qn (ka) Un +f, (hka) Un = =Vîn (ha) U.......................... (28). Again, it is easily seen that the continuity of the tangential components of electric and magnetic force implies the continuity of the vectors (yZ-zY, zX-xZ, xY-yX) and (yy - z/3, za - xy, x/ - ya), respectively. Hence from (18), (19), and (24), (25), we have, in addition 'kaJ (a ( 1),' + ( T f' (a) + (n + 1) (k l)fn ('Ca)} Tn = {haC ' (ha) + (n + 1) n (/ha)} 2...........................(29), and {kaa,,r (Ia) + (n + 1) (/ca)}n U' + kaf,,' (aa) + (n + 1) f,. (aka)} Un` = {han,,' (ha) + ( + 1),, (ha)} U,,'...........................(30). Hence Tn _ Kn (ha) {karn,, (ka) + (n + 1) n, (Ca)} - {haS,,' (ha) + (n + 1),n (ha)} ', (ka) Tn ~ K+,n (ha) {af,' (kIa) (+ +)fn. (ka)} - {ha,,'(ha) + (n + 1),n (ha)} fn (ka).....................(31), Un p î,n (ha) {ka4rn' (ha) + (n + 1) f, (ha)} - {han' (ha) + (n 4+ 1) în (ha)} -n (ka) Un urn (ha) {kaf,' (ka) + (n + 1) fn (ka)} - {han,,' (ha) + (n + 1),,n (ha)}f,, (ka)..................... (32). We shall suppose that the wave-length of the disturbance in the aether is large compared with the circumference of the sphere, so that ka is a small quantity. If we were further to assume that K and, are not greatly different from unity, so that ha is also small, we should obtain at once approxiiimate expressions equivalent to those given by Prof. Love, viz. nK +(n+) {1.3...(2n-1)}2 (2n, + 1) n.. (n + 1) (/ - 1).... (34). n= + (n + ) '{1. 3... (2 )(2- z + 1)' U.... -- It is our present object, however, to examine the case where K is large. For simplicity we shall suppose that /u = 1, so that K = h'/k2. It will be found that the first factors on the right hand of (33) and (34) must be replaced by (n + ) (K- - ) n, (ha)-... (nK+n + l)i n(ha)+... VOL. XVIII. 45 354 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY and ha' (/ha) +...(36) r-,' (hta)..................................... (36), Jr_ (ha) +... respectively, where the terms omitted are of the order k2a2 compared with those retained. It appears that there will be nothing abnormal in the amplitude of the scattered waves, except when ha is nearly equal either to a root of,,(ha)= 0, or to a root of -fr_(ha)=O0, in which cases the preceding approximations cease to be valid. 3. If the extraneous disturbance consists of a system of plane waves, then, assuming that the direction of propagation is that of x-negative, and that the electric vibration is parallel to y, we may write, symbolically, X =0, Y=eik, Z=0.................................... (37), = 0, 8 = 0, y= - eik.............................. (38). If this be resolved into a series of disturbances of the types (14) and (20) we must have, by (13) and (19), _n (n +1) 5k~ (kr) r T~ = yek( n (n + 1) n r) r Tn =ye,................................. (39), n (n + 1) ( r) rn U - zek............................. (40). Now if we put x=rcosO, y = r sin 0 cos, z= r sin sin..................... (41), we have ikyeikX = > (2n + 1) (ikr)'" n (k1r) sin 0 cos o P,' (cos 0)............... (42)*, where Pn (cos O) is the ordinary zonal harmonic. We infer, by comparison with (39), that 2n++1 nT ( = + (ik)-l sin 0 cos o Pn' (cos 0)........................ (43). Similarly, we find 2n+ i Un - n( + 1) (ik)l-l sin O sin w P,,' (cos )........................(44). In particular 3 3y 3 sin cos =....(45), T~= s 2 r............................ o3~ 3z U =- sin sin o - - (46). * Proved most easily by differentiating with respect to cos O the known identity eikr cos O= z (2n+ 1) (ir)a n V (lka) P, (cos 6). OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 355 If we substitute the above values of Tn and Un in the formulæe (31) and (32) we obtain the expressions for the scattered waves. 4. We have now to examine the form which the scattered waves assume at a great distance from the origin. When kr is large we have -....................................... Hence, in the first type of solution, analogous to (14), we have )nZ — i d / n S Y - - _ p-k ( r nxrN"- - | (kr)n e —ikr n - Zrn-2 T, ) n-i C -2 Y = e r T - yr T......................... (48), (ïkr)n eik2 n nyrCl ln) "* (kcr)n (C n d )),kry - z ) n Tn knrn+l dz dyr =n-inr d c\ i = in+ e-'kr -- r Tn\ ( d =brn+l e-ik ( ~d -Y dx ) rtrX!.. We notice that X, Y, Z are ultimately of the order 1/r, whilst the radial electric force (X + y Y + zZ)/r is zero to the present order of approximation. It is really of the order l /r2. The radial magnetic force (xa + y/3 + zy)/r is accurately zero. If the contour-lines of the harmonic Tn' be traced on a sphere of large radius r, for equal infinitesimal increments of Tn, the (alternating) magnetic force is everywhere in the direction of these contours, and its amplitude is inversely proportional to the distance between consecutive contours. The electric force is everywhere orthogonal to the contours, and its amplitude is in a constant ratio to that of the- magnetic force*. For instance, in the case n = 1, if T\` be of the type (45), the lines of electric and magnetic force have the configuration of meridians and parallels of latitude, the polar axis being represented by the axis of y. In the second type, analogous to (20), we have a - (k -ikr n __ n r = -,e akr (Sd n UT\ - Zrn-2 Un ) * Cf. Proc. Lond. Math. Soc., t. xIII., p. 194. 45-2 356 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY X =-k + d d - z U, iZ=- = _ __yni - id d\ Z= knrn ---1 x -yd r! U j, with a similar interpretation. The contour-lines of 'U,, are the lines of electric force, and the lines of magnetic force are orthogonal to them. 5. The calculation of the energy carried outwards by the scattered waves leads to some very simple results. By Poynting's theorem*, the rate at which the energy in any given space is increasing is equal to the integral 4 1 {l (yY-/3Z) + rm (aZ- X) + n (3X- aY)} d.................. (52), taken over the boundary of the space, 1, m, n denoting the direction-cosines of the normal drawn inwards from the surface-element dS. The ambiguities which are known to attend a partial use of this theorem will disappear if the space in question be that included between a sphere of radius r, in the region of the scattered waves, and a concentric sphere of radius so great that we may imagine it not to have been as yet reached by the waves. The rate of propagation of energy outwards is therefore given by the integral -:| {(yZ-zY) +/3z(X-xZ) +(xT-yX)} dS................(53), taken over the sphere of radius r. Before applying this result, the values of a, /3, 7 and X, Y, Z must of course be expressed ini real form. To take first a solution of the first type, since T,,, as given by (31), will in general be complex, let us write rnTl2 \ = j1 + i..................................... (54). Restoring the time-factor in (48) and (49), and taking real parts, we find = krl- (c -i - z dyJ {4 cos (at - kr + e) - < sin (aot - kr + ), &c., &c.,.........(55), and yZ- zY= - (y z - z d) [{, cos (o-t - kr + e) - sin (o-t - kr + e)}, &c., &c.,...... (56), where e may be 0, or + T7r, or v, according to the value of n Hence the mean value of the expression (53), per unit time, is found to be c f[ Ç( dIn dbn'Y 2 ) d4nn dT, dÉPn dPt Swkîîîri-2~ JJ d(k-d- dy d z+ zdx - x ~Jz) + xdx) d dzy dy dx Xdz J +\ dy - dxp 3. ~* Phlj. Trans., 1884, p. 343. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 357 which may also be written 8k20r2 f $(dI + (dd)2 + (rD )2 - Y 2 ( 2 87vrkor (2 \ dx \dy \ dz - 2 + dy/ + d2 - 2 dS.........(58). The expression under the integral signs in (58) is equal to the sum of the squares of the tangential components of the vectors (dn/dcx, d),,/dy, d,,l/dz) and (dbn,,/dx, d,,/dy, dc/,dz). Now if S, be a surface-harmonic of order n, we have r27r & \2 d 7. 2rn f ( d^) + ( d-Y ) } sin 1OdOdw = n ( + 1)j SS,' sin Oddoo......(59)*. Hence (58) may be written n ( + 1) 8 2 2n+ ( +,,2) dS, or n (n + 1) 8. 2 dv............(60), where IT,,' denotes the modulus of T,', and d-r is an elementary solid angle, viz. dS = r2dz. In a similar manner, a solution of the second type gives the result (n + 1) 2d.................................(61). It appears, further, on examination, that the parts of the expression (53) which arise from combinations of the two types, or from combinations of the same type with different values of n, will disappear in virtue of the conjugate property of surface-harmonies of different orders. Hence, if Z be a sign of summation with respect to n, the general expression for the rate at which energy is dissipated by the scattered waves is 8 n (: n ) { | T,1 + | U,,,2} d-,.............................. In the case of plane incident waves the harmonies are tesseral, of rank 1. Writing, for shortness, Tn' = BnT^, U=.................................... (63), * Proved easily by partial integration, making use of This involves products of surface-harmonics of orders the differential equation m -1 and n, and will therefore vanish unless m= n + 1. 1 _d.dS N 1 d2Sn But writing it in the form d si + Ls i+ - n+(12 +.1) OS, - 0. sin 0dc dO )sin20 dX2 (+olX z 15? f The integrals which arise from combinations of the J x \ dy dz two types are of the form we see that it also vanishes unless n=Lm+l. Hence it II [ x {......{ vanishes i1n any case. J jn (II d lx dz dXn'y _ \ 358 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY where the values of Bn and Cn are as given by (31), (32), and T2, U, have the forms given in (43), (44), then since ff{sin 0 cos os P s 0)}( dcs = n (n + 1). 2 1.....................(64) the expression (62) reduces to 4k 2 (2n + 1) Bn + 2}................................. (65) The proper standard of comparison here is the energy which is propagated per unit time across unit area in the primary waves represented symbolically by (37). On the scale of our formulæe this is c/87r. Hence, if I denote the ratio which the energy scattered per unit time bears to the energy-flux in the primary waves, we have I 2 (2n +1) B,j2 + |CJI 2}........................ (66). For example, in the case to which the formuloe (33), (34), refer, the constants K and /L for the sphere being not greatly different from unity, we have 2K —1 2/-1 9k= 2 K-3 2 k3a=.. (67), B l — kaL, Co1.+ 2n=............................(67), and thence = 3 7ra2 + ca) 4...........................( 8)t. 6. We may proceed to examine more particularly the case where K is a large number, whilst t/ is (for simplicity) put = 1. The types of free vibration which can exist in the absence of extraneous disturbance are found by making T =O, Un=0 in (31) and (32). In the first type we have haf,' (ha) + (n + 1) %, (th) K af,' (ka) + (n + 1)fn (a) (), (ha) f=K (k)a) solutiowhere,s of this to beation for which is small. Onare speially concerned to fin have solutions of this equation for which ka is small. On this hypothesis we have ha-r4' (ha) + ( + 1) +n (ha) _ Irn,(ha) ~ -(n?d~ ha.*..........................(70), %,, (ha) nearly. This is satisfied approximately by ha =z, where z is a root of n ()= 0............................................(71), and more exactly by ha=......................................(72). * Ferrers, Spherical Harmonics, 1877, p. 86. t This agrees with a result given by Lord Rayleigh, Phil. lMacg., April, 1899, p. 379. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 359 In the case n = 1, the equation (71) takes the form tan z= z; whence z/7r =1-4303, 2-4590, 3-4709,...........................(73)". Corresponding to any one of these roots we have a simple-harmonie electric oscillation of frequency o z C.................................. (7 4 ), 27r 7r 2K2a( and wave-length X = 2K a.......................................(75). 77 To calculate the rate of decay of the oscillations, which is relatively very slow, we should have to proceed to a higher degree of approximation. In the second type, we have, from (32), with / = 1, han',' (ha) kafn' (ka) ++(() =fnff()).................................. (76), 4n(ha) f,,(ka)(7,,,n- (ha) f_ (ka) or....- f (C.................(77). 01 n(ha) fq,(ka) *** ** ** *- *(*), This is satisfied approximately by ha = z, where z is a root of (z) = 0.........................................(78), and more accurately by ha={1-(2n-l)K} z......................(.. 79). When n= 1, (78) takes the form sinz = 0, whence z/ r= 1, 2, 3,.........................................(80). 7. When in the problem of ~ 2 the extraneous disturbance has a period coincident, or nearly coincident, with that of a free vibration, the approximate formulae (33) and (34) will no longer apply. If in the accurate formula (31) we make the substitution f (ka) = (ka)- in (ka)................................. (81), we find that it takes the form T __ g (ha) Til 0G(ha)-ig(ha)...(82), where g(ha) stands for the expression in the numeratort of (31), and G(ha) is derived from g (ha) by the substitution of *, (ka) for, (ka). The modulus of the expression * The lines of electric force in the sphere are for the in Electricity and Mlagnetism, p. 317. most part closed curves in planes through the axis of the t Which may be regarded as a function of ha since the harmonic T1. Their forms are given in Phil. Trans., Pt. ii. ratio of hk to 17 is fixed. 1883, p. 532; and in J. J. Thomson's Recent Researches 360 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY on the right hand of (82) never exceeds unity; but it becomes equal to unity, and the intensity of the scattered waves is therefore a maximum, when G (ha)= 0.................................................(83),,ha,,n' (ha) + (n + 1) n (a) (la) K 1 a,,' (ka) + 1) l r, (a)(84). ~o~~r ~ - k -............(84). When K is large, the lower roots of this, considered as an equation in ha, are easily seen to be real and to be very approximately equal to the real parts of the roots of (71). When the period of the incident waves is such that (83) is satisfied exactly, we have Tn =- iT,............................................(85). If the incident waves be plane, the dissipation-ratio (68) takes the form 2 (2n + 1) r 2 + 1.................................86) k2 2r( If we compare this with (68), we find that in the case n = 1 the effect of synchronism is to increase the dissipation in the ratio 9 (ka)-6. The wave-length of maximum scattering is of course very sharply defined. If we put ha = (1 + e) z.............................................(87), where z is a root of (84), and e is a small fraction, I find gfla(ha) (a.:... (2n- } g (Itct ) =- -,^ e ). G (/ta)= (ACa2n+î _ (ha). 2e1.........(88), 3~. 3.. (21- (ka)~approximately, whence......................... (89). 1+.... iK. (ka)2n+l For example, in the case n = 1 the dissipation sinks to one-half of the maximum when the wave-length deviates from the critical value by the fraction (ka)3/K of itself. The second type can be treated in a similar manner. Writing (32), with /t = 1, in the form U - g (hha) Un G (ha)-ig (ha)...................... the equation G (ha)= O which determines the wave-lengths of maximum dissipation may be written kn-i(ha) _ _ (ha) ~fr (ha) tqL, (ka) *****.*(9.. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 361 The lower roots (in ha) which satisfy this are very nearly the same as in the case of (78). When (91) is satisfied exactly we have = - n.............................................(92), leading to the same formula (86), as before, for the dissipation-ratio when the incident waves are plane. Also, if we write ha = ( + e)z........................................ (93), where z is a root of (91), I find *, (ha) \. 3...(2n - 1) y (ha)= () 2^- (()= i f ((ha). e............(94), ~1. 3... ~~(21 (ka)2napproximately. Hence Un- i Un 1............................(95). U 1 + 1''Lt - )} iKE (ka)2n-1 The definition is now less sharp than in the case of (89), in the ratio k2a2. 8. It remains to examine what sort of magnitudes must be attributed to the quantities a and K in order that our results may be comparable with ordinary optical relations. Since ka(= 2ra/X) must in any case be small, and since ha must in the case of synchronisnl satisfy (71) or (78) approximately, and must therefore be at least comparable with 7r, it follows that if our molecules are to produce selective absorption within the range of the visible spectrum, the dielectric constant K(=h2/k2) must be a very large number. Again, it appears from two distinct lines of argument* that in a gas composed of spherical dielectric molecules the index of refraction (/,L) for rays which are not specially absorbed is given by the formula 3 K-1 4 ^1 -1=2P* Kt 2.....(96), where p = N. rra3.......................................... (97), N denoting the number of molecules in unit volume. On our present hypothesis this takes the simpler form 3 - =....................(98). * Maxwell, Electricity, ~ 314; Lord Rayleigh, Phil. Mag., Dec. 1892, and April 1899. VOL. XVIII. 46 362 PROF. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY Hence if, = 1-0003, we have p = 2 x 10-4. This determines the product Va3, for a gas such as oxygen or nitrogen under ordinary atmospheric conditions, but not N or a separately. If in accordance with current mechanical estimates we take iV= 2 x 19, we find a= 13 x 10-8 cm. Hence if X = 6 x 10-5 cm., we find ka = 1'4 x 10-3, so that, if ha = 7r, we must have K = h2/kt2=5 x 106. In a dense medium composed of the same molecules the formula (98) is replaced by ', -......................................... (99), where the accents refer to the altered circumstances. Comparing, we have ~'-l2-1 2 p' ' = + 2 -1 (~ - 1)..................................(100). PY~+ 2:3 The fact that the refractive indices of various substances in the liquid and in the gaseous state have been found to accord fairly well with this formula shews that the observed moderate values of K'(== -,2) for dense media, taken in the bulk, are not incompatible with an enormous value of K for the individual molecules. The formula (86) for the dissipation-ratio in the case of exact synchronism is independent of any special numerical estimates. It can moreover be arrived at on widely different hypotheses as to the nature of a molecule and of the surrounding medium. Its unqualified application to an assenmblage of molecules arranged at ordinary intervals may be doubtful, since with dissipation of such magnitude it may be necessary to take account of repeated reflections between the molecules. It is clear however that a gaseous medium of the constitution here imagined would be absolutely impenetrable to radiations of the critical wave-length. As regards the falling off of the absorption in the neighbourhood of the maximum, the formula (95) in the case 2 = 1 would (on the numerical data given above) make the absorption sink to one-half of the maximum when the wave-length varies only by -00,000,000,028 of its value. The formula (89) would give a still more rapid declension. The range of absorption in -a gaseous assemblage must however be far wider than these results would indicate. So far as it is legitimate to assume that the molecules act independently, the law of enfeeblerrment of light traversing such a medium is E = E e-NI.........).................................( ) * This is Lorentz' result. Lord Rayleigh's investigations shew that it will hold approximately even if p' be not a very small fraction. 1t Lord Rayleigh, i. c. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 363 WTe nray inquire what value of the dissipation-ratio I would make the intensity diminish in the ratio 1/e in the distance of a wave-length. If we write I =.3 X2....................................... (102), so that I,, denotes the maximum value of the dissipation-ratio for n = 1, the requisite value is given by I 2 I_.7r_....................................... (103). im 3 On our previous numerical assumptions this is about 4 x 10-7. The corresponding value of e in (95) is about 4 x 10-7. This is comparable with, although distinctly less than, the virtual variation of wave-length which takes place, on Doppler's principle, in a gas with moving molecules, and which is held to be sufficient to explain the actual breadths of the Fraunhofer lines. Having regard to the very much sharper definition which we ieet with in the vibrations of the first type, and to the increase of sharpness (in each type) with the index n of the mode considered, it would appear that there is no primcâ facie difficulty in accounting, on our present hypothesis, for absorption-lines of such breadths as occur in the actual spectrum. 46-2 XIX. The Propagation of Waves of Elastic Displacement along a Helical Wire. By A. E. H. LOVE, M.A., F.R.S., Sedleian Professor of Natural Philosophy in the University of Oxford. [Received 4 December 1899.] 1. IT is known that the modes of vibration of an elastic wire or rod which in the natural state is devoid of twist and has its elastic central line in the form of a plane curve fall into two classes: in the first class the displacement is in the plane of the wire and there is no twist; in the second class the displacement is at rightangles to the plane of the wire and is accompanied by twist. In particular for a naturally circular wire forming a complete circle when the section of the wire is circular and the material isotropic there are two modes of vibration with n wave-lengths to the circumference; these belong to the first and second of the above classes respectively, and their frequencies (p/27r) are given by the equations _1 Ec2 n2 (n - 1) l 4 poa4 1 + n2 ' ~~~.and z-2 ~1 Ec2 12 (n2- 1)2 P 4 poa4 1 + Y + 2, where a is the radius of the circle formed by the wire, c the radius of the section, p, the mass per unit of length, E the Young's modulus and îî the Poisson's ratio of the material. These results may be interpreted as giving the velocities with which two types of waves travel round the circle. So far little or nothing- appears to be known about the modes of vibration of wires of which the central line in the natural state forrns a curve of double curvature, except that the vibrations do not obviously fall into two classes related to the osculating plane in the:same way as the two classes for a plane curve are related to the plane of the curve. The equation connecting the frequency with the wave-length when waves of elastic displacement are propagated along the wire has not been obtained; and although this equation would obviously be quadratic when rotatory inertia is neglected, and so would give two velocities of propagation for waves of a given length, it is by no means obvious what would be the distinguishing marks of the two kinds of waves with the same wavelength. PROF. LOVE, THE PROPAGATION OF WAVES, ETC. 365 It seemed to me that it would be not without interest to seek to answer the questions thus proposed in the case of a wire which in the natural state has its elastic central line in the form of a helix. As regards the free vibrations of a terminated portion of such a wire with free ends, or fixed ends, or under the action of given forces at the terminals, it would be possible to form the equation for the frequency, but the equation appears to be so complicated as to be quite uninterpretable; and in fact in the simpler problem presented by a circular wire with ends, which has been treated in some detail by Lamb*, it appears that to interpret the results the total curvature must be taken to be slight, and the results which can then be obtained are such as might be reached by suitable approximate methods. In the case of a helical wire the most important of all the problems of vibration is that of a spiral spring supporting a weight which oscillates up and down; and this can be treated adequately by means of an approximate theory in which the wire is taken to have at any time the form of the helix corresponding to its axial length and to the position of the load. The problem of the propagation of waves along an infinite helical wire remains. I have found that in general for a given wave-length two types of waves are propagated with different velocities; in both types all the kinds of displacement (tangential, normal and torsional) are involved, and there is no rational relation between the different displacements which serves to distinguish the types of the two waves, but these types are finally and completely separated by a circumstance of phase in the different components of the displacement. 2. The helix which is the natural form of the elastic central line of the wire may be thought of as traced on a circular cylinder, and then any particle on this line undergoes a displacement which may be resolved into components u, v, w along the principal normal, the binormal and the tangent to the helix. The principal normal coincides with the radius of the cylinder, and the displacement u is reckoned positive when it is inwards along this normal; the displacement w is reckoned positive when it is in the sense in which the arc is measured, and then the positive sense of the displacement v is determined by the convention that the positive directions of u, v, w are a right-handed system for a right-handed helix. Further there is an angular displacement by rotation of the sections, of amount 3, about the tangent to the helix, and /8 is reckoned positive / when /3 and w form a right-handed rotation and translatory displacement. Now it is found that in general the \ two waves of given length that can be propagated are distinguished according as the displacements v and w are 1 in the same phase or in opposite phases at all points of the helix. If 1/p and 1/o- are the measures of curvature and tortuosity of the helix, and 27r/mn is the wave-length, then in the quicker wave v and w are everywhere in the same phase, and in the slower wave they are in opposite phases, provided mn2 > l/p2 - I/or2, but if m2 < 1/p2 - 1/ao2 this relation is reversed. ' Poc. Lo1nd. Math. Soc., xIx. 1888. 