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Cui sere piace ae : : ae eas ee apex : : a . oe re ? ee ‘ ars S : : : ; oF * ? & nee! 4 Wee Fe marten pee COtrrad 7 poM rump 9 steers, erennn Fae net ese. iS ehedeae de tes FA2 Arts aes needs Siar vat 9478 date Ed at eg dada ees +", 4 Coop hah eres eee ath ara ans atest teat: Seip Tate SRL PE ESS ee ta oes p Pte: ’ PENS 9 Vast S75 Rorinnr ese. POPS apnoea oe ‘ Orr ty Pea aheed Maye eres ne wy Fake oA : se ‘ a . . eA ‘ rai ‘ . % fee Sao eee Eas i aay tre ants Were Pitsende BEG y wey et te By eT ite ene sob SE DOLI RAR PIS egies Wired i ® Sforead enna oman Doiidetgt eee » whe Hoes prea sel ae eats PENS arty rae , a 5 ‘ ee Na ‘ Ce De Bioehs Cate us os UP U wre ats i : be oe bAht weet eae . rit vets Pea sane ogee ind Ae Naru ioieh dalle Digitized by the Internet Archive in 2009 with funding from University of Toronto http://www.archive.org/details/transactions18camb SIR GEORGE GABRIEL STOKES, BART. TRANSACT TONS CAMBRIDGE Serio PHICAM SOCIETY. VOLUME XVIII. as] CAMBRIDGE: AT THE UNIVERSITY PRESS, AND SOLD BY DEIGHTON, BELL AND CO. AND MACMILLAN AND BOWES, CAMBRIDGE. C. J. CLAY AND SONS, AVE MARIA LANE, LONDON. MDCCCC. ADVERTISEMENT. Tue Society as a body is not to be considered responsible for any facts and opinions advanced in the several Papers, which must rest entirely on the credit of their respective Authors. Tue Sociery takes this opportunity of expressing its grateful acknowledgments to the Synpics of the University Press for their liberality in taking upon themselves the expense of printing this Volume of the Transactions. Sm N June 1899 the University of Cambridge celebrated the completion of the fiftieth year of the tenure of the Lucasian Professorship by Sir George Gabriel Stokes. The Memoirs in this volume were presented to the Cambridge Philosophical Society in response to a desire on the part of the Society to commemorate the long and intimate connection of Sir George Gabriel Stokes with its imterests and welfare. April. 1900. sti 2 iit. Wat Will WAMU XA CONTENTS. Order of Proceedings at the formal celebration by the University of Cambridge of the Jubilee of Sir Georce GaprieL StoKes, Bart., Lucasian Professor 1849—1899 The Rede Lecture: La théorie des ondes lumineuses: son influence sur la physique MODEL TED Ye ALERED, CORNU. cose sc nencesene nee none eae ees eee soli als eiisa stasis (senisinwis scree On the analytical representation of a uniform branch of a monogenic function. By a Gee MIPTAG-LERFEGERY h¢2 sissy sais seis od eclace soe Peo eae Sane int agg Side eG Se ees Application of the Partition Analysis to the study of the properties of any system of Consecutive Integers. By Major P. A. MacManon, R.A., D.Sc., F.R.S., Hon. (Merit CaS ioc orcas aciewers cic wisirileltiscienpa ae atastelde Asante nie ee etaeias si daseis netstat stteueie siete oles ssietac(ave cies On the Integrals of Systems of Differential Equations. By A. R. Forsyru, Sc.D., HRS emosdlenan = Erofessor of cure: Mathematiests.-ass-eemseeeeeen eee os eeece sca eses ce Ueber die Bedeutung der Constante b des van der Waals’schen Gesetzes. Von L. IBOATS THiael IDO IW YNolsnok obey AWW Geoegenacoan boc noose acoc aad bodonbohacconobasduebaessaoods On the Solution of a Pair of Simultaneous Linear Differential Equations, which oceur in the Lunar Theory. By Ernest W. Brown, Se.D., F.R.S., Professor of Mathematics at Haverford College, Philadelphia ....................c.ccceeeeseeeeceeeeee The Periodogram of Magnetic Declination as obtained from the records of the Greenwich Observatory during the years 1871—1895 (Plates I, 11). By Arruur Scuuster, F.R.S., Professor of Physics at the Owens College, Manchester ......... Experiments on the Oscillatory Discharge of an Air Condenser, with a Determina- tion of “v.? By Ottver J. Lopes, D.Se., F.R.S., Professor of Experimental Physics, University College, Liverpool, and R. T. Guazusroox, M.A., F.R.S., Director of the National Physical Laboratory ................2.....ceeceseeece eee eeneecees The Geometry of Kepler and Newton. By Dr C. Tayrtor, Master of St John’s Chelll@4#2 sconsenooobeonqaseguracs sonabonddoudoadoenodc7ddcodo00 sosnneanoseodaBndonoonseDcosaoceOREaDsESE Sur les Groupes Continus. Par H. POINCARB .......-.12.--1eeceeecceeeecneeeceeeee eee eetees Contact Transformations and Optics. By EH. O. LOVETT ........... .1.....cceeeeee reece ees On a Class of Groups of Finite Order. By W. Burnsipr, M.A., F.R.S., Professor of Mathematics, Royal Naval College, Greenwich ..................0s2csecseeeeeee eee neeees On Green’s Function for a Circular Disc, with applications to Electrostatic Problems. By E. W. Hosson, Sc.D., F.R.S., Fellow of Christ's College.......................0-+- Vou. XVIII. b PAGE vil 91 94 107 vi CONTENTS. XIII. Demonstration of Green's Formula for Electric Density near the Vertex of a Right Cone. By H. M. Macpvonarp, M.A., Fellow of Clare College ............... 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. Livernc, M.A., FR.S., Professor of Chemistry ................0..2.0c-s-0cenee XV. The Echelon Spectroscope. By A. A. MICHELSON ..........:.0.-cceeneee cencecerecoe eer ececes XVI. On Minimal Surfaces. By H. W. Ricumonp, M.A., Fellow of King’s College, (ORD eG ET) ocascscaesasdosasnacodococac + docosusodace dur sao se beastssisecquceosecondacsaaesenscosOreda> XVII. On Quartic Surfaces which admit of Integrals of the first kind of Total Differentials. By Anrruur Berry, M.A., Fellow of King’s College, Cambridge XVIIi. An Electromagnetic Illustration of the Theory of Selective Absorption of Light by a Gas. By Horace Lams, M.A., F.R.S., Professor of Mathematics at the Owens @ollecemWancheSter s-cc-ccs- so. eee rece tee ces eect tenon coe eee ene eencen date rete eee eaten XIX. Whe 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 une) Whetiyerkny 8 (Ope L ennnconscsossocosdeonjaascagasonnoe ses bodanansonsossaonop9acsaoRese XX. On the construction of a Model showing the 27 lines on a Cubic Surface (Plates XXIV, XXV). By H. M. Tayror, M.A. F.R.S., Fellow of Trinity College ... XXI. On the Dynamics of a System of Electrons or Ions: and on the Influence of a Magnetic Field on Optical Phenomena. By J. Larmor, M.A., F.R.S., Fellow of Sty John’s College eccssckosqsandeasbace aininets sees casinos saeeens deen ey seer ea Re eC 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 R626 (<>. gras >, ni SCR oS RN OAS AIRS 0.7 CARRE ORO BEEECE Ae onu doe otode PLATES. Frontispiece. Portrait of Sir George Gabriel Stokes, Bart. I, 11, 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, PAGE 348 445 » 932. In regard to § 7—11, reference onght to have been made to the results of Lie, Wath. Ann. xiv. pp. 373—378, or to Darboux § 325, from which the special type of surfaces considered might also be derived. vl a ORDER OF PROCEEDINGS AT THE CELEBRATION THE JUBILEE OF Sir GEORGE GABRIEL STOKES, Bart. a? y TRO ‘ ILLW STR ERODE PHILCSOPE) a» { -RGREGIE MERITO / CEQIRCIO GABRIEL STOMES f NEWIFONE CATHEDRAM APYD CANTABRIGLENSES ANNYA LAM Le Cees TL ET SYA ACAD) PYAR ACAD. SaLvTnet APVEISS APA, Sf TAM SISL GQPAM UPS) DE TALL V ; TANTO INGERIO Jf : ea EONTES ORDER OF PROCEEDINGS AT THE FORMAL CELEBRATION BY THE UNIVERSITY OF CAMBRIDGE OF THE JUBILEE OF Str GEORGE GABRIEL STOKES, Barr., 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 Museum. 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 Thornyeroft, 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 Reva ©. He Prior, MA: 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. vill ORDER OF PROCEEDINGS. University of St Andrews University of Glasgow Academies of Upsala, Copenhagen, Helsingtors University of Aberdeen University of Edinburgh University of Dublin Royal Society Académie des Sciences, Paris University of Pennsylvania American Philosophical Society \ Gesellschaft der Wissenschaften zu Gottingen New York, Columbia University Princeton University, New Jersey Imperial Academy of Military Medicine, St Petersburg Bataafsch Genootschap voor Physika, Rotter- dam Académie Royale des Sciences des Lettres et des Beaux Arts de Belgique Manchester Literary and Philosophical Society Royal Irish Academy Royal Society of Edinburgh St Edmund’s College, Ware Ecole Polytechnique, Paris Ecole Normale Supérieure, Paris Royal Institution P. R. Scott Lang, M.A., Regius Professor of Mathematics. Very Rev. Robert Herbert Story, D.D., Prin- cipal, and Lord Kelvin, M.A., Hon. LL.D., G.C.V.O. Professor Mittag-Leffler. Sir Wilham Duguid Geddes, LL.D., Principal. George Chrystal, M.A., Professor of Mathe- matics, and G. F. Armstrong, M.A., Pro- JSessor of Engineering. George Salmon, D.D., Provost, and Benjamin Williamson, M.A., D.Se. 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 of Physics, Royal College of Science. Professor Becquerel. Professor G. F. Barker, Vice-President. Edward Riecke, Professor of Physics. Robert S. Woodward, Ph.D., Professor of Mechanics and Mathematical Physics, Dean of the Faculty of Pure Science. Professor Edgar Odele Lovett. Professor Egoroff. Dr Elie van Rijckevorsel. Professor Alphonse Rénard, Professor G. Van der Mensbrugghe. Reginald Felix Gwyther, M.A., Secretary. Earl of Rosse, K.P., President, George F. FitzGerald, M.A., Professor of Natural and Experimental Philosophy, Trinity College, Dublin. Lord Kelvin, M.A., Hon. LL.D., President, and Sir John Murray, K.C.B., Hon. Se.D. Right Rev. J. L. Patterson, M.A. (Oxon.), Bishop of Emmaus. Professor Cornu and Professor Becquerel. Professor Borel. Sir J. Crichton Browne, M.D. (Edinb.), Treasurer. ORDER OF PROCEEDINGS. 1x Philosophical Society of Glasgow University of Bonn Cambridge Philosophical Society Royal Astronomical Society University of Toronto St David’s College, Lampeter Institution of Civil Engineers King’s College, London British Association University of Durham Solar Physics Committee, Science and Art Department Cambridge Ray Club University of London London Chemical Society Queen’s College, Belfast Queen’s College, Galway University of Sydney Royal College of Science, London The Owens College, Manchester University of Bombay University of Madras London Mathematical Society University of New Zealand Durham College of Science, Neweastle-on- Tyne University of Adelaide University College of Wales, Aberystwyth Physical Society of. Paris Yorkshire College, Leeds Physical Society of London Mason College, Birmingham Lord Blythswood. Professor Kayser. Joseph Larmor, M.A., President. George Howard Darwin, M.A., Plumian Pro- Jessor of Astronomy, President. R. Ramsay Wright, M.A., B.Se., Professor of Biology. A. W. Scott, M.A., Trinity College (Dubl.), Professor of Physical Science and Mathe- matics. William Henry Preece, C.B., President. Archibald Robertson, D.D. (Durham), Prin- cipal. Sir William Crookes, President. Ralph Allen Sampson, M.A., Professor of Mathematics. Prof. G. H. Darwin. Alfred Newton, M.A., Professor of Zoology and Comparative Anatomy. Sir H. Roscoe. Dr T. E. Thorpe. Thomas Hamilton, D.D., President. Alexander Anderson, M.A., President. Philip Sydney Jones, M.D. (Lond.), Fellow of the Senate of the University of Sydney. John Wesley Judd, C.B., LL.D., Dean; W. A. Tilden, Professor of Chemistry. Alfred Hopkinson, Q.C., M.A., Principal. Dr H. M. Birdwood, M.A., C.S.I. Hon. H. H. Shephard, M.A., Puisne Judge of the High Court of Madras. Lord Kelvin, M.A., Hon. LL.D., President. Edward John Routh, M.A., Se.D. Henry Palin Gurney, M.A., Principal. Horace Lamb, M.A., Professor of Mathematics in Owens College, Manchester. Robert Davies Roberts, M.A. M. Henri Deslandres. Leonard J. Rogers, M.A., Professor of Mathe- matics. Oliver J. Lodge, D.Sc., Professor of Physics, University College, Liverpool, President. John Henry Poynting, Se.D., Professor of Physics. x ORDER OF PROCEEDINGS. Johns Hopkins University, Baltimore Firth College, Sheftield University College, Bristol City and Guilds of London Institute for Advancement of Technical Education University College, Dundee Univeisity College, Nottingham Victoria University Royal University of Ireland Royal College of Science for Ireland University College, Liverpool University of the Punjab University College of South Wales, Cardiff University College of North Wales, Bangor Royal Indian Engineering College, Coopers Hill University of Allahabad University of Wales Simon Newcomb, Hon. Sc.D., LL.D., Professor of Mathematics and Astronomy; and Professor Ames. William Mitchinson Hicks, Se.D., Principal. Frank R. Barrell, M.A., Professor of Mathe- matics. Sir Frederick Abel, K.C.B. John Yule Mackay, Principal. John Elhotson Symes, M.A., Principal. Nathan Bodington, Litt.D., Vice-Chancellor. Right Rey. Monsignor Molloy, D.D., D.Sc. Walter Noel Hartley, Professor of Chemistry. Richard Tetley Glazebrook, M.A., Principal. Sir Charles Arthur Roe, M.A., late First Judge of the Chief Court, Punjab; late Vice-Chancellor of the University. H. W. Lloyd Tanner, M.A. (Oxon.), Professor of Mathematics. Henry R. Reichel, M.A. (Oxon.), Principal. Prof. A. Lodge, M.A. (Oxon.), Professor of Mathematics. G. Thibaut, Ph.D., Principal of the Muir Central College, Allahabad. 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. xl 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 Ql hf aN A a a = Sir G. G. SToKEs THE CHANCELLOR The Recipients of the Degree of Doctor in Science, honoris cuusd : 1. Marie Alfred Cornu 2. Jean Gaston Darboux 3. Alfred Abraham Michelson 4. Magnus Gustaf Mittag-Letter 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. Xll 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 Luris et Scientiarum Doctori Regiae Societatis quondam Praesidi Scientiae Mathematicae per annos quinquaginta inter Cantabrigienses Professori Sh 1A Ib) Universitas Cantabrigiensis. Quop per annos quinquaginta inter nosmet ipsos Professoris munus tam praeclare ornavisti, et tibi, vir venerabilis, et nobis ipsis vehementer gratulamur. Jluvat vitam tam longam, tam serenam, tot studiorum fructibus maturis felicem, tot tantisque honoribus illustrem, tanta morum modestia et benignitate msignem, hodie paulisper contemplari. Anno eodem, quo Regina nostra Victoria insularum nostrarum 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 mathematicorum 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 aéris 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 posteram autem famam diuturnam tibi propterea praesertim auguramur, quod, in inventis tuis pervulgandis perquam cautus et consideratus, 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 doctorem honoris causa nominaverunt, sed etiam exterae gentes honoribus eximis certatim cumulaverunt. Hodie eodem doctoris titulo studiorum tuorum socios nonnullos exteris e gentibus ad nos advectos, et 1psorum et tuum in honorem, velut exempli causa, libenter ornamus. In perpetuum denique obser- vantiae 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 exoptamus. Vale. Datum in Senaculo r te ‘ TN mensis Iunii die secundo (rR A. S. MpCCCXCIX. ; ORDER OF PROCEEDINGS. xiil 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 causa) Marie Alfred Cornu (Professor of Experimental Physics in the Ecole 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-Leftler (Professor of Pure Mathematics, Stockholm) 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 hoe 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 dAapradynpopia quam feliciter lampada a suis sibi traditam ipse etiam trans aequor Atlanticum alii tradidit. Duco ad vos ALFREDUM CORNU. Vou. XVIII. € 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 lucis 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. Idem 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@yyeXov éodov ea Timav peyliotay mpayyate mavtt dépew* avferar kai MaOnaors 8 ayyedias dpbas. Duco ad vos MaGNum Gustravum MirraG-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 GeorGiumM HERMANNUM QUINCKE. ORDER OF PROCEEDINGS. XV Universitatem Goettingensem, a Rege nostro Hanoveriensi Georgio secundo conditam, vinculo non uno cum Universitate nostra coniunctam esse constat. Constat eandem etiam per annos prope quinquaginta Caroli Frederici Gaussu, scientiae mathematicae et physicae professoris celeberrimi, gloria esse illustratam, qui cum ingenio fecundissimo disserendi genus consummatum coniunxit. Iuvat inde professorem ad nos advectum excipere, qui scientiae eiusdem pulcherrimam nactus provinciam, etiam lucem ipsam et crystalla mgenii sui lumine illustravit. Sex virorum insignium seriem consummavit hodie WoLDEMAR VoIG?. 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. Cae, LA THEORIE DES ONDES LUMINEUSES: SON INFLUENCE SUR LA PHYSIQUE MODERNE*. Par ALFRED CORNU, DE L’ACADEMIE DES SCIENCES ET DE LA SOCIBTE ROYALE DE LONDRES, PROFESSEUR A L’ECOLE POLYTECHNIQUE. THE REDE LECTURE (1% JUIN 1899). Notre €poque se distingue des Ages précédents par une merveilleuse utilisation des forces naturelles; l'homme, cet étre faible et sans défense, a su, par son génie, acquérir une puissance extraordinaire et plier a son service des agents subtils ou redoutables, dont ses ancétres ignoraient méme |’existence. Cet admirable accroissement de la puissance matérielle de homme dans les temps modernes est di tout entier 4 l'étude patiente et approfondie des phénoménes de la Nature, 4 la connaissance précise des lois qui les régissent et & la savante combinaison de leurs effets. Mais ce qui est particulierement instructif, c’est la disproportion qui existe entre le phénomene primitif et la grandeur des effets que l'industrie en a fait jaillir. Ainsi, ces formidables engins fondés sur |’électricité ou la vapeur ne dérivent ni de la foudre, ni des volcans; ils tirent leur origine de phénomeénes presque imperceptibles qui seraient * En dehors de l’intérét que présente un coup d’ceil d’ensemble sur les progrés et l’influence de l’Optique, cette lecture offre les conclusions d’une étude approfondie du Traité d’Optique de Newton. On verra que la pensée du grand physicien a été singuliérement altérée par une sorte de légende répandue dans les traités élémentaires ot la théorie de 1’émission est exposée. Pour rendre plus claire la théorie des accés, les commentateurs ont imaginé de matérialiser la molécule lumineuse, sous la forme d’une fléche rotative se présentant alternativement par la pointe et par le travers. Ce mode d’exposition a contribué a faire croire que toute la théorie newtonienne de |’émission était renfermée dans cette image un peu enfantine; il n’en est rien, Nulle part, dans son Traité, Newton ne donne une représentation mécanique de la molécule lumineuse: il se borne a décrire les faits, puis les résume dans un énoncé empirique, sans explications hypothétiques. Il se défend méme de faire aucune théorie, quoique l’intervention des ondes excitées dans l’éther lui apparaisse comme fort pro- bable. De sorte que l’impression générale résultant de la lecture du Traité d’Optique, et surtout des ‘‘Questions’’ du troisiéme livre, peut se résumer en disant que Newton, loin d’étre l’adversaire du systéme de Descartes, comme on le représente généralement, est, au contraire, trés favo- rable aux principes de ce systéme: frappé des ressources qu’offrait Vhypothése ondulatoire pour l’explication des phénoménes lumineux, il l’aurait sans doute adoptée, si Vobjection grave relative 4 la propagation rectiligne de la lumiére, résolue seulement de nos jours par Fresnel, ne len avait détourné. XVill ALFRED CORNU, LA THEORIE DES ONDES LUMINEUSES: demeurés éternellement cachés aux yeux du vulgaire, mais que des observateurs pénétrants ont su reconnaitre et apprécier. Cette humble origine de la plupart des grandes découvertes dont l’humanité béneéficie 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 progrés de la Science pure qu'il faut chercher le secret de la puissance croissante du monde moderne. De la une série de questions qui s'imposent a l’attention de tous. A quelle occasion le gotit de la Philosophie naturelle, si chére aux philosophes de |’Antiquité, abandonnée pendant des siécles, a-t-il pu renaitre 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 a d’admirables généralisations, conformément au plan grandiose entrevu par les fondateurs de la Physique moderne ? Telles sont les questions que je me propose, comme physicien, d’examiner devant yous: c’est un sujet un peu abstrait, je dirai méme un peu sévére; mais nul autre ne m’a paru plus digne d’attirer votre attention, & la féte que lUniversité de Cambridge célebre aujourd’hui, pour honorer le cinquantenaire du professorat de Sir George-Gabriel Stokes, qui, dans sa belle carriére, a précisément touché d'une main magistrale aux problémes les plus profitables 4 l’avancement de la Philosophie naturelle. Ce sujet est d’autant mieux A 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 Université de Cambridge, ses professeurs ou ses éléves, Sir Isaac Newton, Thomas Young, George Green, Sir George Airy, Lord Kelvin, Clerk Maxwell, Lord Rayleigh; et le souvenir de gloire qui se perpétue a travers les siécles jusqu’au temps présent rehaussera l’éclat de cette belle cérémonie. I Cherchons donc, dans un rapide coup d’ceil sur la Renaissance scientifique, 4 reconnaitre influence secréte, mais puissante, qui a été la force directrice de la Physique moderne. Je suis porté & penser que I’étude de la lumiére, 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 & considérer l’Optique comme ayant eu sur la marche des Sciences une influence dont on ne saurait exagérer la portée. Cette influence, déja visible dés la création de la Philosophie expérimentale, par Galilée, a grandi dans de telles proportions qu’on prévoit aujourd’hui une immense synthése 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 & notre intelligence la connaissance du monde extérieur est la lumiere. C'est, en effet, la vision qui nous fournit les notions les plus rapides et les plus complétes sur les objets qui nous entourent; nos autres sens, l’ouie, 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. X1x Il nest done pas étonnant que la lumiere, lien perpétuel entre notre personnalité et le monde extérieur, intervienne a chaque instant, par toutes les ressources de sa constitution intime, pour préciser l’observation des phénomenes naturels. Aussi chaque découverte relative & quelque propriété nouvelle de la lumiére a-t-elle eu un retentissement immédiat sur les autres branches des connaissances humaines; souvent méme, elle a déter- miné la naissance d'une science nouvelle en apportant un nouveau moyen d’investigation d'une puissance et d’une délicatesse imattendues. L’Optique est véritablement une science moderne; les anciens philosophes n’avaient pas soupgonné la complexité de ce qu’on appelle vulgairement la lumiére: ils confondaient sous la méme dénomination ce qui est personnel & homme et ce qui lui est extérieur. Ils avaient cependant apergu une des propriétés caractéristiques du lien qui existe entre la source lumineuse et l’ceil qui percoit impression: la lumiére se meut en ligne droite. L’expérience vulgaire leur avait révélé cet axiome, en observant les trainées brillantes que le Soleil trace dans le ciel en pergant les nuées brumeuses ou en pénétrant dans un espace obscur. De la étaient résultées deux notions empiriques: la définition des rayons de lumiere et celle de la ligne droite; la premiere devint la base de l’Optique; Vautre, la base de la Géométrie. Il ne nous reste presque rien des livres d’Optique des anciens; nous savons, toutefois, quwils connaissaient la réflexion des rayons lumineux sur les surfaces polies et l’explication des images formées par les miroirs. Il faut attendre bien des siecles, jusqu’a la Renaissance scientifique, pour rencontrer un nouveau progres dans l’Optique; mais celui-la est considérable, il annonce l’tre nouvelle: cest linvention de la lunette astronomique. Liere nouvelle commence a Galilée, Boyle et Descartes, les fondateurs de la Philo- sophie expérimentale ; tous trois consacrent leur vie & méditer sur la nature de la lumiére, 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 scholastiques; il pose en principe que tous les phénoménes sont gouvernés par les lois de la Mécanique. Dans son systéme du monde, la lumiére joue un role prépondérant*; elle est produite par les ondulations excitées dans la matiere subtile qui, suivant lui, remplit tout l’espace. Cette matiére subtile (qui représente ce que nous appelons aujourd’hui l’éther), il la considére comme formée de particules en contact immédiat; elle constitue done en méme temps le véhicule des forces existant entre les corps matériels qui y sont plongés. On reconnait ld les fameux tourbillons de Descartes, tantot admirés, tantot bafoués aux siécles derniers, mais auxquels @habiles géometres 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 aftirme la liaison des grands problémes cosmiques, et de la pénétration avec laquelle il annonce des solutions dont les générations actuelles s’approchent insensiblement. * Le Monde de M. Descartes ou le Traité de la Lumiére. Paris, 1664. XxX ALFRED CORNU, LA THEORIE DES ONDES LUMINEUSES: Pour Descartes, le mécanisme de la lumieére et celui de la gravitation sont inséparables ; le sitge des phénoménes qui leur correspondent est cette matiére subtile qui remplit Univers et leur propagation doit s'effectuer par ondes autour des centres actifs. II Cette conception de la nature de la lumiére 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 amenérent des défenseurs: les ondes excitées & 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? Lesprit éprouve done une véritable satisfaction & penser que nos deux organes les plus précis et délicats, l’ceil et loreille, sont impressionnés par un mécanisme de méme nature. Cependant, une grave différence subsiste; le son ne se meut pas nécessairement en ligne droite comme la lumiére; il tourne les obstacles qu’on lui oppose et parcourt les routes les plus sinueuses presque sans s’affaiblir. Les physiciens se partagerent alors en deux camps: les uns, partisans de |’émission, les autres, partisans des ondes. Comme chacun des deux systemes se prétendait supérieur a lautre, et l’était. en effet sur quelques points, il fallait en appeler a d’autres phénoménes pour trancher entre eux. Le hasard des découvertes en amena plusieurs qui auraient di décider en faveur de la théorie des ondes, ainsi qu’on le reconnut un siecle plus tard; mais les claires vérités n’apparaissent jamais sans un long labeur. Un compromis singulier s’établit entre les deux systemes, a l’abri d’un nom illustre entre tous, et la victoire fut attribuée, pendant un siécle, a la théorie de l’émission; en voici l’étrange histoire : En 1661, un jeune éléve plein d’ardeur et de pénétration entrait & Trinity College de Cambridge; il se nommait Isaac Newton; il avait déjé lu dans son village l’Optique de Kepler. A peine entré, tout en suivant les legons d'Optique de Barrow, il étudie avee passion la Géométrie de Descartes; il achéte sur ses économies un prisme pour étudier les couleurs et, entre temps, médite déja longuement sur les causes de la gravité. Huit ans apres, ses maitres le trouvent digne de succéder & Barrow dans la chaire lucasienne, et il enseigne A son tour l’Optique. L’éleve dépasse bientdt le maitre et annonce une découverte capitale: La lumitre blanche, qui semblait le type de la lumitre pure, nest pas homogéne; elle est formée de rayons de diverses réfrangibilités. Et il le démontre par la célébre expérience du spectre solaire, dans laquelle un rayon de lumitre blanche est décomposé en une série de rayons colorés comme l’are-en-ciel ; chacune de ces couleurs est simple, car le prisme ne la décompose plus. Telle est lYorigine de l’analyse spectrale. SON INFLUENCE SUR LA PHYSIQUE MODERNE. XX1 Cette analyse de la lumiére blanche amena Newton A expliquer les colorations des lames minces qu'on observe en particulier sur les bulles de savon; l’expérience fonda- mentale, dite des anneaux de Newton, est l'une des plus instructives de lOptique, et les lois qui la résument sont d’une admirable simplicité. Il en exposa la théorie dans un discours adressé & la Société Royale sous le titre: Hypothése nouvelle concernant la lumiere et les couleurs. Ce discours provoqua de la part de Hooke une vive réclamation. Hooke avait antérieurement observé aussi les colorations des lames minces et cherché a les expliquer dans le systeme des ondes: il avait eu le mérite (que Newton lui-méme reconnut sans peine) de substituer & londe progressive de Descartes une onde vibratoire, notion nouvelle et extrémement importante: il avait méme apercu le réle des deux surfaces réfléchissantes de la lame mince, ainsi que l’action mutuelle des ondes réfléchies. Hooke efit été ainsi le véritable précurseur de la théorie moderne, s'il avait eu, comme New- ton, la perception claire des rayons simples; mais ses raisonnements vagues pour ex- pliquer la coloration dtent toute valeur démonstrative & sa théorie. Newton fut trés 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. I] se défend davoir constitué une théorie, il déclare qu'il n’admet ni Vhypothése des ondes, ni celle de l’émission; seulement il est obligé, pour abréger le discours et faire image, d’avoir recours 4 l'une et A l'autre, comme sil les admettait. Et, en fait, dans la XII* Proposition, au II*® livre de son Optique*, qui constitue ce que l’on a appelé depuis la théorie des accés, Newton reste absolument sur le terrain des faits. Il dit simplement: “Le phénoméne des lames minces prouve que le rayon lumineux est mis alternativement dans un accés de facile réflexion ou de facile transmission.” Il ajoute, toutefois, que si lon désire une explication de ces alternances, on peut les attribuer aux vibrations excitées par le choc des corpuscules et propagées sous forme dondes par |’éthert+. En résumé, malgré son désir de rester sur le terrain solide des faits, Newton n’a pas pu sempécher d’essayer une explication rationnelle; il a trop lu les écrits de Des- cartes pour n’étre pas, au fond, comme Huyghens, partisan de l’universel mécanisme et pour ne pas désirer secrétement trouver, dans les ondulations pures, l’explication du beau phénomene 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 premiere fois, expression mathématique de leur vitesse, aussi bien pour les vibrations longitudinales des corps compressibles que pour les vibrations transversales des surfaces fluides. * Prop. XIJ.—Tout rayon de lumiére dans son passage _—réfringente, et entre les retours, a étre aisément réfléchie & travers une surface réfringente est mis dans un certain par elle. état passager qui, dans la progression du rayon, revient a (Sir Isaac Newton, Opticks or a Treatise of the Re- intervalles égaux et dispose le rayon, & chaque retour, 4 flections, Refractions, Inflexions and Colours of Light.— étre facilement transmis 4 travers la prochaine surface London, 1718, second edition, with additions, p- 253.) + Loe. cit., p. 255. Vou. XVITI. d Xxil ALFRED CORNU, LA THEORIE DES ONDES LUMINEUSES: Mais c’est surtout le troisitme 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 2 la doctrine ondulatoire. x Newton aurait certaimement cédé a ce secret en- trainement si la logique inflexible de son esprit le lui avait permis; mais, apres avoir énuméré toutes les ressources dont la théorie des ondes dispose pour expliquer la nature intime de la lumitre, arrivé aux derniéres questions, il s’arréte comme pris d’un remords subit et la rejette résolument. Et le seul argument qu'il donne, c’est quil n’y voit pas la possibilité de rendre compte du rayon lumineux rectiligne *. * Voici, d’abord, un extrait des ‘‘ Questions” qui prouve la tendance des vues de Newton vers la théorie ondulatoire et les idées cartésiennes. “Question 12.—Les rayons de lumiére, en frappant le fond de l’wil, n’excitent-ils pas des vibrations dans la tunica retina? Ces vibrations, étant propagées le long des fibres solides des nerfs optiques dans le cerveau, causent la sensation de la vision... ‘‘ Question 13.—Les diverses sortes de rayons ne font- elles pas des vibrations de diverses grandeurs, qui, suivant leurs diverses grandeurs, excitent les sensations des diverses couleurs, de la méme maniére que les vibrations de lair, suivant leurs diverses grandeurs, excitent les sensations des divers sons? Et, én particulier, ne sont-ce pas les rayons les plus réfrangibles qui excitent les plus courtes vibrations pour produire la sensation du violet extréme ; les moins réfrangibles, les plus grandes, pour produire la sensation du rouge extréme, etc.?... “ Question 18.