366 PROF. LOVE, THE PROPAGATION OF WAVES The fact that there are two waves with different velocities suggests an analogy with the optical theory of rotatory polarization, and leads to the question whether in any sense the two waves can be regarded as right-handed and left-handed. The most obvious possibility of this kind would be that /3 and w should be always in the saine phase for one wave and in opposite phases for the other; it is found however that this is not the case; another possibility would be that the component displacements parallel to the axis and to the circulai section of the cylinder on which the helix is traced should be everywhere directed like a right-handed system of axial and circular translatory displacements for one wave and like a corresponding left-handed system for the other; this also is found not to be the case. It appears that up to the degree of approximation which is usually included in the theory of elastic wires there is no rotatory effect involved. In three particular cases it is found that the equation for the frequency of waves of given length breaks up into two separate equations. This happens (a) when n2=l/p2+l/o-2, (b) when m2 = l/p - /o2, (c) when the helix is very fiat or 1/o can be neglected. In case (a) one of the modes of deformnation is equivalent to a rigid body displacement of the helix at right angles to its axis, and the corresponding speed of course vanishes; in case (c) the types correspond to the two already known for a circle; in case (b) the two types are distinguished by the vanishing of the flexural couples in and perpendicular to the osculating plane; this case occurs only if the angle of the helix is less than 47r. 3. The wire is taken to be of uniform circular section (radius c), and of homogeneous isotropic material, and in the natural state the line of centres of its sections forms a circular helix of curvature l/p and tortuosity 1/o. The displacement of a point on the central line is specified by components u, v, w along the principal normal, the binormal a.nd the tangent in the senses already defined, but it is necessary to fix the meaning of the angular displacement fi. For this purpose we suppose a frame of three coorthogonal lines to move along the helix so that the three lines always coincide with the principal normal, the binormal, and the tangent; if the origin of the frame moves with unit velocity the lines of the frame will rotate with an angular velocity which has components i/p about the binormal and 1I/o about the tangent. We can construct a corresponding frame for the strained wire by taking as origin the displaced position of a point on the strained elastic central line, as one line of reference the tangent to the strained elastic central line through the point, and as one plane of reference the plane through this line which contains the tangent to that line of particles which in the natural state coincided with the principal normal; when the displacement is everywhere very small the lines of this frame very nearly coincide with those of the firame attached to the unstrained wir,, and the plane of reference just defined makes a very small angle with the osculating plane of the helix at the corresponding point; this angle is /8. The "twist" of the wire is expressed by 1 h ci e 1 eIv ou \ -+ - - l + o as p \as o where cls is the element of arc of the helix. OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE, 367 4. The action of the part of the wire for which s is greater upon the part for which s is less, across any section, can be reduced to a resultant force at the centre of the section and a couple. The force mnay be resolved into components N1 along the principal normal, V2 along the binormal, and T along the tangent, in the senses in which u, v, 'w are reckoned positive. The couple may be resolved into two flexural couples G1, G2 and a torsional couple H about the same three lines. The couples are expressible in ternis of the displacements by the equations G =A -_ (+ -- 1-( _+ W] ap as\(s (- as O- a P G a a(u v +\ (a v \u G,2=A Las-+ p)- a S + -)........................ (1), as\as o-pi o-kas o] H - CLafl a ) +-] |a+ p as OJ - in which A, = lErc4, is the flexural rigidity, and C, = Erc4/(1 + v), is the torsional rigidity. Further, the displacements u, w are connected by the relation of inextensibility of the wire azu _ as p -(2) When rotatory inertia is neglected the stress-couples are connected with the stressresultants and with each other by the three equations of moments aG, Go H -+ -- - V2 = O, as o- p 2 + G0,...(:3). as o aH G, __ _ G= t) as p The equations of sinall motion are the three equations of resolution 811 NV 2 l \ ds- + p- = po0)o t as o- p at + =PoW,l.................(4), as ataT i aV,-w a-s = po)O a in which po is the density of the material of the wire and w, = rc2, is the area of the cross-section. 5. We shall now suppose that simple harmonic waves are propagated along the wire, and take as expressions for the displacements = - mp W sin (ms -pt), v = Fcos (ms -p(t), w = W cos (ms -pt),); = B cos (ms- pt). 368 PROF. LOVE, THE PROPAGATION OF WAVES in which ic and w have been adjusted so as to satisfy the equation (2) of inextensibility. Further, we shall take the forms of G6, G2, H to be G1 = Ag1 eos (ms - pt), G2 = Ag1 sin (s - pt),. (6) H = (np)-1 Ag, sin (ms - pt) in which G1 and H have been adjusted so that the third of the equations of moments (3) is satisfied identically. We then find by (1) g=- +_ _p Y(-+) 2 A&=_+ B + — F -p a2 p o( + C2=2 _ + -) +, 2........................ 2 8, Ag1 I( V W = - C yB~+ - +omp p p c / of which the first and third give g(l+.+2..............(8) and the second is ^ = r - V t2.W+ f 2 -1. +.).................................. mzp paP p c2 If 2 - __ - 2- does not vanish we can solve for V and W and obtain p2 = 2 v(in21~+M + 1$ =_ __ - (7- - 2 = g 1 + Cm22) p Cr2 mr( (10) p p2 - C32) pa- 2 jmp\ p2 -2) ) The first two of the equations of moments (3) now give us N1 = -A (- g + mg2) cos (mîs - pt), r.................. (11). {( =- A {(m- 9 g~ + - 2g, sin (ms -pt) We eliminate T from the equations of motion (4) and obtain a S+ 1 - 1 - = po p2 ( + 1O2p2) ( /p) cos (ms -pt), a -s ( ), +- - p0oplVcos (ms -pt), or, on substituting for Ni and.2, l 2 (m - ) + mg2 -+ = poOP2 (1 + m2p?) W,) A P, /J? _ l............ (1 2 ). OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE. 369 6. We may now substitute for gq and g, in terrns of V and W and obtain by elimination of V and W the equation for p2 IÀ + 1m + A1 < _+ ) ( A )(~ i2p2)( CM2p2 A [(2 +s2p2 +C?,2p2) + p2 2 + m2 + - +) -(i- (i C2p = ~o.........(13) If m2n- - - does not vanish this equation can be written p2 {1 -( + o) } -(1 + 1) X2 (1 - /- - K2)2 8K (2 + K- K1-)=O............(14), by putting /CK=A/Cm2p2, ic, = l/m2p, = 1/,n2o-2) x = A - (1 - -2 p p2 -4.......................... Since A/C =1 + 7, where?v is the Poisson's ratio of the material, CK0 - K1 = 77K, and this is always positive; so that, if for x in the left-hand member of the above equation (14) we substitute the values oo, l/(l+eK), 1/(1 + K), 0, the expression has the signs + - -+, and thus one of the two values of x exceeds l/(l + i) and the other is less than 1/(1 + - o), both values being positive. It follows that there are two possible velocities for waves of given length, the speed of one exceeding r A mp2 (qn2- _ 1/p2 _ 1/U2)2 2 Lpoo 1 + m2p J and that of the other being less than A m2p2 (m2 - i/p - l/o-2)22] Lpo ~1+,+fm2p j' these two expressions become the speeds of the corresponding waves round a circular ring by writing n for mp and omitting 1/o-. The left-hand member of the equation (14) for x breaks up into factors rational in Ko, K1, K2 if [(2 + Ko + K1) (1 - l - fK)2 + SK2 (c2 + K - K)] - 4 (1 + K0) (1 + K1) (1 - KC1- -K_4 is the square of a rational function of K0, K1, K,. This is the case when_.2 + K/ - K = 0, or when 1- Kc - K = 0, or when K = 0, or when 1- I + 2, = O, for in the last case the expression becomes 16 (1 - K,)2 [(1-,) (KC- )- 2 (K + 1)]2. Of these cases the first cannot happen since Ko > Ki, and the third is the limiting case in which the helix becomes a circle; the two remaining cases will be discussed later. VOL. XVIII. 47 370 PROF. LOVE, THE PROPAGATION OF WAVES 7. With the view of determining the character of the motion corresponding to one value of p2 we observe that by the second of (12) combined with the first of (10) g{ _(+ mpA (m2 — +f 2) +g l + 1- + 2p2) -........(16), where x is given by (15), and it has been assumed that mnL- 1/p2 - I/o-2 does not vanish. Hence we find /,( 1 1) 2m 22 =g (m2-p + -) + g2 -and therefore = {î ~2P V(mî +4)} r {n2{ - ( ),lp.}/(.-.-L)...(17). It follows that in the wave for which x (1 +Ko)< 1 we must have + 2 P Vm2+2)<- (18) Again, by combining the first of (12) with the second of (10) we find g- L(j 1- )+)(l+(Ko)(l+~ +K)]+g2 (m2- 2+ ~) {-= (+)}0 (19). Hence + 2l-m+o) 2m ( and therefore 1-æ (1+~0=- pW + -1 mW p2 ~*7 p2-2) 2g 2+ (~!2 -+ I( 1/( in lN (v~Y 2 1 1 2 =-{(-2 + (m-p2)} 1 {2+ ( C- 1) mp}/ (1 +Ko)( --- -2)....(20). It follows that in the wave for which x (1 + K) > 1 we must have / a — >- <0.....................w........... ). T2p +- m X (m _.( + 2) + Kim2-.2) >.... (21). The two inequalities (18) and (21) are not mutually exclusive for all values of The to inekialiIes (w a2 (T2 U r p2utal ecuiv o alvluso OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE. 371 V/ W, but in the present case V and W are not independent. The equation connecting them is obtained by eliminating x from (17) and (20) in the form (Ko-K) (mr2 — 2)%= (2 +Â/pî) {4m2 (i * p2) [1+ -Po (m2 - + ) 2 = 12_ 1V + 1 + 2p( 1 ) Po- W p2 0 olCr+p2J or 2 {V2- W2 ( + 2p2)} (m2- 2 + -wv -P 2m;p2 m /C — (m =......(22), which gives two values for V/W, having a negative product, and thus showing that in the two waves the values of V/W have opposite signs. We now substitute for V/W the values +~o0, o, p m2 _+, _ 2_,m2_ 1+ 1, placing these values in order of decreasing algebraic magnitude. For shortness we write - (m2 1 + 1 2 2 -P+a......................... (23), (n7 p2) p2 1/2) J and then a >/ according as (m2-l/p2+1/o-2)<0. There are three cases depending on the signs of 2 -l /p2 and of m2 _ l/p2 + 1/o2. In any case when we substitute V/ W =, the left-hand inember of (22) becomes p_2_ _ 2 \2 2( P22l2 r -a2p + A!C - 1 - v p-2 - 1 + o 2 and when we substitute V/W==/ the left-hand member of (22) becomes -4W2 Mo2( +A/C) ( 1 /,2 _ 1 1 ~*" 2 2Y+AIC- 1 P2 0'/ p/ 2 1\ Now in the slower wave we have W 1-a-< 0, which shows that if m2' - l/p2+ 1/o2 is positive 0 > V/W>a, and if m2 - 1/p2 + 1/o- is negative a> V/W > 0. In the quicker wave we have (w2,2 - /p2 + 1/o-2) (-Ç- f) >0. 47-2 372 PROF. LOVE, THE PROPAGATION OF WAVES Thus when m2 - l/p2+l/o-2 and n2 - 1/p2 are both positive we have V/W> /3, when m2 - 1/p2 + 1/o2 is positive and m2- 1/p2 is negative V/W> /3>0, when m2 - l/p2 + I/o-2 and m2 - llp2 are both negative O >/3 > / W. When m2 - 1/p2 and m2 - 1/p2 + 1/o2 are both positive we obtain after substitution in the left-hand member of (22) the signs shown in the table V/W = oo 0 / a -oo + - -+ + When m2 - 1/p2 is negative and m2 - l/p2 + l/o2 is positive we obtain V/W = oo ( O a -o + - - + + When m2- /p2 and m2 - 1/p2+ 1/o-2 are both negative we obtain V/W = oo a O 8 -oo -_ - + - By comparison of these results we see that when m2 - 1/p2 + 1/o-2 is positive V/W is positive in the quicker wave and negative in the slower one, but when mn2 - 1/p2 + 1/2 is negative the reverse is the case. When V/W is positive the displacements v and w are in the same phase at all points of the helix, and when V/W is negative these displacements are everywhere in opposite phases. 8. If the helix of angle a is wound on a cylinder of radius a the displacement parallel to the axis is a sec a (w/o- + v/p), and the displacement parallel to the circular section is a sec a (w/p-v v/), and the wave is in a certain sense right-handed or lefthanded according as (W/p -V/a) (W/l + V/p) is positive or negative. We write e for this, and then the values of: in the two waves satisfy the equation 2 [(1/p- /)2 - (1 + r2p2) (e/p + I/o)2] (m2 - /lp2 + 1/0o2) (8 / [P A/C- 1 (2 1 1) 2 (p O- (p U )I- [p 9mnp2 +- 4/C -1 p2 -2).and the two waves will be respectively right-handed and left-handed if the roots have *opposite signs. To show that this is not always the case it is sufficient to take rz very great and substitute for: in the left-hand member the values O - p/o- - o; the signs are - + -, OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE. 373 showing that both values of e are negative, and both waves are left-handed in this sense when m is sufficiently great. A similar method may be applied to show that there are values of m for which both values of /3/w have the same sign, and thus the waves are not respectively righthanded and left-handed in regard to 3 and w. 9. We have already noted that in three special cases the equation (13) for p2 can be solved rationally in terms of m, p, a. Taking n2 = 1/p2 + 1/o2 = a-2 cos2 a, it is convenient to put ms= 0, and then 0 is the angle turned through by the radius of the helix about the axis of the cylinder in passing along the curve from the point frorn which the arc is measured to the point at which the arc is s. In this case equations (12) become ( l \ 2P poo2 (1 ( 1+ g2- ) ~AP (l+m2p2)W, (g + g2m) - -PP V, and equations (8) and (9) become gl (+ Cmp = V+ W, g2-(V so that mY2 + gl/o = { 2 + (A} v+ Tr P) and thus either V+ Wp/o = O and p=, or else V= W (1 + m2p2), P ans Po~~P2(l +m2p) 4m(2 + )/ (2 + * +p) The second kind of motion is an example of the quicker wave, and the speed p is given by,2 8A (p2 + o2)2 (A + ()) 2 + 2Cp2 1 ~ p0o p2a4 (A + C) 2 + Cp2 (22 + p2) In the displacement for which p=0 the equation V+ Wp/- =O shows that there is no displacement parallel to the axis of the helix; we also have W cos a - V sin a = W (cos a + sin a tan a) = W sec a, 374 PROF. LOVE, THE PROPAGATION OF WAVES, ETC. and thus the displacement along the tangent to the circular section of the cylinder is W sec a cos; the displacement along the radius vector outwards is -a sec2 a a- cos a Wsin ms, or W sec a sin 0, and thus the displacement is W sec a at right angles to the plane from which 0 is measured. The helix is displaced bodily, and there is no deformation. 10. Again, taking m2= llp2 - 1/o2, where a2 is supposed > p2 or a< 47r, we find that equations (16) and (19) show that either _ A (m2 - l/p2- 1/a2) 1 - m2p2 po p1 + A/Cm2p2 1 + m2p2' or g. =, p=2 (m2 _ 1/p2 - 1/-2)2 Sp poco m2p2 The motion for which g2= 0 is an example of the slower wave, the speed p of this wave is given by 2 A 4 p2(o_ p2) p po o4 (22 _ p2) {(1 + A/C) 2 - p2}' and the flexural couple G2 in the osculating plane and the displacement v along the binormal both vanish at all points of the helix. The motion for which g,= O is an example of the quicker wave, the speed p of this wave is given by 2 A 4 p po O 04 o-2 - p2' and the flexural couple G6 about the principal normal, the torsional couple H, and the displacements u and w along the principal normal and the tangent all vanish at all points of the helix. XX. On the Construction of a Model showing the 27 lines on a Cubic Surface. By H. M. TAYLOR, M.A., F.R.S. [Received 27 January 1900.] THE general equation of a cubic surface contains 19 constants: 9 conditions are required to make it pass through a given plane section: 6 more are required to make it pass through a second: 3 more to make it pass through a third. It follows that a cubic surface would be determined by three plane sections and one point on the surface. Any data which determine the surface necessarily determine the straight lines on the surface. It is known that twenty-seven straight lines lie on the general surface of the third degree, and that these lie by threes in forty-five planes, the triple tangent planes to 45 x 32 x 22 the surface. There are 2 x3 sets of three triple tangent planes, no two of which pass through the sane line*. There would be no loss of generality in the form of the cubic surface caused by choosing arbitrarily one of the 5280 sets of three triple tangent planes instead of three ordinary plane sections: among these 5280 sets there are 240 sets such that a second set passes through the same nine lines. If ABC, A'B'C', A"B"C" be the triangles formed by the three planes of such a set, the letters may be arranged in such a manner that BCB'C'B"C", CAC'A'C"A", ABA'B'A"B" are planes. In this paper and in the model, of which a representation is given (Plates XXIV., XXV.), each of the twenty-seven lines on the surface is denoted by one of the numbers 1, 2, 3,....., 27 in agreement with a notation adopted in a former paper*. In accordance with this notation, the lines in these three planes are denoted by Arabie numbers as follows:BC, 1 B'C', 6 B"C", 15 CA, 2 C'A', 4 C"A", 12 AB, 3 A'B', 5 A"B", 7 For convenience of reference a complete list of all the triple tangent planes of the surface, showing those in which each line appears, is given in the following table:* Philosophical Transactions of the Royal Society, Series A, Vol. 185 (1894), p. 64. 376 MR TAYLOR, THE CONSTRUCTION OF A MODEL Table showing the triple tangenzt planes which pass through each line on the surface. 1, 2, 3 1, 6, 151, 1 11 1, 16,19 1, 17, 18 2,1, 3 2, 4, 12 2, 8, 14 2, 20,23 2,21, 22 3,1, 2 3, 5, 7 3,10, 13 3, 24,27 3,25, 26 4,2, 12 4, 5, 6 4, 9, 13 4, 16,27 4,17, 26 5,3, 7 5, 4, 6 5,11, 14 5, 18,21 5,19, 20 6, 1, 15 6, 4, 5 6, 8, 10 6, 22, 25 6, 23, 24 7,3, 5 7, 8, 9 7,12, 15 7, 16,23 7,17, 22 8,2, 14 8, 6, 10 8, 7, 9 8, 18,27 8,19, 26 9,1,11 9, 4, 13 9, 7, 8 9, 20,25 9,21, 24 10, 3, 13 10, 6, 10, 11 12 10, 16, 21 10, 17, 20 11, 1, 9 11, 5, 14 11, 10, 12 11, 22,27 11, 23, 26 12, 2, 4 12, 7, 15 12, 10, 11 12, 18,25 12, 19, 24 13, 3, 10 13, 4, 9 13, 14, 15 13, 18, 23 13, 19, 22 14, 2, 8 14, 5, 1 4,13, 15 14, 16, 25 14, 17, 24 15, 1, 6 15, 7, 1 15, 13, 14 15, 20, 27 15, 21, 26 16, 1, 19 16, 4, 27 16, 7, 23 16, 10, 21 16, 14, 25 17, 1, 18 17, 4, 26 17, 7, 22 17, 10, 20 17, 14, 24 18, 1, 17 18, 5, 21 18, 8, 27 18, 12, 25 18, 13, 23 19, 1, 16 19, 5, 20 19, 8, 26 19, 12, 24 19, 13, 22 20, 2, 23 20, 5, 19 20, 9, 25 20, 10, 17 20, 15, 27 21, 2, 22 21, 5, 18 21, 9, 24 21, 0 10,6 21, 15, 26 22,2, 21 22, 6, 25 22, 7, 17 22 11, 27 22, 13, 19 23, 2, 20 23, 6, 24 23, 7, 16 23, 11, 26 23, 13, 18 24, 3, 27 24, 6, 23 24, 9, 21 24, 12, 19 24, 14, 17 25, 3, 26 25, 6, 22 25, 9, 20 25, 12, 18 25, 14, 16 26, 3, 25 26, 4, 17 26, 8,19 26, 11, 23 26, 15, 21 27, 3, 24 27, 4, 16 27, 8, 18 27, 11, 22 27, 15, 20 SHOWING THE 27 LINES ON A CUBIC SURFACE. 377 In the model the six lines, forming the sides of the triangles ABC, A'B'C', are drawn on the surface of two brass plates which are carefully hinged together in such a manner that the straight line XYZ, which passes through the intersections of the pairs BC, B'C'; CA, C'A'; AB, A'B', is in the line of the hinges. Each of the remaining twenty-one straight lines is represented by a stretched string. On each plate the point at which any straight line cuts the plate is marked by the Arabic number which denotes the line. In the explanation, where it is necessary to distinguish between the points where any line, say 9, cuts the two plates, the point where it cuts a side of the triangle ABC, in the left-hand figure, will be denoted by 91, and the point where it cuts a side of the triangle A'B'C', in the right-hand figure, will be denoted by 9,.. It will be observed that the lines 7112l151, 7,12,15,., in which the sides of the triangle formed by the lines 7, 12, 15 cut the sides of the triangles ABC, A'B'C', meet on the line XYZ. We have now chosen three plane sections of the cubic surface, and we have one more condition at our disposal. This is exhausted by the choice of the point 81, that is, the point where the line 8, which cuts the three non-intersecting straight lines 2, 6, 7, cuts the line 2. This determines the line 8, and therefore the point 8,.. As the lines 7, 8, 9 are complanar the straight line 7iZ8 cuts BC in 91 and cuts the line XYZ in a point such that the straight line joining it to the point 7, gives the points 8,., 9,.. In a similar way 41 and 91 give 13/ 61, 8,, 101,,, 91,, 112, 2,,, 8,, 14,. Since 10, 11, 12, and 13, 14, 15 form triangles, 101 and 121 give 111 11,. and 12,. give 10, 13,, 15l, 14 14,., 15,.,, 13,. Lines 1 to 15 are now determined. The remaining lines 16 to 27 form a double six. Any triple tangent plane which passes through one of these twelve lines passes through two of them, and also through one of the lines 1 to 15. We must, therefore, adopt a different method to find one of the lines 16 to 27. One of themii must be found by some quadratic method, and then all the rest can be found as before. The line 17 was found by a method of trial and error from the facts that 171 lies on BC and 17, on C'A', and that the pairs of Unes 7117E, 7,17,. and 141171, VOL. XVIII. 48 378 MR TAYLOR, THE CONSTRUCTION OF A MODEL 14r17r meet on the line X YZ. All the other points were then obtained by drawing straight lines in the following order, in which the suffixes are omitted because the description applies equally both to the left-hand and to the right-hand figures. 7, 17 give 22 10, 17,, 20 9, 20,, 25 12, 25,, 18 13, 18,, 23 13, 22,, 19 12, 19, 24 9, 24, 21 15, 21,, 26 15, 20, 27 10, 21,, 16 All the lines on the surface are now fully determined. The diagrams represent not only the lines used in finding the points, but each diagram gives the 32 straight lines which represent the intersections with the plane of the triangle of each of the 32 triple tangent planes that do not pass through a side of the triangle. From these 32 straight lines a selection of 8 lines can be made to pass through all the 24 points in the diagram. This selection of 8 lines can be made in 40 ways. The following numbers give such a set of eight straight lines for the lefthand figure (triangle ABC):6, 4, 5; 15, 12, 7; 9, 20, 25; 11, 23, 26; 16, 21, 10; 19, 22, 13; 17, 14, 24; 18, 8, 27. It may be noticed that in the left-hand figure the points 4...15 lie by threes on eight straight lines, two of the lines passing through each point. From these eight lines two sets of four can be chosen passing through all the twelve points. We cannot draw a pair of conics through the twelve points. Of the remaining twelve points, 16...27, no three lie on a straight line, but eight conics can be drawn, each passing through six points, and there are four pairs of conics passing through all the twelve points. There are also 48 conics, each of which passes through 6 points in the diagram, and 12 of which pass through each of the 24 points, and from these 48 conics a selection of 4 can be made so as to pass through ail the 24 points; such a selection can be made in 168 ways. The following numbers give one such set of four conics for the left-hand figure:17, 4, 5, 18, 12, 7; 6, 20, 25, 15, 23, 26; 9, 21, 10, 11, 22, 13; 16, 14, 24, 19, 8, 27. SHOWING THE 27 LINES ON A CUBIC SURFACE. 379 Let us consider the section of the surface made by a plane passing through one of the lines; for instance, the line 1. We shall find five pairs of points, 2, 3; 6, 15; 9, 11; 16, 19; 17, 18, on this line and the other sixteen points will lie on a conic. In this case there are 40 straight lines, each of which passes through three of the points. Through each point on the conic 5 of the lines pass, and through each point on the line 4 lines pass. Next, let us consider a section of the surface not passing through a line. It will be a cubic curve and the points on it where the 27 lines cut the plane lie by threes on 45 straight lines, five straight lines passing through each point. From these 45 straight lines a selection of 9 can be made, to pass through all the points. This selection can be made in 200 ways. There are, also, 360 conics, each of which passes through six of the points, 80 conics passing through each point. From these 360 conics a selection of four can be made to pass through all the points except three lying on a straight line. This selection can be made in 168 ways for each particular set of three points, that is in 7560 ways altogether. 48- 2 XXI. On the Dynctmics of a Systemn of Electrons or Ions: ctnd on the Influence of a Magnetic Fielcl on optical Phenomena. By J. LARMOR, M.A., F.R.S., Fellow of St John's College. [Received 24 January 1900.] THE DYNAMICS OF A SYSTEM OF INTERACTING ELECTRONS OR IONS. 1. IN the usual electrodynamic units the kinetic and potential energies of a region of aether are given by =T (87)-lfa2 + / + 2) d, W= 27r2(f2 +g2 + h2) d, wherein &r represents an element of volume, (a, /3, 7) is the magnetic force which specifies the kinetic disturbance, and (f, g, h) is the aethereal 'displacement' which is of the nature of elastic strain. These two vector quantities cannot of course be independent of each other: the constitutive relation between them is, with the present -nits, fdy di3 da dy d/3 da\ d \dy dz' dz~ dx' dx -dy) =4 dt(f> g h), or say curi (a, /, ry)= 47r (f, g, h), which restricts (f, g, h) to be a stream vector satisfying the equation of continuity: it also confirms the view that (a, /, y) is of the nature of a time-fluxion or velocity. It is assumed that (a, /3, y) is itself a stream vector, which must be the case if electric waves are of wholly transverse type. On substituting in these expressions (:, v, ), the independent variable or coordinate of position, of which (a, /3, y) is the velocity, so that (a, e, y)= d/dt (I, r, ^), the dynamical equations of the free aether can be directly deduced from the Action formula (T- W) dt = 0. It is well known that they are identical with MacCullagh's equations for the optical aether, and represent vibratory disturbance propagated by transverse waves. MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: ETC. 381 It will now be postulated that the origin of all such aethereal disturbances consists in the motion of electrons, an electron being defined as a singular point or nucleus of converging intrinsic strain in the aether, such for example as the regions of intrinsic strain in unannealed glass whose existence is revealed by polarized light, but differing in that the electron will be taken to be freely mobile throughout the medium. For all existing problems it suffices to consider the nucleus of the electron as occupying so small a space that it may be taken to be a point, having an electric charge e associated with it whose value is the divergence of (f, g, h), that is, the aggregate normal displacement f(lf+~mgg+nh) dS through any surface S enclosing the electron: over any surface not enclosing electrons this integral of course vanishes, by the stream character of the vector involved in it. Faraday's laws of electrolysis give a substantial basis for the view that the value of e is numerically the same for all electrons, but may be positive or negative. As our main dynamical problem is not the propagation of disturbances in the aether, but is the interactions of the electrons which originate these disturbances, it will be necessary to express the kinetic and potential energies of the aether as far as possible in terms of the motions and positions of the electrons. The reduction of T may be effected by introducing the auxiliary variable (F, G, H) defined by curl (F, G, H) =(a,, y). dH dG\ dF dH dG dF } Thus T (7)-l (d(d dH 13+ ( G _ 'Y dT Jdy Cy dz d dx \dx dy f( ' d/7 da dy d/i dus = (87r)-ij, w, Z dS-+ G -Z dx + H Ydx - d _ a, /3, y Now it follows from the definition of (F, G, H) that d (dF dG dHI d 3 d dV2F — + -+ j - - =r I - ~- - dx (dx dy dz dy dz) (df dt' with two similar equations. Solutions of these equations can be at once obtained by taking dF/dx + dG/dy + dl/dz to be null: this makes F, G, H the potentials of volume distributions throughout the medium of densities f h g, h, together with contributions as yet undetermined from the singular points or electrons. The most general possible solution adds to this one a part (F0, Go, H0) which is the gradient of an arbitrary 382 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: function of position X: but this part does not affect the value of (a, f/, y) through which (F, G, H) has been introduced into the problem, so that the definite particular solution is all that is required. Now the motions of the electrons involve discontinuities, or rather singularities, in this scheme of functions. One mode of dealing with them would involve cutting each electron out of the region of our analysis by a surface closely surrounding it. But a more practicable method can be adopted. The movement of an electron e from A to an adjacent point B is equivalent to the removal of a nucleus of outward radial displacement from A and the establishment of an equal one at B: in other words it involves a transfer of displacement in the medium by flow out of the point B into the point A: now this transfer can be equally produced, on account of the stream character of the displacement, by a constrained transfer of an equal amount e of displacement directly from A to B. Hence as regards the dynamics of the surrounding aether, the motion of such a singular point or electron is equivalent to a constrained flow of aethereal displacement along its path. The advantage of thus replacing it will be great on other grounds: instead of an uncompleted flow starting from B and ending at A, there will now be a continuous stream from B through the surrounding aether to A and back again along the direct line from A to B: in other words the displacement will be strictly a stream vector, and in passing on later to the theory of a distribution of electrons considered as a volume density of electricity, the strictly circuital character of the electric displacement, when thus supplemented by the flow of the electrons, will be a feature of the analysis. For greater precision, let us avoid for the moment the limiting idea of a pointsingularity at which the functions become infinite. An electron will now appear as an extremely small volume in the aether possessing a proportionately great density p of electric charge. Its motion will at each instant be represented by an electric flux of intensity p(&, y, z) distributed throughout this volume, which when added to the aethereal displacement now produces a continuous circuital aggregate. For present purposes for which the electron is treated as a point and the translatory velocities of its parts are very great compared with their rotational velocities, this continuous flow may be condensed into an aggregate flux of intensity e (x, y, i), concentrated at the point (x, y, z). At each point in the free aether, outside such nuclei of electrons, the original specification of magnetic force, namely that its curl is equal to 47rd/dt of the aethereal displacement, remains strictly valid. It has been seen that the effect of the motion of any specified electron, as regards the surrounding aether, is identical with the effect of an impressed change in the stream of aethereal displacement at the place where it is situated: thus the interactions between this electron and the aether will be correctly determined by treating its motion as such an impressed change of displacement. This transformation however considers the nucleus as an aggregate: it will not be available as regards the interactions between different parts of the nucleus: thus in the energy function constructed by means of it, all terms involving interaction between the electron as a whole and the aether which transmits the influence of other electrons will be AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 383 involved; but the intrinsic or constitutive energy of the electron itself, that is the total mutual energy of the constituent parts of the electron exclusive of the energy involved in its motion as a whole through the aether, will not be included: this latter part is in fact supposed (on ample grounds) to be unchangeable as regards all the phenomena now under discussion, the nuclei of the electrons being taken to occupy a volume extremely small in comparison with that of the surrounding aether*. This principle leads to an expression for the force acting on each individual moving electron, which is what is wanted for our present purpose. But the equations of ordinary electrodynamic theory belong to a dense distribution of ions treated by continuous analysis, and we have there to employ the averaged equations that will obtain for an effective element of volume of the aether containing a number of electrons that practically is indefinitely great. We derive then the equations of the aether considered as containing electrons from those of the uniform aether itself by adding to the changing aethereal displacement (f y, h) the flux of the electrons of type e (, y, i) wherever electrons occur. In the transformed expression for T we can, as already explained, treat the part of the surface integral belonging to the surface cutting an electron out of the region of integration (as well as any energy inside that surface) as intrinsic energy of the electron, of unchanging amount, which is not concerned in the phenomena because it does not involve the state of any other electron. The contribution from the surface integral over the infinite sphere we can take to be zero if we assume that all the disturbances of electrons are in a finite region: the truth of this physical axiom can of course be directly verified. We have therefore generally T= f(Fu + Gv + Hw) dr, wherein (F, G, H)= f(, v, w) r- d: and in these expressions the total electric current (uî, v, w) will consist of a continuous part (f, g, h) which is not electric flow at all, and a discrete electric flux or true current of amount e (X, y, Z) for any electron e. When the electrons are considered as forming a volume density of electrification, this latter will be considered as continuous true electric flow constituted as an aggregate of all the different types of conduction current, convection current, polarization current, etc. that can be recognized in the phenomena, each being connected by an experimental constitutive relation with the electric force which originates it. The orbital motions of the electrons in the molecule cannot however be thus included in an electric flux, but must be averaged separately as magnetization. Neither the true current nor the aethereal displacement current taken separately need satisfy the * For a treatment on somewhat different lines cf. Pihil. Trans. 1897 A, or 'Aether and Matter,' Ch. vi., Camb. Univ. Press, 1900. t It may be formally verified, after the manner of the formula for T in ~ 2, that this amount tends to a definite limit as the surface surrounds the electron more and more closely. 384 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: condition of being a stream, but their sum, the total current of Maxwell, always satisfies this condition. 2. The present problem being that of the interactions of individual electrons transmitted through the aether, it will be necessary to retain these electrons as distinct entities. The value of (F, G, H) at any point is therefore of type F= 1l df dT exe F=Ji dT~, in which r represents the distance of the point from the element of volume in the integral and from the electron respectively. Thus 2T = o- ((i2 + giY2 + hih2) r12- dTdT2 + e f/2 ri2- 1 dr2 + >ey g r~2-1 dr2 + fez hi2r2-l dr2 + Z.elei (, 2 + j1Y2 + Z 2) r12-, in which each pair of electrons occurs only once in the double summation. Also W = 2rc2 f(2 + g2 + h2) dr. In omitting the intrinsic energy of an electron and only taking into account the energy terms arising from the interaction of its electric flux with the other electric fluxes in the field, we have however neglected a definite amount of kinetic energy arising from the motion of the strain-configuration constituting the electron and proportional to the square of its velocity: this will be the translational kinetic energy To= Le2 (î2 + y2 + z2) or we may write T,= '2 7( 2+ y2+ 2\) where m is thus the coefficient of inertia or 'mass' of the electron, which may either be wholly of electric origin or may contain elements arising from other sources. This transformation has introduced the positions of the electrons and the aetherstrain (f, g, h) as independent variables. It is necessary, for the dynamical analysis, thus to take the aether-strain as the independent variable, instead of the coordinate of which (a, 3, 7) is the velocity, which at first sight appears simpler. For part of this strain is the intrinsic strain around the electrons; and the deformations of the medium by which it may be considered to have been primordially produced must have involved the discontinuous processes required to fix the strain in the medium, as otherwise it could not be permanent or intrinsic. If the latter coordinates were adopted AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHIENOMENA. 385 the complete specification of the deformation of the medium must include these processes of primary creation of the electrons, and the medium would have to be dissected in order to reveal the discontinuities, after the manner of a Riemann surface in functiontheory*. 3. We have now to apply dynamical principles to the specification of the energies of the medium thus obtained. The question arises as to what are dynamical principles. It may reasonably be said that an answer for the dynamics of known systems constituted of ordinary matter is superfiuous, as the Laws of Motion formulated by Newton practically cover the case. Waiving for the present the question whether the foundations of that subject are so simple as may appear, the present case is one not of ordinary matter but of a medium unknown to direct observation: and its disturbance is expressed in terms of vectors as to the kinematic nature of which we have here abstained from making any hypothesis. Now the dynamics of material systems was systematized by Lagrange in 1760 into equations which amount to the single variational formula f(T- W)dt=O, in which the variation is to be taken subject to constant time of passage from the initial to the final configuration, and subject to whatever relations, involved in the constitution of the system, there may be connecting the variables when these are not mutually independent,-the only restriction being that these latter relations are really constitutive, and so do not involve the actual velocities of the motion although they may involve the time. This equation is known to include the whole of the dynamics of material systems in the most general and condensed manner that is possible. It will now be introduced as a hypothesis that the cognate equation is the complete expression of the dynamics of the ultra-material systems here under consideration. Even in the case of ordinary dynamics it can be held that there is no final resting-place in the effort towards exact formulation of dynamical phenomena, short of this Action principle: in our present more general sphere of operations the very meaning of a dynamical principle must be that it is a deduction from the Action principle. This attitude will not be uncongenial to the school of physicists which recognizes in dynamical science only the shortest and most compact specification of the actual course of events. We have then to apply the Principle of Action to the present case. In the first place the coordinates in terms of which T and W are expressed are not all independent, for when the distribution of (f, g, h) is given that of the electrons is involved. The connexion between them is completely specified by the relation I(df gd dh\, J \dx dy dzj * More concretely, the relation curl (a, (3, y) =4wr (fj, hg, ) kind whose velocity is (a, fi, y), that are required to introinvolves f(lf+mg+nh) dS=O: now (lf(f mg + n+h) dS is not duce the existing intrinsic strain must involve discontinuous zero but is equal to Se: hence the displacements, of the processes. Cf.' Aether and Matter,' Appendix E. VOL. XVIII. 49 386 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: provided this is supposed to hold for every domain of integration, great or small, it will follow that the electrons are the poles of a circuital or stream vector (f, g, h). If then we write fJ2 (dfd +d dh'\ dT- eI, J kdx dy dz) the variational equation will by Lagrange's method assume the form 8 f(T+ To- W+ n)dt=0 in which * is a function of position, initially undetermined but finally to be determined so as to satisfy the above condition restricting the independence of the coordinates. We have to vary this equation with respect to the displacement (f, g, h) belonging to each element of the aether, supposed on our theory to be effectively at rest, and with respect to the position (x, y, z) of each electron. All these variations being now treated as independent, the coefficient of each of them rnust vanish, at all points of the aether and for all electrons involved in it. We now proceed to the variation. Bearing in mind that so far as regards aethereal displacement l i jFdir involves ffif r -d cdrT, that is iY/ifrriç1 T1-jT72, because each pair of elements appear together twice ini the double integral of a product, but only once in a double summation, we obtain as the terms involving f in the complete variation S dt Ffdr - 47rc2 dt fSfdr + dt Ij dfdr, leading, through the usual integration by parts, to f Fsfdr - dt FVfdT - 47T2 jdt fif d1r +\ dt fcfdIydz - fdt d Sfdr. The coefficient of Sf must vanish in the volume integral, giving dF dF P 4vr 'f-= _ d _t dx......................................... Similar expressions hold for g and h. Again, the terms in the variation involving the electron e at (x, y, z) are S dte (xF + yG + zH) + mS (fdt (2 + y2 + Z2) - S fdte', yielding as regards variation of the position of this electron fdte (Fèc + GSy + Hsz + SF+y SG + z8H) + m dt (A&a + ySy + zi) -fdteST in which Ux means the change of the velocity of the electron, so that we have on integration by parts F, ^ H7T d 'D F/2^ DG^ DH^\ 'dF dF dF e FdSx + G3y + -dte DF Sx + - dG y + -DE z - x d x + d- y + d-z dz +... + ~m 3x + Yoy +z tz - m' Jdt ({x& + Y3y + zz)- dte (- x + dt y z+ d z jdx dy dz AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 387 where DF/dt must represent the rate of change of F at the electron as it moves, namely DF dF dF dF dF - - + Xd+YdY+Zd dt dt dx dy dz The vanishing of the coefficient of &x for each element of volume gives m /=e DF d dF dG dX d) (x =e- dt +x-7+Y -d+ZJ(ii). =e(.-/ d7F dc\. =e(.... dt dx). Similar expressions hold good for my and mz. The form of W shows that 47r-C2 is the coefficient of aethereal elasticity corresponding to the type of displacement (f, g, h): the right-hand sides of equations (i) are therefore the expressions for the components of the forcive (P', Q', R') inducing aethereal displacement: thus this force, which will be called the aethereal force, is given by equations of type p. dF d' dt dx The form of equation (ii) shows that the right-hand side is the component of the force e (P, Q, R) inducing movement of an electron e: this force reckoned per unit electric charge is called the electric force (P, Q, R) and is given by p _j - _ dt dx or, in terms of physical quantities only, by P= yY- /Z + 4qra2f. We do not now go into the case of a magnetically polarized material system, for which in certain connexions* (a, b, c) replaces (c, f9, y) in this formula. These expressions for the aethereal force and the electric force, together with a complete specification of the electric current and the experimentally determined constitutive relations of the medium, form the foundation of the whole of electrical theory. MOTION IN AN IMPRESSED MAGNETIC FIELD. When the electrons or ions constituting a molecule describe their orbital motions in a uniform magnetic field (~a, 3o,, y,), its influence is represented by an addition to the vector potential (F, G, H) of the term (oY - oZ, az - yx, ox - coy). * Cf. Ioc. cit. ante. 49 2 388 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: W (~)+ G) + HÎ2)a Thus To (e+ T (2+y z2) + 2e 1(F f+ G + h) d + e Y + 1 f h d. x y z x y z ao 3o /o o7o /o yo As the aether is stagnant, so that the position of the element of volume 8r is fixed, these new terms will not modify the formula for the aethereal force (P', Q', R') unless the impressed magnetic field varies with the time: but they will modify the electric forces acting on the ions by the addition of the term (7oY-/oZ, aoz - o, /3oX - ao). THE SYSTEM REFERRED TO A ROTATING FRAME. It is part of the Action principle, of which the validity is at the foundation of this analysis, that its formal expression is not affected by constitutive relations involving the time explicitly, provided they do not involve the velocities of the actual motion. Let then the system be referred to axes of coordinates rotating with angular velocity (oex, Wy, wez) measured with reference to their instantaneous positions, these quantities being either constant or assigned functions of the time. For the velocity, instead of (x, y, z) there must now be substituted, in the formula for T- W, (x - ywz + ZOy, Y - zox + Xzo, z - Xwy + yOx), and for (J, g, h) there must be substituted (f- goz + hwy, g - hwx -+fOz, h -fcwy + gox), while (x, y, z) remain unchanged. Referred to these moving axes the kinetic energy, which was, so far as it involves the ion e1 (d1, 2Y, z), given by To + T = m, (2 + yl2 + z,2) + el (FF,X, + Gl, + HiZ) +..., where (F1, G1, H1) is the value of the vector potential at the point (x1, y,, zl), has now additional terms which on neglecting the square of the angular velocity are - n,1 n /, il + el xi y, z, + e (3'li l + 9y1'G1 + iSl'H1), X1 yi Z1 F1 G, Hl CIOx Cy CWZ X o Oy oz wherein 'F, - e2- +, r,2 jr12j - E e2(WOyZ 2- Wy) + _h2 - dr. r12 rJ 2 The exact dynamical equations referred to moving axes may now be directly obtained by application of the Action principle. AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 389 As regards the electron el, the first of these terms is the same as that due to an impressed magnetic field given by ( = - 2m\ (a,', p, 7o)=- e (yx, y, z). el The others give rise to terms in the electric forces which are small compared with the internal electrodynamic forces of the system itself when the angular velocity is small: and in. our applications these latter will be themselves negligible compared with the electrostatic forces. MUTUAL FORCES OF ELECTRONS. When a system of electrons or ions is moving in any manner, with velocities of an order lower than that of radiation, the surrounding aether-strain may be taken as at each instant in an equilibrium conformation: thus the positional forces between the electrons are simply their mutual electrostatic attractions. As regards kinetic effects, the disturbance in the aether can be considered as determined by the motion of the electrons at the time considered, so that the kinetic energy can be expressed entirely in terms of the motions of the electrons; and the motional forces between two of them are derived in the Lagrangian manner from the term in this total kinetic energy ele2r~2-l ( 1&~ + yY1 + l02) + elve,2v2d2r1l2/ds, ds2, where ds1, ds2 are elements of their paths described with velocities vL, v2. The Weberian theory of moving electric particles involves on the other hand a kinetic energy term ele2r2-1 (drl2/dt)2: in the field of the electrodynamics of ordinary currents it however yields equivalent results as regards mechanical force, and the electromotive force induced round a circuit, though not as regards the electric force at a point. THE ZEEMAN EFFECT. 4. On the hypothesis that a molecule is constituted of a system of revolving ions, a magnetic field H impressed in a direction (1, m, n) adds to the force acting on an ion of effective mass m and charge e, situated at the point (x, y, z), the term eH (ny - mz, 1 - nx, mx - ly), so that its dynamical equations are modified by change of Y, y, z into x - (ny - nz), y - c (l - nx), z- E (nx - ly), where K = eH/m, e being in electromagnetic units. If the ratio e/m is the same for all the ions concerned in the motion, so is K, and this alteration of the dynamical equations of the molecule will be, to the first order of K, the saine as would arise from a rotation of the axes of coordinates to which the system is referred, with angular velocity -K around the axis of the impressed magnetic field. Hence the alteration produced in the orbital motions is simply equivalent to a rotation, equal and opposite to this, imposed on the whole system. Each line in the spectrum would thus split up into two lines consisting of radiations circularly polarized around the direction of the magnetic field, and with difference of frequencies constant all along the spectrum, namely Kc/27r, together with a third line polarized so that 390 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: its electric vibration is along the same axis while the frequency is unaltered. In fact each Fourier vibration of an ion, which previously consisted of a component disturbance of the type of an elliptic harmonic motion, is no longer of harmonic type when the precessional rotation cK is imposed on it-this precession being imposed additively on the different constituents of the total motion: but it can be resolved into a rectilinear vibration parallel to the axis, and two circular ones around it, each of which maintains its harmonic type after the rotation is impressed and thus corresponds to a spectral line, and which are differently modified as stated. These three spectral lines would be expected to be of about equal intensities *. It is however essential to this simple state of affairs that the charges belonging to all the ions that are in orbital motion under their mutual influences should be of the sane sign, as otherwise e/m could not be the same for all. It is also essential that the ions of opposite sign, or the other centres of attraction under which the orbits are described, should be carried round as well as the orbits with this small angular velocity -/c in so far as they are not symmetrical with regard to its axis. If we admit the hypothesis that the effective masses of these positive ions, or other bodies to which the negative ions are attracted, are large compared with those of the negative ions themselves, this state of superposed uniform rotation of the whole system may still be expected to practically ensue from the imposition of the magnetic field. For under the action of the mutual constitutive forces in the molecule, the orbital motions of the larger masses will take place with smaller velocities. As the additional forces introduced by the magnetic field are proportional to the velocities, they will thus also be smaller for the positive ions. Let us then suppose these larger masses to be constrained to the above exact uniform rotation, with angular velocity o', along with the negative ions, and find the order of magnitude of the forces that must be impressed on them in order to maintain this constraint. The motion of the negative ions will, as has been seen, be entirely free, the forces due to the magnetic field exactly sufficing to induce the additional rotational motion. As regards a positive ion of effective mass mn, the radial and transversal forces, in the plane perpendicular to the axis of the magnetic field, that are required to maintain the motion will be altered from n (-rw2) and m d(r2o) r dt m d to n {r -r (co + o')2} and T dt jr2 ( + o)}. Thus, o' being small compared with w, the new forces required will be -2mrwco' and m d (r2ow); r dt whereas the force arising from the magnetic field acting on an ion moving with velocity v is 2mvo' at right angles to its path. These two systems of forces are for each ion of the same order of magnitude: thus the forces required to maintain the imposed * For more detailed statement, cf. Plai;. g., Dec. 1897. AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 391 uniform rotation in the case of the massive positive ions are small compared with the magnetic part of the forces acting on the negative ions. If these maintaining forces are absent, the system can still be regarded as a molecule in its undisturbed motional configuration rotating with uniform angular velocity, but subject to disturbing forces equal and opposite to those required to thus maintain it. Now this undisturbed motional configuration is a stable one: thus the effect of these slight disturbing forces is to modify it, but to an extent much smaller than the uniform rotation induced by the magnetic field. Our proposition is thus extended to a molecule consisting of an interacting system, constituted of equal negative ions together with much more massive positive ions, and also if so demanded of other massive sources of attraction. It would however be wrong to consider each negative electron as describing an independent elliptic orbit of its own, unaffected by the mutual attractions exerted between it and the other moving negative electrons: for the attractions between ions constitute the main part, if not the whole, of the forces of chemical affinity. But without requiring any knowledge of the constitution of the molecular orbital system, the Zeeman triplication of the lines, with equal intervals of frequencies for each line, will hold good wherever the conditions here stated obtain. It appears from the observations that the difference of frequencies of the components magnetically separated is not constant for all lines of the spectrum: so that this simple state of affairs does not hold in the molecule. The difference of frequencies seems however to be sensibly constant for those lines of any element which belong to the same series, as well as for those lines of homologous elements which belong to corresponding series; a result which cannot fail to be fundamental as regards the dynamical structure of molecules, and which supports the suggestion that inr a general way the lines of the same series arise from the motions of the same ion or ionic group in the molecule, executed under similar conditions. The directions of the circular polarizations of the constituent lines were shown by Zeeman to be in general such as would correspond in this kind of way to the motions of a system of negative ions in a steady field of force. It remains to be considered whether we are right in thus taking the stresses transmitted between the electrons, through the aether, as those arising from the configuration of the electrons alone, and in neglecting altogether the motional forces between them. The former assumption is equivalent to taking the strain in the surrounding aether to be at each instant in an equilibrium state: this will be legitimate, because an aethereal disturbance will travel over about 103 diameters of the molecule in one of the periods concerned, —the error is in fact of order 10-6. The motional forces between two electrons are of type, as regards one of them, ( d d\ -x- + Y,2 + Jz2+ d$ -v 12 (dt dx, dlxj el2 ( ri2 2 v1v2 dseds2) To obtain a notion of orders of magnitude, let us consider the special case of two electrons + e, -e describing circular orbits round each other with radius r. Then qmv2/r=c e2/r2, * Preston, Phil. Mag., Feb. 1899. 