—La chaleur d’un espace chaud n’est-elle pas transmise 4 travers le vide par les vibrations d’un milieu beaucoup plus subtil que l’air, qui reste dans le vide apres que lair en a été enlevé? “Bt ce milieu n’est-il pas le méme que le milieu par lequel la lumiére est réfractée et réfléchie, par les vibra- tions duquel la lumiére communique la chaleur aux corps et est mise dans les accés de facile réflexion et de facile transmission ? “Et ce milieu n’est-il pas infiniment (exceedingly) plus rare et subtil que lair et infiniment plus élastique et actif? Et ne remplit-il pas tous les corps? Et (par sa force élastique) ne se répand-il pas dans tout l’espace céleste?” Newton examine ensuite le réle possible de ce milieu (Véther) dans la gravitation et dans la transmission des sensations et du mouvement chez les étres vivants (ques- tions 19 & 24). Les propriétés dissymétriques des deux rayons du spath d’Islande attirent également son attention (questions 25 et 26). Puis arrive cette volte-face soudaine, cette espéce de remords d’avoir exposé avec trop de complaisance les ressources de la théorie cartésienne fondée sur le plein: il fait, en quelque sorte, amende honorable et continue ainsi: “‘ Question 27.—Ne sont-elles pas erronées toutes les hypothéses qui ont été inventées jusqu’ici pour expliquer les phénoménes de la lumiére par de nouvelles modifica- tions des rayons? “ Question 28.—Ne sont-elles pas erronées toutes les hypothéses dans lesquelles la lumiére est supposée con- sister en une pression ou un mouyement propagé a travers un milieu fluide? ‘Si elle (la lumiére) consiste seulement en une pression ou un mouvement propagé instantanément ou progressive- ment, elle se courberait dans l’ombre. Car une pression ou un mouvement ne peut pas se propager en ligne droite dans un fluide au dela de l’obstacle qui arréte une partie du mouvement; il y a inflexion et dispersion de tous cétés dans le milieu en repos situé au dela de obstacle... ‘«« .. Car une cloche ou un canon peuvent s’entendre au dela d’une colline qui intercepte la yue du corps sonore, et les sons se propagent aussi bien a travers des tubes courbés qu’ travers des tubes droits. Tandis que l’on ne voit jamais la lumiére suivre des routes tortueuses, ni s’in- fléchir dans l’ombre.” Devant cette objection, Newton se voit obligé de revenir 4 la théorie corpusculaire. “Question 29.—Les rayons de lumiére ne sont-ils pas de petits corps émis par les substances brillantes?... “« Question 30,—Les corps grossiers de la lumiére ne sont- ils pas convertissables l'un dans l’autre?... Le changement des corps en lumiére et de lumiére en corps matériels est trés conforme au cours de la nature, qui se plait aux trans- mutations.” La logique le force & poursuivre ’hypothése du vide et des atomes et méme a invoquer (question 28, p. 343), A ce sujet, Pautorité des anciens philosophes de la Gréce et de la Phénicie: on ne doit done pas s’étonner de voir sa per- plexité se traduire par les paroles suivantes : ** Question 31° et derniére.—Les petites particules des corps n’ont-elles pas certains pouvoirs, vertus ou forces, par lesquels elles agissent 4 distance non seulement sur les rayons de lumiére pour les réfléchir, les réfracter ou les infléchir, mais aussi les unes sur les autres pour produire une grande partie des phénoménes de la Nature?” Mais il s’apercgoit qu'il va peut-étre un peu loin et qu'il va se compromettre: aussi ses secrétes tendances, dévelop- pées dans la premiére question, reparaissent-elles un in- stant: SON INFLUENCE SUR LA PHYSIQUE MODERNE. XXL Considéré & ce point de vue, le troisiéme livre de I’Optique n’est plus la discussion seulement impartiale de systemes opposés; il apparait comme la peinture des souffrances x dun génie puissant, tourmenté par le doute, tour a tour entrainé par les suggestions séduisantes de l'imagination et rappelé par les exigences impérieuses de la _ logique. Nous assistons & un drame, a Il’éternel combat de l'amour et du devoir, et c'est le devoir qui a été le plus fort. Telle est, jimagine, la genése intime de la Théorie des acces, mélange bizarre des x deux systemes opposés; elle a été beaucoup admirée a cause de l’autorité du grand géometre qui a eu la gloire de ramener l'ensemble des mouvements célestes & la loi unique de la gravitation universelle. Aujourdhui, cette théorie est abandonnée; elle est condamnée par lexperimentum crucis d’Arago, réalisé par Fizeau et Foucault: on doit pourtant reconnaitre qu'elle a constitué un réel progres par la notion précise et nouvelle quelle renferme. Le rayon de lumiére considéré jusque-la était simplement la trajectoire d'une particule en mouve- ment rectiligne: le rayon de lumiere tel que le décrit Newton possede une structure périodique réguliere, et la période ou longueur d’accés caractérise la couleur du rayon; cest la 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 siécle, et cest le D™ Thomas Young qui, en 1801, aura lhonneur de la découvrir. Til Reprenant l'étude des lames minces, Thomas Young montre que tout s’explique avec une extréme simplicité, si l’on suppose que le rayon lumineux homogéne est Yanalyse de Vonde sonore produite par un son musical; que les vibrations de |’éther, soumises aux lois des petits mouvements, doivent se composer, c’est-a-dire interférer, suivant l’expression qu il propose pour exprimer leur action mutuelle. Quoique Young eft pris Vhabile précaution de se réclamer de l’autorité de Newton*, l’hypothése n’eut aucune faveur; son principe d’interférence conduisait a cette singuliere conséquence que la lumiére ajoutée & de la lumiére pouvait, dans certains cas, produire Jobscurité; résultat paradoxal, contredit par l’expérience journaliére. La seule vérification que Young apportat était lexistence des anneaux obscurs dans l’expérience de Newton, obscurité due, suivant lui, 4 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 experimentum crucis, Young ne réussit pas a lobtenir. “Comment ces attractions (gravité, magnétisme et élec- _initiateur de la théorie de |’émission. En réalité, il hésite tricité) peuyent-elles se produire, je ne m’y arréte pas ici. Ce que j’appelle attraction peut étre produit par des impul- sions ou par d’autres moyens que j’ignore...” Il y aurait encore bien des remarques curieuses 4 faire sur létat d’esprit du grand physicien, géométre et philo- sophe, qui se révéle naivement dans ces ‘‘ Questions.” Les courts extraits qui précédent suffisent, je crois, a justifier la conclusion qui ressort de cette étude, a savoir, que Newton n’avait pas, sur le mécanisme de la lumiére, les idées arrétées qu’on lui préte en le considérant comme entre les deux systémes opposés dont il apergoit claire- ment l’insuffisance et, dans cette discussion, il s’efforce de s’éloigner le moins possible des faits bien établis: voila pourquoi il ne formule aucune théorie dogmatique. I] serait done injuste de rendre Newton responsable de tout ce que les partisans de l’émission ont abrité sous son autorité. * The Bakerian Lecture, on the Theory of Light and Colours.—By Thomas Young. Philos. Trans. of the Royal Society of London, 1802, p. 12. d 2 XXIV ALFRED CORNU, LA THEORIE DES ONDES LUMINEUSES: La théorie des ondes retombait done encore une fois dans l’obscurité des contro- verses, et le terrible argument de la propagation rectiligne se dressait de nouveau contre elle. Les plus habiles géométres de |’époque, Laplace, Biot, Poisson, s’étaient naturellement rangés 4 l’opinion newtonienne: Laplace en particulier, le célébre auteur de la Mécanique céleste, avait méme 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 lillustre Huyghens. Huyghens, en effet, dans son T'raité de la Lumiere, avait résolu un probléme devant lequel la théorie de l’émission était restée muette, 4 savoir, l’explication de la biré- fringence du cristal d’Islande; la théorie des ondes, au contraire, ramenait 4 une con- struction géométrique des plus simples la marche des deux rayons, ordinaire et extra- ordinaire ; l’expérience confirmait en tous points ces résultats. Laplace réussit, & son tour, a l'aide d’hypothéses sur la constitution des particules lumineuses, & expliquer la marche de ces étranges rayons. La victoire de la théorie particulaire paraissait done complete: un nouveau phénoméne arrivait méme tout & point pour la rendre éclatante. Malus découvrait qu'un rayon de lumiére naturelle, réfiéchi sous un certain angle, acquiert des propriétés dissymétriques semblables a celles des rayons du cristal d’Islande; il expliqua ce phénoméne par une orientation de la molécule lumineuse, et, en consé- quence, nomma cette lumitre, lumiére polarisée; était un nouveau succes pour I’émission. Le triomphe ne fut pas de longue durée; en 1816, un jeune ingénieur, A peine sorti de Ecole Polytechnique, Augustin Fresnel, confiait & Arago ses doutes sur la théorie en faveur et lui indiquait les expériences qui tendaient & la renverser; s’appuyant sur les idées d’Huyghens, il avait attaqué la redoutable question des rayons et des ombres et Pavait résolue; tous les phénoménes de diffraction étaient ramenés & un_probléme d’analyse et Vobservation vérifiait merveilleusement le calcul. Il avait, sans les connaitre, retrouvé les raisonnements de Young, ainsi que le principe des interférences; mais, plus heureux que lui, il apportait leaperimentum crucis, lexpérience des deux miroirs; 1A, ° deux rayons issus d'une méme source, purs de toute altération, produisent par leur concours, tantot de la lumiére, tantét de Yobscurité. Liillustre Young fut le premier 4 applaudir au succts de son jeune émule et lui témoigna une bienveillance qui ne se démentit jamais. Ainsi, grace 4 l’expérience des deux miroirs, la théorie du D*™ Young, c’est-a-dire l’analogie complete 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 lumiére se propage en ligne droite parce que les ondes lumineuses sont extrémement petites; au contraire, le son se diffuse parce que les longueurs des ondes sonores sont relativement trés grandes. Ainsi s‘évanouit la terrible objection qui avait tant tourmenté l’esprit du grand Newton. Mais il restait encore & expliquer une autre différence essentielle entre l’onde lumineuse et onde sonore; celle-ci ne se polarise pas, comment se fait-il que l’onde lumineuse se polarise ? La réponse A cette question paraissait si difficile que Young déclara renoncer A SON INFLUENCE SUR LA PHYSIQUE MODERNE. XXV la chercher. Fresnel travailla plus de cinq ans & la découvrir; elle est aussi simple qu inattendue : L’onde sonore ne peut pas se polariser parce que ses vibrations sont longitudinales ; la lumiére, au contraire, se polarise parce que ses vibrations sont transversales, c’est-a-dire perpendiculaires au rayon lumineux. Désormais, la nature de la lumiére est complétement établie; tous les phénoménes 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 enchainement d’expériences et de raisonnements Fresnel est arrivé 4 cette découverte, lune des plus importantes de la science moderne; mais le temps me presse. I] m’a suffi de vous faire comprendre la grandeur des difficultés quil a fallu vainere pour l’accomplir; j'ai hate d’en faire res- sortir les conséquences. IV Vous avez vu, au début, les raisons purement physiologiques qui font de l'étude de la lumiére le centre nécessaire des informations de lintelligence 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 sintéressent aux forces naturelles. En effet, tous les phénomenes qui se passent sous nos yeux impliquent une transmission 4 distance de force ou de mouvement; que la distance soit infiniment grande, comme dans les espaces célestes, ou infiniment petite, comme dans les intervalles molé- culaires, le mystére est le méme. Or, la lumiere est l’'agent qui nous amene le mouve- ment 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 problémes dans une méme conception mécanique: voila le lien secret qui a toujours attiré les physiciens et les géométres vers l’étude de la lumiere. Envisagée 4 ce point de vue, l’histoire de lOptique acquiert une portée philosophique considérable; elle devient histoire des progrés successifs de nos connaissances sur les moyens que la Nature emploie pour transmettre a distance le mouvement et la force. La premiére idée qui est venue a lesprit de lhomme, des |’état sauvage, pour exercer sa force hors de sa portée, c’est le jet d'une pierre, d’une fleche ou dun _pro- jectile queleonque; voila le germe de la théorie de l’émission: cette théorie correspond au systéme philosophique qui suppose un espace vide ot le projectile se meut librement. A un degré de culture plus avancé, 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 matiére, 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 Vorigine de la théorie des ondes lumineuses; sous cette forme, elle ne pouvait représenter qu'une partie des phénoménes, ainsi qu’on l’a vu précédemment; elle était done insuffisante. Mais les géométres et XXV1 ALFRED CORNU, LA THEORIE 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 troisitme mode de trans- mission, tout aussi naturel que le précédent, mais qui offre une richesse de ressources incomparable. Ce sont les ondes a vibrations transversales excitées dans un milieu continu incompressible, celles qui rendent compte de toutes les propriétés de la lumiére. Dans ce mode ondulatoire, le déplacement des particules met en jeu une élasticité dun genre spécial; c’est le’ glissement relatif des couches concentriques a l’ébranlement qui transmet le mouvement et Jeffort. Le caracttre de ces ondes est de nimposer au milieu aucune variation de densité, comme dans le systeme de Descartes. La richesse de ressources annoncée plus haut provient de ce que la forme de la vibration transversale reste indéterminée, ce qui confére aux ondes une variété infinie de propriétés différentes. Les formes rectiligne, circulaire, elliptique, caractérisent précisément ces polarisations si inattendues que Fresnel a découvertes et a laide desquelles il a si admirablement expliqué les beaux phénomenes d’Arago produits par les lames cristallisées. Lexistence possible dondes se propageant sans changement de densité a modifié profondément la théorie mathématique de lElasticité. Les géométres retrouvérent dans leurs équations ces ondes a vibrations transversales qui leur étaient inconnues; ils apprirent, en outre, de Fresnel la constitution la plus générale des milieux élastiques, A laquelle ils n’avaient pas songé. C'est dans son admirable Mémoire sur la double réfraction que le grand physicien émet lidée que, dans les cristaux, l’élasticité de l’éther doit étre variable avec la direction, condition inattendue et d’une extréme 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 sphére et lellipsoide que Huyghens avait assignés au cristal d’Islande. Cette nouvelle découverte excita ladmiration universelle parmi les physiciens et les géométres; lorsque Arago vint l’exposer devant |’Académie des Sciences, Laplace, si long- temps hostile, se déclara convaincu. Deux ans aprés, Fresnel, élu membre de TAcadémie & Vunanimité des suffrages, était élu, avec la méme unanimité, membre étranger de la Société Royale de Londres; ce fut Young lui-méme qui lui transmit la nouvelle de cette distinction avec lhommage personnel de son admiration sincere. V L’établissement définitif de la théorie des ondes impose la nécessité d’admettre l’existence dun milieu élastique pour transmettre le mouvement lumineux. Mais toute transmission & distance de mouvement ou de force n’implique-t-elle pas la méme condition? C'est a Faraday que revient l’honneur d’avoir, en véritable disciple de Descartes et de Leibnitz, ace cacn aaa timieae SON INFLUENCE SUR LA PHYSIQUE MODERNE. XXVIi proclamé ce principe et d’avoir résolument attribué aux réactions du milieu ambiant Yapparente action & distance des systémes électriques et magnétiques. Faraday fut ré- compensé de sa hardiesse par Ja découverte de linduction. Et, comme l’induction s’exerce méme a travers un espace vide de matiére pondérable, on est forcé d’admettre que le milieu actif est justement celui qui transmet les ondes lumineuses, |’éther. La transmission d’un mouvement par un milieu élastique ne peut pas étre instantanée ; si c'est vraiment l’éther luminitére qui est le milieu transmetteur, linduction ne doit-elle pas se propager avec la vitesse des ondes lumineuses. La verification était malaisée; Von Helmholz, qui tenta la mesure directe de cette vitesse, trouva, comme autrefois Galilée, pour la vitesse de la lumiére, une valeur pratique- ment infinie. Mais l’attention des physiciens fut attirée par une singuliére coincidence numérique: le rapport de l’unité de quantité électrostatique & lunité électro-magnétique est représenté par un nombre précisément égal & la vitesse de la lumieére. Lillustre Clerk Maxwell, suivant les idées de Faraday, n’hésita pas A voir dans ce rapport la mesure indirecte de la vitesse dinduction, et, par une série d intuitions remarquables, il parvint a élever cette célébre théorie électro-magnétique de la lumiére, qui identifie, dans un méme mécanisme, trois groupes de phénoménes en apparence completement distincts: Lumiere, Electricité, Magnétisme. Mais les théories abstraites des phénomenes naturels ne sont rien sans le contréle de lexpérience. La théorie de Maxwell fut soumise a I’épreuve et le succes 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 A la Science, empruntant & von Helmholz et & Lord Kelvin leur belle analyse des décharges oscil- lantes, réalisa si parfaitement des ondes électriques et électro-magnétiques, que ces ondes possedent toutes les propriétés des ondes lumineuses; la seule particularité qui les dis- tingue, c’est que leurs vibrations sont moins rapides que celles de la lumiere. Il en résulte qu’on peut reproduire, avec des décharges électriques, les expériences les. plus délicates de lOptique moderne: réflexion, réfraction, diffraction, polarisation rectiligne, circulaire, elliptique, ete. Mais, je m’arréte, Messieurs; je sens que j'ai assumé une tache trop lourde en essayant de vous énumérer toutes les richesses que les ondes a vibrations transversales concentrent aujourd’hui dans nos mains. J'ai dit, en commencant, que l’Optique me paraissait étre la Science directrice de la Physique moderne. Si quelque doute a pu sélever dans votre esprit, j’espere que cette impression s'est effacée pour faire place & un sentiment de surprise et d’admiration en voyant tout ce que l'étude de la lumiére a apporté d'idées nouvelles sur le mécanisme des forces de la Nature. Elle a ramené insensiblement 4 la conception cartésienne d’un milieu unique rem- plissant Tespace, siege des phénoménes électriques, magnétiques et umineux; elle laisse XXViii ALFRED CORNU, LA THEORIE DES ONDES LUMINEUSES, ere. entreyoir que ce milieu est le dépositaire de l’énergie répandue dans le monde matériel, le véhicule nécessaire de toutes les forces, l’origine méme de la gravitation universelle. Voila l’euvre accomplie par l’Optique; c’est peut-étre la plus grande chose du siécle! L’étude des propriétés des ondes envisagées sous tous leurs aspects est donc, 4 Vheure actuelle, la voie véritablement féconde. C'est celle qu’a suivie, dans sa double carritre de géométre et de physicien, Sir George Stokes, & qui nous allons rendre un hommage si touchant et si mérité. Tous ses beaux travaux, aussi bien en Hydrodynamique qu’en Optique théorique ou expéri- mentale, se rapportent précisément aux transformations que les divers milieux font subir aux ondes qui les traversent. Dans les phénoménes variés quil a découverts ou analysés, mouvement des fluides, diffraction, interférences, fluorescence, rayons Réntgen, Vidée directrice que je vous signale est toujours visible, et c’est ce qui fait l’harmo- nieuse unité de la vie scientifique de Sir George Stokes. Que l'Université de Cambridge soit fiere de sa chaire Lucasienne de Physique mathématique, car, depuis Sir Isaac Newton jusqu’a Sir George Stokes, elle contribue pour une part glorieuse aux progrés de la Philosophie naturelle. MEMOIRS PRESENTED TO THE CAMBRIDGE PHILOSOPHICAL SOCIETY. ' m1 a F $ or a a ' :Y _ a i a7 I. On the analytical representation of a uniform branch of a monogenic function. By G. Mirrac-LEFFer. [Received 25 April, 1899.] Ler a denote a point in the plane of the complex variable z, and associate with a an unlimited array of quantities BiG), FOG) pi ON ayer) (a) eee eee (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 (z\a)= = n=0 1 ae AC) HB (Thr eq, COKOOOOOOOE COCO Stafalotelstalsial= 2), ihm (a) (e—a) (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 PG), LEM LIMO) soos 5 LE) (@)y o0e are given. We denote generally by #(#) 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(a#a) and by its analytic continuation within K remains uniform and regular, I shall denote this branch by FA (x). The problem to be discussed here is that of finding * Cauchy, Cours d’Analyse deUV Ecole royale polytechnique, is a finite magnitude. It is known that, if this finite 1** partie, Analyse Algébrique, Paris 1821, chapitre 9, § 2, theoréme 1, 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 1 aoe a Vou. XVIII. 1 magnitude be denoted by = , the quantity r is the radius of 2 Pror. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION an analytical representation of a 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 (a), 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(«|a). 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(«) 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 PK (x)= | % = = de ee heen ete Seen (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 wz, the coefficients of which are expressed by special values of 2 (there being an enumerable number of these) and the corresponding values of FX (#). 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 FA (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), F°(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 FA (2), 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 («) completely. Let C denote the circle of convergence of the series (2), The expression S “ 1 — Fe (a — a) ra (a) (a — a) 0 then gives the analytical representation of FC (x), the equality : 21 FC («)= > — F (a) (e#-a)t w=0h: 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. a? eo mS OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 3 ; ily ee : Ss and of the rational numbers — independent of the choice of the elements: and it is to pw! be remarked that the expression is formed without any a priori knowledge of the radius of the circle @. This radius is determinate, in connection with the elements, by Cauchy’s theorem, and there are various methods of obtaining it from them; but it does not enter explicitly into the expression. Thus Taylor’s series is formed simply by the elements (a), HEN (Gy eELCN (G) ieee, 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 FX (x) an analytical representation similar to that found from the very beginning of the theory for the branch FC (z). For a fundamental treatment of the question which has been proposed, it is first necessary to define a domain A 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 a, let an area be generated as follows. Round a fixed point a let a vector / (a straight lme terminated at a) revolve once: on each position of the vector, determine uniquely a point, say a, 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 (cowpures) in the plane of a 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 LAG), THON LEO) coae L8G); ac satisfying Cauchy’s condition; and form the series P (a|\a)= 5 + Fw (a)(~@—a)'. w=0K: Construct the analytical continuation of P(#\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(#|a) 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(«# a) along the vector. In the latter case, I exclude from the plane of the variable that part of the vector comprised 1—2 4 Pror. 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 elements (1), I speak of the function F(a), 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), F(a), F®&(@), .....- and by FA(a) 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 small as we please. Then it is always possible to find an integer 7 such that the modulus of the difference between FA(ax) and the polynomial GA = Sc F*(a) (a —a)’ for values of n greater than i, 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(a), ..., and of «. It is very important to observe that the explicit knowledge of the star is not necessary for the construction of the function g,(z). When the elements F(a), F(a), F(a), ... are once given, the star belonging to them is definite; but it does not enter explicitly into the expression g(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 Potenzrethen, dated+ 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 inde- pendent variables. The coefficients denoted by ec are given @ priori. They are quite independent of , : : ‘ iis ; : the special function to be represented just as are the coefficients a 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 aoryp. + Ges. Werke, Bd. 1, p. 67. OF A UNIFORM BRANCH OF A MONOGENIC FUNCTION. 9) The formula ne nt nen 1 Grass act Bad fa — a\nthot.-- tha mh — LaT wee in y, WOS 20 Sol oS (@(= (4) hy=0 hg=0 Iin=0 Ma: bye «es Mn: nv gives an expression for g,(x) which perhaps is the simplest of all as regards the mere form. There are other forms in which the coefficients c®” are rational numbers, or are numbers depending in a special manner upon the transcendents e and 7, 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 IE@), L2O(@\, IBEX), sooo ; and by FA(«) the corresponding functional branch belonging to the same elements. This branch FA(«) can always be represented by a series where the quantities G(x) are polynomials of the form G, (x) = Se F(a) (w — a)’, each coefficient ¢" being a determinate number (which can be taken as rational) depending only upon pw and v. The series wo py G2), n=0 converges for every value of « within A, and it converges uniformly for every domain within A. For all values within A we have i Ms G,.(z) = Lim g,(2), 0 “ i) where g,(“) ws the polynomial in my main theorem. In what precedes, a definition has been given of the star belonging to the elements TEE (Gi) WHOM (Ga). CIC) ee RMR see Ce (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(a@|a)= = a F(a) (a@— a). 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 corre- sponds an analytical expression P(«x|a) which has the property of representing F'A (2) within C, of converging uniformly for any domain within C, and of ceasing to converge outside 6 Pror. MITTAG-LEFFLER, ON THE ANALYTICAL REPRESENTATION C. Between the circle and the star, intermediary domains C, (w=1, 2, 8, ...),—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™, there is an analytical expression representing F'A(#) within C™ which converges uniformly for every domain within C™ and ceases to converge outside C™. 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 («) 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 imvestigation 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) can easily be chosen so that C® becomes this domain K: so that M. Borel’s domain AK 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+ published without any acquaintance with my Swedish Notes of the same year, the author says} :— “Pour résumer les résultats acquis sur le probleme de la représentation analytique “des fonctions uniformes, nous pouvons dire§ que nous en connaissons deux solutions “completes; l'une est fournie par le théoreéme de Taylor, l’autre par le théor’me de “M. Runge}. Ces deux solutions ont une trés grande importance & cause de leur “généralité; mais chacune d’elles a de graves inconvénients dont les principaux sont, pour “Ja série de Taylor, de diverger en des régions ot la fonction existe; et, pour la repré- “sentation de M. Runge et celles de M. Painlevé, détre possibles d’une infinité de “manieres 4). “Le but idéal a 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++.” * Journal de Mathématiques, 5™: Sér., t. ii. (1896), “‘Fondements de la théorie des séries divergentes som- mables,” pp. 103—122; ‘Sur les séries de Taylor admettant leur cercle de convergence comme coupure,” pp. 441—454. + Lecons sur la théorie des fonctions, Paris, 1898. t pp. 88 ff. § All that follows on the analytical representation of uniform functions can be applied, mutatis mutandis, to the functional branch FA ( pee > he (8), see; De iB er, SE can be assigned a priori, independently of a, of F(a), F” (a), F®(a),... and of x, 80 that the series possesses the following properties: it converges for every point within A® and converges uniformly for every domain within A®. If convergence takes place for any value, the value necessarily belongs to the interior of A® or is a point of the star A®. When 6=1, the series becomes Taylor's series. The equality FA (2) = Ps (c\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 CM, C®, C®,... form, so to speak, a discontinuous 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 knowledge simply by the study of the elements F(a), F® (a), F®(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 aj. 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 trés-grands nombres” and which aim at the development of the criteria whether a point on a circle belonging to the elements F(a), F(a), F” (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” (a), F(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® (a),..., and if € ws a positive quantity sufficiently small, it is always possible to choose a positive number & so that, o being a positive quantity as small as we please, a positive integer X exists such that |h,™ (8) F (a) (1 + €) (@— a) + hy (8) F® (a) (1 + €) (a@—a) 2+... +2%(a) F(a) {(1 +6) (a2—-a)}\ |<, provided+ A>. If on the contrary, « does not le 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 problem can be brought within my theorem. In fact, knowing the elements F'(¢,), F" (t,), # (t,),... at a given epoch ¢,, we can obtain a development which represents the function and is valid for all real values of ¢>¢, 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 from 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 Math., 3™° Sér., t. iv. (1875), + The quantities 5 and h” (6) have the same significance pp. 5—54. : bn as in the formula (5). Wor. SVAN 2 10 Pror. 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-Leffler’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 Fie. (r) Ps = Bs 2x Asn PxPrs 11 where a” +a=0. Since in this case a finite strip enclosing the real axis is contained in the stars of the functions p,, the centre being ¢=0, 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 9 being the angle which the plane ma makes with a fixed plane through x, and r being the distance of m from the axis x, we have the areal integral 7°$ = C = constant, and the integral of vis viva 7—P=h=constant, where T =}m (#7 + °¥ + 2), a T being the vis viva and P the potential: in the latter expression the masses of the fixed point are denoted by M; and their distances from m by 7;. It is at once obvious that 7 cannot vanish. In effect, if for t=¢,, 7 can become indefinitely small, let us take this quantity as an infinitesimal of the first order. On account of the areal integral, $ would be infinitely great of the second order, and consequently °° (= C3) would also be of the second order: Z 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 7; are greater than 7; hence if 7 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 mm, h, 33m; (@2+ 97 +27) +5 i ts Tiss * 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 2;, y;, 2 are the coordinates of the moving points, m, their masses, 7;,, their distances, and / 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 ¢ into ¢,/—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 mn 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 Kirchhoft’s Mechanik for the differential equations of the motion. The interest of this development is manifest. I remark, however, that the main im- portance of my theorems so far as concerns mechanics appears to me to be that they provide a means of finding a real and positive poimt of my star, and of determining whether this pomt 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(t), F(t), # (é),.... It appears to me that this point of view may become useful in applications to mechanics. PERUGIA, April, 1899. II. Application of the Partition Analysis to the study of the properties of any system of Consecutive Integers. By Major P. A. MacMaunoy, R.A., D.Se., F.R.S., Hon. Mem. C.P.S. [Received 15 May, 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 method of Partition analysis. I will first explain the problem, and afterwards the analysis that will be used. In the binomial and multinomial expansions, the exponent being a positive integer, every coefficient is an integer. This fact depends analytically upon the circumstance that the product of any m consecutive integers is divisible by factorial m; we have ese an integer for all values of x. The present question is the determination of all values of a, a, 4, ... %m for N+1\% (n+ 2\2 (n+3\% (/n+m)\% a ae 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. which the expression There is a parallel theory connected with the algebraic product é = =)" (7 = =)" 1 meee as ¢ = —) 1—2@ 1-2 ue “"\ 1l—a™ : where a, %, 43,-++. Gm 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 that the algebraic product merges into the arithmetical product for the particular case «=1, so that all algebraic products which are finite and integral produce in this manner arithmetical products which are integers. This, however, is as much as can be said, for Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS, ere. 13 otherwise the theories proceed on widely divergent limes; as might be expected the arithmetical products form a more extended group than the algebraical. Denote, for brevity, fg Aa n+s =e daa ie 1-2 by X, and JN, respectively. The principal X theorem, that has been obtained loc. cit. is to the effect that con- structing any X rectangle AG X» AG XG eA X, XG -con AGA XG NG, Xx; AG con AGES XxX, Xx, O00 Xi+s a 5 5 5 G Mon AGrne Xinis XGrets cee Xitm+ 1 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 Xx 1X 7X 3 OUD Xitm—1 ) 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 DCD. EX.ED.G IG. which are not expressible in the rectangular lattice form, the theory of which is not yet complete. We see therefore that the product of V’s contained in the rectangle N, N, NG eee N, N; NG eee Vs N; N, NG eee Nie Nin Nina Ninte alse Nim is an integer for all values of n. It will appear moreover that no product exists which is free from N,, 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 rela- tions of the type AQ, + Ak. + Asks + .-. + AsOls 2 4B, + foo ar Mss te ees ate ern the coefficients X and p being given positive integers, and it is required to find the general values of a, a... , o,-.., bemg positive imtegers, which satisfy the one or more relations, 14 Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO This is accomplished by constructing the sum Da, 7,274% ... Te YaPryPrysPs ... yeh for all sets of values of a, a ...8;, B2,... 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 .@) 1 > 1—m*a,.1—m*2,... l—m™ a,.1—m™y,.1—m "yp... 1 —m™ ¥ where the symbol of operation Vo is connected with the auxiliary symbol m in the following manner :— The fraction is to be expanded in ascending powers of a, 22... %;, Yo) ++»; 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 a, > 8; this leads to Q 1 > 2 1 1—ma,.1—-~ and observing that a 1 os 1 ‘- m } Ll l—ma.1l—2« it = 1—ma.1-— : ss 1-7 w-l-ay we find e f i — 3 ihe tee —2,.1-—amy, ma,.1 sell a also os ee SS : Aaa Bay wigs which is the solution of a,>Bi+s; THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 15 so that the solution of a >P, 1 is given by v ue az ; l-a,.1-—my, 1l—ma,.1—- a Yy Again, if a, > 8, +P, we have the solution. Also the solution of a>B,+P. 1s 1l—a.1—amy,.1— ay. Lastly, m+ %> Pi gives, by repeated application of the above simple theorem, 1—a4,22%, 1l—a2,.l—a.1—my,.1— my, 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 1m. of the Memoir on Partitions which may appear shortly in Phil. Trans. B.S. To come to the object of the paper I commence with ORDER 2. m+ 1\% /n+2\" mien (er ) (“3 = Bete; this product is an integer when n is even, but when m is uneven we must have A, > Ay; i eae Naa and LV aN 1 oN, 1= 0; 16 Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS T shewing that the ground products are ,, N,N., or (ras) — (0) en): ORDER 3. n+1\" n+ 2\% /n+ 3\% ee ( 1 ) 2 ) eg erie When n is of form condition is 4m + 1, a + 2a; > a (a), 4m + 3, 2a, + a; > M% F 3m + 1, A > a; (bd), 3m + 2, a > A (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), (6), (c) respectively we write down the © function @) 1 > b 1—acN,.1—7 N2.1- N; a> be as the expression of the sum >N,"V."N,™. It must be observed that the operating symbol ©, 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 ¢ will introduce a*. It should be remarked that operation upon one letter may 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 Q 1 Sn a a 1 —acN,.1-=N,.1-2 N,N, Observe that this expression would have presented itself if for the two relations a SE as 2 a, a 2As, we had constructed the sum > N41 Ni (N,N;)*. THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. Wf The fact is that we can reduce the three relations (a), (b), (c) to two by writing a+ 4; for 4, a tranformation that the relation (6) permits, and then we have to write N,N, for N, in the sum > NN N°. We next eliminate c, obtaining @ 1 > 1 1—-aN,. 1—_ .. 1—a@N,N,N; an expression that would have presented itself if we had been summing LNV,2 Ni (N,N,N) for the single relation a, + 2a, > a, obtained from the relation O% + A; 2 A, by writing a,+a, for a, a transformation permitted by the relation O > as. 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 1) 1 1+ ay — xyz — xyz? ~T=2@.1—y.1—yz.1—a2? 2 1 1—a’r.1—ay.1——z and thence we derive 1+N,N2N,— N2N2N,— NY NSN, 1—N,.1—N,N..1-— N,N.N,.1— NiNSN, = 1— N?2N2N,— NYNSN, — NPNANS + NENA + NENONS 7 DSnge Tl SSN ENN SINE © 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 Nig, Why ARMING IEINING, INCI connected by the simple syzygies (A) = (M1) (NPN s) — NN2) NNN) = 0, (B) = (MN) NiNSNe) — (NN2) NiNENs) = 0, (C) = (N,N.N,) (NNEN;) — (NNN)? =0; Wore poe lililr 3 18 Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO and the compound syzygies (N,)(C) — (N,NN;) (B) = 0, (N,N2N;) (B) — (NNEN;) (A) =0; indicated by the numerator terms: ad N;N2N;, 7z N?N2Ns, a NyNSN;, +N, Py, oN, a5 NEN, oN; respectively. The generating function takes also the suggestive form :— 1— NV;7NSN, 1—N,.1—WN,N..1— N,N.N;.1 — N,NSN, : N,Ne2N, 1— N,N,.1—N,N.N,.1—N,NON, 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 @ posteriori that this could have been secured by impressing the additional condition O = a3; 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. (" : *)" ee 2\" (“ : a (* 2 “\" = NaN aN es When 7x is of the form condition is 4m +1, a, + 2a, > a, + 2a,, 4m + 2, AO > A, : 4m + 3, Qa, + a; > a. + 2a,, 3m +1, A > Ay 5 3m + 2, a, + ay > As ; The © 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 G23, a, >a. Case 1. a, >a;. The conditions reduce to a, > as (a), a +2a,>a+2a, (b), Ay > Oy (c), a, > as (d), THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. iL) and it is convenient to add the implied condition a, > a, (e). We obtain, for [N,N N aN 4, 9) 1 > ? ee ee a ee, b ad b?ce and, eliminating d and e, this is @ 1 > co be : Ge 1—abN,. 1 —,- 1 — a Nes. 1 — po NN: which, eliminating ¢, is ” = ; V.eN.N, z 1 iy ie b a P NLN.N,.1—N,N.N3N, ee : V,N,.1— and, eliminating a, this becomes O 1—N2N2N,N, > ae I 1—bN,-U— -N,.1 == ? NiNLN,.1—2N,N.N,.1 — NV,NN,N, 1 Ser F the term 1 — p NWS NA , disappearing. This is equal to 1— NEN N,N. ol — are Tae | 1—N,N.N.N, " —0N,.1-NNa” ee in | 1 a x 4 \ —— — + fe — }2N.N_N. — N2N2N_N Ie BNNLN,.1 — N2N2N.N, ee : V.N.N,. Ne¢N2N, wy, 1 Neal. LN VN 1 A® 5: A lhe zs NT Lay, yea, Vane > 1-0N,NWN,.1— 5 W, 1—N,N,.1—N,N.N.N, ANN, : 42 : SANG LNLNLNDN: 2 1—bM,.1- 5 WW. IE ANN gg Uh ANAL IA 20 Mayor MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO u as sae. a ae a ee ee N,N2N, + VLNSN, Lt b= N,N,.1—N,N.N;.1—N,N.NN,.1— N,NSNs "T—W,.1—N,N,.1—N,NN.N,. 1 — N20, Cese 2. a;>a@. The conditions become as > a; (a), 2a, +a; > a + 2a, (b), A 2 Oy (c), A D> As (d), a, + a,> 4; (e); to which it is convenient to add the implied conditions a 2a; (5 as > a, (9): the © function is 1 fe) a > bef ail abg ,, Qa iL eee eo op ere 1 AO a 7S ea = cds in oe ia 1——JN,.1 ee 1 — ae Ns: 1 ~ oq Ns b =o de® 7 3 aha 2 V,N,.1- : Nee = N,.1—-eN,N,N,N, b W.N.N AN. _ 1 ie) d 1 2 3 4 acer ah mateo 7 M,N,.1— € N..1-3N,N.NN, _ MNeNeN, 0 1 ENN N, > arene + Ny.1—W,NeN;M, " N,N 2N2N, (1 + N,N2N,) ~1—N,N,N,.1—N,NeN,.1—N,N,N,N,.1—N,NeNe Ne THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 21 Hence the complete sum TN NNN, : il 3 P=, Sve, ANNAN, i N,N2N, + N,NSN, 1—N,N,.1— N,NLN,.1— N,N.N;N,. 1 —N,N2Ns * NNN, 1-WN,.1-—WN,N,.1—N,N,N,N,.1— N3NLN, + an Z ™, NPN, ¢N,(1+ N,N2N;) 1 N, N.Ns a N, NSN, . 1 re N, N.N;N, .l- N, NZNZN, and we have three ground products of order 4, viz.:— NNN, ENGNG Nas N,NZN2N,, 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 N,N,N, is one of a kind that always presents itself for an even order. The system is : Nia NN No... Nos, and may be separately examined. For the order 6 the ground products N2N.N,, NYN,NN,, NSN.NZN,, NEN.N2Ng, and for the order 8 NYN.N,N.N;, N3N.NN2N;,, NENSNN,, NUN NANGN,, NN MeN Ne, 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 NiNLN,N,N,;, N,N.NZNN;, N,NZNNNs, N,N2N2ZN,N,, N,NZNZNZN;, NiNZNSN2N,, N,NSN2N,N,, N,NZNSNEN;, NiNANZNN:, NeNENINENZ, NANZN,NZN;, NENINZNEN-, NYNANSN SE. 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 con- venient, I denote NNN NN by (0 ,420%30,0). Further, if a portion of the generating function presents itself, which involves merely ground products already obtained in the previous work, I enclose it in brackets [ ] and thencefurward omit it. For example, I write A=[B]+C=C=[D]+E=E; and so on. N+I\™ (n+ 2\% (n+ 8\% (n+ 4\% (rn + 5\% Hor GSU eSrGa hear = = NaN NN N25 = (0 0100,2,0;). When x is of form condition is 4p + 1, a, + 2a; + 4; >a,+ 2a,, 4p + 2, A, > @,, 4p + 3, 2a, +4, + 2a; >a. + 2%, 3p+1, Hy + As Ss, Bp + 2, a + a > 4s, dp +1, a, >a, Dp + 2, a; > as, 5p +3, a >a; omit, 5p + 4, % S45: 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. % SM, %>4s; Case 2. O, 22, As > Ay; Case 3. Wy >, FG, % +4534; Case 4. @ >), G2 >a, a, >a, +4;; Case 5. >%, Ag>%, %+a,>4;; Case 6. CHSSCK, CHSSCH. as > Oy + G5. THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. For Case 1. The conditions become for which the generating function is 10) 1 > 7 5 - 7 iG ee ee ip ey 7 Se a Cc “if which, eliminating b and ¢, is Q 1 men ANAN. aap NNeNS ee ee e 4yi- ef 14Vo- Z 14VolV3. af de W's and eliminating d, e, and f, 9) 1 —(2211) 5, a ea eS ae 1—N,N,.1—aN,N,N,.1 =i Nae ae ese (na yee nid) 1 0 Now = — 1—a@a.1—ay.1—az.1——=w we oc ee ees ~ 1—aw > [" —wu.l—ay.1 ae? 1—ay.1—az.1—* | 1 as 2w + yzw l—a2.1—y.1—z2.1—aw 1-—aw.1—z.1—2w OQ wy? as Z s> 1=ay.1-az.1-—.1 — 2w ne 1 ie _ Fw + yew ~ l-#.l-y.1—z.1—aw’ 1l-aw.1—-2.1—Zw yw yw (pw + yzw) ar ; + = as = l—-y.1l—2.1—2w.1—aw 1-y.1—yw.i—2w.1— cw 24 Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO Hence, putting c= NV,N.N;, y= N,, z=(11111), w=N,N.N,, we have aw = (2211), yew = NE N.N,, zw = (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. Qys>Cs, Oy > Aye The system of conditions reduces to Ob gears alah elcve arch whovwapsltopera rs sateroa wen omoetarnm qeseces (a) Opa, Menace ee ne seicots foeeis wosnes efi oadac mentee (0), OBE Og eecre este ance sche ose eee ieee Oaeane whee eeeee (c), Gy AE Opes Crate nse aon sticioaienae snes cece obkoc eee coe Ree (d), Ogee Og Wis amwnse sahis ene eeaminebisesmeot eens dears eee (e), and TN ANN Ns 1 _2 b * 1 =aN,.1-2 N12 N,.1—2,. 1 a a 5 Sr Ree ee as a e * NNN, Se ar =< ree 1 —N,. 1 —cN,N.N;. 1 — GN NW. 1-- M. 1 es Es (11211) LN. 1— NNN Lao (ae) F yielding the new ground product (11211). THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 25 fal an Case 3. A, >, 24, &U+A,>4;. The reduced conditions are CU tea OC Time retereleletorelstokelstnlalelatatelstsisis\eleisis{sleleleieieieiewereisievee sera ccieeatels (a) Gis SSC ht qa cadsdadnbononedaaCouS UE NSeCOOSHOO EEE meCEere (bd), CLT MCLs peal umes etctatleletedeelelctetelelctetlelertetsteistecistetecietacismtaciceiteraeieieticee (c), Cinta Zs terres rapate ne Olapersiastacome ict aieeae ante eiseke aeiace re cieisiocis nigitsiainteres (d), GECEUAPICtY | Guadondadddooneccous dodcuOrne caTc a mnoeUor RAE eee (e), Cred OL a mer atererelatetelonekel Vateaele(elerotersisicleraisiescielsaversicle aierarsieteietersvoistereteteie (Ga)s Gin 20h, cacndondsqc noes apocondodsscodoboonosnDaqqooouosooaC (9), Oly SPU Se aciawerecnw ace sinelemeernemaseseatamieueaerumeeecnns (h) Obs Oh ger marcrataresoVoratatoteravovoherovatatassVeietereletararalateteretelevafaraveralotavnvatae frejeresare (2) of which the generator is a : = N,.1 Se N, aN a I 0 an a 1 — beef N,N..1— ie 1— ae N,.1- at 1 —ced N,N.N,N,N; ; d bee df the result of eliminating a, g, h and 7. It might be thought advisable at this stage to eliminate b or f, but experiment shews an advantage in proceeding with d. Consider Pp .@) d > w tee Pan” d y w fei Dre eri il di 1 d ~ 21-ay.1—zw\ 1- de re 1 —dz 1_@ a2 aed i Ly _a d 19) gee alls aoe al deen w ~ L-ay.1—2w.1—dix.1—5 1 ,2 qs PY 2 1l—-—ay.1—zw.1—dz.1— y d? Wom, OVID. 4 Mayor MACMAHON, APPLICATION OF THE PARTITION ANALYSIS = : iY ee ee pz _ " [=—e#.1l—ay. l—-zw 1—2#.1—2.1—ay.1—zw prw | 1+ rw) pyz + = l—z.l—ay.l—zw. l—aw” 1—z.1 — zy 1l—ew.1- y2 Hence the generator is :— Jt’ — | | jes 4 1 e ine —bfN, r>=—N,, 2 =-N,, Z=> h 1111 ( putting p=bfN., «x a N3, y fa. ced ( 1)) j / JL N.N, ce nn i = , ae —~N,.1— N,N,.1— 12111).1 —bcef N,N. ie bef Vy 1 beef ( ) cej Vi LV, beef (12111) ewe T 1 — — N,.1—ce(11111).1 — — N,N. 1 —bcef (12111). 1 — beef N,N, bce bef : : | be? (34343) N Ow 1—c(11111).1— =. NM,.1—bof (12111).1 =, (22282) .1—bef NN, bef of ven, (1 +L N, N,) | = ce | eS Ie tae ; je ey ae SS = 9 aE — he ‘6 1 bce Ne: bof NaN} beef (12111). 1 * N2N,.1 beef NN] A+B+C+D, suppose. Zh ay ae ce QO A= 7 1—bdee N,N, .1— bee (12111).1—e (12221).1 Sie N, co) bees sae in — T 1—bceeN,N,.1—e(1111).1—e(12221).1 Fits N; ce Wes (1- M.NeN, (11111)} >1—ceN,N,.1—ce(12111). 1 —e (12221). 1 — NV, N,N, . 1 — (12211) oD Oh. be "> ben,N,.1—e (lil). Se ae22)) pS nnei (13211) {1 — (121) (11111)} 7 P21). 1 — G21) A Sees) n (121) {1 —(121} (11111)} 1—(11).1—(111). 1 — (42111). 1 — (12211). 1 — (12221) #] (121) (1111) 1=(1). 1 —@1) AS) TO THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. a result which indicates the new ground forms (12111), (12211), (12221), (13211). B is easily shewn to have the expression (12111) i— (1) -1— Gi) a) td) = GT) i (12211) (Del dit pr = Ot) i= Geom): C, by elimination of 6 and c¢ (in one operation), becomes (34343) o 1 =((aN aa EL) aan San CRC TL 7 an eee) PT) 1=f(12111).1=f(11).1 = 5 (22282) 1 " (34453) co) ‘K Seb) = 1-(2221) > PUEDE SEES ub Gurr) 1 — f(42141)..1—7, (22282) 2 (34343) {1 — (45348)} = 7—-(l).1—(lll.1—(iliil). 1 —(22111). 1 — (4348). 1 — (33232) = (46564) Senin tyr = (ei) el =(1229 1) = era 4943), 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 D,+D.. For D, we require the result Ww e w e 0 2 Zz 1-ew.1—ey. 1-7 .1— yw cw aw? rll w lool 7. lee Mee ae aw Vy % YP Zw / l-y.l—az.1—yz.1—yw’ 28 Major MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO = which (putting x=bef (12111), y=bef (11), z= — = Ns» — NY W.) brings it to @) bey (131) 2 1 = N;N,.1—f (12211). 1 — ?f# (131). 1 — bef (11) Q bef? (14211) esa ; 1- bop eM. 1 —f (12211). 1 —b2f* (131). 1 — bof (11). 1 — bef (12111) " 9) bi 75 (28422) 2 a ; = pep eM. 1 —f (12211). 1 —b2f8 (131). 1 — bef (12111) .1 — 82/3 (14211) fa) befs(121)2 aa i= pM V,.1—f (12211). 1 —8 2181). 1 = bef A) .1— FA11) id (131) eh) Seas yea) (14211) {1 — (11) (12221)} + 7-11. 1— (lll). 1—(3i). 1 (12111). 1— (42211). 1 — (12231) (14211)? . * 7-31). 1 — (2iil). 1 — (12211). 1 — (12221). 1 —(14211) (121) (stl) A Cae SdamaL=solte yielding the single new ground product (14211). For D, we require the result p 9) e > l= ex). ley ee e e 7 p \ieveeec tere 8 xe ow 2 ~ 1l-a2z.l—yw \l-y [7 Tarlo Seta Sy ee , . Of sro : and putting p= = NEN, «=bef (12111), y= bef (11), 1 : bf? z=— N,, es N2N,,; THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 29 and observing that we may put b=f=1 and that moreover vz=(12111), yw=(131), py =(121) (131), pry = (121) (14211), px? = (13211) (14211), pakw = (13211) (14211)2, pyz =(121)8, aw=(14211), yz=(111), while operation upon the remaining letter ¢ produces no new form, it is clear that no new form arises. Case 4. a)> 0), O > dy, & >a, +45. The reduced conditions are VU imeoletl iotioocqocana nan CORO OSE OODEEOOG ComUnUGC ae Eaereaacn: (a), CRE HE aE LaG SONI SA CeO ER EEE COE EE CER CoE TT Eee (b), Og ONTO we car ncustciecs sce conc nincn acta nion tenet oeeeeeeren (c), 7 Hier a ye ed Pa: bx iS. ree a neon do Cac SOS CaR BEAD OEGLHORCETG OHS (d), Omar CR S564 cosoaocancono sas scm0onobbonGoUDEBDS oaooDbOOOKbOS (e), OE Og h Samtlectarsrsisestseocee scelsisstsiesicectactacaececaltenec (7), CRESCE ocoacodoonsdnos abscnanogdscopoosnooucdasGboose Rasen (9), CR ESCH sonbogdnagou0c soasCa0DNGSR0806 MagaanooBaDuES Adobe (h); leading to {N.> N27 N= NN ,* = _o ac ; ee eet Sy ae ac d be d*f egh and this by elimination of a, b, c, e, g and h becomes 1 = zB N,N,N, V2) Se EEUU EEEEEE UE ESE RRR eeeeeeeeeeeeeemeeemmemmeeerieemeeeeneeee= te 71 —a@f(ui).1—a@f(1221).1-F 7,1 al Me NN, ; c@) 1 and since 2 ; : : 1 De ee aa i 1 a yw ~ l=-a@.1l-y.l—2.l—2w l—y.1—yw.1 —zw fs LW a aw Va loy.l—yu. 1 —2w 1l—z.1—aw.1—yw.1—zw 30 Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO this becomes :— eee. |. i ae (putting g=d?(111), y=d?(12211), z= GN w=5M,) 1 9) ples. 1 2! WW, 1) = (111) a 2). + Noo ey Cp j ; Gris aie. * 12221 . (i222) 1 — d2(12211).1 — 5, NM. 1 — (12221) (1111) ; =F T ; 1 —d*(111). 1—d?(12211).1—— N,N, . 1 — (12221) as the fourth fraction being omitted as obviously contributing nothing new. Now writing #=(111), y=(12211), z=V., w= N.N,N,, p=N.N, w Qa d? a yw(1+y"p) > w = <= , = ; Spey Ra Ree ae eS d? d* w co) d? ‘ yw(1 + yp) 2 p 1l—y.l—«w.l—-yw.1—y¥p eee es w@e®(l+y+y?)1+p) aw l—a.1—ap?.1—yp? 1-—aw yA+p) ww l—-«w.1—y.1—ysp** 1—aw aC K aw +] ap. 1— yp 1—aw’ where K=1l+et+yt+et+ayt yr eyt ay + ry + (a? + ay + y?+ ay + xy? + xy") p + (a°y + wy? + ay") P* Ww (0) d? > 1-—d*x.1—d*y.1— all est pe ee _ yw (1+ yp) (1+ yz) Ee yowp Sl=—y.L—aw.1—yw.1—yp?.l—y2 1—aw.l—yw.1—yp*.1—ye THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 31 vw (1+az)(1+y°?p) aw (yz +y?2zp) aes #.1—y.1—aw.l—a2.1—yp? l—y.l—aw.1-yp? sf ww (xyp + xyzp + xp? + ayp® + xizp® + xyrp® + ay? op? + wiy2*p? + Ye) l—2.1—aw.1—a22.1—<2'p’. 1 xp? In verifying these laborious calculations the relations i 1+ dyz d Zz 1—y#.1—dy ae ae ae 1-d'y.1—- 1—y2?.1 q PB, YE" = 1 1+dy'p Op p 1—yplody” - 1-dy 1-4 yt y agp i= 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. @>4%, dg>%, @ +4;>43. The reduced conditions are Oy > Oy (a) As; > A (bd), 4 + 45> 4s (c), A > Oy (d) A, > Os (e), Oh 2 &, (7); from which eu oO ab 2 Chee == - i) =o e Cae e ap Ad ia ee 1 Neon N, cael 1 quel eae Gy = Vee e le (ULLAL) ot - © N.N,. 1 -£Ny.1 dt). 1 —5N, e d (23422) 71 _ (12211). 1=d (11). 1-5 N,.1-(@111).1- (11211) 32 Mason MACMAHON, APPLICATION OF THE PARTITION ANALYSIS TO _ (12211)(11211) | (12211) ~ J—(11111).1 — (11211) (1 — (42211) . 1 —@11). 1 —Q 11) (12221) ) \ — (12221). 1— (1110) ’ +7 —(2211).1 so that new forms do not arise. Case 6. a) >%, d3>G, @>4,+4,. The reduced conditions are As > A> (a), A; > a, + 4; (b), 2a, + a, + 2a;,>a,+ 2a, (c), Os > Oy (d), a+ a; > 4; (e), a, +a, > a, a); a 2 a; (9); leading to 2 @) ab sie are - EM. (ae -=N, which is readily thrown into the form é = OF eee ae ae ‘5 pW, (1 — 5 MANN ) > ; [are om (a Ona aaa eee 1-e(11l).1—-{ WLNN,.1—7 N,.1—-7 NNNN;.1—> NNNs.1— 551 i4); and eliminating 6 this is oO ¢ (12321) — 3 (11) (12321)? > aes 1 e811). 1-5 NLW.N,.1- (1111). 1 eo? (12211), 18 (11211) .1 = (12221) Q c (12321) > 1 —(1111).1 —(12221).1—c8(11211).1—e(111).1 - 5 NNN, e§ (12211) (12321) 1 — (12221).1—¢§(11211). 1 —¢c?(12211).1—c(111). 1-5 WWW. (Q) c 2 1 l-—cw.l—cy.1—=2 e 1 az+a2°+ a°25 l-#.1—-y.1 yet I —“.1—yz.1 — gz" THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 33 and OD é 2 T 1—¢w.1—cly.1—cw.1— 2 1 Zz Tle lae Peele gel— wl we 2 a ae i ya" l-@.1l—y.l—wz.l1—-yz 1-y.1—wz.1—-yz “e+ azt+ a2° l—a«.l—wz.1—yz.1—a@2*" a Putting now «=(11211), y=(111), w=(12211), 2=(0111), we can examine the generating function. It is clear that a2 = (12321) is a ground product. Also az? = (12321) (13431) = (25752) is a ground product, (13431) not being a solution of the conditions. Further (12211) (12321) z = (12321) (13321), wz = (13321), (12211) (12821) yz? = (121) (25752), (12211) (12321) x2? = (138321) (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 gud 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 a=1 are necessarily ground products. This accounts for 9 of the 13 and it is easy by actual experi- ment to convince oneself that the remaining 4, viz.:— are, in fact, irreducible. Vou. 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 i ney" n+ 3\% ah (“Ey ( 1 ( 2 ( 3 / ( + 5 = (%, &, As, Ay, as)’ n being any integer whatever, that all integral forms are expressible as products of function {) order 1, {(11) order 2, (111) fos order 3, (131) (1111) {aio order 4, (1221) (11111) (11211) (12111) (12211) (12221) (12321) (138211) order 5. (13321) (14211) (25752) (32121) (33232) '(34343) Ill. On the Integrals of Systems of Differential Equations. By Professor A. R. Forsyra. [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 Z = =f (a, Y; 2), naa, (2, Y, Z); then Cauchy’s existence-theorem shews that, if c=a, y=b, z=c be an ordinary combina- tion of values for the functions / and g, so that f and g are regular in the vicinity of z=a, y=b, z=c, there exist integrals y and z of the equations, which are regular functions of # and which acquire values b and c¢ respectively when «=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 #) determined by the assigned conditions. If however a, b, ¢ 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 them expressible in the form IA GS Gy iy B=) Q(a@—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 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF n variables by Kénigsberger*, and in the case of two variables by Goursat?. For our system, the most important reduced equivalent form is dU : t 7, =U + Bd +mtt...=6(U,V,t)| = by cree eee cna (A), t= aU + BV + yt... = (0, Ve t) | where @, and @, are regular functions of their three arguments each of which vanishes with U, V, ¢. The relations between the variables are x—a=t¥, y—b=( +0), z—c=(4+V)*, where 6, ¢, % are positive integers with no factor common to all three, and 6, and cq are appropriately determined constants. The new conditions attaching to the dependent variables U and V are that U=0 and V=0 when t=0; these correspond to the initial conditions that y=b and z=c when r=a: and the matter to be discussed is the determ- ination of integrals of the equations (A) subject to the condition that U=0 and V=0 when t=0. The integrals, so determined, are either regular or non-regular functions of ¢: their existence and their character are affected by the nature of the roots of (E — a) (E — Bs) — a8, =0, 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 ¢ 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 ¢ satisfymg the required conditions. Also Poincaré) and, following him, Bendixson{, have discussed the integrals of n equations of the form du, : dt =O; (tz, Uasiores tn) op (is ces 2) the functions 6, being regular functions of their arguments and vanishing when 1% =0, us =0,..., Un=0: these can be made to include the system (A) by writing »=3, and taking the third equation in the form du deme with the condition that u,, ws, vs all vanish with # Im this case, there is a critical * Lehrbuch der Theorie der Differentialgleichungen, 743—745; see also his Cours d@’Analyse, t. m1, ch. 1. Leipzig (1889), pp. 352 et seq. § Amer. Journ. Math., vol. x1, p. 342. + Amer. Journ. Math., vol. xt (1889), pp. 340, 341. || Inaugural Dissertation, 1879. + Comptes Rendus, t. Lxxxvit (1578), pp. 430—432, § Stockh. Ofver., t. ut (1894), pp. 141—151. DIFFERENTIAL EQUATIONS. 37 cubie 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 Kénigsberger’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 Horn+ 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 Jordan? 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 cB? a me AGE ARE rey 88 yee 2 | , tM tle, Y,; ¥al | 4 A? ae Vaasa Wee sof — (2+ BR+BBza+... aY, T40-Ag 7 My=ay 2 ; x =h Y,+[2, ¥,, va) ie a when the real parts of \, and \, are positive, are the unexpressed terms being 2, §, &, and &, & denoting only way in which z\ ean be terms of higher order in Ay 2, 2 yespectively. The a factor of x is by haying Y,=0™" =e git Aw tAavie | ? Yo=a™ Delgo t Avant Aaven B=0, and then z® is not a factor of y; and similarly as that is, c™ is a factor of Y, and x a factor of Y,. But regards Zand 2’. + Crelle, t. exv1 (1896), pp. 265—306, ib., t. cxvi (1897), pp. 104—128, 254—266. + Cours d’ Analyse, t. 111, §§ 94—97. the integrals of li 2 z = =hye+azx + bzy + cy? | az d 9 at = hoy + azx + Bzy +yy" 38 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF It is convenient to transform the variables. When the roots of the critical quadratic (E—a) (E—B2) — a8, = are unequal, say &, and &,, we introduce new variables u and v, such that u=r~AU+pV, v=NU+yp'S,7 where (4 —&)A+ ap = 0) (a, — &) XN’ + aap’ ay, f BA+(B.— &)u=0) , BW’ + (Bo— &) pw’ =0 y the ratios ©: and 2’:p’ are unequal, and consequently the new variables u, v are distinct. The equations become ot = But gil v, t) dv : ia 0 + dh, (u, 2, t) where ¢,, ¢ are regular functions of their arguments, vanishing with them; except for a term in ¢, 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 & say, we introduce a new variable wu such that u=rAU +z, where (4, — &)X+a"=0, BiA+(B.— &) nw =0. Then we have du. ta Eu + o,(u, v, t), por Sutil tg, (u, V, t) =Kut+té&V + ¢.(u, V, bt), say. It therefore appears that the equations corresponding to the cases I(a), 1(b), I(c), are oS = Fut d, (u, 2, | a ov + dp (u, 2, o| where £, and & are unequal to one another: and that the equations corresponding to the cases II (a), II (6), are Ot = Ent $y (u, v, t) = = Kut Evt do (U, v, p| DIFFERENTIAL EQUATIONS, 39 In both forms, the functions ¢, and ¢, are regular functions of their arguments and vanish with them; and the only term of the first order in ¢, and @, is possibly a term in ¢. For both forms, the initial conditions are that w=0, v=0, when t=0. For brevity, integrals, which are regular functions of ¢, will be called regular integrals: and integrals, which are not regular functions of ¢ but are regular functions of quantities that themselves are not regular in ¢, will be called non-regular integrals. The results are obtained for the transformed equations in uw and v; since U and V are linear homogeneous combinations of w and v, the results apply to the original equations. REGULAR INTEGRALS. Case I (a): the critical quadratic has unequal roots, neither being a positive integer. 1. If the equations du dv te = Fut ¢, (u, Vv; t), ty = be + he (u, Vv, t), possess regular integrals vanishing with ¢, these integrals must have the form 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 t”, we have (n- &) An ine (n — &,) = Gop where f, and g, are the coefficients of ¢” in ¢, and @, respectively after the expres- sions for uw and v are substituted. From the forms of ¢, and @,, it is clear that f, and g, are linear combinations of the coefficients in ¢, and ¢,, that they are rational integral combinations of the coefficients a, d:,..., 6, b.,..., and that they contain no coefficient a after a,, and no coefficient b after 6,. in the respective sets. Since neither £ nor &, 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 ¢, and ¢y, and the denominators are products of powers of the quantities 1-&, 2-&,..., n—-1—-&, n-&4, tes, PAE cane n—-1—&,, n— &. 40 Pror. 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 ¢, and ¢, are regular in the vicinity of u=0, v=0, t=0: let their domain of existence include a region |t}|2. Clearly X =Y; and each of them is given by M Ti+j 4k pu rk Xt XaU i+ and therefore si M\ / Ge 2 X(e+27)(1-) = =u (1-2) 5 In this cubic equation, the term independent of X vanishes when t=0, and the term involving the first power does not vanish because € is not zero. Hence when t=0, 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 ¢ and which is a regular function of ¢ for values of |t| less than the least modulus of a root of the discriminant, that is, for a finite range. DIFFERENTIAL EQUATIONS. 4] To obtain the expansion of this root as a regular function, it is sufficient to determine the coefficients in the power-series X=At+Al+...+Ant"+..., so that the equation M M as Ss Ti+] ke eX = = Ot 2 itp Xt is identically satisfied; because the root which vanishes with ¢ 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 where F,, is the coefficient of ¢ on the right-hand side of the equation for eX. When this value of A, is used for successive values of n, and the new expressions for A,,..., An4 are substituted in F',, the ultimate formal expression obtained for F,, is the quotient of an integral algebraical expression in the coefticients aap by a power of e. p= d Comparing the quantities |7f,| and F,,, we note that a quantity greater than | f,| is obtained when in its numerator every term is replaced by its modulus; that this greater quantity is further imcreased when the modulus of the coefficient of w'v/t* in ¢, or in d M Se re ae: : : ¢, is replaced by arr and that this increased quantity is still further appreciated when every factor of the type |m—é&) in the denominator is replaced by e«. But, on s clear that these three changes turn = comparing the two coefficients a, and A,, it Fn into F,,; accordingly if Se Similarly for g, and F’,, so that a \Giall <5. Also \n—&|>e, |n—&,|>e; hence lee 4le, Dalle Alp. The series Awt+A 0+ Ae +... converges absolutely within a finite region round ¢=0; therefore also the series at + al? +a +..., bt + 6.02 + b+... , 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. Wott, S (u, v, t) = td, (2h, UY, t); but @,° and ¢,’ do not necessarily vanish when ¢, w%, v% vanish. The equations for the new variables are du, a = pan (E-1)p+B+(E—1)y4+td (uw, v1, 0) t (m—1)X+a+(m—1)u,4+ tO) (w, %, 6) Now as wu, % are regular functions of t, the expressions on the left-hand side vanish when t=0; hence (m—1)X¥+a=0, (E-1l)n+8=0. If 4,(0, 0, 0)=m, ¢,'(0, 0, 0)=£,, the equations are | t ues (m—1)u,+4,t + t0,(u,, %,, | ars = (€-1)v,+A,t+td,(u,, 2, ») where @, and ¢, are regular functions of their arguments and vanish when u,=0, v,=0, 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 8 a \ a i Vis al + % - 7 Cs F=T + n) > It is manifest that the equations in », and », can be subjected to a similar trans- formation with a corresponding result; and that, as m is a positive integer while & is DIFFERENTIAL EQUATIONS, 45 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 uw’, v’, we have equations in the form du’ ; Ba Ca u fat thw, v, t)| t s =x + bt+k(u, v, ‘)| where «, =&—m-+1, is not a positive integer; h, k are regular functions of their arguments, vanishing with ¢ and containing no terms of order less than 2. The relation between the variables wu, v and w, vw’ is of the form Ure Ce Oe le where Py. and Q,,. are algebraical polynomials of degree m—1 vanishing with ¢; and u’=0, v'=0 when ¢=0. The coefficients a@ and 6 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 b=pb+ peP+..., VU = gibt got? +...; substituting these, remembering that h and k& are then of the second order at least in t, and equating coefficients of ¢ in the first of the equations, we must have P=p—pt a, which is possible for non-infinite values of p, only if @ is zero. Suppose now that @ is zero. Since uw’ and vo (if they exist as regular functions of ¢) vanish with ¢, we can assume ; : uU=tn, VY =, the sole transferred condition being that », and », are regular functions of t, which now need not necessarily vanish with ¢t. We have pd oa h(tm, tm, th=CH (m, m, 0) 2 We (e—1) tm, + Ot + kom, tm, t) =(k—1)t.