392 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: while Zeeman's measurements give e/m= 107: thus v2= ~c2e2/mr, so that, taking r to be 10-8, e= 10-21, -we obtain v = 10-3c; thus the orbital period cornes out just of the order of the periods of ordinary light, which is an independent indication that the general trend of this way of representing the phenomena is legitimate. With these orders of magnitude, the ternis in the motional forces between two electrons are of orders ele2i/r, ele2.c2/r2 as compared with their statical attraction of order 02ee2/r2 and the forces arising from the impressed magnetic field H of order edHI; the ratios are thus of the order of 10-6 to 1 to 3.10-9E. Thus when H exceeds 103, the forces of the impressed magnetic field are more important than the motional forces between the ions; and in all cases the effects arising from these two causes are so small that they can be taken as independent and simply additive. THE ZEEMAN EFFECT OF GYROSTATIC TYPE. 5. Sensible damping of the vibrations of the molecule owing to radiation cannot actually cone into account, because the sharpness and fixity of position of the spectral lines show that the vibrations subsist for a large number of periods without sensible change of type. In fact it has been seen above that the motion of the system of electrons, on the most general hypothesis, is determined by the principle of Action in the form (T'- W) dt = 0 where T = m (2 + y2 + 2) + Y z y x y z I m n thus it cones under the sarne class as the motion of a dynamical system involving latent constant cyclic momenta, the Lagrangian function for such a system, as modified through the elimination of the velocities corresponding to these nomenta by Routh, Kelvin, and von Helmholtz, being of this type. The influence of the impressed magnetic field is thus of the same character as that of gyrostatic quality imposed on a free system: and the problem cornes under the general dynamical theory of the vibrations of cyclic systems*'. In the special case above considered of massive positive ions, we can thus assert that the motion relative to the moving axes is the same as the actual motion of the system with its period altered through slight gyrostatic attachments to these positive ions. It is moreover known from the general theory of cyclic systems that each free period is either wholly real or else a pure imaginary, whenever the unmodified system is stable so that its potential energy is essentially positive: thus on no view can a magnetic field do anything towards extinguishing or shortening the duration of the free vibrations of the molecule, it only modifies their periods and introduces differences of phase between the various coordinates into the principal modes of vibration of the system. In the general case when ai is not the same for each ion in an independently vibrating group in the molecule, the simple solution in terms of a bodily rotation fails, and it might * Cf. Thomson and Tait, Nat. Phil., Ed. 2, Part I. pp. 370-416. AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 393 be anticipated that the equation of the free periods would involve the orientation of the molecule with regard to the magnetic field. But if that were so, these periods would not be definite, and instead of a sharp magnetic resolution of each optical line there would be only broadening with the same general features of polarization. To that extent the phenomenon was in fact anticipated frorn theory, except as regards its magnitude. The definite resolution of the lines is however an addition to what would have been predicted on an adequate theory, and thus furnishes a clue towards molecular structure. A POSSIBLE ORIGIN OF SERIES OF DOUBLE LINES. The definiteness and constancy in the mode of decomposition of a molecule into atoms shows that these atoms remain separate structures when combined under their mutual influence in the molecule, instead of being fused together. Each of them will therefore preserve its free periods of vibrations, slightly modified however by the proximity of the other one. For the case of a molecule containing two identical atoms revolving at a distance large compared with their own dimensions, each of these identical periods would be doubled*: thus the series of lines belonging to the atom would become double lines in the spectrum of the molecule. It has been remarked that the series in the spectra of inactive elements like argon and helium consist of single lines, those of univalent elements such as the sodium group where the molecule consists of two atoms, of double lines, while those of elements of higher valency appear usually as triple lines. In other words, a diad molecule consists of the two atoms rotating round each other with but slight disturbance of the internal constitution of each of them. Their vibrations relative to a system of axes of reference rotating along with them will thus be but slightly modified: relative to axes fixed in space there must be compounded with each vibration the effect of the rotation, which may be either right-handed or left-handed with respect to the atom: thus on the same principles as above each line will be doubled. If the lines of a spectral series are assumed to belong to a definite atom in the molecule, those of a molecule consisting of two such atoms would thus be a system of double lines with intervals equidistant all along the series, but in this case without definite polarizations. But if the constituents of the double lines of a series were thus two modifications of the same modes of the simpler atomic system, it would follow that they should be similarly affected by a magnetic field. This is not always the case, so that * In illustration of the way this can come about, revolution is different, and each single undisturbed period consider two parallel cylindrical vortex columns of finite becomes two adjacent disturbed periods. Analogous consection in steady rotation round each other. Each by siderations apply to the interaction of the two atoms of the itself has a system of free periods for crispations running molecule, rotating round each other. round its section: when one of them is rotating round the According however to Smithells, Dawson, and Wilson, other, the velocity of the crispations which travel in the Phil. Trans. 1899 A, it is the molecule of sodium that gives direction of rotation is different from the velocity of those out the yellow light, that of sodium chloride not being that travel in the opposite direction: thus the period of effective. VOL. XVIII. 50 394 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: this kind of explanation cannot be of universal application: it would be interesting to ascertain whether the Zeeman effect is the same for the two sets of constituents of a double series such that the difference of frequencies is the same all along it. At any rate, uniformity in the Zeeman effect along a series of lines is evidence that they are all connected with the same vibrating group: identity of the effect on the two constituents of a doublet is evidence, as Preston pointed out, that these belong to modifications of the same type of vibration. NATURE OF MAGNETIZATION. 6. The proposition above given determines the changes in the periods of the vibrations of the molecule in the circumstances there defined. But it is not to be inferred from it that the imposition of the magnetic field merely superposes a slight uniform precessional motion on the previously existing orbital system. That orbital system will be itself slightly modified in the transition. For instance, in the ideal case of the magnetic field being imposed instantaneously, the velocities of all the electrons in the system will be continuous through that instant: hence the new orbital system on which the precession is imposed will be the one corresponding to velocities in that configuration which are equal to the actual velocities diminished by those connected with the precessional motion. On the usual explanation of paramagnetic induction, the steady orbital motion of each electron is replaced by the uniform electric current circulating round the orbit which represents the averaged effect: the circuit of this current is supposed to be rigid so that the averaged forcive acting on it is a steady torque tending to turn it across the imposed magnetic field. This mode of representation must however à priori be incomplete: for example it would make the coefficient of magnetization per molecule in a gas increase markedly with length of free molecular path and therefore with fall of density, because this torque would have the longer time to orientate the molecule before the next encounter took place. It appears from the above that the true effect of the imposed magnetic field is not a continued orientation of the orbits but only a slight change in the orbital system, which is proportional to the field; and in the simple circumstances above discussed is made up of a precessional effect of paramagnetic type, accompanied by a modification of the orbital system which is generally of diamagnetic type, both presumably of the same order of magnitude and thus very small. The recognition of this mode of action of the magnetic field also avoids another discrepancy. If the field acted by orientating the molecules it must induce dielectric polarization as well as magnetic: for each molecule has its own averaged electric moment, as revealed by piezoelectric phenomena, and regular orientation would accumulate the effects of these moments which would otherwise be mutually destructive. But there is nothing either in the disturbance of the free orbital system into a slightly different free system, or in the precession imposed on that new systemi-nor in a more general kind of action of the same type,-which can introduce electric polarization. AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 395 The polarization of a dielectric medium by an imposed electric field is effected in a cognate manner. The electric force slightly modifies the orbital system by exerting opposite forces on the positive and negative ions. In this case these forces are independent of the velocities or masses of the ions. The fact that the polarization is proportional to the inducing field shows that the influence produced by the field on the orbital system is always a slight one. Yet the numerical value of the coefficient of electric polarization is always considerable, in contrast with the very small value of the magnetic coefficient; which arises from the very great intrinsic electric polarity of the molecule, due to the magnitude of the electric charge e of an ion. Taking the effective molecular diameter as of the order 10-8 cm., there will be 1024 molecules per unit volume in a solid or liquid, and the aggregate of their intrinsic electric polarities may be as high as 1024. 10- ee electrostatic units, where ec is 3. 10-11. Now the moment of polarization per unit volume for an inducing field F is (K - 1) F/87r; thus even for very strong fields this involves very slight change in the orbital configuration. A similar remark applies to the polarization induced by mechanical pressure in dielectric crystals. It would be unreasonable to expect any aggregate rotational effect around an axis, such as constitutes magnetization, from the polarizing action of an electric field; in fact if it were present, reversal of the direction of the field could not affect its total amount considered as arising from molecules orientated in all directions. The possibilities as regards the aggregate intrinsic magnetic polarities of all the molecules are of the same high order, viz. eAn/T, where A is the area and r the period of a molecular orbit, which is elnv or 10-5v per cubic centimetre, where v is the velocity in a molecular orbit whose linear dimension I is 10-8. Thus the superior limit of the magnetization if the molecules were all completely orientated would be of the order 10-5v, which is large enough to include even the case of iron if v were as much as one per cent. of the velocity of radiation. In the case of iron a marked discrepancy exists between the enormous Faraday optical effect of a very thin sheet in a magnetic field on the one hand, and the slight Zeeman effect of the radiating molecule, as also the absence of peculiarity in optical reflexion from iron, and the absence of special influence on Hertzian waves, on the other: which must be in relation with the circumstance that at a moderately high temperature the iron loses its intense magnetic quality and comes into line with other kinds of matter. This suggests the explanation that the magnetization of iron at ordinary temperatures depends essentially on retentiveness, owing to facility possessed by groups of molecules for hanging together when once they are put into a new configuration. This is the well-known explanation of the phenomena of hysteresis, which can be effectively dirninished by mechanical disturbance of the mass. In soft iron the magnetic cohesion would be less strong and more plastic, and thus readily shaken down by slight disturbance in the presence of a demagnetizing field, so that retentiveness would not be prominent. It is conceivable that the primary effect of an inducing field is to slightly magnetize the different molecules: that then the molecules thus altered change their condition of aggregation, and so are retained mutually in new positions independently of the field, 50- 396 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: the effect persisting if the field is gently removed: that the field can then act afresh on the molecules thus newly aggregated: and so on by a sort of regenerative process, the inducing field and the retentiveness mutually reinforcing each other, until large polarizations are reached before it comes to a limit. For hard iron these accommodations take place more rapidly than for soft iron, when the field is weak, and thus are of sensibly elastic character over a wider range: cf. Ewing, Magnetic Induction, 1892, ch. vi. ON THE ORIGIN OF MAGNETO-OPTIC ROTATION. 7. The Faraday magneto-optic rotation is obviously connected, through the theory of dispersion, with the different alterations of the free periods of right-handed and left-handed vibrational modes of the molecules, that are produced by the impressed magnetic field. The ascertained law (infra) that the mean of the velocities of the two kinds of wave-trains is equal to that of the unaltered radiation, shows that the phenomenon in fact arises wholly from this difference, and is not accompanied by temporary structural change in the molecule such as would involve alteration of the physical constants of the medium. The general relation connecting the refractive index,/ of a transparent medium with the frequencies (pi, p2,... p^)/27r of the principal free vibrations of its molecules, which are so great that radiation travels over 103 molecular diameters in one period, is of type 2-1 = Ar = 5,~2 + 2 p2 _. pr2 in which A, is a constant which is a measure of the importance, as regards dispersion, of the free principal period 27r/Pr. The quantity on the right-hand side of this equation, of form f(p2), is a function of the averaged configuration of the molecule relative to the aethereal wave-train that is passing over it. Now consider a circular wave-train, say a right-handed one, passing along the direction of the magnetic field: on the hypothesis that the spectrum consists of a single series of lines for all of which /c is the same, the influence of this train on the corresponding right-handed vibrations that it excites in the molecule will be to superadd a rotation of the molecule as a whole with angular velocity 2Ic. This will modify the configuration of the vibrating system relative to the circular wave-train passing over it in the saine way as if an equal and opposite angular velocity were instead imparted to the wave-train. Thus the actual effect of the magnetic field on the light will be the same as would be that of a change in the frequency of the light from pl/2rr to p/27r+ c/47r, the latter terni arising from this imposed angular velocity: the value of the magneto-optic effect may therefore in such a case be derived from inspection of a table of the ordinary dispersion of the medium. The velocity of propagation of the train of circular waves will, on this hypothesis, be derived by writing p- K or p + c for p according as the train is right-handed or left-handed, thus giving when K2 is neglected, /_2 12 _ 2 Ar 2 + 2 -p p - 2. AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 397 For the case when there is only a single free period this result coincides with FitzGerald's formula (Roy. Soc. Proc. 1898), which has been shown by him to give the actual order of magnitude for a Faraday effect as thus deduced from the Zeeman effect. If we were to consider that each system, of lines in the spectrum arises from an independently vibrating group of ions in the molecule, as (supra) there may be some temptation to do, then the value of (2 - 1)/(/2 + 2) in this formula would be obtained by addition of the effects of these independent groups: thus if the value of the Zeeman effect were known for each line of the spectrum of any substance, and the law of dispersion of the substance were known, the Faraday effect could be deduced by calculation. To our order of approximation we should have _ _- 1 ~ E + K4rArP 2 + ) (p2-p2)2* the circumstance that the mean of the velocities of propagation is unaltered points to the A coefficients being unaffected by the magnetism, thus suggesting absence of change in the mean conformation, as already remarked. For the case in which the free periods that effectively control the dispersion all belong to the same series of spectral lines, so that K is the same for all of them, the formula for the dispersion need not come into the argument. The influence of the impressed magnetic field on the index of refraction of circularly polarized light is then the same as the change of p to p + 'KC according as the polarization is left-handed or righthanded. Because that influence is equivalent to rotation of the optically vibrating molecule with angular velocity Kc, the molecule will now be related in the same way to a wavetrain with angular velocity p ~ i/c as it was previously to one with angular velocity p. dV Thus light corresponding to angular velocity p is now propagated with velocity V + - dp instead of V. Now if X be the wave-length in a vacuum and /u the refractive index, we have V= c//p, p = 27re/X: and the rotation of a plane of polarization for a length I of the medium, being ~p multiplied by the difference of times of transit, is (l/YI,-l/V,). 27ro/X, which is 7rlc. 6V/V2X, where SV= /cdV/dp = 2 d-l/dx~-, so that the result is 2-X dx' This expression, - X d-, for the coefficient of magnetic rotation as a function of the wave-length, has been given by H. Becquerel* and shown by him to be in good agreement with actual values as regards order of magnitude, and also with Verdet's detailed observations along the spectrum in the cases of carbon disulphide and creosote. The restriction on which it is here based, namely that the dispersion is controlled by free periods for all of which the Zeeman constant is the same, can be neglected for the case of the anomalous dispersion close to an absorption band, because there the dispersion * Comptes Rendus, Nov. 1897: it was based on the assumption that the magnetic field involves rotation of the aether with velocity tK. 398 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: is controlled by that band alone*: thus the Faraday effect is there very large and of anoinalous character, in correspondence with the experimental discovery of Macaluso and Corbino. From another aspect of the same effect, we can conclude that light of any given period, very near a natural free period of the medium, will travel in it with sensibly different velocities according as its mode of vibration corresponds to one or other of two principal types, elliptically (or in a special case circularly) polarized in opposite directions, and thus will exhibit phenomena of double refraction. THE INFLUENCE OF ROTATIONAL TERMS ON OPTICAL PROPAGATION. 8. The purely formal, i.e. non-molecular, theory of the magnetic influence on optical propagation may be developed in a simple and direct manner, by use of the device of a revolving coordinate-system as above employed. In a non-magnetizable medium the exact relations connecting the magnetic force (a, /3, y), the electric force (P, Q, R), and the electric current (u, v, w), are of types dy d8 dR dQ da _ - _ = 47f-7u, - dy dz dy dz dt -Thus V2p c d /dP dQ dR) du Thus v2- - — h+ + --- =4,7r dx \dx dy dz dt' which will lead to the differential equations of the propagation when in it (u, v, w) is expressed in terms of (P, Q, R) by means of the constitutive relation connecting them. Now for the aethereal elastic displacement we have (f, g, h) =(47rc2)- (P, Q, R). To determine the nature of the most general formal connexion between the material polarization (f', g', h') and the electric force, that we are at liberty to assume without implying perpetual motions, we must make use of the method of energy. The energy of this electric polarization in any region is W = f(Pf + Qg + Rh') dr, where &r is an element of volume: thus its intensity per unit volume is a quadratic function of (P, Q, R), and possibly also of d/dt (P, Q, R) and of the spacial gradient of (P, Q, R), and it may be of gradients of higher orders as well: if the first time-gradients alone are included we thus have the expression F2 (P, Q, R) + a,,PdP/dt +... + alPdQ/dt + a2, QdP/dt +..., F2 denoting a quadratic function. The variation of this energy must from the definition of (P, Q, R) as the force moving the electrons, be W= (Pf ' Q' + t RQ' îh) dr = S f(Pf' + Qg' + Rh') dr- f'P + g'Q + h'SR) dr, * Cf. Proc. Camlb. Phil. Soc., Mar. 1899: for similar who, by introducing dispersional terms of a certain simple explanations but restricted to anomalous dispersion, cf. type including a frictional part into the equations of optical Macaluso and Corbino, Rend. Lincei, Feb. 1899. propagation in a rotational medium, finds that each abReference should also be made to the converse pro- sorption line is tripled, but with an asymmetry introduced cedure of Voigt (cf. Anlalen der Physik I. 1900, p. 390), by the frictional term. AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 399 so that, transposing, s W = f fP + g'îQ + h'SR) dr, in which the independent variable is now (P, Q, R). On conducting the variation in the usual manner, and reducing from d;P/dt to S1' by partial integration with respect to time (such as necessarily enters in the reduction of the fundamental dynamical equation of Action) this leads to a relation of type, dF2 a3 dQ a2 dR dP 47rC' dt 4vT2 dt' where (a1, a2, a3)/4rc2 = (a - a, a - a-3, a, - a.,). When the system is referred to its principal dielectric axes, F - = K - 1 + K- 1 Q2 + KS- 1 R2' 87rC2 87rC2 87rC2 This analysis shows that rotational quality in the relation connecting (f', g', h') and (P, Q, R) can come in through terms in the energy function that involve the time-gradients: or, as may be shown in a similar manner, it may enter through terms involving the space-gradients: but not otherwise. The latter terms introduce rotational quality of the structural type, with which we are not now concerned. The former terms lead to the magnetic type of rotation, here related to the vector (ai, a2, a3), which must be determined by the impressed magnetic field or other exciting cause of vector character: the existence of such mixed terms, involving (P, Q, R) and d/dt (P, Q, R), in fact adds to the polarization a part at right angles to d/dt (P, Q, R) and to this vector (ai, a3, a3), and equal to their vector product divided by 47rc2, which is in all cases entirely of rotational character. Terms of the form of a quadratic function of the gradients of (P, Q, R) by themselves would merely modify the form of the function F2 so that its coefficients depend in part on the period of the vibration, that is, they would be merged in optical dispersion of the ordinary type. The question also arises whether the ordinary dielectric constants, namely the coefficients of the function F2 (P, Q, R), are sensibly altered by an impressed magnetic field. This point can be settled by aid of the principle of reversal. When the electric force and the impressed magnetic field and the time are all reversed, the effect on the induced electric polarity must be simple reversal: hence a reversal of the magnetic field cannot affect the coefficients in F2(P, Q, R): hence these coefficients must depend on the square or other even power of the impressed magnetic field: but the rotational terms depending on its first power are actually very small, therefore any terms depending on its second power are wholly negligible. This is in accord with Mascart's experimental result. The right-hand sides of the equations of propagation in the material medium, as above indicated, can thus, for light of period 27r/p, be expressed in the form?3-2 ( p -2 d2P dQ dR p20- lp-2 dt -a3 dt + a dt Kpd Q dR dP d2R dP dQ\ K2p- a - + a3 d p- a+t)' dt3a1dt+ dt ' dtP 400 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: In the case of an isotropic medium for which K1, K2, K3 are each equal to K, these equations of vibration can be restored to their normal form, when the square of the magnetic effect is neglected, by employing a coordinate system rotating with angular velocity K'-lp2(al, a2, a3). Thus the effect of the impressed magnetic field is that the vibrations of the electric force, propagated as if that field were absent, are at the same time carried on by a motion of uniform rotation around its axis: so also, in virtue of the second of the above circuital relations, are the vibrations of the magnetic force. The electric force is not exactly on the wave-front because under the magnetic conditions it is not exactly circuital: the magnetic force is exactly on the wave-front. Thus we have the direct result that a plane-polarized train of electric vibrations, of wave-length X, travelling along the direction of the impressed magnetic field H, is rotated around its direction of propagation through an angle proportional to EH/KX2 per unit time, so that the rotational coefficient per unit distance is proportional to EHl/K22, where e is itself affected by dispersion and is thus to a slight extent a function of the wave-length. When the wave-train is not travelling in the direction of the magnetic field, it is the component of H along the normal to the wave-front that is effective: the other component of the rotation, around an axis in the plane of the wave-front, then gradually deflects the front so as to produce curvature of the rays, but so excessively slight as to be of no account. The magnetic effect is thus a purely rotational one whatever be the direction of the wave-train with respect to the field: and the phenomena in an isotropic medium may be completely described kinematically on that basis. When the medium is crystalline, its rotational quality is mixed up with its double refraction: yet in ordinary crystals the differences between K1, K2, K3 are slight, so that the phenomena are still approximately represented by each permanent wave-train, polarized in the manner corresponding to its direction of propagation, rotated around that direction with velocity proportional to the cosine of the angle it makes with an axis which need not now be the axis of the impressed magnetic field. This direct method of exhibiting the nature of the effects may also be applied to the case of structural rotation, in which by an argument similar to the above, but dealing with energy-terms involving space-gradients of the electric force, we obtain for the material medium a constitutive relation of type dQ dR 4rc (f', g', h' )= (KP + a3 d - a2 d ' dR dP dP dQ K2Q + a - -a3 d, K3R + a2- a- d-, doc c~d-dy dx when the principal axes of the rotational quality coincide with those of the ordinary dielectric quality. For a plane wave-train travelling in the direction (1, m, n), for which (P. Q, R) cc exp t -2x, (lx + ~ny + nz- Vt), p = 2r V/', V=cK 2~ AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 401 this may be expressed in the form 4a2(f, g', h')=- (KP1 - d2P + na3 dQ mao dR ) + V dt V dt ' ' so that, when K1i, K2, K3 are each equal to K, the equations of propagation are reducible to the normal form for a non-rotational medium by imparting to the coordinate axes a velocity of rotation 2r20K -X-2(lal, mna3), which implies a coefficient of rotation of a plane-polarized wave equal per unit distance to 27r2K-1X-2(la1, ma2, na,) where X is the wave-length in vacuum. This is the law of rotation for wave-trains travelling in various directions in a simply refracting medium with aeolotropic rotational quality. This law also applies approximately to crystals such as quartz, inasmuch as the difference between the principal refractive indices is not considerable: in quartz the vector (ai, a,, a3) must by symmetry coincide with the axis of symmetry of the crystal: thus the coefficient of the effective component, that normal to the wave-front, of the imposed rotation for a wave-train that travels in a direction making an angle 0 with that axis is proportional to cos2 0, not to cos 0 as in the magnetic case. In this case the rotational effect is superposed on the double refraction, so that a plane-polarized wave instead of being simply rotated will acquire varying elliptic polarization: it is however a simple problem in kinematics* to determine the types and the velocities of the two elliptically polarized wave-trains that will be propagated without change of form under the two influences, each supposed slight. It appears from this discussion that magneto-optic rotation is a phenomenon of kinetic origin, related to the free periods of the molecules and not at all to their mean polarization under the action of steady electric force: it is therefore entirely of dispersional character. Again the intrinsic optical rotation of isotropic chiral media is represented by a constitutive relation of type f K P+ (dQ dR f,rC2 dz dy)' showing that the rotational term is proportional to the time-gradient of the magnetic field this effect would therefore be entirely absent in statical circumstances, and only appears sensibly in vibratory motion of very high frequency. In this case no physical account of the origin of the term has been forthcoming: we have to be content with the knowledge that the form here stated is the only one that is admissible in accordance with the principles of dynamics. As the rotatory power, of both types, is thus connected with the dispersion as well as the density of the material, it is not strange that attempts, experimental and theoretical, to obtain a simple connexion with the density alone, have not led to satisfactory results. The existence of a definite rotational constant for each active substance hbas formed the main experimental resource in the advance of stereochemical theory: but the present considerations prepare us for the fact that no definite relations connecting rotational power with constitution have been fonnd to exist,-that the quality, though definite, is so to speak a slight and accidental one, or rather one not directly expressible in terms either of crystalline structure or of the main constitutive relations with which chemistry can deal. * Cf. Gouy, Joutrln. de Phys., 1885; Lefebvre, loc. cit., 1892; O. Wiener, Wied. Ann., 1888. VOL. XVIII. 51 402 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: GENERAL VIBRATING SYSTEM IN WHICH THE PRINCIPAL MODES ARE CIRCULAR. 9. We are entitled to assert, on the basis of Fourier's theorem, that any orbital motion which exactly repeats itself with a definite period can be resolved into constituent simple elliptic oscillations whose periods are equal to its own and submultiples thereof. Such a motion would therefore correspond to a fundamental spectral line and its system of harmonics. The ascertained absence of harmonics in actual spectra shows either that the period corresponding to the steady orbit is outside the optical range, or else that the steady motion emits very little radiation as in fact its steadiness demands. The radiation would then arise from the various independent modes of disturbance, each of elliptic type on account of the absence of harmonies, that are superposed on the steady orbital motion. To ascertain the nature of the polarization of the vibrations when in a magnetic field, we have first to decompose each orbital motion into its harmonic constituents, which are elliptic oscillations: each of the latter can be resolved into a linear oscillation parallel to the axis of the magnetic field, another at right angles to it, and a circular oscillation around it; and of these the second linear oscillation can be resolved into two equal circular oscillations in different senses around it. Now when the uniform rotation around the axis is superposed on the components they all continue to be of the requisite simple harmonic type, but the periods of the two circillar species,-which as has been seen are of amplitudes different as regards the various molecules but equal in the aggregate,-become different: they are the three Zeeman components. Nothing short of complete circular polarization of the constituent vibrations of permanent type in each molecule will account for the corriplete circular polarization of each of the flanking Zeeman lines. If these vibrations were only elliptical, but propagated with different velocities according to the sense in which the orbit is described, each would be equivalent to a circular vibration together with a linear one: and as the total illumination is the sum of the contributions from the independent molecules, the circularly polarized light would then be accompanied by unpolarized light of the same order of intensity. This restriction of type of vibration suggests the employment in the analysis of variables each of which corresponds to a circular vibration, as do the:, Vî variables in what follows. For simplicity let us take the axis of z parallel to the impressed magnetic field, and let (X, Y, Z) represent the statical forces transmitted by aether-strain from the other ions in the molecule to a specified one. The equations of motion of that ion are m (x -y) = X, m(y + K) = Y, m = Z. We now make no assumption with regard to the magnitude of the electric charges and effective masses of the various ions, which may differ in any manner. In this ion let us change the variables to I:= x + Y, rY]= x- ty, AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 403 so that 2x= + 77, 2ty= —, d d d d d d and therefore 2 = -_ -, 2- =d +- d; da dx dy dc dx dy the equations become m (+ + u4)=X LY, m (r - )= X - Y, mz =Z. If therefore X + Y is a function only of the e coordinates of the electrons, and X -Y a function only of the q7 coordinates, and Z only of the z coordinates, these groups of coordinates will be determined from three independent systems of equations. On our hypothesis of ions moving with velocities of an order below that of radiation, the mutual forces acting on them are derived from a potential energy function: thus (X, Y, Z)=-k (d d d W dx' dy' dz where k may be supposed to vary from one ion to another, being equal to the electric charge when the mutual forces are considered to be wholly of electric origin. Then 1 2k dTW 1i 21c dW (X+ tY) =, - -(X -Y)... m mv dr ' m m d: The solution of the coinplete system of equations, three for each ion, will in any case involve the expression of e, r7, z for each ion as a sum of harmonic terms of the form eLPt each with a complex numerical coefficient; but when the coefficients of one of them are assigned those of the others are determined. The vibration for each ion is thus compounded of a systein of elliptic harmonic motions of definite forms and phases. Their components in the plane ~, 7 will be circular vibrations only when the t and V coordinates vary independently of each other, that is when dW/drq is a function of the t coordinates of the ions alone and dW/de a function of the q coordinates alone. This condition can only be satisfied, W being real, when it is a linear function of z2 and of products of the form g.r, or rs: it may thus be any quadratic function of the coordinates which is invariant in forni as regards rotation of the axes of x, y around the axis of z. Under these circumstances the free periods for e coordinates, 77 coordinates, and z coordinates will all be independent, and either real or pure imaginary*: in an actual molecule they will be real. For example a permanent vibration of | type will be represented by:r = >AreLPt+Lar ar being chosen so that Ar is real: thus. = Ar cos (p,.t + ar), y, = SA, sin (pt + ar) representing a series of right-handed circular vibrations, each series having definite phases and also amplitudes in definite ratios for the various ions. Again for the 7 type we have 7lr = CBr.e''t+s t, * Routh, Essay on Stability, 1887, p. 78; Dynamics, vol. n., ~ 319. 51 2 404 MR LARMOR, ON THE DYNAMICS 0F A SYSTEM OF ELECTRONS OR IONS: so that X=r = BrCOs (qrt + /3y), yr = - ZBr sin (q.t + /r), which represents similarly a series.of left-handed circular vibrations. The vibrations of z type will of course be linear in form. Thus supposing the effective masses and charges of the various ions to be entirely arbitrary, the effect of an impressed magnetic field will be to triple the periods and polarize the constituents in the Zeeman manner, provided the potential energy of the mutual forces of the ions is any quadratic function of the coordinates of the vibrations which satisfies the condition of being invariant in form with respect to rotation of the axes of coordinates around the axis of the magnetic field. The essential difference between the type of this system and that of the one previously considered will appear when the latter is derived on the lines of the present procedure. The equations are... 2k dW }- + lK,'__ _2k dW rï - r] —,IC, m d ' k dW m dz On writing:'= e'Kt, '= e- Ktr, 2.,. 2k dW they become ' + c - - 2 dW m d'' k 2 dTWI m dz The form W will be unaltered when it is expressed in terms of:', V', provided it depends only on the mutual configuration of the ions, and K is the same for all of them; hence when KA2 is negligible compared with unity, (:', V', z) are determined by the sane equations as would give (e, V7, z) on the absence of a magnetic field: and from this the previous results follow. 10. We have thus reached the following position. Let the coordinates (x, y, z) of an ion be resolved into two parts, namely (x1, yt, zj) which are known functions of the time and represent its mean or steady motion, and (x', y', z') which are the small disturbance of the steady motion constituting the optical vibrations. When this substitution is made in the dynamical equations the quantities relating to the steady motion should cancel each other, as usual; and there will remain equations, of the original form, involving (x', y', z') from which the accents may now be removed. The forces relating to these new coordinates will still be derivable from a potential energy function: AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 405 and as by hypothesis the vibrations are all 'cycloidal' or simple harmonie, this function must be homogeneous and quadratic in these coordinates. The total potential energy must be determined by the instantaneous configuration of the system, and will therefore remain of the same form when referred to new axes of coordinates. This confines the quadratic part representing the energy of the disturbance to the form given above: the vibration of each ion will then in general consist of a system of elliptic oscillations of all the various free periods, equal in number to the ions: and the effect of an impressed magnetic field will be to triple each vibration-period and to polarize the constituents in the Zeeman manner. The steady or constitutive motion of the system must be so adjusted that it does not sensibly radiate: otherwise it would gradually alter by loss of its energy. As the axis of the magnetic field may be any axis in the molecule, the function which represents the potential energy must thus be such that the vibrations resolved parallel to any axis form an independent system: hence it is confined to the form W = - _,, (X,.r'- Xs)2 + (y1' - y1)2 + (z. ' - Z,' /)2} +,Brs (x,'Xs, + yr'yS + Z/Zs), - Ars {(Rt' - es') (]r' - r7s) + (Zr - Zs')2} + IB1 (' s'4 + - s's'/ + 22v/Zs'). Thus in the absence of a magnetic field the vibrations of the x coordinates, of the y coordinates, and of the z coordinates of the ions will form independent systems of precisely similar character. It is in fact only under this condition that it is possible for the components, parallel to any plane, of the elliptic harmonic vibrational types of the various ions, to form a system of circular vibrations with common sense of rotation. If n/k = X and mrc/k= X', the equations of motion are of type dW dW ' drV d^ d The periods of the right-handed circular vibrations, of type xc eLpt, period 27r/p, will be given by the equation - p-X2 - X'p -A1,, 1 C,3, C14,... Cn = 0, C21, - x2p2-\p-x An,., 23, C24,... C0 C31, c3, - X2 - X3p - A, C34,... C3......................................................................................................... in which Cs= A,. + B,,: those of the left-handed circular vibrations by changing the sign of each X' in this equation: those of the plane-polarized vibrations, which are the natural periods of the molecule, by making X' null. On account of the great number of the constants, compared with the number of free periods, simple relations among the periods can only arise from limitations of the generality of the system. The duplication or triplication observed in the constituent Zeeman lines would on this theory arise from the presence of two or three equal roots in the period equation 406 MR LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: for natural vibrations of the system, which would be differently affected and therefore separated by the impressed magnetic field. This analysis is wide enough to apply to a system consisting of a continuous electrical distribution, whose parts are held together in their relative positions either by statical constraint or by kinetic stability: for then the potential energy still depends on the relative configurations of the elements of mass of the system. We have however not arrived at any definite representation of the dynamical system constituting a molecule, except that it consists of moving electric points either limited in number or so numerous as to form a practically continuous distribution: but reasoning from the definiteness and sharpness of the periods in the spectrum, and the facts of polarization of light, it has been inferred that the vibrations of the molecule form a 'cycloidal' system and therefore arise from a quadratic potential energy function: the total potential energy function must therefore consist of two independent parts, that belonging to the steady motion, in which the coordinates of the vibrations do not occur, and this part belonging to the disturbance which is quadratic in its coordinates: as a whole it must depend on the configuration of the system and not on the axes of coordinates, hence this quadratic part is invariant with regard to change of axes: this confines it to the form given above,-which had been found to be demanded by the existence of the Zeeman phenomena. It has thus been seen that the fact that the vibrations belonging to the Zeeman constituent lines are exactly circular, and not merely elliptic with a definite sense of rotation, requires that the right-handed and left-handed groups of vibrations shall form two independent systems: as the magnetic field may be in any direction as regards the molecule, this requires that its vibrations, when the magnetic field is absent, can be resolved into three independent systems of parallel linear vibrations directed along any three mutually rectangular axes. This again involves that an electric force acting on the molecule will induce a polarization exactly in the direction of the force, and proportional to it*: that in fact notwithstanding its numerous degrees of freedom the molecule is isotropic. Thus the source of double refraction in crystals or strained isotropic substances would reside in the aeolotropic arrangement of the molecules and not in their orientation: but there can also be an independent intrinsic electric polarity in the molecule depending on its orientation and not on the electric field, such as is indicated by piezoelectric effects in crystals. If the molecules were not thus isotropic as regards induced electric polarity, the electric vibration induced in the molecules, when a train of radiation passes across a medium such as air, would not be wholly in the wave-front. In the theory of optical dispersion the coefficients t would then be averages taken for a large number of mole* Cf. Kerr's striking result, Phil. Mag., 1895, that in velocity of propagation affected. the double refraction produced in a liquid dielectric by an t e.g. K, cl, C.,... cl', c2',... in Phil. Trans. 1897 A, electric field, it is only the vibration polarized so that its p. 238. electric vector is parallel to the electric field that has its AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 407 cules orientated in all directions, such as may be considered to exist in an effective element of volume of the medium: and this averaging would constitute the source of its isotropy. But there would remain a question as to whether, when a plane-polarized wave-train is passing, those fortuitous components of the polarization of the molecules that are not in the direction of the electric vibration of the wave-train would not send out radiation as independent sources and thus lead to extinction of the light. The definite features of polarization of the light scattered from a plane-polarized train by very minute particles or molecular aggregations seems also to suggest in a similar manner that the individual molecule is isotropic. XXII. On the Theory of Functions of several Complex Variables. By H. F. BAKER, M.A., F.R.S., Fellow of St John's College. [Received 9 February 1900.] THE present paper is primarily a reconsideration of the paper of M. Poincaré in the Acta lfMathematica, t. xxII. (1898), p. 89; and depends for its interest on the remarkable discovery of the expression of an integral function by means of the potential of the (n - 2)-fold over which the function vanishes, which is virtually contained in M. Poincaré's paper in t. in. of the Acta Mathematica (1883), pp. 105, 106. The following points of novelty may however justify its publication. (i) By means of a generalisation of the theorems of Green and Stokes, for the transformation of multiple integrals, the imaginary part of the function of the complex variables is introduced concurrently with the real part; (ii) and thereby, as would appear, the coefficients in the quadratic function used by M. Poincaré (Acta Mlath., t. xxii. p. 174) are shewn to be zero. (iii) The theory is put in connection with Kronecker's formulae (Tferke, Bd. I. p. 200), whereby it follows that the imaginary part of the logarithm of the integral function is a generalised solid-angle, just as M. Poincaré has shewn the real part to be a generalised potential. In general Kronecker's integral, unlike Cauchy's, does not represent a function of complex variables unless the (n - 1)-fold of integration is closed; in the present paper there arises a Kronecker integral which is an exception to this rule (the integral,r-1, ~~ 1.2, 17). (iv) The definite formula here given for the integral function is not limited to the case of periodic functions; though on the other hand it has not that general application which belongs to the theory of M. Poincaré's earlier paper, in the Acta Math. t. iI. In that paper there remains in the resulting formula an integral function of which the existence is proved, for which however no definite expression is given; in the present paper, in order to have a definite expression, I have hasarded a limitation which may be regarded as a generalisation of the notion of the genre of functions of one variable. This limitation arises by regarding the (nz-2)-fold integral which enters here as a generalisation of the sum which is obtainable by taking the logarithm in Weierstrass's general factor formula for an integral function of one variable. The paper is divided into two parts, of which the former contains a formal proof of a theorem constantly employed in the theory developed in Part II. MR BAKER, ON THE THEORY OF FUNCTIONS, ETC. 409 PART I. PRELIMINARY. Formal proof of the general Green-Stokes theorem. 1. In Euclidian space of n dimensions we can take near to any point P whose coordinates are (x1,..., xa) the n points P1 with coordinates (x1 + d1x,..., xS +dixn),..............................,...,............. P, with coordinates (x1 + dnX,..., x + dnx,), it being supposed that the determinant, M, of n rows and columns, whose (r, s)th element is drXs, is not zero. At each of the points P1,..., P,, we can similarly take n independent consecutive points, those at Pr being Pri, Pr2,..., Prn; at each of these points of two suffixes we can take n others of three suffixes, and so on. Making the convention that the sth satellite point of Pr, namely P,.,, is the same as the rth satellite point of P,, or Psr, or in other words that the suffixes shall be commutative, we can associate the determinant M with the 'cell' which is defined by the 2n points P, Pl,.., nP n) P12,... > Pin,)., P12... ) whose suffixes consist of all the combinations of not more than n different numbers from 1, 2,..., n. We may suppose space of n dimensions to be divided into such cells, and call the absolute value of the determinant M the element of extent of the space, denoting it by dSn. Similarly if we have in n dimensions a space of (n-r) dimensions, defined suppose by r equations fi (X1,..., f) =0, fr (Xl,..., Xj)-=0, with a certain number of inequalities, we can associate with every point P of this space ('n-r) satellite points, P1,..., P,_-., also lying in this space, the coordinates of these points being denoted by x + dkx, *..., Xn + dk k 2,..., (n-r), and with each of these (n-r) others, and so on; and so we can suppose the space of (n- r) dimensions divided into cells, each defined by 21-'' points; with each of these cells we can as before associate an element of extent for this space, which we denote by dSn_-.; this is defined as the positive square root of the sum of the squares of all the determinants of (n-r) rows and columns which can be formed frorn the matrix of n columns and (n-r) rows [dkxi, dkX2,..., dkXn, = 2,..., (n-r), or, what is the same thing, as the positive square root of the determinant of (n - r) rows and columns which is formed by multiplying this matrix into itself, row into row, in the ordinary way. VOL. XVIII. 52 410 MR BAKER, ON THE THEORY OF FUNCTIONS 2. In what follows we call the aggregate of all the points of a space of (n - r) dimensions, limited or not, an (n - r)-fold. We also use ( quantifies, called the direction cosines of the normal to the (n-r)-fold; let the (n- 1)-folds fi =,..., f= 0 be always supposed taken in the same order, given by the suffixes; let b, d, e,...., h, k be any r of the numbers 1, 2,..., n, no two of them equal; then the ratio of the Jacobian T ( fi, **,f r) Jb,d,..., ) Jb,...,hk a (xb, x..,k) to the positive square root of the sum of the squares of all the possible () such Jacobians is denoted by Kb,d,e,...,h,k, and is one of the direction cosines in question; we suppose in general the suffixes taken in their natural ascending order; from each of the (n) direction cosines - 1 others can be formed by permutation of the suffixes, every interchange of two suffixes causing a change in sign in the direction cosine. We have then the following theorem: Suppose that a finite portion of the (non-singular) (n - r + l)-fold given by fi= 0,...,/ f = 0, is completely bounded by a closed (non-singular) (n - r)-fold given by fi=,..., fr- 0, / = 0, and that throughout the limited portion of the (n-r+ 1)-fold we have fr< 0; let P be any function of xl,..., xn for which it is supposed that itself and its first differential coefficients are finite and continuous (and single-valued) throughout the space considered; then f r( a ap aD K(-1 Cde...hk bX + (- l2 Kbe...hk a- + + Kbd...h X dSn-r+i aXb îax aXk = Kbd... h P. dSn-, wherein the second integral is taken over the complete closed (n -r)-fold, and the first integral over the enclosed portion of the (n - r+ 1)-fold; in the first integrand there are r terms, the suffix in any one of them consisting of (r-1) numbers in their natural ascending order. If we introduce (r) functions such as P, and make the rule that an interchange of two numbers of a suffix shall entail a change in the sign of the function, we can put the result in a clearer form ffXKde.....aPde.hkm - =Sfn_+l bd......hkSn-r. OF SEVERAL COMPLEX VARIABLES. 411 where on the left under the integral sign the first summation extends to every combination of (r-1) different numbers d, e,..., h, k from 1, 2,..., n, and the second summation extends to the (n-r+1) different values from 1, 2,..., n which mn can have so as not to be equal to any one of d, e,....,,; on the right under the integral sign the summation extends to every combination of r different numbers b, d, e,..., h, k from 1, 2,..., n. 3. Of this result it will be sufficient to give a proof for the case r= 3, the general case being similar. We suppose then a finite (non-singular) portion Hn- of an (n - 2)-fold, which is given by the equations fi (x,..., xn) = 0, f2 (!x,..., xn) = 0, to be bounded by a closed (non-singular) (n - 3)-fold H,_3 given by fi (i,..., xn) =O, f2 (x,..., xn)=0, f3 (Xi,..., )=. We can imagine H,_2 divided into cells in a manner before indicated, the satellite points of P, whose coordinates are (x1,..., oe), being denoted by Pk whose coordinates are (x1 + dkXi,..., X + dkX.), k =, 2,..., (n - 2). In general the differentials dkXr are arbitrary, save that the determinants of (n - 2) rows and columns formed from their matrix must not all be zero; but we shall ultimately find it convenient for our purpose to suppose that of the differentials dn-_2x, dn-2x2)... dn-2xn all but three, say all but dl-2xb, dn-2xe, dîn-2Xh, are zero; the ratios of these three will then be determined from -f dn-2zb +f dn_2Xe + -- dn-2h = 0, axb axe aXh af dn-2_ + -f dn-_2Xe + f dn- h = 0; Xb axe, aXh it is clear, in fact, that we can draw on Hn-2 through every point P a one-fold (or curve) along which all the coordinates except Xb, Xe, Xh are constant; taking then any point P and taking (n-3) of its satellite points P1,..., Pn3 arbitrarily, we can draw such a curve through P and each of P1,..., Pn3,, and take for the satellite point P,_2 a point near to P along the curve through P; we thus arrange the cells into 'strips,' each strip having (n-2) curves, such as those through P, P,..., Pn-_, as edges. 