+bt+lK (m, m, 0), where H and K are regular functions of their arguments. The second equation shews that, when ¢=0, then («—1)7,+6=0; accordingly taking b ieee ep b, 44 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF we have & vanishing when t=0. As regards 7, there is, as yet, no restriction upon its value when t=0; denoting it by A, we take m=A+&, where & vanishes when t=0. Both & and & are regular functions of ¢ When the values are substituted, 4 remains undetermined by the equations; and therefore an arbitrary (finite) value can be assigned to A. The equations for & and & now are mS Gat (A+S, oe) b 1a (eI) 4K (446, mth t) with the condition that & and & must be regular functions of ¢ vanishing with t. Let them, if they exist, be denoted by BSC G=s lies n=1 n=1 substituting in the equations which must be satisfied identically, and equating coefficients, we have relations Ndn=fn, (N—K+1)b,= gn, similar to those in the Case I (a). These equations are treated in the same way as in the Case I(a). Since « is not a positive integer, no one of the coefficients of 6, 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 & and & converge absolutely for a finite region round t=0; 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=0 be satisfied, the equations do not possess regular integrals vanishing with t=0. Lf that condition be satisfied, the equations possess regular integrals vanishing with t=0 and tnvolving an arbitrary constant: in other words, they possess a single infinitude of regular integrals vanishing with t=0. The condition represented by a=0 can be obtained from the original equations d t oe =mu+at+0(u, v, t) dv ta = fu + 8t+(u, », t) as follows. Let m—-1 u= > f,t?+t"U, p=1 m-1 = & gpt? +i"V; p=l DIFFERENTIAL EQUATIONS. 45 substitute in the equations, and determine (by comparison of the coefficients) the values of fi, -»-, finas Gis +++) Ima. Then the condition is that the coefficient of ¢” im m-1 m-1 at + 6 ( Sif age’, t) p=1 p=l / 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 du to =mu+at+@(u, »v, t) | fale: du ty et Rt+ ou n,t)| These equations can be transformed, as in the Case I1(b), by m—1 substitutions in turn; and ultimately they acquire the form du’ aot bee u fabs ie a ON co =k + bt+k(w, v, » | where «,=n—m+1, is a positive integer greater than unity, vw’ 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 = OS fit, Us > opie zu 1=1 substituting these values and equating coefficients, we have h=Pp+a, g=«n+, (l—1) p,= coefficient of ¢’ in h(w, v’, t), (U = K) Gy = -recererecneeee Gainer): It is clear that, if @ is different from zero, the first equation cannot be satisfied; and therefore as one condition for the possession of regular integrals, @ 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 J=1, 2,..., «—1 in successive sets, each set being associated with one value of J, we have the values of py,..., Dea, Qis++) Gea; all these in general involve a. In order that g, may have a finite value, 46 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF so that (J—«)q, vanishes for 1=x«, we must have the coefficient of # in k(w, 2’, t) zero. If this coefficient be zero, the value of g, is undetermined; let a value 8, provisionally arbitrary, be assigned to it. For the remaining values of J, the equations determine formal expressions for the remaining coefficients, involving a and 8: and no further formal conditions need to be imposed. When the values of py,..., Pea, Qir++ Gea are inserted in k(u’,v’, t), the coefficient of t* in that quantity may be an identical zero; involve two arbitrary constants a and 8 so that, if the functions actually exist, there is a double infinitude of regular solutions vanishing with t. , in that case, the functions w’, v 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 ¢: 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 @ be satisfied; this relation is then to be regarded as determining a, and then for each value of @ so determined, there is a single infinitude of regular solutions vanishing with ¢. These results are stated on the assumption that the power-series, as obtained with the coefficients p and qg, converge: the assumption can be justified as follows. Let A, = pit + pol? +... + Deal, Ba = Qt t+ Gt + ... + Gat, the coefficients p and g being known; then if functions wu’ and v’ exist of the specified form, we can assume hh SEE v=Bi+t°V%, where U’ and V’ are regular functions of ¢ that vanish with ¢ Thus, assuming a= 0, we have dA,_, dU’ =a) ay ae ape ee 1’ =A, tO + h(Aa tt, Batt Vv, 6). t Now the quantity dA, my td ee is equal to the aggregate of the terms involving ¢, @,..., #7 in Ce I 0) Also in h(w’, v’, t) there are no terms of dimensions lower than 2 so that, in Ae te U5 Bea eae) (Ape eb enraut)s the coefficients of t#7"U’, t#1V’ are of dimension at least unity, and therefore this expression may be taken as equal to Ean (Ul aVaient)s DIFFERENTIAL EQUATIONS. 47 where H, 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,,, B,4, t) of order higher than «—1 be Ob + Cea, OO + LS then aU’ ity, ania tas TES) OM Sire oC ge ab 655 Te EE (OM, WS ay, and therefore dU’ . 1) ta 2a eU +c¢,t+ H(U’,V’, t), on absorbing the other powers of ¢ into H, and denoting by H the new function which has the same character as H,. Similarly dV’ ta =V' +b¢+ K(U’, V’, 2), where the terms in k(A,_,, B,,, t) of order higher than «—1 are aie Sona and K is a function of the same character as H. As « 1s a positive integer >1, 2—« 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 1 (0b). There is no regular integral vanishing with ¢ unless b,=0; 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 b,=0 satisfied, it is known from the preceding result that the equations in U’ and V’ possess regular integrals, which vanish with ¢ 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, =0. To obtain them directly from the original forms, we can proceed as follows. Let w= Zit, v= zag, t= t=% be substituted in the original equations: and determine p,, ..., Pm, Gi, +++» Gna. The first condition is that the coefficient of ¢” in m—1 m-1 @( Spt, = gts t) t=1 l=1 48 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF shall vanish. Take p,,=a; and then from the original equations determine Pnii,---, Pn: Gms+++, Yn. The second condition is that the coefficient of ¢” in n-1 n-1 Co) ( > pit’, > qt, t) = t= 1 1 shall vanish. It is not difficult to verify these statements. Summarising the results, it appears that, wnless one condition be satisfied, the equations possess »o 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 tt 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 ta but hunt) | dv _ DFE eu + E+ gu (2, t) where £ is not a positive integer; the functions ¢, and ¢, are regular and (with the possible exception of a term in ¢) contain no terms of order lower than 2. If they possess regular integrals vanishing with t, they must have the forms w= > pal D= Bat n=1 n=1 Substituting these expressions and equating coefticients, we find (n— &) Pn=fn (n— é) Qn = Gn + KPn where f, and g, are the coefficients of ¢” in ¢, and ¢, respectively, when the series for and v are substituted. It is clear that 7, and g, are linear in the coefficients of ¢, and do, that they are integral algebraical combinations of p,, Ps,+--, dis Qe ++: and that they contain no coefficient p or g in the succession later than pj, and q,4. As & is not an integer, the foregoing equations, taken for successive values of n, determine formal ex- pressions for the whole set of coefficients p and q; in particular, p, and g, are obtained as sums of quotients, the numerators of which are integral functions of the coefficients in ¢, and ¢,, and the denominators of which are products of powers of the quantities 1—& 2-&..., n-& To discuss the convergence of the power-series for wu and v with these coefficients, we DIFFERENTIAL EQUATIONS. 49 introduce an associated set of dominant equations. Let a common region of existence of ¢, and ¢, be determined by the range |= |uj2. 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 M Me 0 oie \ o From these we have NeX = M(eY —cX), so that Wax (at +‘); M when this value is substituted for Y in either equation, say in the first, we have (+5 +54 +)x(1-3) 1-= (G+ ) Sey (Soe t Pp. Mimae/\e ilps = a cubic equation in X. The term independent of XY vanishes when t=0; and the term involving the first power of X does not vanish when t=0, because ¢€ is not zero. Hence the cubic has one (and only one) root vanishing when t=0. Wore ovale “J 50 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF lt follows, as before, that this root of the cubic vanishing with ¢ can be expressed as a regular function of ¢ in the form of a power-series, which converges absolutely for values of |t| 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 Sue / IN) Ve Y=X (57+): a i | Y= Qt+ QO+...+ Qnrt™+...)7 In the second place it can, as before, be shewn that the analysis, effective for the determination of p, and g, in connection with the original equations, is effective for the determination of P, and @, in connection with the dominant equations by merely making literal changes, and that these literal changes are such as to give WealkeZeas | Qal< Qn for all values of n. It therefore follows that the series pitt pol +psh+..., htt got + gl+..., converge absolutely within a not infinitesimal region round ¢=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,t at + O(u, v, t) dv : t 7, = Kut mut Bt + o(u, v, b)| where m is a positive integer, the functions @ and ¢ are regular, vanishing with uw, »v, ¢, 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 w and of v after each operation. We take u=t(X+u), v=t(wtr), choosing > and pw so that uw, and v, vanish with ¢: then uw and » 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, KN +(m—1)u+B8=0, DIFFERENTIAL EQUATIONS. 51 so that a Ka B Ta EGP mel? and the new equations are d ; : t—2 =(m—1) my +at+ (um, %, 8) | d , a = ky +(m—1)%4+ Bt+o.(%H, uw, a) 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 , dw aa =u +at+h (wv, v, t) dv’ to anu +0 bt +k wv, | where h, k are regular functions, vanishing with w’, v, t, and containing no terms of dimensions lower than 2; also w’, v’ are to vanish with ¢. There are two sub-cases to be considered, according as « is zero and « is different from zero. First, let « be 0; so that the equations are _ =u +at+h(u, v, t) dv’ ; Fonds are +bt+k(u,, v’, t) It is easy to see, by substituting expressions of the form “’=pttpelt+..., v=gqt+gt+..., that the equations cannot possess regular integrals vanishing with ¢ unless =O, =O. Assume, therefore, that a=0, b=0. If the equations then possess regular integrals vanishing with ¢t, we can take oes vet where now the only transferred condition to be imposed upon U’ and V’ is that they are to be regular functions of ¢. Substituting these values, we find t° sath =h(¢U", tV’, t) =e (U’, V’, 0), i > =k (tU’, tV’, t) =eK (U’, V’, 8), 52 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF so that dU’ aye +s -=H(U’,V’,t), = =K(U’, V’, 6), dt dt 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 ¢ and assume assigned (arbitrary) values when t=0. Accord- ingly, the equations in wu’ and v’, in the case when the conditions a=0, b=0 are satisfied and when the constant « is zero, possess a double infinitude of regular integrals which vanish when t= 0. Secondly, let « be different from zero. If the equations possess regular integrals, they are expressible in the form w=at+tat?+.., v=bt+be+...; substituting these, and taking account of the first power of ¢ on the two sides of both equations, we have a =a+4, b, = xa, +b, +b. Hence we must have a=0; then b, is undetermined, and b a — K a finite quantity because « is not zero. Assuming that the condition a=0 is satisfied, and assigning an arbitrary value A to b,, let so that 7, and », are to be regular functions of ¢ vanishing with ¢; the equations for m, and 7, are d pM ah (to + tm, tA = th, t) dt K = CH (m, No» t), In. b bop = atm + k(t + im, ta + ty, t) t K = xtn, + OK (m, m, 0), that is, they are d to = tH (m, M2, t) dno : t= aN, + tK (m, Nos ol 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 & is made zero. Accordingly, all the analysis of that earlier discussion applies when in it e is taken equal to unity. The equations in 7, and 7, possess regular integrals vanishing with ft, and their expression involves A, the arbitrary constant; and therefore the original equations in uw and v possess no regular integrals vanishing with t unless the condition represented by a=0 be satisfied; but if that condition be satisfied, they possess a simple infimitude of regular integrals vanishing with t. The conditions represented by a=0 and b=0 in the sub-case when « is zero, and the condition represented by a=0 in the sub-case when « is different from zero, can be expressed as before. For the former sub-case, we determine coefficients a and b so that U = Ot +... + Oma US +...) v=bt+...+bn 14+ i satisfy the equations pot mu + at + @(u, », t) | =mv+ Pt+ o(u, 2, a} and the conditions are that the coefficient of ¢” in m—1 m1 at + 0 ( Scat, be t), il 1=1 and the same coefficient in m—-1 =1 ™ Bt+ $( = af l=1 l=1 byt? t) shall vanish. For the latter sub-case, we determine the 2(m—1) coefficients in w and v so that the equations S = mu + at+ @(u, v, t) - =Ku+mv+ Bt+t ¢ (u, », t) are satisfied; and the single condition is that the coefticient of ¢” in m—-1 m-1 at + 0( = at, = byt! t) T=1 i=1 shall vanish. This completes the discussion of the regular integrals vanishing with ¢, with the respective results as enunciated in the various cases. 54 Pror. 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 = SW eB at Se EE | dV yn ai tp = Ut BV + yet +... = 800, V, 2) | possess regular integrals which vanish with ¢: and these are unique as regular integrals. Denoting them by w%, %, let =e, ca aay so that if functions « and y exist, different from constant zero, they are non-regular functions of ¢, and they must vanish with t because U, m, V, % all vanish with t= Then = a(uta, wU+Yy;, t)— (wu, %, t) i 0 =~ 2 - ar = (2 aun, +ye)" 6, (1, %, t) 2 _ = a = tit =25(¢ Ou, fed ae 0, (%4, VN; os are equations to determine « and y. On the right-hand sides there are no terms involving ¢ alone; the only terms of the first order are 42+ ,y, a+ 8.y respectively; and the coefficients of the other powers of « and y are functions of ¢ and of %, x, that is, after substitution of the values of m4, 1%, these coefficients are regular functions of t. Hence we may take the equations in the form dx to = met By +(e, Y, é)| d toe = ar + By +%(@, y, t) | where 3, and S, are regular functions of a, y, t, vanishing when «=0, y=0, and con- taining no terms of dimensions lower than 2 in 2, y, and ¢. 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 ¢ which vanish when ¢=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 (E — a) (E — By) — a8; = 0. DIFFERENTIAL EQUATIONS. 55 When the roots of the quadratic are unequal, say & and &, we take new variables t=rAc+py, t=Ne+p’y, where (a — E:)%+ Ape = 0) (ay — &) M+ tap’ = 0) BrA+(B.—£)p=0f BA +(Bi—E)ui= OJ” the equations become at, t dt an Eh or di (h, ty, t) > ty top = bate + bet t,t) where the regular functions ¢, and ¢, vanish when 4,=0, t,=0, and contain no terms in ¢,, t, t of dimensions lower than 2. When the roots of the quadratic are equal, the common value being &, the cor- responding forms are dt, _ ty = bat tilt, t,t) dt. ) ta = Kt + Etat bolt, to, t) with the same characteristic properties of the functions ¢, and ¢, as for the former case; here t,=y and t=Ar+py, where (a — €)AX + ap = 0, BA + (8B. — —) wh =0, and the constant « is given by cA= a. 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 d t z = At, + att, i dt, t a ty + btt, integrals (if they exist) are required which vanish when t=0. The solution of these equations, which are linear, can be made to depend upon that of a linear equation 56 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF of the second order having t¢=0 for a singularity: it appears that the integrals are normal in the vicinity of f=0. Their full expression is abe (ably \ n= AM tag Rad sp GEA a ( abt? (abt?) — Ber il+s. + ; SE ets +T=p | 7 2@-p) 2.48—p)G—p) b res | abt? (abt?)? ; = 1 = cee Sater l+5@40124G+GF—* abt? (abt?) | +Br {i+ sat et ets 2(1—p) 2.4(1—p)(3—p) where p=A— wz: 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=0 are t= AP + — Be] 1—p b (aa 4£=— A+ Be 1 =P /D the quantities A and B being arbitrary. If the real part of X and the real part of , be both positive, then, when the variable ¢ approaches its origin not making an infinite number of circuits round that origin, 4, and ¢ ultimately vanish when ¢t=0, that is, as X and pw are not integers, there is a double infinitude of non-regular integrals vanishing with ¢ If the real part of X be positive and the real part of ~ be negative, then, when t tends to zero as before, t, can tend to zero only if B be zero: and if B=0, then t, and ¢ ultimately vanish when ¢=0, that is, there is a single infinitude of non- regular integrals vanishing with ¢. Similarly, if the real part of X be negative and the real part of m@ be positive, there is a single infinitude of non-regular integrals vanishing with t¢. If both the real part of 2X and the real part of mw be negative, then ¢, and & vanish with ¢ only if 4=0, B=0: that is, non-regular integrals vanishing with ¢ 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 ¢* alone, when they exist, are of the form 4 =tp,, t= po, DIFFERENTIAL EQUATIONS, De where p; is an arbitrary finite quantity and p, is zero when t=0; and that the parts depending upon ¢ alone, when they exist, are of the form Z=to,, t= a, where o, is another arbitrary finite quantity, and o, is zero when t=0. These particular results are general and, in this form, can be established by an appropriate modification of Goursat’s argument (/.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 dt, t GP = Ah + att) dt, t dt = ki, + At, | integrals (if they exist) are required which vanish when t=0. 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 t, = Aat1(1 + daxt+...)+ Ba fant (1+ dant+...) log t+ (1 — fare? — ...) A}, t= At (1+ axt+...) + B {act (1+ act + ...) log t+ (ak — wet —...) }. When the real part of X is positive, these integrals vanish with ¢; and there is a double infinitude of them. When the real part of » 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 ¢. When B is zero, then the integrals become of the form L= tpn, ,= t po, where p, is an arbitrary finite quantity, and p, is zero when ¢=0. 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 «f, in the second equation. The special results here obtained are included in the theorems relating to the equations in their general form: they suggest that integrals exist which are regular functions of ¢, #4, and # log ¢, when the real part of X is positive. Vou. XVIII. 8 58 Pror. 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 ¢: and therefore, in order to consider the non-regular integrals (if any) that vanish with ¢, we transform the equations as in § 6, and we study the derived system dt, t= Eth + gilts ts d) dt _ th = Et. + ho (h, te, t) where ¢, and @, are regular functions of their arguments, vanishing when ¢,=0, t,=0, and containing no terms of dimensions less than 2 in 4, tf, t The integrals 4, and t% are to be non-regular functions of t, required to vanish with ¢. The main theorem is as follows :— When the roots of the critical quadratic & and &, have their real parts positive, and are such that no one of the quantities A-N&+p&+y, ARt+M-1E+y, vanishes for positive integer values of 2X, mw, v such that X+p+v>2, then the equations possess a double infinitude of non-regular integrals vanishing with t, these integrals being regular functions of t, t&, t. Immediate corollaries, when once this theorem is established, are as follows :— If the real part of & be positive and that of & be negative, there is only a single infinitude of non-regular integrals vanishing with t: they are regular functions of t and t®. Likewise, if the real part of & be positive and that of & be negative, there is only a single infinitude of non-regular integrals vanishing with t: they are regular functions of t and t. Tf the real part both of & and of & 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 # and ¢ by new variables, say thoz, th=z, CANS) so that, by the general theorem, regular functions of 2, 2, ¢ are to be established as DIFFERENTIAL EQUATIONS. 59 solutions of the equations. Accordingly, regarding ¢, and ft, as functions of these three arguments, assume GeO ert) > => Omen zyMos"tP) a where the summation is for all positive (and zero) values of the integers m, n, p, with the conventions oon = 9, Deon = 0. Moreover He a a Oa tay Ye iag Orage? Hence the differential equations are ot ot ot t at a Ez 22, + E25 3a, =Ft+o G, ts, t) ts at, at, t at ate Ez az, =F E25 a, = Ents Te de (4, t, t) Substituting the assumed values of # and ft, and afterwards equating coefficients of 2,{"z."tP, we have \((m—1) & + n&+ p} Gnnp = al {m&, +(n — 1) & + p} Bmnp =P mnp where @,,, 1S a rational algebraical function of the coefficients in gi, of the coefficients Gmin'y’ 1 t such that m2, vanishes. Then when the equations {(m —1) E+ nEs+ P} Gnnp = @'mnp | {m&, + (n— 1) & + p} bmnp = Pas | are solved in groups for the same value of m+n+p, and in successive groups for increas- ing values of m+n-+p beginning with 2, they lead to results of the form Amnp = Imnp> Des = Pawns where @mnp, Smnp are rational integral functions of the coefficients that occur in ¢, and ¢y, these functions being divided by a product of factors of the forms (m—1)&+n&+p, m&+(n—1)&+ p, for m+n+ p22. It has been seen that dq, =0, bo, =0: we easily see that ao,=0, bo,=0 for all values of p. For every term in ¢,(, ft, ¢) and every term in ¢,(4, t, £) involve 4, or #, or both: and the equations for ap, Boop are (Ga &,) Qoop = Ap, (Ga= &,) boop = Boop; where A,o,, Bo, are integral functions of the coefficients m ¢, and ¢,, and of coefficients pop’, Poop Such that p’1. Hence when the coefficient- equations for k and J 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 om =Ymn> lennon where y and 2 are integral functions of the coefficients that occur in y, and y,, each divided by a product of factors of the forms n+mé&,, n+(m+1)&—&.. Moreover each of the coefficients k and J, thus determined, contains 4 as a factor. It now is necessary to prove that the series for p and +, 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 y, and y, be defined by the ranges |t|<7r, |al1, there must be a least value for the moduli of the quantities for the various combinations of m and n; let this value be m, so that n+mé, aie, T= LE inn ert” 64 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF converge absolutely: that is to say, the formal expressions p and 7 have significance, being regular functions of z, and ¢. The equations accordingly have integrals of the characteristics indicated. This completes the first stage of the proof. 11. For the second stage, let ms aae i eras’ ihe the equations for 7, and 7, are r, ee Cee 72, +7, t) — b,(p%, TA, t) =£7,4+,(fL,, AS Ay t) | dT, , are = &T, +.(Ti, T2, &, t) after substitution for p and 7. Here y, and y, are regular functions of their arguments vanishing when 7,=0, 7,=0; they contain no terms of aggregate dimensions lower than 2 in 7,, T., z, t. In accordance with the statement in § 9, it has to be proved that these equations possess solutions of the form T,=20,, T,=2.8:, where ©, and ©, are regular functions of ¢, 4, 2: it will appear that ©,=8B (an arbitrary constant) and ©,=0, when ¢=0. Substituting these values for 7; and 7,, we find aes, 56, , Sone er ee , i +&2, =F =f ee S (& &,) 0, =h (©,, @., 21, 22; t)| We 00, 00, 00, , : t Et ae oF E325 ae =f (©), @,, 421, Za, o| the functions f, and f, are regular in their arguments, every term involves ©, or ©, or both, and a term involving ©, and ©, in the form ©,'@," has also a factor z,\t#™. If quantities ©,, ©, exist, being regular functions of ¢, 2%, 2 and satisfying these equations, the substitution of expressions of the form 0, = LD=pPimn ZZ", 0, = 2 =2dimn 2,20", in these equations must lead to identities. Accordingly, equating coefticients of 2,!2.”"t” on the two sides of both equations, we have (nm at (l st 1) & 3F (m SF 1) £,} Pimn = Gir are (n ar lE, 3° més) dimn = Kim , DIFFERENTIAL EQUATIONS. 65 where 7Timn, “mn are linear functions of the coefficients in f, and f,, and are integral functions of the coefficients pymn and grmn, Such that Vel, mem, wen, V4+m +n’ 2; hence in the equations for Pimn, Yimn, DO One of the coefficients of Pynn, Ymn Vanishes when n+1l+m2>1. Hence these equations can be solved for all the coefficients p and q after poo, Goo. They are most conveniently solved in groups for the same value of n+l+m, and in succeeding groups for increasing values of n+1+m, beginning with 1; the results are Pimn = Tima» Yimn= Kimn> where Timn, Kimn are sums of integral functions of the coefficients in f, and fi, each divided by products of factors of the types n+(l—1)&+(m+1)&, n+lé&+mé,. 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 f,, which are regular in their arguments, be defined by the ranges |¢| 1; and let B\= 5’. Then the dominant equations to be considered are nP, = 7 (®, — B’) ai N 2 (cera) iB Pi P2F/ P2F2 : N _ We, No) (1 ey (1 *) P21 P2 \ #/ pr Vou. XVIII. 9 66 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF The common value of ®, and ®,—B’ is determined as a root of the cubic equation / N N NB’) / 2D ZB 2,P,) {o,(n+ = + <_) + Lee 1-2 )\(1- es) P21 P22) pan) \ aa P2F> Pon r > y \P271 P2% P21 p'a,02) Gaga) \ Lp Pi When t=0, 2,=0, 2=0, the term in this equation independent of ®, vanishes: but the term in the first power of ®, does not then vanish, because 7 is different from zero. Hence there is one root, and only one root, of the cubic which vanishes when t=0, z=0, 2 =0: and it is a regular function of t, 2, 2 in the immediate vicinity* of t=0. Actually solving the equation for this root, we find ®, = she (- tke 22) + terms of higher orders; P22 \T Pi 2a and then we have ®, = B+ AAD (- aes oe ) +terms of higher orders. NP2F2\T Pi Px As in the preceding stage of proof of the main theorem, we can obtain the expression of these particular quantities ©, and ®, otherwise. Knowing that ®, and ®, — B’, equal to one another, are regular functions of #, 2,, Zo, let OD SS by) SS 2 A substitute in the dominant equations, expand the right-hand side in the form of regular series, and equate the coefficients of z,’z,"t" on the two sides. We find ener = 1D Tere But instead of actually deriving [jn, from the equations so obtained, we can utilise the analysis that leads to the quantities 7imn, imn, as follows. Construct | im»! and, in its analytical expression, effect the following changes in succession :— i. Replace every modulus of a sum by the sum of the moduli of the terms: u. Replace each denominator-factor |n+(l -—1)&+(m+1)é, by 7: iii. Replace the coefficient of 0,%O,™2,%2,~t? in fi and f, by N+oa,)™oa.™p,"p.™r?, for all values of m,, ms, %, ts, p: iv. Replace |B} by B’. and |n+l&,+mé,| The final expression, after all these modifications have been made, is Ijn,. But the * It remains a regular function so long as |t 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 |7mn|, of each of the modifications is to appre- ciate the value; hence taking the cumulative effect, we have | Timn |< Wrens Similarly | ee || < Wee Now the series for ®,, when Pin, is replaced by Tn, converges for a finite region round the origin; hence the series @,= — LESminn ata") ©, = B+ 222 kimnZ'Z"t") also converge for that region. Consequently the modified equations have integrals of the character T,=2z0,, T,=2z@,: and therefore the original equations have integrals h=p4t+20,, t=72,+ 202, where p and 7 are regular functions of ¢ and z,: and ©,, ©, are regular functions of Un Pay Zoe 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 @ positive integer. 12. Let the integer root be denoted by m, the non-integer root by &; the equa- tions can be taken in the form du tT = mu + at +0 (u, v, t)) 1G = bv + Bt+ (1 v, d| 5) where @ and ¢ are regular functions of their arguments, vanish with wu, v, ¢ and contain no terms of dimensions lower than 2. The same transformations as were used in § 2, viz. “ : C= (-—*, +4), v=t(- “7 +n), can be applied m—1 times in succession: and ultimately we have equations dt t= htat +filh, t, ) dt. B (hg t dt = kt, + bt +e (4, ty, D)) 9-9 68 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF where x, =E—m+l1, is not a positive integer, the functions f, and f, are regular functions of their arguments of the same type as @ and @ above, and the integrals t, and ¢ are to vanish with t. It has been proved that there are no regular integrals of the equation vanishing with ¢ unless @ is zero: and that, if a=0, 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 ts not zero, there exists a simple infinitude of non-regular integrals vanishing with t, these integrals being regular functions of t and tlogt: and when a is zero, these non-regular integrals do not exist. To establish this result, we proceed from equations (Oo mon att A(a, y t) 1 = xy + bt + 0, (a, y, t) where o 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 @, and @, are required, we suppose A(x, y, t) = LDaypxtyt?, ((+j+p22). A(x, y, t) = TEXDjpaiyit?, With these equations, we associate a set of dominant equations. Let \Qijp| = Aaj, |Dijp| = Bip, |a|=A then the dominant equations are —— oX + At=@, (X, Y, »| ae tS eV + B= 0,(X, ro where @,(X, Y, )=2azAye ee @,(X, Y, t|)= 22> By, X* Vite If « be real, not being a positive integer, we choose that sign for the term + Bt, which makes B k-—1 a positive quantity; if « be complex, we choose a term + Bt, such that B ck—1 is a real positive quantity and |B) >|). DIFFERENTIAL EQUATIONS. 69 By the theorem of § 10, we know that solutions exist, which vanish with ¢ and are expressible as regular functions of ¢ and ¢%. Let a new variable @ be introduced, defined by the equation t?—t=(l—o)@; and, in the solutions indicated, replace t? by t+(1—oc)@; they then become regular functions of ¢ and 6, expressed as converging power-series. To obtain their coefficients in this form directly, let Ne nmol VES (0 (YE where d=0, b,,=0; then since dé t dt = of = t, we have dX t a LZainn {NO™t” + mb" t” (cA — t)} = TE {(n +om) Ot” — MO" t")} Ann, and OV ts = ty = {((n+ om) Ot? — MO" Bins Substituting in the differential equations and comparing coefficients, we have (n+ oM—C) Onna —(M41) Onn, ra= Hm, ‘| (n+ om — kK) lire a (m at 1) (Dra, n—1 — Am, n where H,,, and K,,., are sums of terms of the form Sr 3 Fee = = ‘A sin Om, se Omni; Dmy'ny ode Dmny ’ and similarly for K,,,,, such that t+j+p22 M+... +M; +m +...+mj =m ptmt...Fu+n/+...+nj =n N beimg 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=0, so that m=0, n=0; then Gon = 0) (6. — 03 For m+n=1, so that m=1, n=0: and m=0, n=1; then OB aay — OM (G—16))0 9) 108 (l—o)d,-—aQ,=—A, (l—«)b, —b,.=F B; 70 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF so that Qy=(l—c)an+A, (l—«)bnx =FB, 1, =0; thus a, is undetermined and therefore can be taken arbitrarily, say =C, where C is positive. Thus @, yp, b, are positive. For m+n=2, so that m=2, n=0: m=1, n=1: and m=0, n=2; then Cla Aenean (26 — kK) bay = By? , Ay, — 2Aoy = 2A 299 Ay Aor + ArroGr000 + Arn Go (1+ ¢—£) by — by = 2Boo Gidea + Byodioba + Birr dr (2 — ©) Ay. — Ay = Ag Gy) + Apo Dan + A o29 bd? + Ain Gn + Aon dor + Aco (2—k) be — by = Boyan + Brodaba + Boob a + Bindn + Bobo + Boos 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 x beginning with zero. The result is to give Oran = Orns) Oren = bmn; where Om, and mn are sums of a number of terms; each term is a quotient, the numerator being a positive integral function of the coefficients of @, and @, and containing a,” as a factor, and the denominator being a product of quantities of the form nN+om—a, n+om—kK. 