4. A set of (nz-3) neighbouring points Q1,..., Qn- in which the curves drawn on Hn-2 through P...,.. P- intersect the (n-3)-fold Hn- may then be taken as the satellite points on Hns of the neighbouring point Q in which the curve through P intersects Hn-3; we have thus a possible basis for the division of Hn- into cells, which it will later be convenient to adopt. We assume that the curve on H_-2 which is drawn 52 —2 412 MR BAKER, ON THE THEORY OF FUNCTIONS through P intersects the closed Hn-3 in an even number of points; and to shorten the proof we shall speak only of two, say Q(o) and Q(i). Then if the differentials dn_2xb, dn_2Xe, d-_2xh be always taken in the same direction along this curve Q'o)PQf(), the expression dn-2f3 = 'af dn-2Xb + '- dn-2Xe + 3' dn-2Xh aXb aXe aXh will have different signs at Q(0) and Q(i), and in fact, since f < 0 over Hn-, the expression will be positive at the point, say Q(1), where the curve through P leaves H,n2, and negative at the point Q(0), where the curve enters Hn-2. 5. Considering now any point P of Hn2, and its satellite points (x1 + dkxl,..., X( + dkX2), = 2 in regard to which we do not until special mention is made of the fact introduce the convention that all but three of the differentials (dn-_ lx *, dn-2_xn) are zero, we have f dkX +... f- dkXn = 0, É^-é 80fi' c=, 2,..., (n-2),,f2 dkx.. +. f dkx = 0, and hence easily find J12 _ tirs _ M12 - - M.S - = e say, wherein T - afi af axr' ax, af2 af2 ax,., ax and Mr, denotes a determinant of (n - 2) rows and columns, obtained by taking the determinant which remains when in the matrix of n columns and (n - 2) rows | dkX1,..., dkXn, k=1, 2,..., (n-2), the rth and sth columns are omitted, and prefixing to this determinant the sign (- l)r+s-1 or (- 1)r+ according as r < s or r > s. We require now to make it clear that we can suppose the sign of the ratios en- to be the same for all points of the limited H,_2; for this purpose suppose Pn-X, with coordinates (xw + dn-xz,..., x + dn-_xn), and Pn, with coordinates (xz + dx,..., xn + dnxn ), OF SEVERAL COMPLEX VARIABLES. 413 to be satellite points of P of which P,,- is on fi= O but not on 12=0, and Pn is not on either of the two f= 0, f= 0; then we have en-2 M ~ - ' M fl Jri d _n- xi +... + Jrn dn Xn ri i dn-i Xi +.+ Min dn-ixn' odf, - ~f2d f1 m/r where M, denotes the value obtained by taking the determinant left when in the matrix of n columns and (n - 1) rows dkxi,..., dkXn, k=1, 2,..., (n-1), the rth column is omitted and prefixing the sign (-l)n+r to this determinant; hence as dn,,_f = we have n_-2 MXr ax, _ ax ~Ar D 1r -M =..... ' M I dfi xnX +...+ fi dx n Mi dn x + @ @+ Mldnxnl dnfi where M is as before the determinant | dkXi,..., dkxn, k= 1, 2,..., n. Thus on the whole we have J12 Jrs dnfi. dn-if en-2 M12= - -" = 8 -M" We now mace the assumptions (i) that for all points P of Hn- the satellite point Pn is taken in space on that side of the (n - l)-fold f = O0 for which fi > O, so that dnfi is constantly positive, (ii) that the satellite point P,,-_ is taken on fi = O on that side of f, = 0 for which f2 > O, so that dn-if2 is positive, (iii) that the satellite points P1,..., Pn-2 are constantly taken on f' = O, f = O in such a relation to Pn_ and Pn that M is constantly positive over Hn-2. Under these assumptions each of the ratios Jrsl/Ms. maintains a positive sign over 414 MR BAKER, ON THE THEORY OF FUNCTIONS H2, and the direction cosine Kcr of the (n -2)-fold t =O, f,=O, which by definition is given by Kcrs = Jrs. V J2rs 1, and is therefore equal to sgn e-2. M rs ] /SM, where sgn e2_- means + 1 or - 1 according as E,,-2 is positive or negative, has throughout Hn_- the sign of Mr,. Thus KrsdSn-2 = Ms sgn (r /) = Mi, sgn (- df `) = M. 6. Next we consider any point Q of Hn_3, and its satellite points (xl+dkXl,..., x+-dk X), k=1, 2,..., (n-3). From the equations fidk + + +. fldkX, = 0, f2dkx +... +f2dkn = 0, 3dkX +.. +f3dkXn = 0, = 1, 2,..., (n-3), we find as before J12 _ __ Jrt M123 rs- Mt., y, where "rst = afi afi af8 axy ax' axt xr' axs' axt and Mrst is the value obtained by omitting the rth, sth and kth columns in the matrix of n columns and (n - 3) rows I dki,..., dkX, k= l, 2,..., (n-3), and prefixing a certain sign to the resulting determinant. This sign is supposed to be given, as for the two previous cases and as in general, by the following rule; consider the determinant of n rows and columns whose first (n - 3) rows are formed by the matrix just described, whose (n-2)th row is A1,..., An, whose (n -l)th row is B1,..., Bn, whose nth row is C1,..., C,; then the expansion of this determinant is lton,Ç ArBsCtlM,,st; thus when r, s, t are in ascending order the sign to be prefixed to the determinant OF SEVERAL COMPLEX VARIABLES. 415 by which Mrst is formed is (- l)+~+s+t-l. Hence, taking the satellite point Qn-2, of Q, upon fi = O and f= 0, but not upon f3= 0, we have E =rsi _-_rs2 i - rsn _n-3 = Mi.l = M = M- = M... s Mrsa Mrsn Jrsi dn-2x1... + J rsndn-_xn M.rsi dn_2-x1 +... + + drsn dn-2xn here the numerator is afi afl axr ' 8s '-2, af2 af2 d, ax ' axs ' and therefore, since d_2f = 0, dn-2f, = O, is equal to Js dn-2f3; the denominator can be seen to be exactly equal to M;rs thus we have en-3 -= M - n —J3 and so obtain KiCst = Jrst - I /J2rst | Mrst (Jrs d8n^,-3 VMrs 2 the element of extent dSn_3 being by definition equal to \/ZM2,st1. 7. At this point it is convenient to recall the connection which has been established above (~ 4) between the division of HIn- and Hn11 into cells. With that arrangement the J,.s and Mr now arising in the consideration of iCrSt may be regarded as identical with those that arose just previously in considering Kcrs. We proceed to utilise this. Consider the determinant of n -2 rows and columns di P-St, dx,..., dxi x, dl_-2Pst> dn-2li,..., dn-2_n wherein (x + dxx,..., x+ dn_,x) are satellite points of (x), upon H,2,, the columns containing xr, xc, xt are omitted, and Pst is a function which is single-valued, continuous and finite upon Hn- and Hn_, and possessed of single-valued finite differential coefficients. The suffix of Prst is not necessary for our present purpose; but it is convenient, as enabling us to define functions derived from this one by interchange of two elements 416 MR BAKER, ON THE THEORY OF FUNCTIONS of the suffix, with corresponding change of sign of the function. The determinant, if r, s, t are in ascending order, is equal to (y_ )++ ra2,,ap.pst aP, prt ( _ l)r+X+t LaPt - aPS Mrt +at M On the other hand, supposing as in ~ 3 that all the differentials d,_,x,..., dn-2x, except* dn_2xr, dn-_2, dn-2xt are zero, the determinant is equal to (- 1)r+s+t dn-2Prst. M.rst * Hence finally we can evaluate the integral fY aPstr aPrts aPrst\ II Kcst + Krt + Krs X ) dSn-2 taken over H,-2. Suppose Hn- divided into strips as in ~ 3, and find the contribution of one of these strips. The integral is sgr/ -r s ax tax, which by the identity just found, and because we can suppose sgn (JslMrs) to be the same over the whole of H,_2, is equal to sg (rs) fJ rstdn-2 Pst; as we pass along the strip under consideration the determinant M,.st is constant; thus the integration along the strip gives sgn ( Mrs st L[st - Pt-st where the single integral sign indicates an integration extending to all the strips, and P(t, Plt are the values of Prst at the points Q(), Q(0) where the curve of integration through the point P, along which only the three coordinates Xr, Xs, xt vary, respectively leaves and enters Hn_3. We have seen that KrstdSn-3 = Mrst sgn (' s dn-_2f3), and moreover that dn_2f3 is positive at Q(1) and negative at Q(0); hence the elernent sgn ( M).r M t [P() - P()] sgnM ~' s L" rst " rst is the sum of the two elements of the integral over Hn- which is expressed by Krst Prst dSn-3, which arise corresponding to the cells at Q(~) and Q(0). Thus we have proved that the * In ~ 3 the differentials not zero were denoted by d_2 Xb, d_ x0, d_ Ax^h. OF SEVERAL COMPLEX VARIABLES. 417 latter integral, over H,_,, is equal to the integral from which we started which is taken over Hn-2; and this is what was desired. We can then by a summation infer that I S Kt Prst dSn-3 = K r, s. d Jrs, t.rs t which is a convenient way in which to state the result. There are () terms in the lefthand integral, and (n - 2) () =3 () terms in the right-hand integral. A similar argument will be found to lead to the general result stated in ~ 2. 8. If we put, in the case for which the proof has been carried out, _ P.st Xrs =- C -t t axt we have >X = 0, s a$s as n necessary conditions that the integral f|rs Krs Xrs dSn-2) taken over a finite portion of an (n - 2)-fold may be capable of being represented as an integral over the closed (nt-3)-fold bounding this portion. If these conditions are satisfied, functions P,st satisfying the equations aPrst Xrs = Y t axt must be found, in order that the expression rnay be possible; but it is necessary that the functions Prt so found should be finite on the (n -3)-fold (cf. ~ 28). The equations ax,., have been given by Poincar (Act Math. x. (1887) p. 37) from a somewhat different have been given by Poincaré (Acta Math. Ix. (1887), p. 337) from a somewhat different point of view. We can as an application generalise Cauchy's theorem to the p-fold integral j.....J (e,..., p) cie... p where:,..., p are complex variables. For example, for n = 4, the integral ~~ SfVOL. XVIIT. 53 VoL. XVIIIT. 53 418 MR BAKER, ON THE THEORY OF FUNCTIONS taken over a closed (n-2)-fold, which (see ~ 9 below) may be interpreted as (el)/(1 2)(x13 + it14 + iK23- t24) dSn-2, is equal to the integral f L( + iK2) ( + i -( / + i(4) \ + 0 72) J taken over the (n-1)-fold bounded by the (n-2)-fold; and this vanishes identically on account of (a t 2 z a (3O, +%i^ f = 0. It is supposed that the original (n - 2)-fold of integration is not one given by the vanishing of a single equation involving the two complex variables, since otherwise (cf. ~ 9 below) K13 = I24, K^14= - C23, and therefore i13 + iKl4 + iK23 - c24 = 0. PART II. The expression of an integral function whose zeros are given. 9. In what follows we consider a space of n dimensions, n being even and equal to 2p. The points of this space being as before given by the n coordinates xl,..., x, we define from these p complex variables by means of the equations,. = 21-i + ix2r, (r = 1, 2,..., p). As it is desirable to take the various points separately we begin by supposing that we have defined in this space an (n - 2)-fold, given in sufficiently near neighbourhood of any point (x1(),..., xn(0)) of itself by the vanishing of an ordinary power series in the quantities - (, 0).,.. -p(), where ~) =- X(L + ix(O) We proceed to shew that the (n - 2)-fold can be given throughout its extent by the vanishing of a single-valued integral function of l,.., - p (~ 15). Such an (n - 2)-fold, given by relations involving only complex variables, may be called a complex (n - 2)-fold; its direction cosines satisfy particular relations, as we now prove. It is determined in sufficiently near neighbourhood of any point of itself by the two equations arising, say, from + (1,..., ^)= + = O, where u, v are real functions of the n real variables x,.., xn, which satisfy the equations au av au 3v OXî_r-, =x-r' X =r - 2_r-; OF SEVERAL COMPLEX VARIABLES. 419 thus if we denote au/axs and av/axS respectively by us and v, the direction cosines of the (n-2)-fold, defined in ~ 2 ante, are given by Kr, s = (UrVs - UsVr)/h, where h is the positive square root of 2 (L,.vs - utsV8)2 = (12 +... + Un2)2 = (12 +... + Vn2)2; ', s now we have Zt2r-1 V2s-1 -- - U,2s-1 Va? ' - 'vr s + V2s U2r = - '2r-1 U2s + U2rU2s-I, U^2r-_V2s - U2sV2r_1 - V21'U2s1 -1 V22Sl-22r = U2r-1U2S-1 + U12îU2s, so that UG2r U2s- -- U2r-1 '2 2r-1, 2S-1 = r, 28 = —9.,n 2r-1,2S-1K2,2S 2 U12 +..+ + u U'r_-i'L2s-1 + U 2r 2s 2r —1, 2s =-C2,2S-1 = _... + n2, and K12 + <34 +.. + KCnl,n = 1. These relations are of importance to us. They of course require modification at any singular points of the (n - 2)-fold; the present paper is so far incomplete that the consideration of the effect of the singular points is omitted; the final results obtained are expressed in a form which is believed to be unaffected by the existence of such singular points. 10. Consider now a limited portion of an (n - 2)-fold, bounded by a closed (n - 3)-fold. Denote by xi,..., x, the coordinates of a point on the (n - 2)-fold or on the (n- 3)-fold, and by (t,..., tn) the coordinates of a finite point of space not on either of these, the corresponding complex variables being as before given by = t2r-t + it2, r=1, 2,..., p. Let L1,..., Ln and R2,..., Rn be single-valued functions of xi,..., xn and of ti,..., tn which are continuous and finite, with their differential coefficients, so long as (x1,..., xn) is upon the (n - 2)-fold or (n - 3)-fold under consideration, and the point (t1,..., tn) is in finite space and not upon the (n - 2)-fold or (n - 3)-fold; further suppose that these functions are such that aLM aRi at (i, s=a, 2,..., n), and aR aR2, Rn. +- +... +' + 0. axl ax2. x. Consider the n integrals r = f(irîL +... + KrnLn) dSn-2, (r = 1, 2,..., n) 53-2 420 MR BAKER, ON THE THEORY OF FUNCTIONS taken over the limited portion of the (n - 2)-fold; we have afr a8$ f ( aR1 aRl aRi aMRn at, -"ag =-J 1 F(Kri + +* rn a., — A/cs a - Ks n a J dn -2, at8 atr ax8 ax axax and therefore, adding to the right hand the vanishing quantity laRI aR2 aRn rax ax, a, ( Raxanx(\ 7 we have ar as - f(ir aRi +rsaR aRi)S ats atr i-iJX 1 axi +Ksj where i does not take the values r, s. Thus by the formula proved in Part I. we have aar _ a-8a 1=E t sriRi. d Sn-3) where the integral is taken over the (n - 3)-fold bounding the limited (n - 2)-fold over which the integrals j,..., n are taken. 11. It follows that if the (n - 3)-fold integral vanishes, the expression dt1 +... +,ndtn is a perfect differential; on grounds further considered below (~~ 12, 22) we suppose that this (n-3)-fold integral does vanish; we suppose also that L2r -iL2r-l and that the (n - 2)-fold is a complex (n - 2)-fold; then from the equations /C2r-1, 2-1 = 2r, S l, 2 Cr-l, 2 - -C2r, 28-1 it follows that Ç'2r = f {(K2r, + igK2r,,) L1 +4... + (K2.,2,_r-2) r-i +... + (Ka:r, n-. + iK2r,n) Ln-} dS_-2 -= i f {(Krc_, * iC2r-_, 2) L +... + (iCKr-i, 2r) L2r-.i +... + (tCr-l, n-i + k2r1-l, n) Ln-i} dS.-2 so that 2r-= i2r-1, and therefore a~a2r-l = a = a i 8s-= i a2r-_, at2s atr2r-. at-2r-i at2s-1 which gives (at, +ia)2-l=o 0. Thus, under the hypotheses introduced, all the functions C,,..., n are functions of the complex variables Ir,..., rp, and there is a function < = f((dt2 + ^A2 +... + dtn) = f(idTi + ^d72 +... + n-ildTp) OF SEVERAL COMPLEX VARIABLES. 421 for which ar - = 2r-1~ 12. We now give a more special value to the functions L1,..., Ln. Let 1 n-2 (lt) = T- [(- tl)2 +. + (n - t)2] 2 and Hm=,(x lt)- P(xO()+ jt )(xO)-... + - t () (i 0o), m being a finite positive integer or zero, and t(t D- = t + 3... + t', When r2, =t +... + t2, is sufficiently small in regard to R2, = 1+... + x2, we have (- 1m+l / a \mf+i (_ 1)m+2 / a \M+2 -(n+)!t ) ( +I )! t; g() 0(o)+... to o, rm+i y R-l Kin+i () + En K1+2 (+ ) +. to 00, where u denotes (xt, +... + xt,)/Rr and is numerically less than unity, and K, s(U) denotes, save for a factor independent of,/ but depending on n and s, U 2n-3 ds r n-3 -( -) —()h )~ 2- i As we can find a real angle 0=cos-1/, we have 1. 1 / r \-"rn-2 n-2 (xlt>) 2 R=-2 - Qei) 2 (1-R ( e i) 2; by expanding the binomials and considering the explicit expression for the coefficient of rk+/1Rn+k-l it is immediately obvious that this coefficient is not greater than if 0 were zero. Thus when r < R the absolute value of Hm is of the form rm+l (- _ 2) (n- 1)... (n + n- 2) 1 Rn+m-i lm+ 1 1 - e' where, supposing n, m, r fixed, e is only unity when R =r, but for R>r has zero for limiting value as R increases indefinitely. It follows that a value R0 can be chosen such that for fixed n, m, r and all values of R > R, we may have I TmT 1 R with B a finite constant independent of R. 422 MR BAKER, ON THE THEORY OF FUNCTIONS It is easily seen that aHm n Hm-_ at=i aDxi an equation which holds also for negative m provided Hn be then understood to mean p (x 1 t); and thus / a2 ' a2 a2 \ ~ /a2 a2 a2 \ rr (l+ a +... +a = (l0+a + a.+2) Hm_=0. Hence if we put 7? -.P -(aHmIl.aHm-i L21, = iL2.l = -.a2- - - / a _ ax2,f1. axe.j where m is to be kept the same throughout the investigation, we have aLi aRi ats axs ' (a3R2r- + aR2r = i a2 - + a2H1I) = o {R = 1 2r-1 2} r+ = o2-, r=1 ax_2r-l ax7î. r= -l a so that the conditions of ~ 10 are satisfied. We suppose also the further condition, of ~ 11, namely that the integrals [ P /, aH im-1.3m-i (irs,2- + iCr,,9, 2k) (- ' aH-1 d-3, J k=l "',\ 2-1 2k taken over the closed (n - 3)-fold, bounding the (n - 2)-fold under consideration, are all zero, to be satisfied. This hypothesis arises as follows. We suppose the (n - 2)-fold, over a limited portion of which our (n - 2)-fold integral is taken, to extend to infinity; when (tj,..., t,,) is in finite space and (x,,..., x,,) is very distant, the function Hm,_ is a small quantity of the order of (x,1+... - x22)-f (+m-2); we may therefore suppose that if the (n - 3)-fold be taken entirely at sufficiently great distance from the finite parts of space and m be sufficiently great, the (n - 3)-fold integral can be made less in absolute value than any assigned quantity. A particular examination is given below (~~ 20-24); it can be definitely shewn that the hypothesis is verified, even for m= 1, for a large class of cases, which includes the case which arises in the consideration of periodic functions. The application of the present paper is limited to the cases where some finite value of m is sufficient; as will be seen this is a limitation which we may regard as analogous to that, for functions of one complex variable, to functions of finite genre. Connected with this hypothesis is a further one; supposing the (n - 2)-fold integrals C,..., to extend over the whole infinite extent of the (n-2)-fold, we suppose that they and their differential coefficients in regard to t1,..., tn are convergent. OF SEVERAL COMPLEX VARIABLES. 423 Then we have the result; the p functions =-ji f_ i (^'-1, 28-1 +TK2i-1, 28s) (aH n- i ) dSn-2 J 5=1 + a 28-1 OX _ 2 extended over the infinite complex (n-2)-fold, are functions of the complex variables r1,..., rp, and are the partial differential coefficients of a function: =f (i ~ +4- 3dr-2 +... + T n'-dTp). 13. We proceed now to put this result into a new form, from which it will appear that the real part of the function 1' is equivalent with a result given by Poincaré, being a generalised potential function, and that the imaginary part is a generalised solid angle function. Putting,2r-1 = 2r + 8.21-1, we have, clearly, tc2î, - 3I aHM aH+ i? +.2 s 82r-1 -— [ KC2r-1, i aX+ K-' a -l, 2 x- +.. + K2r-i, %n a) dSXn-2 8)2r = j\ - a-lH2 f74A2 - - _.. + K.c2r-l, n-l 8 _ dSn 2; of these the latter, in virtue of the equations C2i'-1, 2S-1 = K21', 2S, C2r-1, 2S - - K2r, 2S-1 can be written 2ar = JkK2r 1 a + KC2. 2 +2. + K2r'n an sn-2 We proceed now to shew that in fact - afr dSn-2,2) -ja-, fn-2) these integrals, like the others, being extended over the whole infinite complex (n - 2)fold, and supposed convergent. Take the first of the two forms given for 2,., namely ^2 _ [` /m 7^, ^ i( ^H, aHH aH 2r = - f H dnK2r-1, 2r 9H dS 2 m + KC2 -1, 2k-1 St-2k )2> J C2r-1 k<=1 a%^2^1 > aX2k where k does not assume the value r; over a finite portion of the (n - 2)-fold, bounded by a closed (n-3)-fold, the integral r( Ï alHM aWm aHm J K2k, 2 -1. f_ + "K2r-1 2k-i - + -k, 2k d n2r- d 2 is by the results of Part I. equal to jfCk-1, 2k, 2-i _ Hm dS&3_, 424 MR BAKER, ON THE THEORY OF FUNCTIONS taken over the closed (n-3)-fold; assuming now, what is a similar assumption to those already made, that this integral diminishes without limit when the (n- 3)-fold passes to infinity, we have s2r= - 2r-i, 2r d Xn-2 k=- ji, 2k -2 — M-2 J 2r-1 k=IX2d-i which, since KC12 + C34 + * + 4n-1, n = 1, is the result stated, namely that 32r = -f a dS 2. The corresponding form for s2,-i can be proved in a similar way, starting from çs _~- [(~- ~ ~ SlT;Hm aHm a L^ 7 2r-1 = J (2r, 2 Ka r + -2r, 1 a + K2r1, n-l ) Sn-2. __ xJHm aHm Thus also (= a- - - da-S and, if we put (Poincaré, Acta Math. t. xxII. p. 168) V f= Hm+ dSn-2, i2r =- 2r-l - - t at2i7-1 Ct2r so that V, which may be regarded as a generalised potential, of the (n - 2)-fold of integration at the point (tl,..., tn), is the real part of <b, or differs therefrom by a constant. 14. Supposing that the integral is taken from the point t = O = t2=... =tn, which is supposed not to be on the (n - 2)-fold of integration, we may write q) =/ (adT1 + dr2, +... + p-_dTp), = I (2d1 - l dt2 +... + S2pdt2p-l - 2p-lt2p) + i f(S1dt1 + 82dt2 +... + 82p-dt,2P-1 + 82pdt2p); of this, in virtue of the results of ~ 13, bearing in mind that Hm+,, and therefore also V, vanishes for t =O,..., =t,=O, the real part is exactly V; the imaginary part is il2 where OF SEVERAL COMPLEX VARIABLES. 425 Q=f(8 dt +... + 8)dtn) = fdSn-2 2 {dt2,1-1 (2r-1, 1 S- + H + )2r-1, n 9 - JOJ r=i' ax +... + KH. 3a\ {( aHm Hm f o/7 +dt2r I2r, 1 a, +... + Kc2, n - j = dS1_n-2 s K, idt - dts,1 J oJ r, s \ uJ^sUax, the summation extending to all pairs of different numbers r, s from 1, 2,..., n; now we have seen (~ 5) that if M,, denote, for r < s, the product of (- l)+S-l and the determinant obtained by erasing the rth and sth columns in the matrix dkXi, dkX, dkX, x, | dkacl, dk$2, *..,n djc^n |n C= 1, 2,..., (n-2), then KIrsdSn_2/M1s is independent of r and s, and equal to + 1 or - 1; denote it here by c; thus Q can be put into the form Q =f Je dt,, dt2,..., dtn aHx aHn aHm ax81 ' 82 a??a dlx,, dlx2,., d.x, dn_2-x, dn-_2x.,. dn-2xn This shews that Q may be regarded as a generalised solid angle, namely that subtended at the point (t,,..., t,) by the (?n-2)-fold of integration. (Compare Gauss's well-known form given for instance in Maxwell's Electricity and Magnetismn (1881), Vol. ii. p. 39; or' Gauss, Werke, Bd. v. p. 605.) The same will appear anew from a transformation of Q into an (n-l)-fold integral (~ 17). The result is not merely curious; the function Q2 is in fact a single-valued function of t,..., tp save for integral additive multiples of the quantity n 27r _ 2.rI~rtF (p-1)!' which is the complete solid angle in n dimensions, namely the total extent of the (n - 1)-fold t2 + t2+... + t21. As the function V is clearly single-valued it follows that the function 2rr 27r e =e is a single-valued function of (71,...,,p). VOL. XVIII. 54 426 MR BAKER, ON THE THEORY OF FUNCTIONS 15. The interest of the results just obtained arises from the following fact. Suppose that, in the neighbourhood of any point (1(0),..., ~p(O) of itself, the equation of the (n- 2)-fold of integration is expressed by (el - e(O),..., p- p(0)), the expression b being an ordinary power series of presumably only limited range of convergence; then, with a proper signification for the logarithm, the deifrence 2w,( _..., _r)- log (T, - ),..., T - (0) remains finite and continuous as (1r,..., p) approaches indefinitely near to the (n - 2)-fold. This capital result is taken from the paper of Poincaré, already referred to, Acta Math. xxII. (1898), p. 169. Other proofs, themselves important to us for another purpose, which shew how the result follows from Kronecker's theory, are given below (~ 16); the most natural method of verification is however by direct evaluation of <P in the neighbourhood of the (n - 2)-fold of integration (~ 25). The result itself is a direct generalisation of wellknown facts for p = 1. If for the present it be assumed, it follows, putting 27r O (71,..., ) = e that the ratio =. e ()-10(; (r) wherein ) (r) is written for (-r, -:),..., Tp - (), is not infinite or zero in the neighbourhood of the (n-2)-fold b (r) = 0. It can be seen however from the form of the integrals by which P('rl,,..., Tp) is defined that, for finite values of ti,..., t,, this function can becorne infinite only when (ti,..., t,) approaches the (n - 2)-fold of integration. Thus we have the main result of the enquiry. The equation of the arbitrarily given (n - 2)-fold of integration is obtained by the vanishing of the integral function o (71,,..., T)= 0, which is equal to exp {- - r d7rf a im-t n dSn-2 ( Or=l J 2r-1 2r/ {27T- p JH f/ +l - i aHmf dj } or exp { f dr j-^ - î- d:~.H 7Y =1 at2 16. Denoting as before by s0 any one of the series by the vanishing of which the (n - 2)-fold of integration is defined, we have, as already quoted from Poincaré, the theorem that the difference 27r ( V+ im) - log remains finite and continuous even indefinitely near to the (n -2)-fold on which s5 vanishes. OF SEVERAL COMPLEX VARIABLES. 427 To the proof given by Poincaré we may add the two following, both of which make use of some results in Poincaré's paper. (a) Denote the (n-2)-fold of integration by I; let (x1,...,,,) be the coordinates of a varying point on I, so that (xi,..., xn) are functions of (n - 2) parameters; then if (x11,..., xn') be current coordinates, and e a fixed small quantity, the envelope of the spheres (X/ - X)2 +...+ (x- xn)2= 2 is an (n-l)-fold S, surrounding I, of which, when (x1,..., x,,) is not a singular point of I, the points are given by i = + (ui cos 0 + vi sin 0), (= 1, 2,..., n), where 0 is a variable quantity, so that (x1',..., x.') are functions of (n - 1) parameters; here ui denotes u//axi, vi denotes av/axi and h is the positive square root of I12+... + un2 The point (x1',..., x,/n) lies on one of the single infinity of normals which can be drawn to the (n - 2)-fold I at (X1,..., xI), and is at a distance e from I. The direction cosines of the normal to 2 at (x1',..., x') are the quantities (uz cos 0 + vi sin 0)/h; the element of extent of Z at (x/',..., x') is dSni = edOd&S2, ultimately, squares of e being neglected, where dSn_- is a corresponding element of extent for I. If. = x2-1 - ix2, yr = X'2r + ix2r, we have ()1 ' " ()' - h 2' where (P,.)' is the conjugate complex of 9~/9r, and therefore equal to 21._- i ~ -v2_1; and, what is permissible to the first order of small quantities, ' is written for 1 (e'l- e1) +. + ~P (ep - p). With these results we combine now the following, which is a particular case of a theorem of Kronecker's. Let f(ri,..., Tp) be a single-valued function finite and continuous upon a certain closed (n - l)-fold, whereof 1i,..., K, are the direction cosines; consider the integral f(,, p) (cl + iK2) (-L - ) ( 1 t) +.. + ( -l + e) ( - Ïi.