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 imto the denominator product for each of the terms in @n, and mn 1s ~€m+2n-—1. On account of the theorem of § 10, establishing the existence of the integrals as regular functions of ¢ and ?%, it follows that the series Llama", ZLOmnO™” converge absolutely. Now proceed to the limit in which o increases to, and ultimately acquires, the value unity; then @ becomes —tlogt, the differential equations become > a ee (== X+ At=@,(X, Y, é) | dy i E [> er? + Bt=0,(X, ¥, t)| and the integrals change to Ded ant, ODO an Os, where @m_ and b’n, are the values of Gn and bm, when o is replaced by 1. DIFFERENTIAL EQUATIONS. (fil In @,n, let LT be any one of the terms, and let 7” be the value of 7 when co is replaced by 1. As regards the numerator in 7, it is the sum of a series of positive quantities: and it is unaffected by the change of o, except that a, is replaced by A, that is, by a diminished quantity; hence the numerator of J” is less than that of JT. As regards the numerical denominator, each factor n+om—co is replaced by n+m-—1, which is a greater quantity than the factor it replaces, unless m vanishes; but when m=0, then because then n>2. Also every factor n+om—x is replaced by n4+m—x«; 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=0, and then they are the same. The number of factors in the denominator is not greater than m+2n—1: hence N+om—-c n+om—K —, < 9 — o)mten-1 | n+m—1 n+m—k ( ) <4 (2 = Gp) piace The changes made have diminished the numerator of 7’; thus <\ Intom—cao n+t+om—k | a < é = T | n+m—1 n+m—-k | < (2 es: Gp) pu Remembering that @,,, is a sum of terms 7 and bearing in mind the character of 7, we have / | @ mn | < (2 poet ag )tnran. | Ann Similarly / b mn | < (2 = Gp) are | Brora Now the series > 2Genn gm t, Dian (GE converge absolutely for a finite region round the origin. Let this be defined by |t|0O and all values of x. Similarly Dinn = 0) for the same combinations of m and n. In this case, @ disappears entirely from the expressions Siena Oe Ores so that the integrals become regular functions of ¢, which are known to be solutions of the equations when a= 0. DIFFERENTIAL EQUATIONS. 73 13. The main theorems as to the equations dt, ; Ez ot tthG, t,o] dt, : ais topo kt bit fal te t)| 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 « 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, t* and tlogt. If « have its real part negative, there is only a single infinitude of such solutions ; they are regular functions of t and tlog t. When a is zero, so that the equations possess a single infinitude of regular solutions vanishing with t, then if « 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 t*; but if « have its real part negative, the equations possess no non- regular 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= a0 + Att EES Ginn Ot” b > =, _,t+Be+ 22 hime COME where the summation is for values of J, m, n such that 1+m+n>2, the coefficients A and B are arbitrary, € denotes ¢* and @ denotes tlog t. When a is zero, all the coefficients gym, hyn for values of m>Q0O vanish; so that @ disappears from the expressions for #, and ¢,. The resulting expressions then can be resolved each into the sum of two functions: one a regular function of ¢ which involves A, the other a regular function of ¢ and € which involves B, and vanishes when B=0. It may be noted that a slight degeneration occurs in the solutions when « is the reciprocal of a positive integer; a regular function of ¢ and ¢* is then merely a regular function of ¢* When the equations in their first transformed expression are d ie =mut+at+O(u, v. t) 5) dv tr = &+ Bt + o (u, v, .)) the general results are the same as above; the value of « is £—m+1, and the critical condition, which is represented by a=0, is stated at the end of § 2. Vora NaValilile 10 74 Pror. 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— =mu+at+0 , t Ar mu + at (u, 2%, 0) dv ee 525m) They can be modified by substitutions similar to those adopted in the preceding case ; such substitutions can be applied m—1 times in succession, leading to the forms dt . bo htat tilts ts) dt, _ > | t ai = xt, + bt + fo(h, te, t) where x, =n—m-+1, is a positive integer greater than 1, the integrals 4, and #, are to vanish with ¢, and the functions f#,, f. are regular functions which vanish with their arguments and contain no terms of dimensions lower than 2 in &, fy, ¢ combined. It has already been proved (§ 3) that the equations possess no regular integrals vanishing with ¢, unless two relations among the constants be satisfied; one of them is represented by a=0, the other by (say) C=0, where C is a definite combination of a, b, and the constant coefficients in f, and f,. The theorem as regards the non- regular integrals is: The equations in general possess a double infinitude of non-regular integrals which vanish with t; they are regular functions of t, and tlogt. 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 «x 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 lu t a =mu tat + 0(u, v, | dv | eer: =nv+ Bt+ o(u, v, t) represented for the modified forms by a=0, C=0, 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 ¢: and therefore, in order to consider the non-regular integrals (af any) that vanish with ¢, we transform the equations as in §6, and we study the derived system dt, th = &t, 3 di (4, to, t) dt. Ca = xt, + &t. + bo(t, tr, t) where ¢, and @¢, are regular functions of their arguments, vanish when ¢,=0, t,=0, and contain no terms of dimensions less than 2 in 4, t,, t combined. The integrals t, and f, are to be non-regular functions of ¢, required to vanish with ¢. The non-regular integrals are given by the theorem: When the repeated root — 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, #, # logt. When the theorem is established, there is an immediate corollary : If the real part of the repeated root & 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 theorems for the case of I (a) when the roots of the critical quadratic are equal to one another. If &=£& +6, where 6 is infinitesimal, then t= t(1+4+6 logt+...), so that a function of ¢, t&, t becomes a function of ¢, t, #log¢; but further investi- gation 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 f=8, —n=# logt. If therefore integrals of the character indicated in the theorem exist, they can be expressed in the forms G= 222 Ginn Sit”) ; to = DEE dimn Ent”) * 10—2 76 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF and these values must, when substituted, satisfy the differential equations identically. Now dg dy ‘= j-— = —& so that d ta (f'n™t”) = (v + lé + mé) Ghee =, me t1_m—1 yn, Hence equating coefficients of ¢'7”t" on the two sides of both equations after substi- tution ef the assumed values of ¢, and t,, we have {n Ta (l +m-—1) &} Aimn — MQ, ma, n = nan } {n aS (J +m — 1) E} bimn — mbys, m+, n — Oaimn + Bimn where @’mn, Bunn, being the coefficients of ¢’n™t” in ¢, and ¢, respectively, are linear functions of the constants in ¢, and ¢,, and are integral functions of the coefficients Qymin's Ormin', Such that V2. If l=m=n=0, then @’mn=0, B’imn=0; hence Qooo = O, ery ele (0) For values such that /+m-+n=1, we have Ol Gog =O} that Is) ap = AG, C—O) ON Gs =O wb hateis orgs SOs = Ost = UG la SO Ope), 0. Diogo = 8. Gino = OL. In order therefore to obtain finite values for the coefficients a and b, we must have ie=@), J5=(): and then by, by are arbitrary; that is, we have Cpe, tha SO, Chae? ay el, lea SU, Vig = Ot To obtain the terms of dimension two in € », ¢ m # and ¢, we require the explicit expressions of ¢, and ¢,: let them be gb; = att, + btt. + ct? + etto + he db. = att, + Btt, + yt? + eht. + xt?4+... DIFFERENTIAL EQUATIONS. 77 The terms in ¢, and ¢, of dimension one, obtained as above, are %=0, t~=—Ct+ Bn, so that, as far as terms of dimension two in ¢, and @, after substitution, we have gs = bt (By + CE) + k (By + CS), $2 = Bt (Bn + CE) + « (Bn + CEY. Accordingly, for 1+ m+n=2, we have Edhong = KB*, dy, = bB, (2 — &) dum = 0, Gy, = OC, (1+ €) ayy — dom = 2kBC, Edog9 = KC? ; ED 09 = KB? + Oo, Don = BB+ Odor, (2 — E) Bow = Adon, Bin = BC+ Ody, (1 + &) dro — Boon = Oday + 2KBC, EDaq, = KC* + Odin 3 and therefore the terms in ¢, and ¢,, of dimensions two in the arguments ¢, 7, ¢, are - 2kBC + ae z z 026+ ae n+ eon + bC&t + bBnt, in 4: and \ 02 ‘ / 0k ‘ Wai (e+ Ges +(e+ 5) E+ (3 +O) Che + (B+ 6b) Bat iS (1+ =) BG e (eBc+ 2B)! eS ee on in t. And so on. The equations, when solved in groups for the same value of 1+ m+n _ beginning with a zero value of J, and solved in successive groups for increasing values of l+m-+n, give values of Ajnn, Fim, which are sums of integral functions of the literal coefficients of ¢, and @,, 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 = mn» Dimn => Binns As in § 9 for the former case, it can be proved that Gop =0, Doop = 0; for all positive integer values of p, so that there are no terms in f or in f, involving t alone; every term involves either € or 7 or both € and 7». 78 Pror. 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 ¢, and ¢, all the terms which are free from 7; as each of them involves ¢, their aggregate can be taken in the respective forms p, ¢7r; and the remaining terms then have 7 for a factor, so that we may write 4= op te 1, t. = r+ 7O,. It will be proved, first, that solutions of the form 4= fp, t. = Cr exist, where p and 7 are regular functions of ¢ and & p vanishing at t=0 and 7+ 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 t=f+7®,, t=fr+7O,, it will be proved that functions @, and ©, exist, so that they are regular in their arguments ¢, 7, t, they involve an arbitrary constant C, ©, vanishes at ¢=0 and 0, 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 =p, t,=¢ in the equations for 4 and #, we find p and + determined by dp ty a (P- cou) ro dt ty = Op + Yo (p, 7362) where the general character of y, and w, is as before. If these are satisfied by regular functions of ¢ and ¢, their expressions p= pp) ae (ele t", T= inne t”, must, when substituted in the above equations, satisfy them identically. Accordingly, comparing coefticients of €”¢" on the two sides of both equations, we have (n oo mé) Koen = Thao (n+ ME) jmn = J mn + kmns DIFFERENTIAL EQUATIONS. 79 where Km, J’m, are limear in the literal coefficients of p and 7, and are integral functions of Kyyy, jmn', Such that m’ Jmn = 'mn> where kmn, mn are sums of integral functions of the coefficients in y, and yw, divided by products of factors of the form n+ mé. The dominant functions are constructed as before. Let e denote the least value of |n+mé€| for integer values of m and n, so that € is a finite (non-vanishing) quantity ; and let |@|/=0, |C\=C’. Also, let a common region of existence for the functions y, and w, be given by the ranges |t}> Kin om ”, 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 Kmn, tmn can be used to obtain the value of Kyp,, 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| < Kee \Garel < Iie 80 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF and therefore as the series Keen Grune converges, the series LakmnSMt”, C+ Ll S et", also converge. The existence of the integrals, connected with the first stage, is therefore established. 17. Now writing t=Cp+70,, t=fr +7®., where p and 7 are the regular functions of ¢ and € as just determined, the equations for @, and ©, are dO, ta =A, @., & n, t) | e) | . aS G0, + f2(@r, Os, & m, t) |} where 7, and f, are regular functions of their arguments, vanishing when ©,=0 and 0,=0; the coefficients of the first powers of ©, and ©, vanish when ¢=0; and any term, involving ©, and ©, in the form 0,0“, contains 7***~ as a factor. The method of proof and the general course of it are the same as before (§ 11). The regular functions of £ 7, t, which are the formal solution of the equations, are proved to converge, by being compared with the functions which satisfy the dominant equations a AWeane —— Day = e ate 7 \ / 121) at / 1 TA, SAW ee = a = 2:3 (2 r) ( ;) (2 oat, ) \e a ( Sl | 1 M e®, = €|C| +|6|2,+— | "SE eee ee oes : 3) (1 5 A oH : oa,/ (ae = r/ p and are such that, when t=0, €=0, »=0, then ®, is zero and ®,=|C|. There exists a single quantity ®,, satisfying these equations and vanishing with ¢, which is expansible as a regular function of ¢, 7 in a non-infinitesimal region round ¢, the power-series which is its expression being consequently a converging series within that region. And therefore ®,, being given by |4| ®,=|C|+(1+ 1), is also expressible as a regular function of t, & » which, when t=0, acquires the value | C|. DIFFERENTIAL EQUATIONS. 81 A comparison of the coefficients of ¢€'/nt” in ©, and ©, with those of the same combination of the variables in ®, and ®, 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 specitied 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 ag met at + O(u, v, t) d t s =Kku+mu+ Bt+ du, v, t) where the functions @, @ are regular, vanish with wu, v, ¢, 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 m—1 times, leading to new equations of the form dt ty mh tatt A(t, tt) | dt, (ie tO) = Kh + t+ Ot + Os(ty, te, 6) where ¢, and # are to vanish with ¢; and @,, 6, are of the same type and_ properties as 6, ¢ in the first form. There are two sub-cases according as « is zero, or « is not zero. 19. First sub-case: «=0. The equations can be taken in the form da ta erat+ A (a, y, t) | dy the integrals are to vanish with ¢; and the functions 6@,, 0, are regular functions of their arguments, which vanish when 2=0, y=0, t=0 and contain no terms of order lower than 2 in a, y, t combined. The integrals vanishing with t are defined by the theorem: The equations possess, in general, a double infinitude of non-regular integrals vanishing with t, which are regular functions of t and tlogt; and it is known that there are no More aval 11 82 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF regular integrals vanishing with t. If, however, both a=0 and b=0, 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 = —oX + At=32E Ay, Xi Vite 5 pee Seat ol PIS Cap —pl + Bt=>>> Bijn X* Vit? where 1+j7+p2>2 in the two triple summations. The quantities o 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 ¢, these integrals being regular functions of ¢, ¢7, ¢?. Let two new variables 6 and ¢@ be introduced such that i? =t—(c—1)0+(c—1) 9, et (pi Leads we easily find ty tt 8a -p)(1 2) = 86 | . t+ =(c+p-1) $=ag where @ and § are constants which, when p=1, c=1, are equal to 1 and 0 respectively. The regular functions of ¢, ¢7, t? are expressible in the form of absolutely converging power-series ; when ¢? and ¢? are replaced by their values in terms of @ and ¢, the new functions are regular functions of ¢, @, ¢. To obtain their expressions in this last form directly from the differential equations, we substitute X = D>E hinn tO” a V=22> Kinn to" in the equations which are to be satisfied identically. Now Gy 0) dd8 dp 0) Um at) deo aped exe = >rz {(1 +m + an) hin, t! O™ $” = Mhimn tin er o” bd Nhamn t! ern Ge ae Bimhinn t! gra p+} : s ae DIFFERENTIAL EQUATIONS, 83 hence, comparing coefficients of ¢/@”" on the two sides, we have (+m +an— oc) himn—(m+ 1) hpamun—(n +1) hy, m—,n4a + (m + 1) Bhi mia,n—a= Oimn- Similarly (L+m + an— p) kimn —(m + 1) kta, mian—(n + 1) ki,m—a,nu + (m+ 1) Bhi mi1,n— = B’imn- : e : 4 : : MS: : Here @ mn, B’mn are expressions which are linear in the coefficients Ajj, Bip respectively, being an aggregate of terms of the form N, Ag Upper. nec lorenere Kum’ m, Le Kuyamg nj» N, Bijy hi, m,n, weet mene Kitt my nm! soo lianas respectively; the subscripts are subject to the relations m+... +m +m’ +...+mj =m m+...+ m+ +...+n =n |; pt+tht+..t+ 4+ 1/+...4 Fal and the numerical factor NV, is the number of integer-solutions of these equations. In particular, we have hoo = 9, — Kroog = 0. When het: m+n=1, the equations for the coefficients in XY are (1 = ca) hisoo = lOc =-— A, (1 —o) how — hoo = 0; (a —a) hoo + Bho = 0, which are satisfied by hor = ad = a) leap ar A | les =(1—c) how | and fy is arbitrary. Similarly, hoo =(1 = p) Keon sF . kon =(1 — p) Keo and k, 1s arbitrary. When 1+m+n=2, the equations for the coefficients in X are (2 = co) lien = Tees = Caren (l+a-—c) hor +2 Bh 020 — hoor = Aon fF? (2a —) hoo + Bho = 4 oe (2 — &) hyo — 2hoe0 — han = & v1 el Sp eh co) hin = Dawe =r Bho = Cro r (2 =) hoo a hiro = A099 I= 84 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF The first three equations, when solved, determine he», hon, Aow; When the values of li and hy, are substituted in the next two equations, they determine hyo, hy; the last equation then determines the form of Aug. Similarly for the coefficients in Y. For values of 1+m-+n>2, the equations can be solved in a similar way. They are solved in groups for the successively increasing values of 1+m-+n. In each group, say that for which 1+ m-+n=W (so that the coefficients hym'n, Krmn, such that Vim +n ~2, we have lim-—o eee teas be ge where y is a positive small quantity of the first order, unless /+m=2, and then y’=0. Hence ies il 1 Bin eye ea 1 Pil Se = 1 > the difference between the two sides being a small quantity of the first order. Also Qs WEP is a small quantity of the second order, that is, a quantity of an order less than the foregoing difference; consequently Ey # 1 IPs Q8 (2 = Fp) EL : The changes depreciated the numerator of 7 into that of 7’: hence fi Tea ys) 7 Ww Ps < (2 — Bp < (2 — Gp) ATI This result holds for every term in hijnn; hence , ' Winn hamn | < (2 = BPEL, Similarly, | ie | < (2 — o)stsmtan, Let the region of convergence of the power-series VV Thinnt OG", LEZ kimnt oO" gh” be defined by the ranges tre Ol an pra DIFFERENTIAL EQUATIONS. 87 and let M,, M, be the maximum values of the moduli of the series respectively within this region; then M, himn < repre ) M, Kimn < rpm rit 5) consequently M, h imn S ? l Ty yam r, n? Q=cyJ (26) ((2@—c) ' M, k mn S = Hence the series LDN mnt! O™ db", Se Eat OL GE. converge absolutely for values of ¢ such that |t| <7. The existence of integrals of di BS a tae =a + at+ TUL ajypa'yit? d a: say =y t+ bt+ LIV; 2 yt can be deduced from the preceding result, by choosing JaJ=A, |b|=B, |aip|= Ay, | dip | = Bip, as the quantities A, B, Aj, By, for those dominant equations. The expression for the integrals is ll £ = >>> Hin ang”), y = SEEK inn FONG") | where Hym, is derived from h’jm,, and Kin from Kim,, by changing A into —a, B into — b, Ajj, into ayy, and By, into bij,. The effect of these changes is to give | Zim < Rape Greer < Keirants and therefore the series for # and y converge absolutely. The actual values are x=atlogt+Ct+ 22> Amn tmp") y = bt log t+ C+ EEE Kimn tl!) ’ where 0=—tlogt, ¢=4¢t(logt), the summation is for values of J, m, n such that 1+m+n>2, and the coefficients C,, C, are arbitrary constants. 88 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF But the formal expression is more general than the actual value. The equations determining the coefficients are (I +m+n—-1) Hina — (m+ 1) eee a (n a5 1) H,, m—1,n+1— Pee (I +m+n-—1) am —(m +e 1) Ki-1, m+, i (n Si 1) K;, m—1, n+1 = Lipo : with Hyw=C,, Hu =—4a, Hm =0, Kyo = C2, Koyo =— 8, Ko = 0. It is clear that, when 1+m+n=2, nai: nO if n=l, 2; hence Hin, Kimn both vanish for 1+m+n=2 if n=1, 2. Thus for /+m+n=3, 0! Lip (0), if n=1, 2, 35 hence also Hin, Kimn both vanish for 1+m+n=3 if n=1, 2, 3. And so on: all the coefficients Hynn, Kian vanish if n>0; that is, the quantity @ does not actually occur in the expressions for « and y which accordingly are regular functions of ¢ and ¢ logt. The theorem is therefore established. Note 1, Any term in @ and y is of the form Kt (t log t)", that is, Ke*(logt)"; and therefore the index of logt¢ is never greater than the index of t. If, however, the equations were te aat at + ct log t+ TETZaijpg wiy’t? (t log or) dy G SS sd ty Jt? (t | al tay HY t bt + ct log t + SEZZbiing w'y't? (¢ log t) where i+j+p+q>2 for the summations, then the values of 2 and y satisfying the equations are x=— tet (log t+ at log t+ Cit + TEL Aimnt'O"'G") y=— et (log t? + bt log t+ C.t+ DEE Kimn tang”) ? where ¢, 6, ¢@ have the same values as above, and the summations are for values of DIFFERENTIAL EQUATIONS. 89 l, m, n such that 7+m+n>2: and the coefficients Hyp», Kim are determinable as before. Any term in @ is Heltmrn (log (ay earetes, that is, the index of logt is not greater than twice the index of t. Note 2. If a vanishes but not 0b, 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 ¢ are possessed by the equation. If a=0, b=0, then H,,=0, Kim=0, if m>1: that is, tlogt disappears from the expressions for « and y, which then become regular functions and are the double infinitude of regular integrals that vanish with ¢. In this case, the regular integrals are the only integrals vanishing with ¢ that are possessed by the equation. 20. Second sub-case: « not zero. The theorem is: The equations possess in general a double infinitude of non-regular integrals vanishing with t which are regular functions of t, tlogt, 4t(logt); 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 tlogt; 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 « be different from zero: otherwise some of the coefficients in the second set are infinite unless 0 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 t,=a0+ At+..., t,=xap+(KA + b)0+ Bt+..., the unexpressed terms being of higher order in ¢, 6, @: here A and B are arbitrary, 6=tlogt, and d=4¢t(logt). Any term in the expansion of ¢, or ft, which involves ¢ contains « in its coefficient; the disappearance of the terms in ¢@ from the integrals in the first sub-case is thus explained, for « then is zero. Concluding Note. 21. Some sub-cases still remain over from Case I(a), when the roots &, and &, of the critical quadratic do not satisfy the conditions that (§ 8) prevent some one (or more) of the quantities A-lE+pE+y, AE+u—-1)E +», Vor, XVI 12 90 Pror. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF EQUATIONS. from vanishing for integer values of A, w, v such that A+m+v>2. The real parts of £,, &, are supposed to be positive. The instances that can occur are obviously for X=O in the first set and ~=0 in the second set; both are included in the form E=pnt+y, where & and 7 are the roots of the quadratic, and w+v>2. The cases p=0, p=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, t, tw’ log t, where w and v are integers. It can be established in the same manner as the similar theorems in the preceding sections. IV. Ueber die Bedeutung der Constante b des van der Waals’schen Gesetzes. Von Pror. BottzMaANN und Dr Macugs, in Wien. [Received 1899 August 14.] In dem Buche von Professor Boltzmann “Vorlesungen iiber Gastheorie, 1. Theil” wurde die van der Waals’sche Formel aus der Vorstellung abgeleitet, dass die Gasmolekiile Anziehungskrafte auf emander ausiiben, deren Wirkungssphiire gross ist gegen den Abstand zweier Nachbarmolekiile. 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 kénnen, falls die Anziehungskraft gleichmassig nach allen vom ny. Es gilt aber allgemein fiir ein Gasgemisch aus n, und n, Molekiilen verschiedener Art die Beziehung pV =— (m+n) = MRT (n, + n.). 12—2 92 Pror. BOLTZMANN cxnp Dr MACHE, UEBER DIE BEDEUTUNG Nennen wir a die Zahl der Molekiile bei vollkommener Dissociation, so ist “= + 2ny=N, + KN? Hingegen ist die Zahl der freien Molekiile im betrachteten Zustand at+NH N=NM+hm=—>—- Durch Elimination von n, und Entwickeln der Wurzel findet man hieraus den Naherungs- : WK , - wert n=a—- und folglich auch weiters pveaunr—-oURE Ist aber m die Masse eines Molekiils, » das Atomgewicht, v das _ specifische m am_ 1 ie aa ? Volumen, endlich 7 die Gasconstante des betrachteten Gases, so ist M= Bee ae : endlich —=r und es wird auch mv oder wenn man auf den Ausdruck fiir « zuriickgeht wae ot Maat cal hes py ang Opa {e® [(co +1)? +1] — [(eo +8 +1)? +1)} ‘a Hiebei ist aber in v noch der von den Deckungsphiiren der Molekiile ausgefiillte Raum p= See abzuziehen. Wir erhalten also als Zustandsgleichung nL a rT A Pvp py Zur Discussion dieser Formel finde noch folgende Betrachtung Raum. Es ist, wie man sich leicht durch Rechnung tiberzeugt, ease) —— 1 ie e®[(co +1) +1] -[(eo +8 +1 +1] =e 'o%8 5 Ss S (e5)")= mie = + : 1 1 2 2 2 (i 5 oO \o Ferner ist Al = == Qro*br TS = ~ (eB) )- =, ian =saTHT| + aot: Es gilt weiters die Beziehung -¢-=2hC= S 5 uy 7 ; : ) mr” T Setzt man endlich Ln 27076 = 4, oe =f, ve €, m mr é so ist auch p= | ee peg ae a ee ee Se. DER CONSTANTE 6 DES VAN DER WAALSSCHEN GESETZES. 93 und es liasst sich die obige Zustandsgleichung in der Form schreiben: oe ill arT = (5) (1 2 2 = y ae v—2ae (v— ae)? 24 T) In! n+1lle n+2!2 Die Constanten dieser Gleichung haben folgende Bedeutung: Es ist a gleich dem halben im Volumen der Masseneinheit vorhandenen kritischen Raume, Y Br= 2 gleich dem Potential der Anziehungskraft auf der Oberfliche der Deckungs- sphare, endlich e=5 gleich dem Verhiltnis aus dem Durchmesser des Molekiils und der Distanz, auf welche die Anziehungskraft wirkt. Da die Gleichung 233, von welcher wir ausgegangen sind, voraussetzt, dass die Anzahl der Tripelmolekiile gegen die Anzahl der Doppelmolekiile 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 himaus wachsen. Dies wird um so weniger der Fall sein, je grésser e ist. Aus den Versuchen von Amagat und Andrews iiber die Compressibilitét des Kohlendioxyds berechnet sich e€ fiir dieses Gas zu ungefahr 100. Nach dieser Vorstellung scheint also der Anziehungs- bereich sogar noch relativ klein zu sein gegen den Durchmesser des Molekiils. Wir haben bisher unsere Zustandsgleichung abgeleitet, indem wir fir f(r) ein bestimmtes einfaches Abhangigkeitsverhiltnis eintihrten. Liisst man f(r) ganz will- kiirlich, so ergibt sich leicht, dass dies den Typus der Zustandsgleichung, auf welche man kommt, in keiner Weise verandert. rl v—p (= a) Es wird stets p= und es ist nur noch A von f(r) abhangig. Dies gilt freilich nur solange man die Anzahl der Tripelmolekiile und der noch hdheren Congregationen vernachlissigen 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 héheren Potenzen enthalten. Es ergibt sich dann fiir p eine Potenzreihe, wie sie fihnlich auch schon Herr Professor Jager von anderen Betrachtungen ausgehend aufgestellt hat. Leider begegnet die Auswertung ihrer weiteren Coéfficienten kaum zu iiber- windenden Schwierigkeiten. V. On the Solution of a Par of Simultaneous Linear Differential Equations, which occur in the Lunar Theory. By Ernest W. Brown, Se.D., F.R.S. {Received 1899 July 14.] In the calculation of the imequalities in the Moon’s motion by means of rectangular coordinates a certain pair of differential equations is continually requiring solution. The left-hand members 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 com- puter can use easily and is besides peculiarly lable 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 xr dy A — In’ —2 = ag ag t Lx+L’'y=R, d*y ,dx / (TE ey of qe to” ar lert Sle in Sere Gas on Mr BROWN, ON THE SOLUTION OF A PAIR OF EQUATIONS, ETC. 95 where E L’ Th h, of the forms S,q;°0° {i(t-—t,) +7 (¢—4t,)} (n—w) RP. “ii afin 0 i)5 ) tv, &, 7, , n’, q; bemg known constants, and 7 taking all positive and negative integral are of the forms iq; ae (20+1)(n—7')(¢-1t), values; 7 is either an integer, in which case #,=¢,, or is incommensurable with an integer. The corresponding particular integral required is, in general, Zs pi cos i =; Bain {¢(t—t) +7 (t—t)} (n—7). 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 7 increases, Nevertheless, it is frequently found necessary to proceed as far as 7= +5, demanding the determination of about 20 unknowns from the same number of equations. In the determination of the latitude the equation dz ; det 2-2 ) occurs; L,, R” are of similar forms to L, R’, respectively. If 2,, z. be two particular integrals of it is known that the particular integral required’ is Be C=a, [a Rat — 2, |z,R" dt where C is a constant given by dz, dz, C= 4, Ft 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, * Mem. R. A. S., Vol. ui. 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 wu, s, where U=L+ Yl, s=@2—YL. We also put €=exp..(n—w)(t—t), d 1 d a afar ay aren TS n nS sr (=) She The generality of the results is not affected by the last supposition. The simultaneous equations then take the form (D+m)+Mu+ Ns=A | f FS Saiconddnuou te necoercaso dee sant nec (1), (D—-m)y+Ms+Nu=4 J where M, N are of the form =p;¢, jc : 2 , L2=0,+1 A is of the form =p,f%+* + Sp, ) M= WM. The bar placed over a letter or expression denotes here and elsewhere that « has been changed to —z, that is, €+ put for ¢ f To obtain the particular integrals of equations (1), it will first be necessary to obtain four independent particular integrals of (D+m)yu+ Mu+ Ns=0 ) = Liisi eia as cteaio en sta ainsi eeaseeee (2). (D—m)?s+ Ms + Nu=0 } Denote these integrals by W=G, §S—= 53) J — 12585) 45 so that if Q; denote an arbitrary constant, the general solution of (2) is u= >; Q; Uj, s= 2; Q;5;, jek mh, 3} 4, By supposing the Q; to have variable instead of constant values we can then proceed i 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 Du + 2m Du OE Te +s) i . 2 (us)3 : 3 8 : D's — 2m Ds +5m*(wts)— 7 =O, | | DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 97 with their first integral, 2k : (us)? 5 3 P= Du. Ds+ 7m (ut s+ admit a particular solution, Uy — Oe eS — Sa — > ee Cot containing two arbitrary constants; these constants are the quantities denoted by n, t, above. The coefficients a; are functions of n and the known constants present in the differential equations. Put U=Utwy, S=SH+S, and, after expansion in powers of %, s,, neglect: squares and products of these quantities. Omitting the suffix, and giving proper meanings to M, JN, the resulting equations become those denoted by (2) above. The first integral = C becomes b= oh U = Ee =i()) OU 08) 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 inde- pendently 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, w= Du,, s=Ds,, and obtained the forms of the other two; the coefficients of the latter have been obtained by myselft. It is therefore only necessary to find a fourth solution, linearly independent of the other three, in order to obtain the generai solution. ue The Fourth Integral of the Equations. (Da sin HEI Mite INS 0) caoascdenoddodooococapssoqnandgas (3), (Di hies ee! tb INNS (eankey One Renee BOE Cea REECE (3’). The known integrals may be denoted by Uy = pe Cte, = Dd, Gaue Ue = Dee gor, CR Pa an So oe (4). UU, = > (22 == 1) TG. 83 = > (22 = I) pee * Acta Math, Vol. yur. pp. 1—36. t+ Mem. R. A.S. Vol. x11. p. 94. Vout. XVIII. 13 98 Mr BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR If Q,, Qo, Q; be three arbitrary constants, then w= >i Oju;, $= >;Q)5;, i— de DF BOs release neeeaisisiceieenies eeeee (5) is a solution of the equations. Owing to the introduction of Q,, Q,, Qs, we can consider Uy, ... 8; completely known; ¢ 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 u, DQ, + Ue DQ. + U;DQ;= 0. By substituting (4) in the differential equations we find Du,. DQ, + Di. DQs + Duz. DQ; = 9, DF (s;D?Q; + 2Ds;.DQ; — 2ms;DQ;) =0 viajeealele eeleie oe s)sioleisiels vaawieisle cic (6). Put Uo Dug — Uz Dis = 0, ete. Then DQ: _ Ds = DQ: = [, suppose. a, As As Substituting in (6), the equation for Z may be written, (Sas) D+ 20-D(zas) — L (SsDa + Qmdas) = 0... occ. nee eeoeceneene (6), where Das = 0,5, +48, + 4,53, ete. The last term of this equation can be shown to be zero. Substitute w,, s,; and wz, s, successively in (3): multiply the resulting equations by s., s, respectively and subtract. We thus obtain (D + 2m) (s,Dw — 8, Duz) + (m? + WM) (sy — U2S,) = 0. Also, treating (3’) in a similar manner, (D — 2m) (Ds, — uw, Ds.) + (m? + MW) (2.8, — 2%) = 0. The sum of these two equations is integrable and gives 8. Du, — Wy Ds, + Uy Ds, = 8, Duty + 2M (S2ty — US8,) = Cy, where (C,, is a constant. It should be noticed that this constant is not arbitrary since the values of 2%, S,, V2, Ss, were definitely fixed, so that C), may be treated as a known constant. Denote the last equation by Multiply these three equations by wm, ww, us and add. Noticing the meanings attached tO 4, Q@, a, we obtain U, Cos + UpCy + Us Chg = Las. Similarly 0 = 4 Dfrs + Df + UsDfro = YsDa + 2m Sas. DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. Substituting the last result in (6’), we find DI D (as) _ ais ae ase) a 0, which, on integrating, gives L, I, L= = = ; (Sas)? (Cy + UsCy + Us C2)?’ where JL, is a new arbitrary constant. a : etc., (2 Cy + UC + UsC 2)?’ Thence Q, =(Q,)+L,D> 99 in which (Q,) is a new arbitrary constant and D— denotes an integration, 1e. the operation inverse to D. If, finally, we now let Q,, Q, Qs, Q, represent four arbitrary constants, the general solution of (2) is U= Qty + Qos + Qyus + Qyus, — Q: 84 ar Q. 8. ae Oe, + Ojst 5 (U, Coz + Uo Cy + Ug Co)?’ where Us = ju; D> i) 1h Py By This result is true whatever particular solutions are represented by Tes SS Whe BH 2 Tha Se as long as they are linearly independent. As, however, the expression for wu, 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 C,=0=C,,. For, looking at the forms assumed, we see that uw, s, contain the factor €°, wm, s, the factor €~ and ws, s, have no such factor. Hence f, has the factor €°, f, the factor €~*. As ¢ is supposed incommensurable with unity, the equations (7’) are only possible if C,=0 and C,,=0. Hence we have Uy Dus, — UzDu u; Du, — u,Du U, Dus — u.Du 2 3 = 3 etl UD 3 1 = 1 SAE OD 1 2 2 oe Us" Us Us? Wg Ca ee The first two terms of the right-hand side are integrable and become that is, zero. Whence considering (C;,? as absorbed in the arbitrary Q;, we have Us = Uz D> ( - Us? aD) Ree ei) ti ptt (8). We may similarly show that if ae ea —— 4: = 93 = i an Sia oS $3" 100 Mr BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR III. Although this is probably the simplest form obtainable for u,, it is unsuitable for calculation. The values of w,... are all of the form sum of cosines+e (sum of sines). To adapt u, to calculation it is best to express it in the form us (P + Qu) where P, Q are real. I shall show that % _ pa (a = — Us Us? S,Uy — Uy So (Che Silo — Uy So Du, Ds 27 CaS eh pa a eee (2m reer ) ee (9). a UsS3 (UsSs U3S3 Us Ss Since f;=0=f_ and fi2=Cy, we have 1 Gis) Ua fis — thifas Wee Uy Duy — Uy Du, Z. UyDs, — Uy, Ds. + 8; Due — $2. Diy ” Us83 Us Ss Us3S3 us” 2U383 $,Us — US. Dus S$ Uy — Uy Se Duy — Us Du Sy Us — Uy So - —m ——— ; a 2 UsSs U3S3 Ug Us85 Us __ 81Ue — WSs (= =e) — yy 8 UsSs Submitting this to the operation D~ and transposing we obtain the required expression. It is easy to see that (9) is of the required form. For when we put —zc for «, that is, 7 for &, the expressions Uy, Uz, $1, Se, Us, Ss, D+, D respectively become Shy iohn hy Ung Sos ae —-D, -D; the first term of (9) is therefore unchanged, while the second term simply changes sign. Hence the first term is real and the second a pure imaginary. Tae It is necessary to examine the four solutions and especially the one last found a little more closely. Write u;=u;(P + D“P,). The expressions (4) show that P and P,, being both real, will be expressible as sums of cosines of multiples of the angle 2(n—7n’)t. As P, contains a constant term B, DP, contains a term of the form ¢Bt(n—n’), and therefore wu, is of the form us {uBt (n —n’) +a power series in £3}. It is therefore of the same form as us, except for the part iBtu; (n— 7). = DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 101 We saw earlier that the equations (2) admit of a first integral p=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 @ the solutions wz, 8, or ww, 8: OF Us, 8. For the solution ws, Ss, the constant takes the value C, which is not zero. Hence though (u,, s;) belongs to the linear equations (2) it plays no part in the non-linear equations from which these were derived. The solutions ~%, s,; and w,, s, 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 u,, s; and w,, s, are not used in the development. The particular solution of the original equations of which use was made was U=Uy, S=Sp, where Up = Tia, F*1= Ta; exp. (21+ 1)(n—7n’) (t—t,). If we add a small quantity 6¢, to ¢, (which is an arbitrary constant of this solution) the resulting expression will still be a solution, Expand in powers of 8, neglecting squares and higher powers. The additions to w, s) will be oe Beis, e éu= 2h, 6t,=—Duw. dt, ds = 7k, bt) = — Ds,. Sto. These values when substituted for wu, s in (2) must satisfy them independently of the value of 6f. Hence w=kDu, s=kDs is a solution obtained merely by altering the arbitrary t, and is therefore unnecessary for the development of the Lunar Theory. The other arbitrary constant in % 1s n, and the coefficients a; are functions of n. If we make a small addition 6n to n and proceed as before we see that Weiony wits Ieee TN ie ae is a solution of the linear equations (2). It is only necessary to identify this with w, s,. The forms for both are evidently the same. For we have & S im + (2641) (tt) as} exp. (21+1)(n—n)(t—h) >; S exp. (20+1)(n —7’) (t—t,) +(t-—t) Dw. The terms with ¢ as factor agree (f, was put zero in the expression for u,) 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 { 7 — poe u,D> ( u, Dus “0u) on \ us This relation is a somewhat remarkable one. In investigations where the arbitrary : eMELIs (ae? Posy constants are varied—and there are many such—we have a means of obtaining ant oe (which are the most troublesome to find) when the numerical value of the ratio n’/n has been used in finding 2, 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)?u+Mu+ Ns =0, (D —m)s + Ms + Nu=0, in the form = ZO es = Ors py — le Zsa, the next problem is to find the solution of (D+myu+Mu+ Ns = A, (D—m)s + Ms + Nu= A, where A, A are functions of the time. Following the usual method of varying the arbitraries we have >Du;.DQ;= A, =Ds;.DQ,=A Yu; DQ; = 0, Xs; DQ; =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 IL show that we can derive s,/s; from u,/u; by putting => for € and changing the sign. For w, s, interchange as do w%, s,, while D changes sign: u, becomes —s,. Since Us = Us (P + Qt), we have 8,= 8, (— P+ Qu). Hence U4S3 — SyUz = 2UsS,P ig Cie stl GRINS AER sine wNaic anal saieis wistele etetae (11) by the result obtained in III. * Proc. London Math. Soc. Vol. xxvur. pp. 143—155. DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 1038 Again, as the first integral obtained in II. is equally applicable to w,, s,, we have Os, = fos = 8, Dus + uy Ds; — Uy-Ds, — 8; Ditg + 2m (8,3 — UyS3), which, by inserting the expressions for u,, s, just given, becomes Oy, = — 2 (s3Dus — us Ds;) P — 20383 DQ + 2m (82, — UoS;), or, using the values of P, Q obtained in IIL, Sy Uy — Uy 82 Os, = — (8; Du; — uz; Ds) ‘12 U383 Du, Ds + (8, Uy — US») (2m —— _) + 2m (SU — S; U2), \ Us S83 whence Gr Ohh cooncos ono oseunG esac ence ORE ee ee eerrreee (12). We can show as in II. that O,=0=(4,. Solving equations (10) we obtain where N= Dun Di Dieu, | kas D3 Ds;, Ds, Oh, Si Oly Yr S), Soy S83, Sy A= |7A5 Du Du Du, AR Ds) Ds sues 5, GS: 0) 0, Som Sen Br In the determinant A the first minor of Du, is D3z (Us84 — 83s) + Dss (W482 — 8p) + Ds, (Uo; — Sottg), = 8. fra +S3 fio t Si fos, = 8, 0x, + 83.09 + 8,Co5. Also, the first minor of Ds, is similarly = (Uy Ogg + Us Cn + Ug Cr). 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 C,,=—C, and that all the other constants Ci; are zero, we obtain (6,4 a AO A= (Aa Aes, A,= (s,A+1u,A) Cs, A= — (GAS A) OL and A= —(s, Du, — 8, Du, — 8,Dus + 8, Dus) Op. 104 Mr BROWN, THE SOLUTION OF A PAIR OF SIMULTANEOUS LINEAR But the effect of putting {7 for ¢ in A 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, Ds, + u. Ds, — u, Ds; + 8, Dus) Cs. Whence, by addition, 2A = —{ fio — 2m (s2%y — UoS,) — fog + 2M (8,Us — UySs)} Cho =— (C,.— Cy) CG. = — 26,7, in virtue of (7) and (12). Hence A=—(C,,%. Ay sA+ UA Finally, AQ, = 7 C. ete. and Oh = a D> (s,A + U,A), ete. And the particular integral corresponding to the right-hand members, 4, 4, is u= a {uD (s,A + uA) — UD (s,A + u,A) —u,;D— (s,A +u,A) + u,D> (s;A + u,A)} aen) (13), : =o {s,D- (8,4 + tA) — D> (8,4 + m4) —s,D(s,A + u,A) + 8,D7 (s,A + usA)}. It is easy to see that s is derivable from wu (as it should be) by putting €~ for € In fact, the coefficient of w, in the first term is conjugate to that of w. in the second term, that of u, in the third term is a pure imaginary and that of uw, in the last term is real. VI. In the applications of this result to the Lunar Theory 4 is always an expression of the form Digit + Ot Gomes uO; aE ile 35 2, tery where 7, q:, g: are known constants; A is derived from A by putting {> 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 w,, s, contain ¢ in a non-periodic form. I shall now show that in general all the non-periodic parts disappear from the particular integral. Put Uy = Us + Bust (n— n’), 8, = s¢ + cBs,t (n — 1’). DIFFERENTIAL EQUATIONS, WHICH OCCUR IN THE LUNAR THEORY. 105 Then w,’, s, are periodic. The sum of the third and fourth terms of (13) becomes — u,D— (s,/ A +uj A) + uj D3 (s,A + u,4) —[u;D> {(s,A +u;A) t} + uj¢D>(s,A + u,4)] cB (n—1’). The first line of this expression is in general periodic. The second line becomes, on inte- grating its first term by parts, u;BD~ (s,A + uA). The non-periodic part thus disappears. When we perform the double integration involved in this last expression, we obtain Us (Cy + Cie(n—n') t + periodic part} where C,, C, are arbitraries. The terms containing C,, C, are simply parts of the comple- mentary function and may be considered as contained in Q,u;+Q,u,. The particular integral may therefore be written u= = [uD (s,A + UA) — UD (s,A + u,A) + uj D- (8,4 +u;A) 12 2 —u;D— {s/A + u/A — BD (s,A + u;A)}]......(14), which is its final form. WAL In general this particular integral consists only of periodic terms. There are, how- ever, two cases in which non-periodic terms may arise. If t=an odd integer, that is, if A is of the form Yq", the integrals multiplied by wu,’ and wu, might give rise to terms of the form at where a is a constant. In this case, s,A4+u;,A is of the form =; (¢% —{-) and therefore its integral will be periodic. The last term of (14) is of the form — u,D~ (coust. + power series in €?), =—u,;(tk + k’ + power series in 7), k, k’ being constants, the former definite and the latter arbitrary. The terms — wu, (tkh+hk’) may be written 5 kus B r= 7 e(n—n') Be(n=n’) The first two terms of this may be considered as included in the part Q;u;+Q.u, of the complementary function; the last part is definite and periodic. Hence no non-periodic —k'us— \ustBe (n— nv’) + uy} part remains. The second case of non-periodicity occurs when A= DiGi Gaui ae Bigs CE, Here the first two terms of (14) may give rise to the non-periodic part {ute (n —n’) [8.4 + UA], — uote (n —n') [s,A + uA], + Cp, Vor, XVII. 14 106 Mr BROWN, THE SOLUTION OF A PAIR OF EQUATIONS, etc. where [yy], denotes the constant term in the expansion of Ww as a sum of cosines. Now sA+uA and s,A+u%,A are conjugate. Hence [s.A + wA],=[s,A + A], = [(s, +5) Ah. Thus the non-periodic part is (ate) Sass) PAN ce — 0) 6 Gin ne eoceoeeeaeceten escent (15). In the applications to the Lunar Theory, the part of the complementary function used is obtained by putting Q,=0=Q,, and the constants in ™m, uw. are so adjusted that we can put Q,=1=Q.. I shall show that (15) is equivalent to a small addition éc to c in the index of ¢€ in Uy + Us = TpegSFtt’ + Sie, squares and higher powers of 6c being neglected. Put e+6éce for ¢ in the last expression. It becomes Uy C8 + Uh EPC, Remembering that €=exp.c(n—n’)t and expanding in powers of S¢ we obtain Uy + Uy + (Uy — Uy) See (n — nr’) t. Comparing with (15) it is evident that we can put 6c =[(s, +8.) A], + Co. This is nothing else than the general form of the expression which I obtained in a paper, “Investigations in the Lunar Theory*.” For Co =fio = [fxlo = 2) (27 + 1 +m +c) 62+ 5) (27 -1—-m+c)e;7, on substitution of the values (4) in f,. Also s,+s, is the same as the expression there denoted by s,. 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, etce.t.” No useful purpose will be served by giving further details of the comparison of the two forms for &¢e. 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 6c—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 terms. In the general problem the non-periodic terms will remain. * American Jour. Math. Vol. xvu1. p, 336, equation (16). + Mem. R.A.S. Vol. ur. p. 75. VI. The Periodogram of Magnetic Declination as obtained from the records of the Greenwich Observatory during the years 1871—1895. By ArrHur Scuuster, F.R.S., Professor of Physics at the Owens College, Manchester. [Received 1899, Aug. 1.] I. [xrropuction. 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 must 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 all 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 fluctua- tion 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 im definite periods, such as the lunar and solar variations, from the more complex changes on which they are superposed. Il. THrE PERIODOGRAM. Let f(t) be any function of t, and consider the quantity R determined by the equations ttuT ttnT ynTA=[ f(t)cosetdt, 4nTB= I Faysman as (1), T where «=27/7 and n is an integer. In these equations 7 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 7 with a constant value of n and 7’ will cause R to fluctuate round some mean value. Let S? be the mean value of R? which, still keeping m constant, will in general depend on 7. With 7 as abscissa and S* as ordinate, draw a curve, which may be called the “Periodograph.” I define the “Periodogram” as the surface included between this curve and the axis of 7. 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. Casz 1. Let f(t) be a simply-periodic function, so that we may put T(t) =cos (gt + 8). FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 109 The integrals A and B are easily calculated and expressed in the form nTA=2 E (aD) peas | sin $gnT, J+K g-K ji nTB=2 ee gl) + ass =| sin dgnT, g+K =k ahd where 2a=aq4+ &, 2b=6,+ B., and a, = KT, 8,=97 + 6, m=K(T+nT), Br=g(r+nT)+6. 2 Hence nTR= ¢ San sin }gnT' [2 (g? + «°) + 2 (g? — kK”) cos 2b} 5 If the average of R® is formed for different values of 7, the term containing cos 26 will disappear, and therefore writing y=4t (g—Kn)nT=7 c— & i it follows that ee. J2(g +e) siny gtk y S= If n is large, S will only have appreciable values when g and « are very nearly equal, and in that case we may put with sufficient accuracy goth af 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 27/« which has its principal maximum in that direction. If the incident lght has a period 2/9 the expression for the distribution of amplitude is sin[a V (g — «)/«]* aN (g—K)/K ? which is identical with S if NV, the number of lines on the grating, is equal to n, the number of periods included in the integration. In obtaining the “ Pericdogram,” 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. xxxvit. 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” m is increased, but they approach nearer and nearer to the principal maximum. They are therefore dis- tinguished 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 A,, A,, B,, B, respectively. Hence R?=(A, + A.) + (B, + By. The products A,Ad, and £B,B, will vanish in the expression for S* when the average is formed for varying values of 7. Hence S= 424+ B2+ A2+ B2= R2+ Re, 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(f)=ct, and per- forming the integrations, it is found that Cun 2c A=—sin«t; B=——cosa«r, K K R= 8 = 42/2 = CT?/ 2°. 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 ¢, 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, V being the total number of ordinates, the : : : SNORT Hee. number having a value intermediate between 8 and @+d8 is asi e-"® dB. I have 7 shewn+ that in this case 2 Sg Ni?’ * In my paper ‘On hidden periodicities” (Terrestrial change to the latter form is apparent from the above. Magnetism, Vol. ut. p. 13) I defined the ordinate of the + On the investigation of hidden periodicities, loc. cit. Periodogram to be S instead of S*. The advantage of the FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 111 so that the periodograph is a straight line, parallel to the axis of ZY, the distance between the two lines being inversely proportional to the number 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 A should he within assigned limits; this probability being independent of the initial time 7. 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 dA, and B, be taken to be components of a vector defined by the equations ~T+mT t+mT hnPA,=[ — f(eosatdt, $n TB,=| f(t) sinetdt Se ar 2 -T+2mT -T+2mT Similarly tn7A,= | J (t) cos xtdt, 4nTB, =| F(t) sin xtdt, “rt+mT timT -t+smT ptt+smT and so on until 3x74, =| F(t) cos xtdt, 4nTB,= | F(t) sin xtdt, r+ (s—1) m7 “r+(s-1)mT with the condition that sm=n, m not being necessarily an integer number. We may choose m7’ sufficiently large to secure complete independence of successive vectors, all directions of the vectors bemg 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 R® is proportional to the number S of vectors; hence keeping m constant and increasing S, the ordinates of the periodograph will vary inversely with nZ’. 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 | re) cos xtdt fluctuates for the functions under consideration about /0 some value which is proportional to 7 when f(t)=cosx«t and proportional to J/7 when f(t) contains no real periodicity of periodic time 27/x. * Phil. Mag., 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. CabLcuLaTioN 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 me 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 0°1 minutes of are. 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 1771 and 15’8, the intermediate values were put down as 16°8, 164, 161. 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 sub- division 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. 115 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 meaning the 356th day of 1870. The unit in the first three Tables is 0°1 of a minute of are; in the remaining Tables, unless otherwise stated, it is the minute of are. 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 imdependent 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, Founer’s theorem applied to the interval 0 to T gives us CIV CH 5 Pm A ge Sats. I . Garts 2 —ct=— a= sn Pp +5 sin . +5 sin aed ; a oeaieseeserne (3). The effect of such a uniform progressive change would be to leave the cosine fe 3 : : terms unaffected, and to add ©" to all sine terms of period 7’. 7 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 2°52°7 less than during 1869, giving during 25 years a change of almost exactly 3°. Throughout this investigation the magnetic declmation has therefore been assumed to be made up of a uniform progressive diminution of 72 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 0-1 of a minute, the correction is made by adding to successive columns, successive multiples of 1800/n, where nm 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. 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The Fourier coefficients were obtained by means of Coradi’s Harmonie Analyser, belonging to the City Guilds of London Institute, which Prof. O. 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 a, cos Kt + a, cos 2xt+...... +6, sin xt +b, sin 2et+...... TABLE IV. Days in No. of Period eet Oia es b, “, b, a, b, a, | 6, | Periods | | dint 24 |+7-52 | + 8-92 | + 6-48 | + 0:08 | — 2-32 | — 4-48 | — 3°32 | + 5-44 | 41-24 | + 3°08 | 380 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 HI. 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. r°=a?+ b?; 72=a2+b,2; etc. Table V. gives the reduced squares R,*, R2, which correspond to R* in (2), 7 being the Ist of January, 1869, 7 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 R? are subject to considerable variations, R,° 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. IAL ZS 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 S? of the Periodograph corresponding to a period of 27 days. The value of S, or the amplitude of mean square, ae. the square root of the expectancy of £&,°, 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 R2 R2 | Re | Re Re | 24 946-9x10-® | 290-7 x 1078 | 176-5 x 10-* | 281-9 x 1078 76-5 x 107° 25 17 205°1 189-6 416-4 | 146-0 26 | 1392-4 434-5 | 125:3 134-2 121-7 27 1099-7 | 7443 | 448-7 204-2 23-5 28 657-6 | 396-2 80-2 284-4 | 89:8 29 225-0 299-9 | 1338 39-0 1711 30 2705'8 2933 | 234-0 74 | 143-9 Mean (8?)=1005:6x10-" | 380-6 10-°| 198310") 195-4 10-8 | 110-4 x 107° 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 7, cos (Kt — dy) + 72 cos (2at — fo) +....-. ; Table VI. gives the values of r,2, r.2...... rs for the 26 day period, Table VII. the same values for the 27 day period, and Table VIII. the angles ¢, and ¢, for the same periods. 118 Mr SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION TABLE VI. 26 Day PERIOD. Year rn re ry re rs Periols 186 10-574 3-480 1-116 0-122 0-930 14 1870 1:082 1-478 1150 0-824 0-580 14 1871 1-992 4°526 1-682 2-497 0-284 14 1872 13-744 4-802 0-268 07128 2-401 14 1873 12:109 0-603 0-404 3°803 0-716 14 1874 8-488 0-601 1-892 0-504 0-213 14 1875 3-624 0-284 1-016 0-678 0-692 14 1876 4-004 2-511 2-949 0-771 1:604 14 1877 1-297 2-339 2-180 0-569 0-008 14 1878 1-758 0°328 | 0-811 0-216 0-179 14 1879 5-673 0-578 | 0-433 0-876 0-194 14 1880 1-403 0-305 | 1151 0-265 1:300 15 1881 2-448 1504 0-392 0-216 0-149 14 1882 7-092 2-932 0-014 0-532 0-005 14 1883 2°500 2-938 0-758 1341 0-190 14 1884 3°379 0-437 0-464 1386 0-052 14 1885 0-315 2-512 0-041 0-592 1-632 14 1886 3-118 1-850 0°503 0-116 1-182 14 1887 4-180 4-640 1638 0-763 0-058 14 1888 1-946 2:269 1550 0-847 1-847 14 1889 2-064 2-694 0°354 0-184 0-029 14 1890 1-790 0-240 0-573 0-531 0-834 14 1891 0-715 0-090 0-303 0-132 0-382 14 1892 0-784 1:271 0-784 2°726 O-754 14 1893 9-143 9-541 0°362 0-583 0-471 14 FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 119 TABLE VII. 27 Day PeErRiop. Ne —, | 1869 | 3-667 0-923 1-205 0-457 0 339 ee | 1870 | 13-875 4-326 3-968 1375 1-800 14 1871 18-320 1181 | 1-978 L-071 0-301 13 1872 22-859 1-256 0-447 0-113 0-088 14 1873 3-963 6-470 0-708 2609 | 1-871 13 1874 5-044 1-073 1-260 1330 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 4 1877 0-673 0-548 0-105 0-509 0-148 4 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 | 5-722 0-794 1-456 0-269 0-331 13 1881 2-258 3-589 2-960 2-662 0-305 14 1882 | 12-857 1561 0-693 0-163 0-146 13 1883 | 2-198 9-844 0-155 1-992 0-680 14 issa | 4-348 1-619 0-098 0-875 | 1-060 13 1885 | 1-190 0:519 0-816 0-009 | 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 0427 14 1889 1-270 2-347 0-804 0-270 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 1177 3:533° Pl 0-307 0-035 14 1893 14-654 3-059 0-908 | 0-252 1-190 13 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 | l a] i] =a $, td || 4, $, r, r r, ry 1869 73° | 155° soz | 318° || 325 | 1:87 192 | 96 1870 | 271 354 || 56 280 104 | 1-21 372 | 2-08 is71_ | 293 275 141 51 1-41 213 || 428 | 109 | is72 | 257 | 201 297 303 371 | 219 || 478 | 112 | 1873 | 253 263 289 268 3-48 | 0-75 1:99 | 254 1s74 | 250 | 349 280 | 143 291 O75 225 | 1:03 1875 | 261 193 135 70 1:90 | 0-53 66 | 0-86 is76 | 135 | 43 84 338 200 158 230 | 0:69 is7z7_ | 253 | 158 226 271 114 | 153 || -82 | 0-74 1878 30 133 78 115 1:33. | 0-57 2:29 | O14 1879 51 89 151 251 238 | 0-76 1-44 | 0-13 1880 7 344 317 298 118 | 0-55 2:39 | 0-89 iss1_ | 238 | 55 110 346 157 | 1-23 150 | 1-90 1882 | 283 294 261 133 266 | 171 3:59 | 1-25 1883 67 | 356 267 34 158 | 171 148 | 1-69 1884 | 163 177 221 267 184 | 066 2:09 | 1:27 1885 87 110 || 238 245 0:56 1-58 109 | 072 1886 | 185 290 || 198 297 1:79 | 1:36 293, 118 1887 31 233 | 126 54 204 | 215 217 | 0-41 1sss_ | 290 | 134 | 258 45 1-40 | 1-51 132 | 0-06 1889 51 288 161 342 144 | 1-64 113 | 1:53 1390 | 287 359 272 49 134 | 0-49 11s | 1-03 1891 | 208 206 266 153 0:85 | 0:30 256 | O84 1892 64 249 350 350 089 | 113 228 | 1-09 1893 | 233 211 310 65 3:02 | 3:09 383 | 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 S? in accordance with the previous notation represents the expectancy of the square of amplitude : The values of S? 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 zs Period in | Average of Days S? Days S two Periods ai =| 26=1 02147. | 27=1 “03145 -026460 26+2 “Olan | weet 32 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 S? 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. : k Soe in Days for one year for 25 years Ratio Final Mean Secular Variation 27 26460 x 106 1006 x 10° 26:3 1052 x 10°° 28704 x 10° 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 S? which have been found from the 25 years curves (Table V.) Vou. 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 265 days, but for our purpose it is sufficient to neglect the difference of half-a-day. Considering that the value of S* 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 Pp. G. made up of the sums of Columns vi. and Il. or Il. 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 8 7 6 5 4 3 2 1 0 5 10 15 20 25 30 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 138'4x10-* 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 indepen- dent 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 ampli- tude 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 Period- TABLE XI. Period in Months Ss? S2 S37 S2 S$ | 11 | 04591 00475 “00158 | “00079 ‘00054 12 08828 | -01610 700842 =| -00287 -00218 13 09344 | “01082 00891 | :00237 | 00196 Average 07588 =| = -01055 | *00630 00201 | -00156 Period in Months 12 | 6 | 4 3 21 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. 11. 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 dis- cussion. The average daily value of magnetic declination, leaving the secular variation out of account, oscillates round some average value. If 8 is the difference between any observed value and its average, there will be some function #(8) such that f(8)d8 will represent the number of cases in which the value lies between 8 and 8+d8; for instance, if the ordinary law of errors holds, the number of cases in which the deviation from the average lies between 8 and 8+df will be un T e® 18, where h is a constant and WN the 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*"** p dp. This expression holds on the assumption that the values on successive days are entirely independent of each other. The expectancy (Z) of the square of Fourier’s coefficient is in that case + 2 2 Nh2e—-t Nh’? ndp = |p . Whe?" odp Whe’ and the probability that p? should exceed a value «# is simply e*. This latter ex- pression 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 eg. the case when the causes which produce the deviations from 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 ordi- nate 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 «=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 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 Caan K Orts 05 | 9512 6 2-48 x 1078 10-9048 8 3°35 x 107* 20 ‘8187 10 454 x 1078 ‘40 | 6703 12 6-14 x 1078 60 5488 14 8°32 x 1077 80 4493 16-113 1077 100 | 3679 18 | 152x107 1:50 2231 20 2:06 x 107° 200 | +1353 95 | 1-39x10-" 3:00 | -0498 | 30 | 9:36 x 107" 4-00 0183 | 40 | 4:25 x 10-8 |. 5-00 00674 | 50 2 x10 | | 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 «# is given by e~* and find the probability that their sum exceeds 2p#. If the first lies between «H and («+dx)H the second must be greater than (2o—«)# as long as « is smaller than 2p, if greater the second may have any value. Hence the required probability becomes 2p ew? + | e*e-@—") de =e” (1 + 2p). 0 By a repeated application of the same process it is found that if there are n quantities, the probability that their average exceeds «H is i 1 —NkK if 22 3 3 mn—1 .~n-1 e | + nk + dn?« +53 0e sr Gani K I; which is equal to n” ——— 2—1 p—NK sole e—”™ dk, 126 Mr SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION so that the probability that the average of n values should lie between «# and («+dk) FE is n 2—1 p—NK Game e™ dx. If nx is large, we may simplify the numerical calculation by putting approximately according to Stirling’s theorem log (n — 1)! =(n —4) logn —n + $ log 27, from which it follows that n n Pe RIE ty ee (n—1)! s Qa 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 n* 4 (n=1)! gr (Free 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 lme HK gives the position of the expectancy, and the curve 4A,4,, 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 much depends on the law of distribution of accidental Fourier coefficients. According to the theory the probability that the square of any coefficient exceeds « times the expectancy is e-*; 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 «. Table XIII. shews the comparison between the TABLE XIII. | Caleulated | Observed | Range of « number of cases | number of cases | | | | | Above 3 12:5 14 Between 2 and 3 a 25 | eS Mabe | 220 19 | 3 i140) 65. dle 36:2 D2) - Bie aaenil (i 20-4 20-5 | eS 6. 8 24-8 23 | i Aes 6 30-4 27:5 | 5 Sere ok: BH/(eil 48°5 Under -2 45:3 40 Altogether over | 92-0 90°5 fe under | 158-0 | 159-5 calculated distribution of these values of « 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 « agreed to 2 decimal places with a limiting value, it was considered as being half-above and half-below that value. Thus «=-60 was entered as one-half into the compartment including the values of « between ‘6 and ‘8, and as one-half into the compartment including the values of « 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 A,, Ag, ete.; B,, B,, etc. be defined by the equations nT -2nT snT ie | f (t) cos gtdt, A,=| f(t) cos gtdt, An=| f(t)cosgtdt, 0 “nT (s-1l) 27 nT 2nT nT Ba | F(t) sin gtdt, B= | F(t) sin gtdt, Bn= i] J (é) sin gtdt, 0 bar i (s-1)2T where g = 27/T. It is required to find air eo ee, d —— t) cos «tdt, ‘=—. t) sin x«tdt, |, fo pr), F sine where « = 27/7". If « and g do not differ much from each other we may put approximately mnt mnT J (t) cos xt = | F @) cos (gt + om) = Am COS Am — Bm SIN Ay, ....2.2-200000 (4). (m—1) nT J (m-1) aT The greatest approach to equality is assured when the curves cosx«t and cos (gt+ an) 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—4)nT, nt=gt+ an. This gives a4, = Qn ( — 7) (m — 4 yn ] a NE Te = E We may now put snT m= m=s 7 s v7 | J (t) cos xtdt = } A» COS Gm — > Bm Sin Om, 0 m=1 1 m= snT . m=8 : m=s | Ff (@sin ctdt =X Am sin 4m + = By COS Om. “0 1 m=1 m= The coefficients which we suppose to have been calculated are 2 2 a=—A b,=—,B,, ete. 1 nT 1> 1 nt 1> ’ 2 -snT so that aT | F(t) cos xtdt = - Y (An COS Oy, — Vy SID Ap). 0 : _ FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 129 If snZ=pT’ and p is an integer, the left-hand side would represent the coefficient of period 7” 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 *(pt+e) T’ | F(@) cos etdt = =) "F(6) 008 eid + | FCS Rta reercrerrrnse (5), oa0) 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 0’ Fie Bi LS a= pT” » = = (am COS Gp — bmn sin am) = ~ pT’ xy m COS (Am ats dm), | eee (6), pe Le (LL — yale = = (am sin am 2 Din cos bm) = 3 iva 27 m SIN (@m + $m); where @m =7m COS dm, bm =Tm SID bm- The fraction n7’/pT’ may generally be taken to be equal to 1/s. The values of a are those given above, so that ee T—T' TT Oy = 7 — Fs a, = 377n T° Am =(2m —1) 712 To ittteseeees (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«t, so that mnT ae nknT An, =] cos xt cos gtdt = - 5 —4) nT, “(m-1)2T and as Om =(K —g)(m—4) nT, Am=+t =singd«nT cos ap, e—g where the lower sign is taken when n is odd. Similarly Bae 2g ,sin$«nT sin ap. Kg > By substitution it follows that, using equations (6), y 2 : F a =P a= 9 sin danT ¥ (« cos? a, +9 Sin? Gp) 2 1 : & =o ag sin $anT ¥ {(« +9) + (« —g) cos 2am}, Wor, SOV 17 130 Mr SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION or writing y= = nT’, nT sin .K— a=—, = bat ead, cos Baal pr ys | ekg j ESS ad SIN 2. k+g Similarly b=. The factor sin y/y only having appreciable values when y is smali, the value of ae 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 siny BESTT p is defined as the nearest integer to ns7/T’, and as ns, 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 gaan UY The original function vestigated cos xt, 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 sin? y/7*. A Table of sin*y/y? is given in Mascart’s Optique, Vol. 1, 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 between 7 and lo? T’, for in that case =7™m — Zid a pe Sh or & fae Saeed If 7 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 2647 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—T’=tday 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 come to the conclusion that several meteorological and magnetic phenomena shew a periodicity having a time 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 am- plitude of ‘7 minute of are 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 Schmidt+ was led by a discussion of Hornstein’s results to a duration of 25°87 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 1s 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 capita K Semi-period Amplieade K | 25°87 “001001 “95 12:935 000316 83 25-929 001027 93 12-965 “000200 oy) 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. uxtv. p. 62 (1871). + Kongl. Svenska. Akad. Vol. xxxt. No. 3 (1898). + Ibid. xcvr. p. 989 (1887). 