- p (w I t)} dS,,, where (x1,..., x,) denotes a varying point upon the (n - l)-fold. By Green's theorem it is immediately clear that this integral is unaltered by any deformation of the (n-l)-fold of integration which does not involve a crossing of the point (t1,..., tn) or of any point where f(7-1,..., Tp) ceases to be finite, continuous and single-valued. For the condition for this is simply (Part I. of this paper) (a a a +i f(, *.*.) ) - i p ( i t) +..0, namely (a2 + 2~ +.) ( ) = 0. 54 2 428 MR BAKER, ON THE THEORY OF FUNCTIONS Hence, if f(7r,..., rp) be single-valued, finite and continuous for the whole interior of the (n- )-fold of integration, the integral is zero when (t1,..., t) is outside, and, when (t,...,tn) is inside the (n-l)-fold, it is equal to /(7,..., Tp), as we see by supposing the (n - 1)-fold of integration to be deformed to (Xi- t )2 +... + (xn- t)2= r2, and then taking r to diminish without limit. Now consider, in the region of convergence of the series b, a (multiply-connected) closed region, bounded by (i) part of the (n-l)-fold Z surrounding the (n- )-fold b =O which has already been described, (ii) part of a closed (n - )-fold S described in the region of convergence of (b, the part being limited by the (closed) (n - 2)-folds in which S is intersected by E; and take /(r) = log, a'Tr where r is one of 1, 2,..., p. Then when (t,,..., tp) is interior to the (multiply-connected) region above described, we shall have ()= /J +i (1 -a ) ( t)+ *} dSn-, where the integral is taken over the two partial (n-l)-folds denoted by (i) and (ii). The part (ii) of this integral is finite for all the positions of (t,,..., t,) under consideration; consider the limiting value of the part (i) as the (n - )-fold 2 is taken nearer and nearer to the (n - 2)-fold I, namely by the decrease of the quantity denoted above by e. By what has been stated above we may ultimately put K*2s-1+ itS=-(^S) l 2 e, ' f(f)=, dS=n-l = eddSdn&_; then, if (ti,..., t,) have some fixed position at finite, not infinitely small, distance from the (n - 2)-fold I, we obtain, for this part of the integral, - 1 d f (f edSn,-, (: (, e -2s atzs) ç (t} O J (JE) { =l() )t ( -_ i 28-1) ~ (x ) )}_ n; [s=i a - a, ) now the direction cosines of the (n - 2)-fold I are (~ 9) such that U2. -i. V2s-i - U-2S-1 V2r-i + i (2.r-1 V2S - U2, Vr -1) s2r-1, 2s-1 + I~ 2'-1, 2S - h2 = 2 (I2r-1 + iv2r-1) (u2S-1- iv2S-1); OF SEVERAL COMPLEX VARIABLES. 429 thus this integral becomes 27ri f (K2r-1, 2S-1 + iK2r-l, 2s) ( --- - — a (x t) dSn-2. Therefore it follows that f(r) differs from this integral by a function which is finite and continuous for all positions of (t1,..., tn) in the region of convergence of b, except actually upon = 0. Recalling a previous notation (~ 12) this is equivalent to the fact that log (T)- -2 - remains finite and continuous as (r1,..., rT) approaches the (n-2)-fold I, so that also log _ (T) -2- (1dT +... + -i drp) remains finite and continuous as (1i,..., Tp) approaches the (n-2)-fold; as was to be shewn. (b) By using Kronecker's integral in a different way we can obtain the same result otherwise. Consider the region of convergence of one of the series b by which the (n - 2)-fold I is defined; describe in this region as before a closed (n - l)-fold S, containing in its interior a portion of the (n - 2)-fold I; about I suppose as before an (n - 1)-fold 2 satisfying the condition that every point of it is at a small distance e from some point of I. Then the portion of n-fold space inside S and outside E is multiply-connected; but it can be rendered simply-connected by supposing an (n-l)fold diaphragm P to be drawn, bounded partly by the (n - )-fold E and partly by the (n - 1)-fold S, each of which it intersects in an (n - 2)-fold. Within the n-fold simply-connected space so constructed the function log b is singlevalued. Hence, if ('71,..., Tp) be a point within this space, we have, as explained above, log g (,) = - J log / ( l) [(~ + i) ( - i a) p ( It) +... dSn-,, where the integral consists of three parts:(i) that over the part of S lying outside the closed (n - 2)-folds in which S is intersected by Z, and excluding the (n - 2)-fold in which the diaphragm intersects S; (ii) that over the part of Z lying within S, excluding the (n - 2)-fold in which P intersects Z; (iii) that over the two sides of the (limited) diaphragm P. The part (i) remains finite and continuous for all positions of (r,,..., Tp) within the n-fold space under consideration. The part (ii) ultimately vanishes when the quantity E dirninishes indefinitely; for we have seen that dSn-i = edOdSn_, and it can be shewn, as by Poincaré (Acta Math. xxII.), that as the point (/',..., 'p) on a normal of I, at 430 MR BAKER, ON THE THEORY OF FUNCTIONS a distance e from I, approaches indefinitely near to I, the limit of e log (e'), when e and therefore (e') vanishes, is zero. The part (iii) of the integral is equal to 27ri f[(K + i2) (a, - iZ a,) p (x +...) S taken over only one side of the (limited) diaphragm P; for the values of log 4 at two near points on opposite sides of P differ by 2ri. Consider now the real part of this integral, namely 27r /C a +.-C2 a,)dSby the theorem of Part I. of this paper we can replace this by an (n - 2)-fold integral taken over the (n - 2)-fold which forms the boundary of the diaphragm; this (n-2)fold lies partly on S and partly on S; the (n- 2)-fold is (2 + 34 + * +... Ken-1, +n) (x | t) dSn-2, as is immediately obvious on applying the theorem. If we now suppose that the diaphragm is so chosen that the bounding (n - 2)-fold is a complex (n-2)-fold (~ 9), we can infer that, when (ir,..., Tp) is within the region considered, log 4 (r) differs only by a finite and continuous function from a function whose real part is equal to -2 f (x t)dS2, where the integral may be supposed to be taken only over the part of I which lies within S; for we have seen (~ 9) that for a complex (n - 2)-fold ^i12 + C34 + *. + ^Cn-i, n = 1. The theorem to be proved can then be immediately deduced. 17. Incidentally we have remarked in ~ 16 that if a finite portion of an (n-1)fold be bounded by a closed complex (n - )-fold, then, under certain conditions of continuity and single-valuedness for the function U, we have UdSn-2 = J'1 a - -2 a +... dSn-,, the first integral being taken over the closed (n - 2)-fold, and the second over the bounded portion of the (n - l)-fold. We now extend this idea to the (n - 2)-fold I, given by the aggregate of the series ). We imagine this (n - 2)-fold, which is defined only for finite space, to be completed into a closed (n -2)-fold by means of a complex (n - 2)-fold at infinity; and, as before, we assume tentatively, that the part of the integrals under consideration which is contributed by the portion of the (n-2)-fold of integration lying at infinity vanishes (see ~ 22). OF SEVERAL COMPLEX VARIABLES. 431 Then, firstly, we may put V = fHm+ dSn-2 -= J( am - 1C2 Hm+l +... dSn1, ax2 ax, where the right-hand integral is taken over the infinitely extended (n - l)-fold diaphragm bounded by the complex (n - 2)-fold. And, similarly, we have (~ 14) = rl +.+'" K n +n)dS +-2 JJL ( aHln\ ( jH) r. + +-H2n\~ 1 /Cl= JKî / E~- +... + 1 + + +Kn a iK dSni, ax, + '"+ a2H,' a2,H x2 + ax2 ax2 thus, taking tQ to vanish when (t1,., tn) = (0,..., 0), we have 0 aHl affi n^+, Q = m+î1 m + -... +- dSn -1. Thus, as has been indicated in connexion with the definition of Q1 as an (n-2)-fold integral, Q is a generalised solid angle. It is not a single-valued function of (t1,..., t); its values at two near points on opposite sides of the (n - 1)-fold of integration differ by integral multiples of r; this follows, in a well-known way, from the fact that the value of the integral taken over the closed (n - )-fold (X - t1)2 +... + ( - tn)2 = r2 is ultimately r when r diminishes indefinitely. Thus it is obvious that 2(g+ia) O (71,..., T) = e + is a single-valued function. 18. We come now to the consideration of the question of the convergence and vanishing of the infinitely extended integrals used in this paper. Somrne guidance may be sought in the comparison of the general case, when p > 1, with the case of functions of one variable, for which p=l. For this latter case there is no continuous (n - 2)-fold of integration; the corresponding thing is ca series of discrete points, in general of infinite number. We have in this paper found a formula, o (71,..., 'p) = 0, 432 MR BAKER, ON THE THEORY OF FUNCTIONS to represent the equation of a given complex (n-2)-fold extending to infinity; let us apply this to find the equation of the (n -)-fold constituted when n=2 by any enumerable system of discrete points f, 2,..., having infinity as a point of condensation, in regard to which it is assurned that for some positive integer, rn, the series -n-2 + 2-m-2 +... is absolutely convergent; this condition corresponds to the general one that the integral jlm+i dSn_2 is convergent; for instance the points may be those given by a formula a + 2kw + 2k'', where k, k' are integers and o'/o is not real, in which case, as is well known, it is sufficient to take m=1. Taking P (x t) = log [(xi - tl)2 + (2 - t)2], and, as in the general case (~~ 13, 14), V = ifHm+ dSn-2 = H.m+, (-m 1) =2 ~ go(x 1-o( |0)+-t... + e(, + --- (t ~;) g |(0[o)J, where the summation extends to all the points e = xi + ix,, and i = j (Cdit ajm - dt2Hn) dsn2, =, dt - _ dt, -W ) we find easily r\ T 1 T2 1 nt^+1 T+ia= [ log l-)+j+ -+...+ ^^]; hence as r= 27r for n= 2 we have o (T) = I [1 - ) "t (,+l1),n+l] namely the theorem gives the general integral function of finite genre*; whereas in Weierstrass's factor formula for a general integral function the number of terms it is necessary to take in the exponential may increase beyond all limit as we take a more and more advanced factor of the product, our theorem limits itself to the case where the same finite value of m will suffice for every factor. 19. In the case of functions of one variable a simple case of functions of finite genre arises for periodic functions, the value of m for the sigma functions being m==1. And in the general case the fundamental (n-2)-fold of integration may be periodic; * The usual exponential outside the infinite product being absent. OF SEVERAL COMPLEX VARIABLES. 433 in the sense that it is possible to divide n-fold space into period parallelograms, the interior of any one of these being given by the p equations ti = X + 2ei, lX +... + 2oi, 2pX2p (i = 1,..., p) where X is a constant and X,..., 2p are real variables each between O and 1, and to regard the portions of the (n - 2)-fold lying within these various parallelograms as repetitions of one of these portions. Then it can be proved, under a certain hypothesis, that the value mn= 1 is sufficient for the convergence of the integrals. The hypothesis is that the extent of the (n-2)-fold contained in any one such parallelogram is finite; and the truth of this hypothesis is deducible from the mode in which the (n-2)-fold of integration has been supposed to be defined. Of this result, which is given by Poincaré, the proof is included in the investigation below (~ 22); it may be remarked at once however that the formula obtained here is not limited to the case of periodic functions; as we may see by taking a simple example. We apply the formula when n= 4, to form the equation of the complex (n - 2)-fold l1=7; putting y = a + ib this is then the two-fold given by x = a, x2 = b. The matrix dl x dix2 dlx3 dlX4 d2xl d2x2 d2x. d2x4 with the help of which the direction cosines may be defined, may be taken to be 0 0 dx, 0, 0 0 0 dx4 so that Kc.- O= except Ki2= 1, and dSn-2 = dxdx4. As the integral (f dx dx4 - f2rd R rdir J(a2 + b2 2 + x +2)2 J O Jo (D + r2)2 vanishes when,R R0 are infinite we infer that it is sufficient to take n= O, and therefore Hm = p() t)- |0), H,,+ = (x\t) - (\1 0) + (t a) ~; then (~ 12) we obtain, for af/H. aaHm W =V o JL.2 b -l -3 2 + (515 + K4) a -t - i ) dSn-2i. ( aHM m f.Hm = (31 + 32) - + - - )Z K34. [Hr)} dSn-2, the respective values?3=0 =- fId3dx4 {[(x x -tï )2 +i(2 -t ) [+ x-ix } VOL. XVIII. 55 434 MR BAKER, ON THE THEORY OF FUNCTIONS wherein, in the latter, x,=a and x2=b, and the integration in regard to '3, x4 is to be taken for each of them over the whole range - o to + oo. Hence we obtain r, (1_ 1 ) which, as the general theory requires, is a function of the complex variables (in fact only of T = tl + it2). Thence ) = | (Uldr, + rd7r) = 7r + ( -)] and therefore, as w = 27r2 for n = 4, 27r 7T which is precisely right. 20. Transcendental functions of one variable which have no essential singularity in the finite part of the plane of the variable may be distinguished into two classes according as, to speak first of all somewhat roughly, their zeros become indefinitely dense or not, as we pass to the infinite part of the plane. If circles be described in all possible ways, each to contain a certain definite number, say 1V, of the zeros of the function, N being at least two, the areas of these circles may have zero as lower outside value as we pass to the infinite part of the plane, or may have some quantity greater than zero as lower outside value. More precisely, in the former case, however small A may be, and however great R may be, among the circles described to contain N zeros whose centres are at distance at least R from some definite finite point of the plane taken as origin, one or more can be found whose area is less than A; in the latter case it is possible to assign a quantity A finitely greater than zero, and a finite R, such that among the circles described to contain N zeros whose centres are at greater distance than R from the origin, no circle can be found whose area is less than A. The most obvious example of the latter possibility is the case of a periodic function; here a period parallelogram necessarily contains only a finite number of zeros; and this parallelogram is indefinitely repeated to however great finite distance we pass. As example of the former possibility we may take the case of an integral function whose zeros are the real quantities log 2, log3, log 4,.... The length of the streak which contains the N zeros beginning with log R is at most log (R + \T)- log R = log (1 + which diminishes without limit as R increases. 21. Consider now an integral function of one variable of the former of the two kinds, for which circles containing a specified number N of the zeros of the function are formed of as small area as we desire, however great be the distance R of their centres from a finite point of the plane. It is still conceivable that for proper choice of the constant m, independent of R, and not less than unity, the product Rm- C, OF SEVERAL COMPLEX VARIABLES. 435 where C is the area of such a circle, may be finitely greater than zero for all values of R greater than a certain assignable Ro. We proceed to shew that under this hypothesis the infinite series formed by the sum of the negative (2 + m)th powers of the zeros of the function is an absolutely convergent series. The case n = 1 is that of the latter of the two kinds of functions considered in ~ 20. Let concentric circles be described with centre at a finite point of the plane; consider the greatest number of zeros of such a function which can lie in the annulus between two such circles of radii r and r' (r' > r), the circles being supposed to be drawn so that no zeros lie actually upon them. By the hypothesis, if r be taken great enough (and finite), the annulus may be divided into regions each containing a finite number, say M, of zeros, such that if C be the area of every such region rOm-i C B, where B is some quantity greater than zero. Let k be the number of these regions, which is finite so long as r' is finite. Then 7 (rV2 _r) r m-i _ kB; as there are kM zeros in the annulus, the sum of the moduli of the inverse (2 +m)th powers of these zeros is less than kM r2+m, which in turn is less than 7rM (r/2 - r rm r 1 B r2+m which, if r' =r (1 + ), is equal to o-B (1 +)M( +)(r r); we can suppose the successive circles drawn so that e remains constant; then the sum of the moduli of the inverse (2 + m)th powers of all the zeros of the function which lie beyond the circle of radius r0 is less than l l t r he and can be maade as small as we please by taking ro large enough. This proves the convergence of the series. 22. Pass now to consider an integral function of p coinplex variables, and consider the (n-2)-fold over which the function vanishes, this being supposed to extend to infinity. Imagine closed (nz-l)-folds to be described everywhere convex, and as far as possible, for the sake of definiteness, of spherical form, with the condition that the extent of the zero (n-2)-fold contained in any one of them shall be some definite 55- 2 436 MR BAKER, ON THE THEORY OF FUNCTIONS quantity, say A. In regard to the shape of these closed (n-l)-folds the important point is that the linear dimensions shall be always of the same order of magnitude in all directions. In regard then to the n-fold extent, V, of these closed (n- l)-folds two things are possible as we pass to the infinite parts of space. Either V may have a lower outside value B finitely greater than zero, which case arises ini considering functions having 2p sets of simultaneous periods. Or, the zero (n - 2)-fold may become so bent and crumpled upon itself that at sufficient (not infinite) distance from the finite parts of space it may be possible to find an n-fold extent V less than any assigned quantity, which shall still contain an extent A of the zero (n - 2)-fold; or in other words, that the volumes V may have zero for lower outside value as we pass off to infinity. When this latter is the case it is conceivable, denoting by R the average distance of the points of a closed (n -l)-fold from some finite point, that its n-fold extent V may not diminish faster than some positive power of R increases, namely that there may be a quantity m, not less than unity, such that Rm-lV y B, where B is a finite constant, for all values of R which are not too small. Under this hypothesis it can be shewn that the integral i dSn-2 J Rn+m extended over the whole infinite (n -2)-fold, is convergent, R denoting the distance of a point of the (n - 2)-fold from some finite point. For suppose concentric spherical (n-l)-folds to be described, with centre at the finite point from which R is measured, and consider the extent of the (n - 2)-fold lying in an annulus bounded by two of these spheres, of radii r and ri (r > r). In accordance with the hypothesis we can suppose the n-fold content of the annulus divided into regions each containing a finite extent, say M, of the (n - 2)-fold, such that if V be the n-fold extent of any such region rm-l YV B, where B is some constant greater than zero. Let k be the number of these regions, which will be finite when ri is finite. Then (r1_ - r ) rm-i - kB n as the total extent of the (n - 2)-fold lying in the annulus is 1cM, the contribution to the integral dSn_-2 J n+m which arises from the annulus is less than kM rn+m ' OF SEVERAL COMPLEX VARIABLES. 437 and therefore less than r M (rn - rn) rm-1 n B rn+m which, if ri = r (1 + e), is equal to nB (1 + - n BÊ e r ri we can suppose the spheres chosen so that e does not become infinite; it is therefore obvious that the integral is convergent. It is tacitly assumed in this arrangement that the extent of the (n-2)-fold lying in any finite n-fold extent taken entirely in the finite part of space is finite. This follows from the method by which the (n - 2)-fold is supposed to be defined; for it can be shewn that if (71,..., 7-.) be a power series, the extent of the (n- 2)-fold Q = O which lies within a closed (n - l)-fold lying within the region of convergence is necessarily finite*. This generalises the well-known theorem for functions of one variable, that a power series cannot have an infinite number of zeros lying within a region which is actually within its circle of convergence, that is, cannot have an infinite number of zeros with point of condensation actually within the circle of convergence. 23. The investigation of ~ 22 applies to the integral (~ 13) V = fHjm+dSn-2; denote by (xi,..., x,) as before a point of the (n- 2)-fold, and by (t1,..., tn) a finite point not upon the (n-2)-fold of integration; when R2 = x12 + + xn2 is large, that is, for the very distant elements of the integral, and r2 t +... + t,2 is finite, we have rm+2 Hn+i = R-n+m KM+2 (p) +..., and it will (~ 12) be sufficient for the convergence of the integral that for any assigned small quantity e it be possible to find a finite Ro such that the integral fdSn-_ jRfn+mn taken over the part of the (n - 2)-fold of integration, extending to infinity, for which R > R, shall be less than e. We have in ~ 22 proved that this is so under the hypothesis advanced. 24. The method just applied to the integral fHn+i dSn-2 avails to justify the assumptions which have been made in regard to the other (n - 2)fold integrals considered in this paper. * A sketch of a proof is added below, ~ 27. 438 MR BAKER, ON THE THEORY OF FUNCTIONS There remain certain assumptions in regard to (n-3)-fold integrals, and in regard to (n- l)-fold integrals. We have assumed that if a finite portion of the (n -2)-fold of integration be bounded by closed (n - 3)-folds, the corresponding (n- 3)-fold integrals [(a i fm_-1. affm-i\.0 (i Cs, 2k-1 + Kr, S, 2k) (Hm i- aH -) dS-3 (J,.,K,=_=+i, ~,) \a2Âx-l ax2k / ultimately vanish as these (n- 3)-folds pass to infinity. This really follows from what has been demonstrated. The (n-3)-fold integral arose as equal to an (n-2)-fold integral. In the course of the proof above it has been shewn that this (n - 2)-fold integral is such that if taken over infinitely distant portions of the (n - 2)-fold the corresponding contributions ultimately vanish. Thus it is legitimate to regard the (n - 2)-fold as closed at infinity, namely by an (n - 2)-fold for which our hypothesis (~ 22) remains valid. In which case the (n - 3)-fold integrals that arise are mutually destructible. We have considered also the (n - l)-fold integrals v=f (1 Hm+- K2 aHm+i +..) dSn71, V= aq ~,ax, a:~ ~x,~ + '"dS = = ( -aHm+l + 2c am+1...)dSnltaken over the infinite (n - )-fold bounded by the hypothetically closed (n - 2)-fold just considered. It is necessary to see that these are convergent. This follows because the portion of either of these (n - )-fold integrals taken over the portion of the (n - )-fold which lies at infinity can be replaced by an (n - 2)-fold integral taken over a closed (n - )-fold lying entirely at infinity-and by the proof given above this (n - 2)-fold integral ultimately vanishes. 25. Note to ~ 15. In the course of this demonstration we have utilised the fact that as (ti,..., tn) approaches indefinitely near to the (n - 2)-fold of integration the integral 2w r - (x t) dSn-2 becomes infinite like log mod. ), where = O is the equation of the (n - 2)-fold in the neighbourhood. The following direct verification of this fact is of interest. To a first approximation the points of the element dSn-, satisfy the following equations, the origin of reckoning being taken at the point of the (n - 2)-fold, 'tIX1 + -U2X2 +... + UnXCn = 0, vlx1 + v22X.. +... + Vnx, = 0; OF SEVERAL COMPLEX VARIABLES. 439 these give ( (3v2- u23) 3 +... _ (uv3-n 3vl) 3 +... XI 1 -- X2 --. X3 = — X -—, X4... UV'a - U2V1 U1V2 -?2Vl whereby all the coordinates are expressed in terms of the (n - 2) quantities x3,..., 5x. Thence, using the equation K, dSn-2 = M12, we have dSn_2 = -2 +2+2-n dx... dxn. n12 + n2 We can further suppose the axes so chosen that U3 =... = Un = V3 =... = n = 0, so that, for dS,_-, xl = O and x2 = 0; and dS,,_ = dx3dx4... dxn. Also, the origin being at the point of dSn_2 which is to be considered, x3,..., xn are, for dS,_2,, subject to a condition of the form x32 +... + n2 a2, where a is small and fixed; these coordinates are otherwise unrestricted; we can therefore put dSn_- = rn-3 s- sin —3 0. sin5.. sin n-drdO... dO,_,, where the limits are r = to a, 03 =0 to 7r, 04=0 to r..., =0..to r, 2n- = 0 to 27r. The point (ti,..., t,), as it approaches the (n - 2)-fold, can be taken subject to xtl +... + xntn = 0, t +... +tn = 2, where x1 = O, x, = 0 and (0, O, x3,..., xn) is any point of dSn-2. Then to the integral r (x 1 t) d8-2 the contribution arising from dSn_ is 1 27r a, r n-3 dr. - 2 i Jo (r2 + e2)-d f sin-4 03... sin 0n- dO3... dwhich is easily seen to be a rn-3dr (r2 + e2)1n-j ' putting n = 2p, r2 = e2z, z + 1 = t, this is 1 1+ 2 (t- )P-2 d 2Ji ( +-' dt) ( - -a2î) ^ ( -.-*Y - (} r [i3-2 \ l+9~- i 440 MR BAKER, ON THE THEORY OF FUNCTIONS of which the infinite part, for diminishing e and fixed a, is exactly loge. But as we approach the (n- 2)-fold in the way here taken we can put 6 = heei0, (see ~ 16); so that the infinite part of log is also loge. Thus in the limit the difference _ — \x (\1 t) dSn_ - log s6 remains finite, as stated. 26. In this paper we have hitherto supposed the (n-2)-fold of integration to be given à priori, by means of a succession of power series. Some remarks must be made in regard to the problem in which this conception has arisen. Suppose that a single-valued function F(r1,..., r) is known to exist for ail finite values of T1,..., p, and to have no essential singularities for any finite values of ',..., Tp, namely can be represented in the neighbourhood of any finite point (r1(0),..., T( in the form F= o (T1-7,I..., Tp- 7-p() ). (71r- -r7)..., r r - (0)) where r0, o0 are ordinary power series (of positive powers) with a presumably limited common region of convergence. If the series 0o, 4o have a common vanishing factor at (1(0),..., T(0)), that is, are both divisible by another convergent series which vanishes at (-i1(),..., pr)), this factor may be supposed divided out (Weierstrass, Werke, ii. (1895), p. 151). There is then a region about (r1(),..., Tp0>)), within the common region of convergence of 0k and S0, but not necessarily coextensive with it, such that, if (C1 + 7r 0),..., Cp+ -p (0)) be any point in this region, and the series '0, b0 be written as power series with this point as centre, by putting Tk-7k(~) = Ck +Uk, the resulting series in 'ui,..., up have no common factor vanishing at uzl=0,..., p=O (Weierstrass, loc. cit., p. 154). This region we may temporarily call the proper region of (T1(~),..., 7Tp()) for the function F. There may be points within this region at which *o, <so both vanish without having a common factor vanishing there, such points lying upon an (n - 4)-fold at every point of which F has no determinate value. If the series o0, /0 as originally given have no common factor vanishing at (r)),..., r7(0)) there will similarly be a region about this point at no point of which have they a common vanishing factor. This region also we call the proper region of (71(~),..., Tp~) for the function F. By hypothesis there is then a proper region belonging to every finite point. We assume further, what is not quite obviously a deduction from the former hypothesis, that the whole of finite space can be divided into regions, each of finite extent, each having the property of being entirely contained in the proper region of every point of itself. The function F will then be represented in one of these regions K0 by an expression, belonging to an interior point (), F=,o 0o OF SEVERAL COMPLEX VARIABLES. 441 wherein %, 0o have no common factor vanishing at any point of K0; as we pass to a contiguous region K1 we need a representation belonging to a point (T(1>,...) interior to K1 of the form By considering the equality S0o S in the region common to the proper regions of (i().,..., (0)) and (r7( >,..., 7Tp() we are then able to deduce that aill the points for which.0 = O are also points for which 1= 0, and conversely. We thus build up the idea of a zero (n - 2)-fold for the function F, and an infinity (n- 2)-fold. If the former be represented by e =0, and the latter by P =0, the function F can be represented in the form = e, where X is an integral function; and e, <) have no common zero other than points belonging to an (n- 4)-fold at every point of which F is indeterminate. 27. Note to ~ 22. If an n-fold space bounded by a closed (n - )-fold be taken actually within the region of convergence of a power series in the complex variables 1,.... ep, say b (e,..., tp), where n = 2p, the extent of the portion of the (n- 2)-fold given by 0 = O which lies within the (n-l)-fold is finite. For consider the points of this portion for which y2=72,..., p =, where 7y,..., yp are certain definite values; these points are given by the equation in el, s (el, 2,..., p) = 0, wherein e is capable only of a limited range of values determined by the (n-l)-fold; as this range is included within the region of convergence of the l:-power series b (el, 7y2,..-, 7p), there cannot be an infinite number of values of ev within this range for which (e,, 72,..., p) =. Thus on the portion of the (n - 2)-fold + (e1, e,..., p) = 0O lying within the (n - 1)-fold there exists only a finite number of values of el corresponding to given definite values of 2,..., *. Let dS-_2 be an element of the (n - 2)-fold b =0; we have dSn-2 = 1 J~2dSn-2 +... + |Kn-, n dSn-2, the integrals being taken over the portion of the (n - 2)-fold which lies within the (n - l)-fold; to prove that fdS_2 is finite it will be sufficient to prove that every one of the integrals on the right is finite; we prove that the first of them is finite. Take upon the (n - 2)-fold, = 0, (n - 2) independent sets of differentials given by the rows dixi, dlx., dx3, O, 0, 0,,... dx, xd2x, 0, dx4, 0, 0,... d3x1, d3x., 0, O, dx0,,... d4x1, d4x2, 0, 0, 0, dx,... VOL. XVIII. 56 442 MR BAKER, ON THE THEORY OF FUNCTIONS where, for instance, dcr-ixi), d2,r-i2 are determined in terms of dx2~,+ by the equation 01 (d2a-il i + d2-+2) + ~+ldx2:r+i = O, and d2,xi, d21.x2 are determined in terms of dx,,+2 by the equation (i (d2ril + id2rx2) + ifr+'idX2r+2 = Os Then K/cdSn-2 may (~ 5) be replaced by dx dx4... dxC -idx,; since then the range of values for each of x3, x4,..., x,, for points under consideration, is finite, and, as proved, there is only a finite number of points of the (n - 2)-fold for which x3,..., xn have a given value, it follows that the integral fdx3... dxn taken over the whole extent of the (n-2)-fold within the region considered can only be finite. 28. Note to ~ 8. The following example, relating to the transformation of integrals considered in Part- I. of this paper, seems worth preserving. For n == 4 we have for the transformation from a closed (n- 2)-fold to an (n - l)-fold bounded thereby, the equation (C23 23 + K31 P31 + K12P12 + K14 P14 + K24P24 + K34P34) dSnfdfi {Ki (/a(P12 ap+ 3 aP+ 4 dn-1 Ki1 + -- + +- ax, ax, ax taP1, aP3,, aP2 ax, -' +%7- +%- __ ax, -A a/4 Q1, ax, aX3 ax4 ax, Q' aP4, aP 42 aP43 % c+ d + -P = Q; - @$ + 1=Q) ax, jax, ax, ap1 aP4 9 ax8 a^2 a4 ax, ax, ax,4 OF SEVERAL COMPLEX VARIABLES. 443 aQ1 aQ2 aQ3 aQ4 therefore + + + =, ax, 2ax, a3 ax4 which is a necessary condition for the consistence of the four equations just written. It is satisfied for instance by Q2=iQ1= af +i af Q4=Q3= f + a f being any function of x,, x,, x3, x4; corresponding to these values the four equations just written are satisfied by P1 = P34=0, P13 =, P14=f, P2 = if, P24=-f. But it does not thence follow that (C13 + i~14 + tK23 - 24) f. dSn-2 = f{( +iK,) (f - )) ( K- + iK4) (f + i )}dS for the first integral vanishes for a complex (z - 2)-fold, and the second integral does not necessarily vanish, as we see by taking for instance i (1x -tl)2 + (X2- t2)2 f 2 (. - 21) ( 2) - tl)2 + (X2 - t2)2 + (X3 - t3)2 + (X4 - t4)2 when we get + = (a) (lt) af + a a ~ —+^.*/-=.-a ^. =t, +?- v d} (x\tg ax, a/ X 4 ax,3 dx4 vax a/x, whereby the second integral becomes fd&s + {(Kl+i~) ( -i ) ( X+) (X t) +( +i 4) ( -i (1 t)}, which is not always zero. In explanation it may be noticed that on the (n-2)-fold there are points where e: = 72; and for these f is infinite. 25 Julz 1899. 56-2 INDEX. Absorption, selective, electromagnetic illustration of, Darboux, 9, 324 348 Dawson, 393 Analytical representation of a monogenic function of Declination, magnetic, 107 one variable, 1; of an integral function of several Desargues, 204 variables, 418 Descartes, 197 Arrhenius, 131 Didymium salts, absorption spectra of solutions of, 298 Ashworth, 135 Differential equations, Forsyth on the integrals of systems of, 35; of lunar theory, E. W. Brown on, 94 BAKER, on functions of several variables, 408 Disc, circular, Green's function for, 277 Becquerel, 397 Bendixson, 36 Echelon spectroscope, 316 BERRY, on quartic surfaces with integrals of the first Eckholm, 131 kind of total differentials, 333 Elastic displacement, waves of, on a helical wire, 364 Bigelow, 131 Electric density near vertex of a right cone, 292 BOLTZMANN and MACHE, on Van der Waals' law, 91 Electromagnetic illustration of selective absorption, 348 Bonnet, Ossian, 325 Electrons, dynamics of a system of, 380 Borel, 6 Electrostatic problems, application of Green's function Briggs, 203 to, 277 Bromwich, 324 Equations, differential, Forsyth on, 35 BROWN, on differential equations of the lunar theory, 94 Erbium salts, absorption spectra of solutions of, 298 BURNSIDE, on groups of finite order, 269 Ewing, 396 Campbell, 221 Fabry, 9 Cantor, 204 FitzGerald, 397 Capstick, 186 Fontenelle, 197 Cavalieri, 204 FORSYTH, on the integrals of systems of differential Cayley, 328, 333 equations, 35 Chasles, 219 Frisch, 197 Clifford, 328 Functions, of one variable, representation of, 1; of Condenser, oscillatory discharge of, 136 several variables, 418 Coradi, 116 Corbino, 398 Gauss's formula for solid angle, 425 CORNU, la théorie des ondes lumineuses, xvii Genus, numerical, of a surface, 345 Cremona, 346 Geometry of Kepler and Newton, 197 Cubic surface, model of, 375 GLAZEBROOK, on the discharge of an air condenser, 136 446 INDEX. Goursat, 36 MACDONALD, on Green's formula for electric density Green's function for a circular disc, 277; formula for near vertes of a cone, 292 electric density near the vertex of a right cone, 292; MACHE, see Boltzmann formula in integral calculus, 409 MAcMAHON, on Partition analysis, 12 Greenwich Observatory, magnetic declination at, 107 Magnetic declination, periodogram of, 107 Groups, Poincaré on continuous, 220; of finite order, Magnetic field, influence of on optical phenomena, 380 Burnside on, 269; and optics, Lovett on, 256 Maxwell, 137, 143, 292, 361 Mehler, 292 Hadamard, 9 Meteorology, 107 Heine, 292 MICHELSON, on the echelon spectroscope, 316 Helical wire, waves of elastic displacement on, 364 Minimal surfaces, Richmond on, 324 Henrici, 116 MITTAG-LEFFLER, representation of a monogenic funcHill, 97 tion, 1 HOBSON, on Green's function for a circular disc, 277 Monge, 325 Horn, 37 Moore, 275 Hornstein, 131 Hunbert, 345 Newton, geometry of, 197 Noether, 333 Integrals of first kind on quartic surfaces, 333 Ions, dynamics of a systeni of, 380 Optical phenomena, influence of a magnetic field on, 380 Optics, and contact transformations, 256 Jordan, 37 Oscillatory discharge of an air condenser, 136 Kelvin, Lord, 137, 277, 287, 291 Painlevé, 7 Kepler, geometry of, ]97 Partition analysis, 12 Kerr, 406 Periodogram of magnetic declination, 107 Klein, 269 Picard, 36, 333 Klugh, 116 POINCARÉ, on continuous groups, 220 Kônigsberger, 36 Poincaré, 36, 333, 408 Kronecker, 408 Poncelet, 204 Kummer, 345 Poudra, 205 Preston, 391 LAMB, electromagnetic illustration of selective absorption of light by a gas, 348; referred to, 365 Quartic surfaces with integrals of the first kind, 333 Lanthanum, 301 Laplace's equation, with minimal surfaces, 325 Rayleigh, Lord, 109, 139, 147, 348 LARMOR, on dynamics of a system of electrons or ions; Rede Lecture, xvii influence of a magnetic field on optical phenomena, RICHMOND, on minimal surfaces, 324 380 Rotation, magneto-optic, 396 Levy, 266 Routh, 403 Leyden jars, oscillatory discharge of, 136 Runge, 2 Lie, 220, 256 Russell, 301 Light, selective absorption of, by a gas, 348 LIVEING, on absorption spectra of solutions of didy- Salmon, 346 mium and erbium salts, 298 Schur, 221 LODGE, on the discharge of an air condenser, 136 SCHUSTER, on the periodogram of magnetic declination Lorentz, 362 from records of Greenwich Observatory, 107 Loria, 203 Smithells, 393 LOVE, referred to, 348; on waves of elastic displacement Sommerfeld, 277 along a helical wire, 364 Spectra of solutions of didyrnium and erbium salts, 298 LOVETT, on contact transformations and optics, 256 Spectroscope, the echelon, 316 Lunar theory, differential equations of, 94 Stokes, 409 Substitutions, infinitesimal, 245 Macaluso, 398 Surfaces, minimal, 324; quartic, with integrals of the first MacCullagh, 380 kind, 333; cubic, model of twenty-seven lines upon, 375 INDEX. 447 TAYLOR, C., on the geometry of Kepler and Newton, Vitellio, 201 197 Voigt, 398 TAYLOR, H. M., on a model shewing the twenty-seven lines on a cubic surface, 375 Waals, van der, his law, 91 Thalén, 301 Walker, 348 Theory of numbers, 12 Waves of elastic displacement, on a helical wire, 364 Thomson and Tait, 392 Weierstrass, 325, 335 Thomson, J. J., 359 Wilson, 393 Van der Waals' law, 91 Yttrium, 301 Velocity of light (v), determined experimentally, 136 Verdet, 397 Zeeman effect, 316, 389 CAMBRIDGE: PRINTED BY J. AND C. F. CLAY, AT THE UNIVERSITY PRESS. Phil. Soc. Transs XVIII Plate. DIAGRAM OP 26 DAY PERIOD Year: ear0 0 5 10 15 20 25 A1 1440 ____-_ — I\ + + B 9013 20 ++ + /_ 1200 ~o ~ ~~ + 840 ______+_+_l__ -Ill____________ +. + ô Ia / '.. ô ], + s (/ o? 480 360 // 3-~~o" + 240 120.- +. + + O t OO../:.,. ( The points of strongest amplitude are marked thus,, weakest,,,, -,, intermediate,,.,, Phil. Soc. Trans. XVIII. Plate IL. DIAGRAM OF 27 DAY PERIODs Year Ba e40 5 10 16 20 26 $ + + 1560 -- A 0 1440 ___ -_- _ + 13 20.". / i A i -s... e r m e d i e " " " 0 ):: + 100, + / 960 ------- 840 - e / vl 7200 600 - A t / + + 480 120 __________ Q,, weakest -,, 4,,, eaktest,,,, 4 - C(m bridge.1P/zilosoprziclt Transactions, -'o1. X VI, Plate 3. PmLATES 3 -23 IL.LUST.RATI:NG ProFseso IVEINfL S PAPER (p3p 298-315, On th e effects of Dilution, mnTempaturer aad otheî cireicnmstacnces, on the A bsorption zStpectra -tf so/ttio-rns of Dicidyniuwn antid ]Zriu salts.?These plates a-re ail:repodutio enlarged tco double the size, of photographs of some of the spectra friom. which the conlu-etsions in. the text have been deduced. In the processes of enlarlgelnent and:ireproduiction soma of the lf inter dlet ails visible in the original negatives have (perhaps una voidably) been l ost: but they present the salient features of the' changes in -the spectra piodl uced by t;he variations of cirCemnstance Tihe references to thlse plates 1in the text applied tlo -lthe original negatives and were printed before the rleproducti;ns were reaf d t he ]lc la.tt er, being postivoes.i, tare reversed, and:i orderit t-i it lh references may be easily i:nt eligible it haLs been necessary fto place thé red cu-l tds o:f te spectra on the left htan.d UThe figures at the t-opt of each plate tre the approximate wave lengths of tthe bands in. t?é spectra benealith tle:, an:. sl.ficiently indicate thie range uotf[ tle spetruil ph.'totogrraphed. PLATE 3. Absorpltionls of solutions o:f didyuini chlloride, i- four degrees of dilution inl tiiacknesses inversely;as the d(iluti-ons. 'lThi ost concenxltrae ated solution contained 1t40-7 g;ra.s per litre, an.d the absorbent thickness of t his solution was 38 min. — î^'~~~. Ti s.uon S C s tre g tl -i strongest t so lut on - --.....i ~ seolutioz-i 1/4 strenigtl i3i5iiii lo;i 152'5 min. [hink M ~ '"" solution 1/2 stirengh 2-: 760f mini. lhick I:t wi ll. be n-oticed how very nea erly identical these four spectrat aie. r -The original. photograph shews a number of fziint antllds which have not comne ont inl thle reprodiuctiuon They re however as nearly identical ini all four spectra as arae the stronger lands here reproduced. ~: - ` m: rQ tTt |@0V - Gs \; t - 0 m.a 0Lq S 11 1 ^ " a -,-.E l l, "1> ICI >: ' 'S ~,, Ccambridge Philosophical Transactions, Vol. XVIII., Plates 6, 7. PLATE 6. Absorptions by solutions of didymliu nitrate, concentrat e d, and extremely dilute. The most concentrated had 611-1 grams of the salt per litre; the other was part of the same solution diluted to 45-5 times its bulk. c0 T-I O O cs a! Ct^ o oo c 305 mmn. thick There is very little difference between these two spectra except that the band in the yellow is broader with the stronger solution, and those at X476 and 427 more washed out. PLATE 7. Absorptions by solutions of didymniunm chloride of concentrations equivalent to those of the nitrate used for plate 6: the stronger containing 462'9 grants of the chloride per litre. C D O ro ~ t- ~ O î H o c: z S c1 o 0 1 s 7solutico 6'7 min. thick ~~~Te2 tls noo definit1/45-5 strength 305 1min. thick There is no definite difference between these two spectra. ------- 6(tC bri'tCde lM.tkiosop/ica2 l 7'rai.sacttions, 'I:.0 XVtI, Plates 8, 9. PLATE 8, Absorpti)ons of a solution of erbium nitrate containing 467'6 gra ms of the sait 1per litre, and. of a1 S)olutiolt, }ade by U di]lutino t thoe rl'er to 4,5)'5 times its bulk,:0 ïA -4 1 -J ppt 7H: tO i~~~ ~~~~~~~ -l'';:-:~i;' stronger solution 6-7 miin thick 1/45'-5 strength;'305 mm. thickç. The t bads are more diffuse with the stronger. solution, that t about X377 being decidedly broader. Tfl.e bttand at about X44.9 is more distinctly seen in the original and is more diffuse with the istronbger solutionn tlhani wit tle weaker, PLATE 9. Absorptions by solutions of eJbnl chllorîide of concentrations equivalent to thLose of the nitrate used for plat.te 8; thré stronger solution contaiining 363'3 grams t of the salt per litre. cc ct 00 Coa T^ 0 L L î-0 i-% ^ - -^ ---- - --- 1/45' strength 305 mni. thick stronger solution 6'7 min. thick Trlere is hardtlIv a difference betw-een. these two spectra except that the bau)cJ-d about À377 is rat aher stronteÇr citfhi théu more concen.tratedt thaui withi the dilite solution, owing probably to tlie overlapping of the general diffuse absorption of thie concentrated chloride at the more refritangible end. The fainter bands which are visi-ble i the original photo!raph ca.n hardly be traced in the reproduction. <embridge P/ailoesep/ie Treïsactiens, VolN JJIL, P/etI 10L PLATE 10. Absorptions by sol iutio.is of erlilmn chlride and eq ui valenrt solutions of erbtium nitrate, alternately: fut1r degrees of concentration, the strongest having 726'6 grains of t he anh.yrons chloride to the litre, tind the ec-quivalent nitrate 935'2 grarms to the litre. iQ D C C. ' '- G _0 S"t 00 Y' } ï 0 0 L ~ r~1~...no ~,,-~.~~~ - =,~;1st crongest solution of Er C1o,iii: i:i..2~r~~~~~~~~~~~.ii~ ~38mnfl t thick st roniest solution of Eî (N0 '>): 38 mi-n. thick hal tst'ength ch]oruide 176 int. tbick;f:s~ ( ), C, z ia~~~lftll- 6t;e~~l ength nitrate ' ~ ~~~4 ti-1~~ ' ' ' '~. m...7i.. tilick (.aiirter-sti'eogth chloride I! ~~>554;4!2":i15?'305 min. thickly one-eighth-sti-c ng-th nitrate 305 mmn. m thick Thec greater diffuselness of thie bLnds wiih the more concent'ted solutions of theli nitrate is evident, a1id so is t'ihe extension of the gn\enal a bsorption t tbe niore refrcngYle eîd uf tu.he spectrulli with -t-e ouist eoncent'-ra'te solution of the C'lhloîide, The difference between n the absorptions by thie chloride and nitrate din-inishes -ifth. dilution and lias almost, or qitte, disae, ipea lred in. the case of thie w,kest solutions. Cit bridge Piosopidal Toenisaetions, Vol. V Plate 11 PLA TE 1L.Absorptioils by lidyrmiml elide and nitra iite, atenately, iln equivale-lt s(olutionls of four degrees of concentration, egiii.gin tl witl the strongest solution coutaini.g 462*9 g'ram-s of the anllldrous ehtloride to t:h litre foilowe) d.next with theé qui talent solution containing 61.1'1 grams of nitrate to the litre. 0a *ïi 0 cr e t C t OC C W L X. O r ' i |' t ~ - < -C -',3 C;*i:hI.... o. g.. st ro ngest solutionn of.tiCI't as;o::: {h 3niateo 38 m il t hiek inV)~~ th~ ms;;dilj | ) 1testron est solution of '5H' Iia~~~~~~~~~~~~~~i(NK0i) 38 is m thick,sal i.strength chloride 8 ";:?1:.:"3:, _:::::~...x'~-;:3:_;::::;;.30a176 mxin. thick hIalf —strength niutrate 76 mmi thick 5( i.t.J,) `,li gti',.t..)!( Clt -ÉJ#.le 'li>t..-t].<r.t)t.i.e-5ll tuas tes -st isofre irsth chloride 1'3'5 in ni. thick quiaritesr-strength nitrate 1i52' mim. thick ois.-eio lsusrt, length cthicrire 305 imsu. thick t oie-eighthlssti ength nitrate 0115 1mi. thick cosnce.utrated so.ustions. of chloride is evident in the, uppermost fi ire. 'W' tih sucl strro.ng solutions as were -ilsec tr tiese pshot(sgralphs either differences tetweeu the absorptions y1 chls-ride ~'usd nîitrate carl be seis. onlyiv in the wea kear ba.usds sucih as those -urom. X433 to X4.06. Ihesne ire weakened by diffusions in the c.ase ot tihe istitrate, bit there is e tt dierece te bsolirptions by chloride aidn nitrate iin tihe îïsiost.dilute sîoltiiori s. (Ckmbr'i, ge, P/tosop/'c/ Trassatioss, Vol. Xi oII, I.. Pltae 1. 2 PLATE 12. Absorption y solutions )f iï.droeldchl sc l id in a-leohol, 1and ii water, 1otispasred. *with thie absorption )y v:re aL' -,, strongest solution of HCI il alcohol, 38 1am. thick 2 lhalf.stlength, do. 76 in.-tik quarter-strength, do. i 1525 rom. thick pure water 305 ins. thick _Thé::~ _eotéyme b ed st arounest solution o fe CI in wvaler, 38 min. thick ihé gaorptio h lf-t strength, do. 76 ni n thick 7 q ijuariter-stereingthi, do, 12o insis, tMie one-eighth-streîsgth, do. 305 mils, thick Thie ediiet of tise hydicotilorie scd t tile Isoroe nrerllsibde en-d. is isible she ai.u the dolss-îsut,îofi ut thie.bssioisosi w titols Ailrssisslts edesî7eets:sti onol eet etoo t seid sa en s the achlreous solutions u>is. N sà, 6, lsîille Aisssisssksed eosscenistislo hais little or no e5ffet in the c ise of the alcoholic soi autionsiN os t 3, (v:i)u:Jdge Ph/ilosoph.acl..2antsactiols., o. t [''F]L, Pla te 13, PLATE 1 3. -bso13pt3ils.t by >soli.on of erbiu-ri. icldoride, cold. antd l:t alteri.itel}t, iln t-wo degrit: s f on centre itiou. cô~ s f i C C lhalf-strenotli solution i "C~~~~~~LI~~~~76 mi. thick t 23 C. hlf-i strenugtli solution 2 76 ino. thiek aet 97'0 ( stronger solution 3 i)~~~~~~~~.ie mmO:~~38. th. ie 25 ' -C s-ir onger sutlution 38 inu. thiick at 9 C. Tlhe extensionr i then general absolpti( n at thc more refranrile end of thié slpectri iii) hy rise of ter-pen> ture is ueaniest liu these ulotorplihs, and st v the greater diff-senless f tti bands at a-:dbout X449 t-d À488. ;:,.-brib,'</e IP/.gtlosoph.ki/a(l /T7/Ylc~.it t.oP lo!. X II.-, 1?.Plgate.14. PLATE 1'4. Absorpltions by solttolutions of erbium nitrate?, cold ai ho>t alteruately, in four degrees of dilution;, i t.i:k1e.-ss,8s ilt iVe.r:rsol as thle dil.titions, The stlronigest s-ol:.tion hld 566 g'ramts <of eri.) nit-irate.xperx litre, 31. strotstt t o g tuti sool s 38 mi. thick at 22- C,::::}:::dS înore: difuse, a:n" th:: t::t:i: `:thé badn i -lo.ds a, S 9 ' Ca. solution 1/4 sltren hgtc:Th s we o min. thick at 23p Ci,,o. do. at 94-" C, tl.a: th' repr)odtusolution 1/8 streCilgt a. 't.dSti]lt;in in tshé spXt3rai o thtét. cold305 Moin, thlik ait:3 23' t oSMj~ ~do. Cdo. ut 01~ C,. solution 1/2 stre:ngt 76 miut tMiek at 23> C, do. ai 97 Co lt o dll w i-oticcd tIni t totP etlc> of li-tiii % the solution. is il-u oeieral to rendertilr iu absorption baids: dmd e tiffwse, aind tha, t i i t '.e bauds tIi.P: increweo i utffuhseness witlih inicrcasiug' con.cetra.tion ~of te sod:i toin. whicl are imuost aft. lctetd bv tiît}e rise of teimllcat:île, Tflic ourginal slibo\N-,tograîds suer,serni fmier bauds whicùi haxe F nt xi couic ont. ici tlue reprodrictioni, nid lidIso slie.w the lighter ni tcrspaces between tUe. àbsortioîis ii Uic nitra îîiolet niiacl more distintiri t1han i:the ouc~, r tilt, Even i. the:erodu. ci:on ths light_ intersptt:ces iii the dltra violet arei tîore distiilet î hiî-I F lie sc ctit of thte eoid sol.uti.orn:s tbhall iii thiiose oft 'te hiot solutionsl (Cabriïàge L>/iiosphil Transc;tions, Vla. X /` L1, Plates 15, 16. PLATE 15, Absor options 'byL solution of didy.vni.tn sulphate, cold aund ltot, in two decrees of concentrations( Thel t1tbi'Jci' t.ioti', w.( it It, s' tur ticed soi.ti tlin { at )(3 C,} Ô i ~ C: - Ln o stronger solution cf Di( S0 )3 i,.,< i-: - li_-~ 38 min. th ick, at 23& C slinee rsolant ioni and saelet thickiiess, at 90~ C. half-strengt.h solution half-streingth solution 4 ~ ~~:1:i-~.-::.:'-Iii1-:1~-:::::1-:~76 in. t.nn ick, at 92:: C. ThLe oesion cf tae itttil dtsorptio-n. iet ttc e mai-e rtlrarinitlc einil ti oft spec-trunt, aJd ut înercascd dii.ffiîsnerss (ni iho thié bad. i t thé li`nte9, lboi thte rise of' telltcraLture is plainlr secin ini these pi htoroapits ]PLLATE 16. Ai.sortptin'is by s> i-otlit of c ritli chloridic, teuttrl al t aciut i tiwo deg-res of concaetir ation; thle Stroii)ir - I.icttîa l. tml:ti.n liavinur tj'' g76 o 'îis of th, cli.l.oideto to e litre, and tle acid solttiotn ha-vin — tbe:ides an:). ai:mo.t of h.ydroci'.lori acid (.iivat lenrtt to the amount of 'fnelutral salt. o- -i c0 t- ~ t' hstT stronger neutral solution 38 ramo, thick st longer acid solut io i 389 rit. thick one3 third strenigtlh neutral solution, 152'5 mma, thick 1omet4hird strengths acid solîufaon, i 152'5 oins. thick (TVite tliksesi ss tiof t l.sorbetit, ss ollutionu s is.(ott prO)îOttiofnca to the diltIticons, so) ttatt ttihe atbsorpttiots. of itgures 3 c atd 4 a tre pro-ici d by ae qulantity ofi sait ie-third greater than that which t,gt f igtlre'es anlld t, whic.. makes tii iba.nds of 3 aind 4 strongt'er. Tr itect of the itaci i s telofHs tt extend tîl. génerttal alsorpititîtou ts t thé moret.or 't-reainible edit I i f tule spectt'tItn`. (t'am.br, aid7ge J/ïlO-1)ci, /tfc iCR<s aottî s^t2e &t Vrc tti, s.:wt,. xV.I T. Plates 1, 1.8.. PLATE 1.7. Absorptions by sol-utionfls of e, rbium nitrate, neutral, at-nd acid, in two degrees of concentration. Th lstron]r neuttra;l soluti). on had 935-2 glran:s of tihe sait per litre, a nd the acid solution hacd inu it besides ats -T.much nitric a cid as waLs exquiva leri. t t( tihe a.inount of neutral sa.llt r:, " e cw.o j t *f4 - - t — -t- - _- --- Astronger so lustion, neutral l38 mn. tirîck tlronger; solution, aeiid 38 min.. trliik * atfitstreugtii, nueutr.ai 76 inr. thiick haittf,,~streirgtlr sol.fioni, a.ci. 76 nrin, tldick Trhle affect ofi tifre r.cici. i-l rerule ii tl-i.e, n —ids, ci-:or diffcLise is s- iee lu these plr.otog,;iap.s, an.i i- thel extension. of the age-erat tsi:s oti o n t t the more refrxn gible.e enod of -tihe second figit.e. PLAT1:E 18. AB4Lsîjtm~tti on. bu solrîtiorns emf ffidmdryuur lhuacdic -initrl arime acid, inm truo degrees )f ceuumcentrrrtirmrm timem acblfd strloimms cturrmmnmiilg te sutrinet uaou:limmlllît of didynrti;i mm:i per litre, ris tIre iertruri slrut;iolms Bot wit. r. himvdrocliioiic aucid in udin ltiou Oc 7 t- - O f stra..o.nger soltio. cf o)iDC1,3:urentral, 38.rim. thick stronger solution, acid 38 rram. thick hal flstrenrrgth, ileultral 76 minur. thiet. 4~`:A':~:::::~ ~ m iirat~ half-strength, acicl 76 min. thick 'flu(er ci.ietfl e:fit ~f ftie relUd is ti stctd. t-llt gel-cmr'ul uRb,~T)mrî)ltluî it tîro orure re;fnirluîiB elcii cf te s1etrmr (C bide 'P/.ilosophiclc - 7Tansactions, Vol. XI 1/L, Plates 1 9, 20 PLATE 19. A..tbsorptionls by nearly éq. tuiva lent solutions oef d.cidlniun cîhl:oride in water, in alcohol, atnd iln alcohol changed with h 'droclor.'i aceid, Theé aciCl solu.ti on iwas.rprepaird from. tl] e neutral ailcoholie solution.p by t~passin.g hydrochlioric acid gas infto it a,nfd was tfund to lbe a.:)out:li:ne-ttenthis of théi strength in didyminiu. otf 'Xth(e iteuf tra, soln t ioo n, 0'; I:i < ~L *L Cti 0:. > 0. -i ~ > s* + ineuttril agnieous solution. ne utral alcotielio solution a od acl. oli.i solution it/ Tlhté g lra(!'tl 1t>0J)t.io 110 ltt tf.li or r'' tnefhrnil ctd is xteund3cd a litiffe b1y ttle aico011ol ad1 dJ. tiil '0lore iy lihe tlddition tof 0ci.d, Tlcé n'al:balds a'e gelr(ct<lly t'enlretcd imo.re' difiise 100<> tcîcitttl lnld an little sbuift;cd touardss thl ied el:d if t:lo. s]'ectrtl: tii:> shiit! il-treatsina g ns t]':e re:<ifrangibiit?" dcireaices. Tle tl acid seem)is o diftin-se aiwa.y th baltnds in th, blue, t:es strong air at about X-0 aO Lre just visible in t;he spectr'ml of t l.eto a<eiid so>lut.ioni corlsidetrali bl shifted to:wa;rds tlle red. Anid thel strong' gr'ou il. thi. ye l(ow is still. sio t ' cIl ' '<if I > s piea, nd uf:.d ()lit thc..tt S\iveral of thie cotopoinentt bands:e sear at-'cratetd P: I-,A.T[,E "2, <.A bto.l'tptio.l by équia..cetin.t so.ution.s ot tf d.iyntim.tlln te't in wtt:er a, f nd. ln. glycetroi.l ( S t; ry, <s rI -ï-O -. i,:'n '. ":' <s < s S tI _~<< "'0<.':l~ <~aq.ueo t s soluttion glycerol solution.No ietn i.te shlifof the tht l h gtvertil ttt-ppers in thi 'ltototgî )]-I, l)ut there. ist ac xt nson. of thlic grcl-tter'l absctrpttio at;il tlthe itmore refrt<angible ettci of thtl st-pctcftit:tand the bucds are rendered moro difftse by ttfl.ci glycerol. tmb; ïiJQje />/io.sopM/ca/ f anisactions, Fui. N V/7t, P/l/tes 21,:22. Pt)L tA, l~I''TE 2:1,.A.bsoerptuios by gllas o bo rax coloured witbh. 'idlSiun 0xi\de ani by a, sltiotn il water of didYni-niA.nt.i:trate eo:utai ning-l ai qiuantlity tof didymt.iuni. eqoiu to e tt.it n1 tuhe glass. C; çC,e " S - G 0 bora glass coloured wit. didymium aqueoms solution of didymnirum nitra e ThF'o es b are 'phot s ae dis.figu'red witi horizont-ll. lines due teo dust on the sl.it of the spectroseopei It will bt see..i that tie bFandAs are for the in)st part shifted bI- t;le borax Fut verty neoqually se;" also tiat th.o bands are rendered more diffuse byv the borax andi soe alnimost di.ffiused way. PLATE 22. Alsorpftio(ns by oquiatlent su.tiont: omim didamu.i actate in acetic acid at:.d didyinii nita'miite n. 'water. a: ' i i shitt diîinishnis as Ith hoFnd is Ieass taefîmgîragb Fie ut th disp~erion i eth spuectriscuîo aiso Aimninisloes as thc light is less refrangible; so the apparent di.innution of the shit is not altogether real. Taho scetie acid. asi icr'eses the di itseises of th boadsn a s veboisr, '7nif test in tbn cse eof ot bald. at about X471, abnld inia be te iraced i on t hetrss (Can'bridfe Pkl.ilosop}/.cal.Tr'ansactionsf 'oI. XVol JII Plate 23. 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