17—2 132 Mr SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION The column headed « gives the ratio of the intensity (square of amplitude) to the expectancy; and there is a remarkable unanimity in the smallness of this factor, shew- ing 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 declin- ation 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 exist- ence of the periodicity may escape attention. Hornstein’s investigations, as treated by Schmidt, do not claim any great accuracy, and a period of say 25°84 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 corre- sponding 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 A,A,. 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 v Lm — Am— = 2arNn ii ae Bile If 7=26 and n=14 the corrected time Z’ 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 B,B, and C,C,, and to bring the phases along these lines into agreement the corrected periods were calculated to be 26:255 and 26181. 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, 27327 days. The inten- sities 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 26814 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 circum- stances, 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 7 and ZJ” may be con- sidered as independent when T-T’ 1 TI hn? n being the total number of periods included in 7. For Z7=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 564 and 5°86 times the expect- ancy. 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 argu- ment. There are however two considerations which lead me to pause before finally reject- ing 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 IIT. 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 MAGNETIC 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 25°825 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 ‘O77 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 27327 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. | f | | | Period earaee | kK Semi-period io aiae | Kk | 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 « which is 756 is considerably higher than any other given within the whole range of investigated periods. An accidental coincidence is not excluded, for as calculation shews 1t may happen once in every 2000 trials that such a large value should be found for «. 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 are 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 dis- appear 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 considera- tion. If from a periodic function cos xt another is formed by taking averages over a period 27 we obtain 1| t+r 1 ’ — [ cos ctdt = — sin xr cos xt, Q7 J KT t-r s : : é fee P 5 that is a reduction in amplitude of —sinxr. If 27 is one solar day, 27/« one lunar KT day, 7 9953° hence «rt 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 magnetic 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. ona. of. Canb. Dril. Soe. B eT) VII. Experiments on the Oscillatory Discharge of an Air Condenser, with a Determination of “v.” By Outver J. Lopez, D.Sc, F.RS.. and R. T. Guazesroox, 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 » 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 = 2a /(pl, . Kl.), and the formula would be verified if the resulting value for the product of the as yet entirely unknown constants, ~ 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, ere. 137 that the followmg 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 MetTHOD. The first essential is a condenser of capacity directly measurable from its dimen- sions. 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 small- ness 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, ie., 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 and Addresses, Vol. 1. p. 119. Lecture on Electrical Units to the Inst. C. E. Vou. XVIII. 18 138 Messrs GLAZEBROOK ann LODGE, EXPERIMENTS ON THE OSCILLATORY “4,330 yards of No. 22 tinned copper wire covered with 2 coats of pure india- rubber to the diameter of ‘035 inch. This was the only covering. In two parallel coils, internal diameter 102 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 terminals of each coil were led to the outside and finished off on four Q" a | 4" aE = ebonite pillars. 1 y They could, therefore, be connected up in series, or parallel, or used separately; but in practice they were usually joined in simple series. With this coil many preliminary experiments were made at Liverpool. The self-induction of the double coil was estimated as about 5 secohms or “quadrants,” but no attempt was made to measure it with any care at this time, because it was better to do it when all the Bio. 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 micrometric 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. 2 ee ee here a DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “vx.” 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-an- inch 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 whip- cord 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-oilimg 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 lable to fluctuation this was found at certain times in the day or evening regular enough for good observation. MopE 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 electromagnetically maintained Koenig fork, whose loads were adjusted to give 128 vibra- tions 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 ann 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 mechani- cally 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 electro- magnetic 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 214 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 determi- nation, 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 28rd July, 1889, the “eight” pattern being kept steady and every other tap, or every 200th revolu- tion, being timed: TABLE I. h m. s. h m EE mal 6¥4 (Bis) Xl oo 43:0 6 ; 9°5 PN ree 49% SNieen lor ES oats 56:0 mem) 2950) » oh 2:0 5a Re” an Aah) eT 85 ee SoaD Be Fer ena Eh Seen aste eae), (052) Pate Sen AO a ye) AR sy a) op | O86) so Op. (OES yp BARS ~ Ae 55 5 er) ey atie AO) mS 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 ann 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 some- times 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 OF AN AIR CONDENSER, WITH A DETERMINATION OF “‘v.” 143 READING OF THE RECORD. The negative when thoroughly finished was subjected to careful micrometric examina- tion. 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 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 Papers, Vol. 1. p. 549. 144 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY The coil whose coefficient of self-induction Z 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 @. 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 self- induction 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 6Q. In consequence there will be a deflec- tion of the galvanometer needle; let @ be this deflection; let w, x be the currents in the arm P before and after the alteration of Q, X the logarithmic decrement, and let 7 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 11., 1882) L=39* Se 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 are 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 4, B, &c., a, b, &e. ‘ae ar, « 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 1°5 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 66 centims. long, with a screw thread cut on it, was fixed to this stirrup at right angles to the plane of the murror, 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 3847 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 miuror, 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, four wé 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 altera- tion 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 middle point to the centre Wor; .WA00 19 146 Messrs GLAZEBROOK anp 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, «ae, 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, ¢ for the deflection due to the alteration 6Q, and let d be the distance between mirror and the scale. Then tan 2a= 45, tan 20=45, and from this we find 2sinfa_ a fy _ lla*— 8c? tan@d cc | 128d? J’ neglecting higher powers of a/d and c/d. The values of (1la*—8c*)/128d* varied for the different arrangements from ‘00173 to ‘00023. The value of the ratio 2’/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+6Q respectively. e the E.M.F. of the battery, which we suppose does not alter *. Let X and X’ be the resistances between the pomts 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’ =x (Q+ 6Q + 100 + 328Q), E =x (Q+ 100), a | y Udtaoin also r= aa lis, if a term of the order Y8Q/90,000 be neglected. Y is of the order 1 ohm, and 6Q of 4 or 5 ohms. ax Q +100 Hence zw Q+100+3208Q° * 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.r. would only produce a first order change in the independent of changes in the battery E...r., 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 “v.” 147 In a second series of observations the approximate value of Q was 100, and in a @ + 100 x @+100+ 236Q" this case the formula becomes The actual value of the ratio will vary with the value of 6Q in the various experiments; in most cases it is from one to two per cent. greater than unity: 6Q being negative. Introducing these the formula for £ expressed in terms of quantities which can be directly observed is 1=8Q 7-1 +2) 95 1 100 + 33 8Q Be ~ 128d? Q + 100 a{, lla&— =I [The coefficient of 6Q in the denominator is in some of the experiments ?3.] 13° 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 tempera- ture 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. u., 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 vibra- tions 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 there- fore 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 are of vibration (ve. the ditference of successive elongations) be a before reversal, the are due to induction be a, and if b be the difference of zeros, the subsequent vibration is expressed by 4 (a+ aq) sin nt + b cos nt, 19—2 148 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY in which ¢ is measured from the moment of reversal, and the damping is for the present neglected. The actually observed are of vibration is therefore 2/{¢ (ata)? + Bt, or with sufficient approximation 2 Gata)+ f a so that in 26° a=observed are F ad — 7a “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, a, the throw at the time of reversal, and b the difference between the equilibrium positions before and after reversal, we have 2b? ay, a= $A — The sign to be attached to a, depends on the directions of a, and a). After two throws right and left respectively have been observed, and the equilibrium position is taken with the battery key im one position—denoted by R, say, in the table—then Q is altered by 6Q 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 m.) 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 ¢, the deflexion to the right; the battery key is then reversed and a deflexion to the left found; the resistance 8Q is then removed and a second zero reading taken; from these two, we find the deflexion c, to the left. The sum of ¢, and ¢, gives ¢ the double deflexion required. The values of Q and Q+6Q are calculated from the resistances on the multiple are in the arms of the bridge. DISCHARGE OF AN AIR OONDENSER, WITH A DETERMINATION OF “vy,” 149 Thus, on July 18th, for the balance the resistances were 205 and 7750 ohms, for a deflexion 205 and 3950 ohms. Hence 1 il Q 205 ' 7750’ 1 __ palpi Q+6Q 205 ° 3950° Whence Q = 199-713, Q + 6Q = 194884, 6Q =— 4829 “legal ohms.” The temperature of the box was 185. Having obtained a value for ¢ 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 6Q = — 5°4329 5 Ilia? = 8c? ag = 00107, pole 8|8 ll — >) bo paar — 150 Messrs GLAZEBROOK ann LODGE, EXPERIMENTS ON THE OSCILLATORY TABLE II. ] Mean Mean Time | Zero a, a, h a throw deflexion 5 a c 11.10 | L. 75:65 1 | 80:3 05 80-2 80-2 Throw R. 75°6 2 80 80-2 Deflexion |g, ce c | 2 aE e5s6D ilalyy | 41°35 82-5 82°5 Defiexion | R. 75:6 Bate - 41°15 Mo a, 6 a | TESTE RE (ae) | 80 05 80 80 | Throw L. 75-65 1 79-9 80 Deflexion C Cy c TALE Ue |] Late 7aki3)3) 34:5 41-05 82:3 82:3 Deflexion Ibs, (ORL 116-95 41-25 a, ay b a TM beatsy |) 1s eer 2 79-9 “15 80-1 80-05 Throw R. 75:55 1 80:1 80 Deflexion | Gh Gs ¢ TG PH | 1s ASR es 117 41:3 82-4 82-4 Deflexion R. 75°65 34:55 41-1 MN a b a a Raios0D “1 19-7 05 79-8 79:85 Throw Ty 727 “2 79:7 79:9 ———eEE—E——— Heed See) ee ee ae | IL. 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/e 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 “vw.” 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:; 77=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 ‘01524. From thesé observations we obtain for the value of L L= 46488 Legal Quadrants. The result requires a small correction because 6Q was at 17°5 instead of at 17° at which the box is right. Introducing this we find as the value 46494 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 d. TABLE III. Date [tempera a e | 5Q P L beer) July 18th | 18-5* 80-71 73°87 | 10-742 — 4-829 199-684 4-6480 4:6499 July 20th 18:9 7701 71:46 10716 — 4:903 199-909 46460 4-6485 July 29th | 17-5 80-025 82:40 | 10-713 —5-4329 | 198-966 46488 4-6494 July 30th 18-9 80°725 77-25 | 10°723 — 5:0553 199-640 4-6471 46496 | | | Mean 46493 legal quadrants. * There was a slight uncertainty about this temperature. 152 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY It appears that the greatest difference between two results is ‘0014 in a total of 46500, 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 self- induction 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, 46493, is in legal quadrants; ve. the resistance of a column of mercury 106 em. long has been taken as 10° CGS. units. To reduce it to “Henry’s” or “International Quadrants” it must be multiplied by the ratio 106/1063. We then find as the value of the coefficient of self-induction of the coil 46362 Quadrants, or 46362 x 10°, 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 Z,, L, be the two coeffi- cients 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=1,+1.4+2M, L’ =f, + f,=— 2M =2(L,+L.)—L. Thus L=2(£,+ L,)— TL’. 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 : LZ, = 1-405 Quadrants for semi-coil marked A. Tig = 1393 a : = een: LT’ = 0963 . Whence L = 4633 legal quadrants; agreeing fairly well with the true result. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “v,” 153 ACE Tell 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.cem., while the axial depth of the coil is about 52cm. 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_practic- able. 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 1035/64 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 156cm. 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 em. 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 les 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, WoL; XeVINE 20 154 Messrs GLAZEBROOK ayp 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 2Z, 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. Case i. Let a, a be the currents in the wires connecting the positive and negative plates of the first and second condensers, #, «, those in the wires connecting the corresponding plates of the second and third condensers, and so on. Let Q,, Q,... 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, d , =, = — Dy d 2 , , m—m=—- Poa, Ate Ly = Ly, 3 = %4, etc. Now let V,, V;, V; be the potentials of the positive plates, V,’, V.’, etc. those of the negative plates, R, Z the resistance and inductance of the wires joining consecutive plates, Ly be, + Ra, = V,— Vs, Lay’ + Ray = Vy — Vy’. But CS ae DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “v.” , = a. a has -V/= V.— Vz = 9 Q: Q. Sie + 2 Re = Vi= Veo a(Ve— V)=3-9- Now if L’, R’ denote the inductance and resistance of the plates of any two consecutive condensers, then L’=2L, R=2R. Then if R is the whole resistance, Z the whole inductance, R=nR’, L=nl'. And the equations to find the period are L'a, + R’a, = = 2 =: Qs Din + R’tn = i a ae etl 2 Gh 1Qn In ima Hence we have fas fie (1 1 Lit, + Ray + (a+ 7) aa # il FO — a= CO i ie i ii =, 0, + Ls, + R'e 4 = bight si Fy Ln + Ra, + yen= 0. Put a, = Xe, and let S| S’ pel’ — Rr} —2=Q, S’ el’ - RaA}-1=R. s’ {er Bis —1=P, Then the equations become PX,+ X.=0, X, + QX, + X;=0, X,+QX,+ X,=0, Xna+ RXy, =0. two wires joining 155 the 156 Messrs GLAZEBROOK ann LODGE, EXPERIMENTS ON THE OSCILLATORY Whence to find the periods we have the determinant IP ik Oe Wane = (I) iy Ord 40:13 01 Q1.. (to » columns). | 2 vy | Now the determinant EP Orel 2. ee le-U oes Oe | | | 2 ce Q1 ‘(n—1) columns @ 4 i p, (#=Decolumns 1 Rk 1 Ov] =) P| Ry A ss SE (Si) 3 zeal | if @ i Q1 (n — 1) columns — aaa (n — 2) columns. | pear 1 Q Now (eed | =R\|Q 10 +G1)j">|1 0 O.nT | | | oS a0 | (m) columns | . : : (m — 1) columns : ‘ : | 1 Q \iestseraeee =0||@) il Sac SS uF (4) ak 1 Q 1... |(m—1) columns 1 Q 1 |(m—2) columns Ge popbs-sronee = Te — Am-2, if A,, stands for Q 1 0 0... | to m columns. iO, Hue Ore: Gg) ih a} al Q Thus the given determinant = IP {RAs = Anes ae {RA, -3 A, a} =0. Now if Q=2cos 6, then we know that _sin(m+1)0 pp. aa: 4. = — ae (Rayleigh, Sound.) Hence we have as the equation for the periods P{Rsin (n—1) 6 —sin (vn — 2) 6} — {Rsin (x — 2) @—sin(n — 3) Oo} =0. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF Kora ON In the first instance neglect the terms depending on the resistances, then SL’ —2= Q =2 cos 8, \9 / A) SL = 2 (1+ cos @) = 4cos? 5. Now if the whole of the capacity 8, were concentrated at one part of the circuit connected by a wire of inductance LZ we should have 1 1 De SL n(n— PSE 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 \2S’L’ is a small quantity of the order 1/n*. So that 2 cos 5 1s of the order 1/n, and @ is not much different from 7. Put 0=7 — @: ultimately @ will be treated as small, though for the present the solution is general: then SL’ = 2 (1 + cos 0) = 2 (1 — cos op). Whence substituting (S’A2L’ — 1} {(S’A2L’ — 1) sin (n — 1) d + sin (n — 2) op} + (SL — 1) sin (n — 2) $6 + sin (n— 3) ¢ _ a (S’2L' — 1) sin (n— 1) + sin (n— 2) o} =0. Thus 2S°L’ {sin (n — 2) p — sin (n — 1) $} + sin (x — 1) $+sin (n— 3) 6 —2sin(n— 2) - * sin (n — 2) $—sin (n—1) 6+S 21’ sin (n— 1) op} + S*4L sin (n—1) b = 0, “. 28°L’ {sin (n — 2) 6 —sin (n — 1) >} + 2 sin (n — 2) $ (cos ¢ — 1) - = {sin (n — 2) 6 — sin (n —1) $+ S’7L’ sin (n — 1) op} + S?\4L? sin(n —1) 6 =0. But 2 (COS til) tS ONT Pee S85 c ic sesncdnoeieeoeaseeact (A) “. WL’ {sin (n — 2) d — 2 sin (n — 1) >} = 5 {sin (n — 2) $—sin (n—1)6+S8’A2L’ sin (n — 1) $} gS SHU VAY (C1 \C a0) ares. or cs aee ee eee ae (B). 158 Messrs GLAZEBROOK ann LODGE, EXPERIMENTS ON THE OSCILLATORY On eliminating ¢ from (A) and (B) we obtain an equation for X% Now we have seen that 4sin?¢/2 is of the order S’L’/=Z, where = stands for the whole capacity. In the most important cases S, the external capacity, is large compared with §S,, the capacity of the coil; and in this case the whole capacity is large compared with S,. In the general case on substituting in (B) from (A) we find 1’ {sin (n — 2) 6 — 2 sin (n— 1) $+ 2sin(n—1) P(1 —cos ¢)} =; {sin (n — 2) @ — sin (n— 1) 6 + 2 (1 — cos d) sin (x — 1) PF}. YAR es oa sin(n—1)¢ Whence MES Sin (9) = coe RO a _ sin(n—1)¢ sin nd =1-—cos $+sin ¢ cot ng, 5 MES — AS!) SSI COG ND ye mc sence cencas. Comme anyaeeees (C), or substituting for A*L’ from (A) Oran e (SEG) Scotian ee eee (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 S,, and then ¢ is small; S may have any value. In the more important cases S is large compared with S,. In this case ng is of the order (S,/S)!. We may expand cotr@ and cot ¢/2 and use 1/SZ as an approximate value for X° in the small terms. Now we have 2(1—cos gd) =A2L’ 8’. Thus p= MLS’ (1+ Arvel'S’ + ...). b Hence P=rvL'S’ {1 +5454 al where a, b, etc. can be found approximately. Hence expanding in Bernouilli’s numbers, n/n (1-5) (i+< md ) nm nt by {i ‘ 2 Bed? iF Bani! Ib Ts (4 aS By = Bd! { EP TA eee Now, as a first approximation, 549 LS IS, Ns ils Ue = aS 3 ( ). DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF «v».” Hence n*f*? is of the order S,/S, ; Se =) its se Roe == || == or To00 \ a n Co) is of order ne (2) a 1000 & since n= 30 approximately in the experiments. 159 Again, the value of S is in the final experiments 5853 cm. while S,, when the two coils (A) and (B) are used, is about 1600 cm. (See Part IV.) Thus the important terms are those in n°, néf..., while a term such as one in n’p* is, when S, has its largest value, of the order (S,/S)*; and when S, is only 100 cm. it is of order (S,/S)* Hence retaining the most important terms, MES {1 -5 a st = 1 — Se _ eae and B=x mge= (144). Thus MIS —3 Gps -55 +c. Hence MLS = -5 = 1! - ros , 18, 1 =1- 35 !- saga} omitting terms of the order 1/n? in the coefficient of S,/S. Substituting in the terms in n‘¢‘ and introducing the value of n in the last term, we have approximately jertehiy LS 1 4 /8,\? MES =1-3 (1-5 >) +a (=) In obtaining the coefficients of (S,/S) the terms in np have been retained when compared with terms in ¢. Hence for S,/S = -2733, MLS = 1 — -0896 + :0066 =-9169. In some of the preliminary experiments however the value of S,/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 anp LODGE, EXPERIMENTS ON THE OSCILLATORY The equations to be solved are 2 (1— cos ¢) . > ean fan $ (= S— 1) =cot ng; n-1’ S for S=58:53 metres, S,=16 metres. Hence (- —1)=21 Hence 218 x tan $= = cot 31¢. An inspection of the Tables and a trial shews that ¢ is nearly 56’. By plotting on a large scale the values of cot 31g and 218 tan ¢/2 at about 56, we find the curves intersect at 56’ 30”. Hence 6 = 56' 30”. Hence MLS =2(1— cos d) ae = 2(1 —cos ) n(n—1) = S; = 919, substituting for ¢, x and S/S,. 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 $/2 {58 x 60 —1} =cot 31¢. Thus 3479 x tan $ cot 31¢. A similar procedure gives for ¢@ the value 14’ 45”, and substituting in the equation for LS we obtain MLS = "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 l DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “y” 161 original equation; thus, supposing the coil to consist of » parts, so that nS’ = S,=16 metres, we have tan =cotnd, g= 7 2n+1’ 2 MLS, = 2n? (1 — cos $) = Grea ans T 1) : 4 =a {1 = ar neglecting 1/n? = 2°38 if n=31. The case of a continuous coil of uniform capacity s and inductance J 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, » the potential at a distance « from the end at which the potential is zero, Then v= iz ; a The charge on an element da at « is pee as ads a V222 Energy =4tquv=ts 7a dx. The total electrostatic energy of the coil is thus and thus =34x1iVsa=tV°S,, if S, is the capacity of the whole coil. Hence the total electrostatic energy of the coil and condenser is $V7(S + 1S8,). The electrokinetic energy is 4Zu? if w is the current. 1 Nee ee Henee DISSE 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 Vou. XVIII. AI 162 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY used and the value of S, therefore was 16 metres, the values of S were approximately 2, 5, 5°5 and 10°5 metres. The values of @ and X2ZS can be found for these cases in the same manner and we thus get the following Table. TABLE IV. s Si, S/S, ¢ LS 2 16 "125 2° 37' 48” "2449 5 16 “312 2 16 40 4582 As, 16 344 2 12°20 4741 10°5 16 656 1 52, 12 6498 58:53 16 3°659 56 30 9169 0 16 2°38* 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 cor- rections 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 #ZS=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 1, be the coefficient of induction of a wire in which the current is #, due to itself, mo, 73, etc. the mutual coefficients. Then the strict equations for any wire joining two of the condensers each of capacity S’ will be LG + Myo®%o +... + = (a, — X)=0; put DL'G, = hay + MypHe + «-.; and let ig — 2 + Ye, i= 2, + Ys, etc., LH, = (L, + mye + «--) B+ Maio + Mss + «.-- * For this case S, takes the place of S. — DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “v,” 163 Now y/% depends on S’/S, and mj, ete. are all finite and less than [Ks L’=(G+mo+...) + — m, where m is of the order of the arithmetical mean of Ms, Ms, CtC., , S. mee i, Hh + My +... + SM. Thus in the case of a large number of turns, if S,/S is small the equations already used are correct if LZ’ be 1/nth of the whole self-induction, for we may neglect the term Sjm/S compared with the sum J, + mp»-+ ete. There is now the correction for resistance to be considered. In the case of a simple cf 28)} A» being the uncorrected value; thus we may put eireuit i 2 = — (lb: fe Vineet Now so far as the inductance of the circuit is concerned, 1 2 = —— (le) N= op ) And in the more important cases both k” and k’ are small. Pap, Therefore v= SL (l—k’—k’), where k” has the value already found from Table IV., v= In some of the experiments \, is about 2c x 10°, R=200x10° L=5~x 10°, approximately, and the correction is negligible. If the period be 1/120 second as in other experiments, Ao = 2m X 1:2 x 10°= 7-2 x 10? approximately, ky 1 eM na L799 2PProximately, f 1 ° eH ne a3 and k' =— i= 1900" which is also negligible. 21—2 164 Messrs GLAZEBROOK anp 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. Fie. 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 z be the current leaving the main condenser, y, y;, y, the currents between the lower plates of the coil condensers, then the currents between the upper plates of the same are 2—%, —Y;, etc., and the equations are LE G)= Weve EG) ae L'9na = V'n ae Hence n'a = V,— V,'+ Vy — Vy =V,-Vi+ ee DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “vy.” 165 Qs =2(V,— Vi) — Ba’ — 4 6 , ,_ 2Q Q» On (n+2) LZ ae —gte 58 a Yo + Yn (n+ 2) LE=——Q- ave = L' (& — 242) = V2— Vs —(V,— V;') Also _ SS S’ > Le 2%y..) = — Li (— 2%,) = BEM, ries Yo+ 2y3 — ¥ Hence Li (& — 295) Re ; eee eee eee eee ee ee eee eee ee eee) Li (& — 2ijna) = Tat cs For superpose everywhere on the system a Now we may shew that y.=Yn4. And now choose v so that ‘the potentials of potential v; this will not affect the currents. the plates of the main condenser are equal and opposite, we. so that V,+v=—(V/' +»), v=3(Vi+ V1); the distribution is a symmetrical one and obviously in this case Ya = Yn Ys = Yn-2, ete. Hence if n= 2m and we have m—1 equations WAC . 2y.— Li (é — %,) ==, ase es = o+2 =F \ L’ (& — 23) = y “= Le Lk — 2ijm) = = a . ies 3 166 Messrs GLAZEBROOK ann LODGE, EXPERIMENTS ON THE OSCILLATORY put M2, = %— 2%, t3—2,— 2yz, Yo=4(%—%), Ys= 4 (a,— as), etc., Ys — Yo = $ (2 — Zs), ete. Ae 1 (@, — %) gen gag og %— Hy y— Xs Hence { = | OE Lam = 4S" (2m — Lm); 38) Ss 4 for spark on second circle . ; > BOY Ze a for another set ditto . : ; 5 OY Pe o. for spark on third circle. : 5 a Bey x nd for another set third circle : o (Py * 7 for spark on fourth circle . : Ons Or General average for this plate. : i : ‘ on oo ie Frequency deduced, 1766. The speed for the outer circle was steadiest. Vout. XVIIT. 23 178 Messrs GLAZEBROOK ann LODGE, EXPERIMENTS ON THE OSCILLATORY Same date. Cylinder and disc again, with coil connexions reversed, otherwise every- thing the same. Average of readings off all alternations on outer circle 6° 13' 5” second circle 6° 15’5” a third circle 6°11’ 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 disc 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 542 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 “vy.” 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 »2LS =k each combination as given in Table V. in Part III. We thus obtain the following results : TaBLE VI. Both coils A and B where & has the proper value for being used. Capacity in Frequency | Frequency Date Condenser centimetres | calculated observed | | July 25 Five-plate dise 2 2590* 2580 July 24 ; x : z By UNA 2403 July 25 Eleven-plate disc 5 2400 2470 July 22 : = 2060 2160 Fuly 25 Cylinder OPteiles Sono 2370 July 22 Cylinder and disc : vAR 1770 Aug. 1 in parallel Ore nie 1766 Cylinder, dise and (e is a | Aug. | eawiien dee 30°5 | 1170 1180 Weare Aug. 3 Sheet Glass oar | Me | 450 | | July 31 126 Aug. 2 Paraftin condenser 3000 128 127 ” 9 128 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 ann LODGE, EXPERIMENTS ON THE OSCILLATORY TaBLeE VII. using only one coil. | | Frequene | Frequene Date Condenser | Capacity RE Re - ; 232 Aug. 3 Paraftin condenser 3000 233 230 | | 190 26 = | - 2 Glass condenser | 237 827 820 Considering the nature of the experiments the agreement may be considered satis- factory. The capacity taken for the cylinder condenser is probably too high; if it were assumed as 571 instead of 55 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 dise condensers in use, the connexions were as shewn in Figure 9, and the frequency was 1766; this result is given. Spark gap Earth Fie. 9. The connexions were then altered so that the condenser was to the terminals B DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “vy.” 181 and @ 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. Earth Fie. 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 imner, while at the same time the portions of the coils which are nearest to the condenser, and in which there- fore 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 S,=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. Earth Fie. 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 im 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 anp 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 5:1 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 removed and oscillations taken with the coil alone. In this case, assuming the theory developed in Part III, we have 1) _ 238, nN, 8,In= 5 (1— or if we suppose the capacity uniformly distributed along the coil, S, DD? = 3. On substituting for S, the value 16 metres, and for LZ 463 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 formule 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 S, 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. | | [ | ; Frequency Frequency Date Condenser Capacity calculated | observed 2 ia a! | = Aug. 1 Cylinder and disc | 10°5 1860 1830 July 350 Cylinder Bs) 2470 2560 &5 Coil only 4820 4630 DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “»y.” 183 The general concordance of the experimental results with theory appears to shew that the capacity of the coil, layer upon layer, has no marked effect; if it be taken into CY / 3 18 “ 5 account a correcting factor of the form 1 — 60 = must be introduced, where Sj’ 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 result we were led to construct an air condenser of considerable capacity which we could calculate with some degree of accuracy. IPARA Ve THE Ain CONDENSER. We proceeded to make an air condenser of eleven flat plate glass slabs very care- fully 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 zy milhmetre 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 3:157 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 anny 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 ......... 515 centim. » » 3 POOH a MBs Sa Histone 88 v ee ls tees ou SHOOE safe thc nose ace 2°5 » 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. 1. § 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 terminations of thin parallel planes, is b 7, loBe 2: 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 correction for thickness £, cos ue 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. i. §§ 307 and 314.) The total correction is thus 22b + 258 =:11+°18='29 centim. to be added all round. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “vy.” 185 Thus the capacity of the condenser proper is 10 (59°74 + °58) x (59°64 + °58) 47 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 — een = 6-9 centims. 4x 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 4 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 73 metres long and one millimetre thick, with a span of 14 metres between it and the walls. Its capacity 730 was therefore ~_—-—. = 50 centims. 2 log, 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. Vout. XVIII. 24 186 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY PART VL. 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 irregu- larity 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 @. In the spin for plate P the coils were in multiple are, 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 PuatTe U. Coil B only used. Outer circle. Actual readings. Differences. Averages. Spark (1) 194° 0’ 186 49 14° 36 179 24 (14 45) —— a2 4) ide sige PE 2 164 59 (14 24) 157 40 * The record for this plate happened to come 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’ (147 46) 140 53 133 34 126 20 Spark (3) 112° 40’ ; 106 4 ie Sa wes (14 43) oe 0B BF A LAM ee a8? 97 57 | ie) 90 48 : 4G 12)? 83 45 76 36 General mean for this outer circle 14° 24’. 14° 21’ 14 12 1422 Second circle. Spark (1) 136° 23’ 129 4 14° 12’ 122 11 14 11 | Spark (2) 118 29 b 14° 14’ 114 58 14 19 TG 107 52 14° 18’ 104 11 14 10 14° 18’ 96 56 14 28 | 89 43 Spark (3) (Ge QE 168 1 (161 6) 154 2 146 25 Spark (4) blurred. 14° 18’ | 14 0 | 14° 13’ Here there was some simple overlapping, giving no difficulty in sorting out. The general average for this circle is 14° 14’. 188 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY TaBLeE VIII. Pirate U. Coil B. Girolee||e me |) Sa: uae || ing v. Spark 1 | 14°36’ | 14°12’ | 14°33" | 14°17’ 1445 |1411 | 1425 | 14 6 | 14 12 14 25 14 19 14 19 14 25 14 24 2 | 14 21 14 18 14 31 | 14 9 14 12 1412 | 1410 | 14 18 14.5 14 10 14 33 14 28 14 22 14 15 14 11 14 49 1443 |} 14 0 | 14 19 14 20 | 14 20 14 17 14 21 14 16 14 9 14a 4 Nee cee eich The) 11426 | 14 9 14 36 | 14 25 Meantorks|/<) sc 4n hae Seal Poe ee oar ris | 14° 24 14° 14 142 27 14°19 14°18 Mean of | swings for central | y4°y6 | 14° 8° | 14°91’ hee | 14°12 each circle General mean from plate Mean from central swings VI. VEIL VI. I< 14°34’ | 14°39' | 14°46’ | 14°56’ 1426 | 14 40 | 1418 | 14 41 14 24 | 14 43 | 14 24 | 14 37 14 56 14 296° | 1445 | 1411 | 14 16 1414 | 1493 | 1414 | 14 12 14.94 | 14°36 | 1411, | 14-9 15 5 1 Tie | Tbe: Fe lame | eee 14 5 | 1429 | 1420 | 14 10 14294 | 1416 | 14 23 | 14 16 14 14 14 21 CB 14 43 14 23 14 38 14 20 14 39 14 34 14°26’ | 14°31’ | 14°22’ | 14°29 1458" [14° 987 TAS 9" || Waa 14° 23’. eae DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF “vy.” 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 Vis 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...| 1 Il, ET, IV. Mo VI. | Vil. VII. Exe | | | Ee Pope eone'| «ag i ome Mogwai AoW Ste, ine 120) |) 18 it || Gee) oe | 24 | 12 | 17 0 | 19 5 wm SMBS toll WE 9 9 20. -|| Tea) 38) | oon 1 ao 26 | 9 | 14 | 290 | 16 | 23 20 | Averages | 16’ | 8 | 21’ | 14 | 12° | 18 | 98° | 19° | a7 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 PuLate 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 ann LODGE, EXPERIMENTS ON THE OSCILLATORY On the outer circle were taken 4 sparks, speed steady. = second ,, * 3 4 e we) fair 26 third se se a 3 * » Steady. Fr fourth ,, * es 4 Ls fair: . fifth 4 is " 4 . Petar rm sixth a FS 3 4 e5 » quite steady. x seventh ,, x a 4 io » Steady. 4 eighth _,, 2 ¥ 4 A » Steady on average. ” ninth ” ” ”? or ” » ad a ad (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., IIL, 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 le 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’, Le. to about °6°/,. The frequency corresponding to 14°20’ is 64 x 360/14°35_ 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 “v,” 191 TABLE - X. PiatE S. SUMMARY OF READINGS. Coil A. Circle... I. Il. Il. DVS Wo Wal Ta | 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 aero id aie Id | 14 35) | ddesg 14 56 1S Tate elAce lade ale) |S) .5S). | al peedane pI ealioTaeoD 14 11 14 22 14 17 14 9 14 11 14 22 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 1419 | 14 34 14 44 14 26 | 14 47 14 47 14 38 14 45 14 30 14 19 14 26 14 25 14 17 14 50 14 42 oo 14°22' | 14°98’ | 14°98’ | 14°96’ | 14°96’ | 14°38" | 14°97’ ean Mean of central ATI |) Wee IP ey NP et oI” | az GY || TMP age | aiveoye waves General mean 14° 28’. Mean for centre swings 14° 19’, 192 Messrs GLAZEBROOK anp LODGE, EXPERIMENTS ON THE OSCILLATORY Four sparks were taken on each circle. Circle I. speed moderately steady. Circle V. speed _ fair. we - fair. Vie e steady on average. Ii. * quite steady for 3 sparks. \0 eo slightly backing. is . steady. WADE 5 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. eases |) ine in |) NE VI., VIL } EE ee Spark 1 | 26°52’ | 26°53’ | 26°34’ | 26°43’ | 26°47’ | 2616 | 26 22 | 2611 | 2619 | 26 23 26.18 | | 26 46 | | | | 2| 26 44 | 2655 | 2651 | 2652 | 27 1 a : | 26 4 | 2618 | 2615 | 26 20 | 26 23 i I | | 26 51 Bien a 2 | g | 3 26 59 | 26 51 | 26 55 | 26 46 Eo | joa 26 4 | 26 12 26 19 = o 26 50 26 37 6 4 | 26 38 | | 26 4 | eS a a oe || 26°38! | 26°30" 9) 26327 Wl 2Gna%" |) 26°37" Mean > eee Central | 96°19 | 26°15’ | 26°13’ | 26°14’ | 26°21" | Mean | im ' | General mean of plate 26° 34’. Mean from central series 26° 15’. DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF « v.” 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 64x 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 Dy 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 MW N Ss Number of sparks 20 28 7 27 33 Length of wave 14° 14’ 14° 32’ 14° 25’ 14° 18’ TEES TIBY | Won. XeViLEE Mean length of wave 14° 20’. 194 Messrs GLAZEBROOK anp 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 LS v = 2m . frequency x 5a where & is the constant, the values of which are given in Table V., occurring in the 7 OQ EOE ee DISCHARGE OF AN AIR CONDENSER, WITH A DETERMINATION OF «,y,” 195 formula ZS)?=k. In the case in which the two coils were used there is no difficulty in deciding on the value of f. The formula for X is that given on p. 159 (D), elie inst 1 4 /S,\2 fe aia S }-sq=nl+e (3) Tipe? and hence k =:916. If only one coil is used two cases may arise; if the lower coil is completely in- sulated we have the case dealt with in Figure 4; the corresponding formula as far as terms in S,/S are concerned is (F) on p. 166, viz.: fev re], At eT: 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 & 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 in- tentionally 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 Z 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 #S’/S if « 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 M/120. Now the value of Z, is about 1-4, and of M about -91. Hence the value of I, in the experi- ments 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 4636 | 58-53 3009 x 10" | Mean from seventy sparks A ‘978 1611 | 1-409 58°53 2-939 x 10” Unused coilassumedinsulated te | 1611 | 3-409 ese ESTE. JC! |) ene mere | ‘916 1583 | 1-409 58-53 2-984 10" | Plate H, coil uninsulated | ee B ‘978 1607 | 1-393 58°53 2°922 x10" | Unused coilassumed insulated - 1607 1-393 58-53 3°020 x 10° 5 - » uninsulated 916 1595 | 1-393 58°53 2-990 x10" | Plate J, coil uninsulated | | | 25—2 196 EXPERIMENTS ON THE OSCILLATORY DISCHARGE, ete. 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 10", 2°99 x 10" and 3:01 x 10”. 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 10° and 2°92x10", 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 10" and 3:02 x 10" respectively, given in lines 8 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=3009 x 10” 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. VILL. The Geometry of Kepler and Newton. By Dr C. Taytor, 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 brilhant 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 Zuelid, 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 Vitellionem Paralipomena, quibus Astronomiw 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 im vol. UL. 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 PAGE 92. n on 3° PAGE 93. on 10 20 30 TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 4. De Coni fectionibus. Coni varii funt, reétanguli, acutanguli, obtufanguli: item Coni reéti feu regulares, & Scaleni feu irregulares aut compreffi: de quibus vide Apollonium & Eutocium in commentariis. O- mnium promifcue feétiones in quing; cadunt fpecies. Etenim linea in fuperficie coni per feétioné conftituta aut eft reéta, aut circulus, aut Parabole aut Hyperbole aut Ellipfis. Inter has li- neas hic eft ordo caufa proprietatis fue: & analogicé magis quam Geometricé loquendo: quod a linea reéta per hyperbo- las infinitas in Parabolen, inde per Ellipfes infinitas in circulum eft tranfitus. Etenim omnium MHyperbolarum obtufiffima eft linea reéta, acutifima Parabole: fic omnium Ellipfium acutiffi- ma eft parabole, obtufiffima Circulus. Parabole igitur habet ex altera parte duas natura infinitas, Hyperbolen & Rectam, ex altera duas finitas, & in fe redeuntes, Ellipfin & circulum. Ipfa loco medio media natura fe habet. Infinita enim & ipfa eft, fed finitionem ex altera parte affectat, quo magis enim producitur, hoc magis fit fibiipfi parallelos, & brachia, vt ita dicam, non vt Hyperbole, expandit, fed contrahit ab infiniti complexu, fem- per plus quidem compleétens, at femper minus appetens: cum Hyperbole, qué plus aétu inter brachia comple¢titur, hoc plus etiam appetat. Sunt igitur oppofiti termini, circulus & reéta, illic pura eft curuitas, hic pura rectitudo. Hyperbole, Parabole, Elli- pfis, interiecte, & recto & curuo participant; parabole ex zquo, Hyperbole plus de reétitudine, Ellipfis plus de curuitate. Pro- pterea Hyperbole quo longits producitur, hoc magis reéte feu Afymptoto fuze fit fimilis. Ellipfis qué longits vltra medium continuatur, hoc magis circularitatem affectat, tandemque coit iterum fecum ipfa: Parabole loco medio, femper curuior eft Hy- perbola, fi ezqualibus interftitiis producantur, femperque re¢tior Ellipfi. Cumdue vt circulus & reéta extrema claudunt, fic Para- bole teneat medium: ita etiam vt reéta omnes fimiles, itemque & circuli omnes, fic funt & parabola omnes fimiles; folaque quantitate differunt. Sunt autem apud has lineas aliqua punéta precipue confide- rationis, que definitionem certam habent, nomen nullum, nifi pro nomine definitionem aut proprietatem aliquam vfurpes. Ab iis enim punétis reéte eduéte ad contingentes fectionem, punctag; contaétuum, conftituunt zquales angulos iis, qui fiunt; fi punéta oppofita cum iisdem puncétis conta¢tuum conneétan- tur. Nos lucis causa, & oculis in Mechanicam intentis ea punéta Focos appellabimus. Centra dixiffemus, quia funt in axibus fe- &tionum, nifi in Hyperbola & Ellipfi conici authores aliud pun- étum centri nomine appellarent. Focus igitur in circulo vnus eft A. isque idem qui & centrum: in Ellipfi foci duo funt BC. equaliter a centro figures remoti & plus in acutiore. In Parabole Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. vnus D eft intra fe€tionem, alter vel extra vel intra fectionem in axe fingendus eft infinito interuallo a priore remotus, adeo vt eduéta HG vel IG ex illo ceco foco in quodcunque punétum feétionis G. fit axi DK parallelos. In Hyperbola focus externus 35 if F interno E tanto eft propior, quanto eft Hyperbole obtufior. Et qui externus eft alteri feétionum oppofitarum, is alteri eft in- ternus & contra. Sequitur ergo per analogiam, vt in recta linea vterque focus (ita loquimur de reéta, fine vfu, tantum ad analogiam complen- dam) coincidat in ipfam reétam: fitque vnus vt in circulo. In circulo igitur focus in ipfo centro eft, longiffime recedens a cir- cumferentia proxima, in Ellipfi iam minus recedit, & in parabo- le multO minus, tandem in reéta focus minimum ab ipfa rece- dit, hoc eft, in ipfam incidit. Sic itaque in terminis, Circulo & re- éta, coéunt foci, illic longiffimé diftat, hic plané incidit focus in lineam. In media Parabole infinito interuallo diftant, in Ellipfi & Hyperbole lateralib. bini aétu foci, fpatio dimenfo diftant; in Ellipfi alter etiam intra eft, in Hyperbole alter extra. WVndique funt rationes oppofite. Linea MN que focum in axe metatur, perpendiculariter in axem infiftens, dicatur nobis chorda, & que altitudinem often- PAGE 94. pie) PAGE 95. dit foci 4 proxima parte fectionis a vertice, pars nempe axis BR. 5 200 Dr PAGE 96. on TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. vel DK. vel E. S. dicatur Sagitta vel axis. Igitur in circulo fagitta z- quat femichorda, in Ellipfi maior eft femichorda BF. q fagitta BR. maior etiam fagitta BR. quam dimidium BP femichordz feu chord quarta pars. In Parabole, quod Vitellio demonftra- uit, fagitta DK precise equat quartam chorde MN. hoc eft D N eft dupla ad DK. In Hyperbole EQ plus eft, quam dupla i- pfius ES. fe. minor eft fagitta ES. q quarta chorde EQ. & fem- per minor, atque minor per omnes proportiones, donec eua- nefcat in recta, vbi foco in lineam ipfam incumbente, altitudo foci feu fagitta euanefcit, & fimul chorda infinita efficit, coinci- dens fe. cum arcu fuo, abufiué fic diéto, cum reéta linea fit. Opor- tet enim nobis feruire voces Geometricas analogie: plurimtim namque amo anaiogias, fideliffimos meos magiftros, omnium nature arcanorum confcios: in Geometria precipue fufpicien- dos, dum infinitos cafus interie¢tos intra fua extrema, medium- gue, quantumuis abfurdis locutionibus concludunt, totamque rei alicuius effentiam luculenter ponunt ob oculos. Quin etiam in defcriptione feétionum analogia plurimim me iuuit. Etenim ex 51. & 52. tertii Apollonii defcriptio Hyper- boles & Ellipfeos efficitur facilima; poteftque vel filo perfici. Pofitis enim focis, & inter eos ver- tice C. figantur acus in focis A. B. annectatur ad acum A filum longitudine AC. ad B. filum longi- tudine BC. Prolongetur vtrumque filum quali- bus additionibus, vt fi duplex filum digitis com- prehendas, iisque a C difcedentibus, bina fila paulatim dimittas, alterajue manu fignes iter anguli, quem vtrumque filum facit apud digitos, ea defignatio erit hyperbole. Facilius Ellipfis de- fcribitur. Foci fint AB. vertex C. Fige acus firmas in A.B. vtram- que filo ampleétere, fimplici amplexu, vt inter AB filum non interfit. Fili longitudo fit AC duplicata, & capita fili nodo fint connexa. Infere iam Graphium D in eun- dem fili complexum cum AB. & tenfo filo, quantum id patitur, circa AB circumduc lineam, hec Ellipfis erit. b Ctm hec tam facilis effet defcriptio, non indigens o- perofis illis circinis, quibus aliqui cudendis admiratio- “¢ nem hominum venantur; diu dolui, non poffe fic et- iam Parabolen defcribi. Tandem analogia m6- ftrauit, (& Geometrica comprobat) non multo operofiis & hanc defignare. Proponatur A fo- cus, C vertex, vt AC fit axis; is continuetur in partes A. in infinitum vfq;, aut quousq; Parabo- len placuerit defcribere. Placeat vfq; in E. Acus ergd in A figatur, ab ea fit nexum filtii longitudi- ne AC. CE. Teneas manu altera caput alterti fi- li E. altera graphium, citi filo extende vfq; in C. Sit etiam ad CE. ereéta perpendiculariter EF. a Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 201 igitur graphio C & manu altera E difcede equalibus interuallis a linea AE. fic vt manus altera & fili caput femper in EF maneat, filumque DG femper ipfi AE parallelon; via CD. quam Graphio 20 fignaueris, erit Parabole. Dixi hec de feétionibus conicis tanto libentits, quod non tantum hic dimenfio refractionum id requirebat, fed etiam in- fra in Anatome oculi vfus earum apparebit. Tum etiam inter problemata obferuatoria mentio earum erit facienda duobus 2; locis. | Denique ad preeftantiffima optica machinamenta, ad pen- flem in aére ftatuendam imaginem, ad imagines proportiona- liter augendas, ad ignes incendendos, ad infinite comburen- /<¢/:amdé- 2 4 ips. ta Optica dum, confideratio earum plane eft neceffaria. Porte. The headlines of the edition quoted are Joannis Kepleri and Paralipom. in Vitellionem up to page 221, and afterwards Joannis Kepleri and Astronomiw Pars Optica. PAGE 92. Kepler begins by saying that rays from 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. 1x. 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 lne from the pomt 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 quidem complectens at semper minus appetens” (line 5). The hyperbola Vou. 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). Para- bolas 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 Focr (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 w 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 cir- cumference 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, ie. 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 in 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 are 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 }, 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. Xt. 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 7) 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 7 at infinity were the same point. Kepler states expressly that he gave the name Foct 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 his 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. tv. 14—17, 1883), both of which have been referred to by Professor Gino Loria in his writings on the history of geometry. Henry Bricas. Frisch (11. 405 sq.) quotes a letter of Henry Briggs to Kepler dated, Merton College, Oxford, “10 Cal. Martiis 1625,” which suggests improvements in the Paralipo- mena 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 “The Ancient and Modern Geometry of Conics is hereinafter referred to as AMGC. 26—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 ex- pressed by Briggs as follows: c 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 systemes de droites paralléles comme concourant & Vinfini, et qu’il leur appliquait le méme raisonnement” (p. xxxix.). Chasles on the Porisms of Euclid refers to this remark of Poncelet. In his Apergu Historique (p. 56, 1875) he writes that Kepler “introduisit, le premier, usage de l’infint 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 tiber Geschichte der Mathematik writes under the head of Girard Desargues (1593—1662), “Wir miissen einige wesentliche Dinge hervorheben und darunter zunichst die Anwendung des Unendlichen in der Geo- metrie...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 Vorgingern” (11. 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.), referrmg to Poudra’s ee eg: Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 205 uvres de Desargues 1. 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 uvres de Desargues, 1. 210, under the head Traité des Coniques, we read, “ Nombrils, point brulans, Joyers—Crest & 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 a 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 other- wise, 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 econ- structions and propositions. The Lemmas and Propositions of the Principia are quoted by their Roman numerals, II. THE CoNICc THROUGH FIVE Pornts, Prop. A. Given five points of a conic to Jind @ sixth, het AS BG 1D! RP’ be given points of a conic. Through P draw PT7SO parallel to BA across BD, AC, CD. It is required to find the pot 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 J, OK.OP/0C.OD=IA.IB/IC.ID=O8. OT/OC. OD. Therefore OK .OP=OS. OT, which determines K when the other points are given. Inflect PR to CD parallel to AC. Then the point K ws found by drawing CK parallel to RT, and PRT, SCK are similar triangles. Cor. 1. Yo 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, namely 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. Given five points A, B, C, D, EH, two pairs of parallel chords can also be determined as in Prob. Ly. of the Arithmetica Universalis. Let AC, BE cross in H. Inflect DZ to AC parallel to BE, and HK to DI parallel to AC. Then, in order that 7D, EK may meet the conic again in F, G, we must have with Newton's notation for rectangles and proportions, AHC.BHE :: AIC.FID :: EKG. FED. Cor. 2. To determine the conic touching lines 7B, JD at B, D and passing through P. Supposing AC, BD in the figure to coalesce, find A 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 J, and find its vertices, and those of the conjugate diameter. Cor. 3. Hexagrammum M ysticum. 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 C of a conic, a sixth point D on the curve can be found as follows. Draw any parallel to 0K meeting PK in T and meeting the parallel through P to AC 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 im 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, PB: Hence the following construction. Take fixed lines PR, PT; fixed points B, CO; 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 mfinity. 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 7, 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 anharmonic 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} = {D}, or any four points D 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. POPS PT= SC. SA/SK . SP = fg, if f, g be the focal chords parallel to AC, AB. See also below on Newton’s Lemma xvi. Cor. 11. The extension at the end of Cor. 6 follows from a simple transformation of the figure by which the parallels RY are turned into convergents. In the figure as 208 Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. it stands suppose DBw drawn to PQ. Then, the points A, B, C, P being fixed and D variable, (0} = (R}=[T} = {0} But P is the position of O, and likewise of , when RZ vanishes. Therefore Ow passes through some fixed point F. When D is at A the line Ow becomes QS, and when RT passes through B it becomes CH. Thus F is the point (CH, QS), and as Ow turns about # the point D is found by drawing CO, Bo. 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, AC 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, 7 are conjugate points, through P and S (or 7’) draw parallels, and through ZY (or S) and K draw parallels meeting them in AR 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. 2: Locus ap TRES ET QuUATUOR LINEAS. APOLLONIUS OF PerrGa. We shall see that Newton mentions Apollonius of Perga in connexion with the problem of the quadrilinear locus. What Apollonius says of the Tomos emi Tpels Kal Técoapas ypayuds is translated as follows by Dr T. L. Heath in his edition of the Conics of Apollonius in modern notation (p. Ixx. sq. 1896), “Now of the eight 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 dis- covered 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. m1 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, Qir’ be drawn parallel respectively to TQ’, TQ, and if Qr, QR meet in r and Qr’, QR in r’, then Qr . Wr’: QQ? =(PV?: PT?) x (TQ. TQ: QV), 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. mi. 55), for tangents to one branch and two branches of a conic respectively, “give the property of the three-line locus.” The constancy 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 JZ bisects both KK’ and RR’, the intercepts KR, K’R’ are equal, and likewise KR’, K’R. ie L Q K Therefore RK .RK’ (or KR.KR’) varies as KQ?. This is the trilinear theorem as proved by Apollonius. Inflect RD to QQ’ parallel to Q7. Then RK.RK’ varies as RD?, and the theorem may be stated thus, The distance of any point on a conic Srom 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 line. Apollonius does not enunciate the theorem, but he proves and uses it in the course of his propositions mentioned above. Vot. 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 dA, B, C, D being denoted by a, 6, c, d respectively, its distances from AB, BC, CD, DA vary as mean proportionals to ab, be, 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-lime theorem in this way. The Lemmas and Propositions quoted below by number are Newton’s, whose proofs and diagrams in lib. 1 sect. v. of the Principia should be referred to. Lemma XVII. Case 1. AC, 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 8,7. 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 three- line 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 X, which might have been found by drawing the parallel to RT through B. This construction leads to the proof of his Lemma xvi. 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 XVIL, 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 quadri- linear theorem by Descartes, “Atque ita problematis veterum de quatuor lineis ab Huclide incepti & ab Appollonio continuati non calculus, sed compositio geometrica, qualem veteres querebant, in hoe corollario exhibetur.” a 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 PO: PS draw CRD, BDT to any sixth point D of the conic. Then will PR/PT be a con- stant ratio, and conversely, Case 1. The constancy of PR/PT is deduced from the four-line theorem proved in Lemma xvi. Interchanging P, D in this paragraph only, let A, B, OC, D in the figure of Prop. A be fixed and P variable. Through D draw a parallel to AG meeting CP m 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 PTB 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., IIR IONE (p. 206) zs a constant ratio. Hence the line RT is given in direction. See Prop. XxI. and Prop. xxutL, where Pt/Pr is made equal to PT/PR by drawing tr parallel to TR, “actA rectd tr ipsi 7'R parallela.”" 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 ( turn about those points as poles, one pair of the lines or bars containing the angles Pp intersecting at M on a fixed line or director which cuts BC in MN. Then the other pair will cross at a point D lying on a conie 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 CNM. Then PCR, NCM are similar triangles, and PR/NM = PC/NC. Inflect PZ to BD, making the angle BPT equal to the constant angle BVM. 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 V 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 B7 at B, make D coincide with B. See Prop. xxi, Corol. 1. In other words, make the angle NCM 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 joinmg them. For Newton's construction see Prop. XXVIIL Scholium (p. 216). Prop. B. Jf two angles AOB, AwB of given magnitudes turn about poles O, a, 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 O; and since any line through O (or @) meets the locus of B in n other points, the locus is of the order 2n. 4, INVENTIO ORBIUM. Prop. XXII. Pros, XIV. To describe the conic through five points. This is done by Lemma Xx., and again by Lemma XxXt1. Prop. XXIII. Pros. 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. a Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 2S Case 2. In the general case, HZ being the given tangent and BOCDP the given points, draw HAJ, ICPG, GBDH, and make the ratio compounded of JEVLEEHD) STEMS CARE CID GND 5 (COIS » KO Ue LARs a ratio of equality. Thus HA/ZA 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. Pros. XVI. To describe a conic through three given points and touch- ing 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 BR?/DR?=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. XXIII. Imaginary Points. In the second case of Prop. xxi. and in Prop. xxiv. Newton uses an auxiliary line which is supposed to cut the conic in points XY 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 XY and Y are imaginary. LemMA XXII. Figuras in alias ejusdem generis figuras mutare. Here Newton gives a method of homographic transformation, in which the loci of points G, g correspond so that the coordinates XY, Y of @ and «, y of g are connected by relations of the form, _OA.AB ya 04-y x 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. Pros. XVII. To describe a conic through two given points and touch- ing three given lines. Transform the given tangents and the line through the given points into the sides of a parallelogram. Let these sides be hei, idk, kel, lbah, 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 he/m =ic/id = ke/kd = le/n, and each of these ratios is equal to the given ratio of hi+4l, the sum of the antecedents, to m+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 le 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. Prop. XVIII. Yo 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 0. Draw OP to the third diagonal, and take @ a harmonic conjugate to P with respect to O, o. 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 0; and finds Q the other end of the diameter PO. In the retransformed figure Q would therefore be found by the previous construction. Prop. XXVIII. Prop. XIX. To describe the conic touching five given lines. This is led up to by three Lemmas, one of which, with a transformation as in Prop. XXv. or Prop. xxVvL, 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. bo = or Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. Then AM: AF=BQ: BI, and FI, MQ meet on the diameter AB. We can now solve the problem as follows, F Complete the parallelogram JKIM by drawing the tangent KL parallel to 7M. Then ZZ, 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 (FJ, MQ) on the chord of contact AB; and with 7M, KL they determine a point (JZ, 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 Xx. 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 ITEM, IQK be fixed tangents to a conic and MK the diameter parallel to their chord of contact, then, HQ being any third tangent, the rectangle KQ.ME, or (1K -—IQ)(IM—TIE) is constant, This leads to a tangential equation of the form, a.JH.IQ+b.JE+c.1Q+d=0, 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 ZK, EQ, eq, LM determine equal cross ratios on the tangents 7K and JM. 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 XxXIv., and the conic described by Prop. xxm. 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 xx1. 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 eircle through B, C. Draw its diameter KZ 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 ZH; and when X is at LZ, 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 7 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 & parabole in quam hyper- circulum transit,’ the points IJ and m being the inter- bola mutatur, dum puncta m, M coeunt; atque etiam sections of the director MN and the circle. —iii_,2.0 Se ee Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 217 5. PERSPECTIVE AND CONTINUITY. In Lemma xxi (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 wt 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 Hnumeratio above mentioned, in which curves of the mth order are called curves of the (n—1)th genus or kind, the straight line in this way of speaking not bemg 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 “Parabole quing; divergentes” generate by their shadows all other cubic curves, and so from “Curve quedam 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. lv., and the writer’s note on Perspective in vol. x. of the Messenger of Mathematics (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 wz, y, with Xz and Ya/y constant. Page 200. 21 gquantumuis absurdis locutionibus] Poncelet used “ce quwil appelle le principe de continuité,” which is Kepler’s principle of Analogy under a new name. This principle Kepler formulated in terms suitable to its later applications. Including normal Vor. SVE. 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 analogiam 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 « 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. Lut. 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 must 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 Focormps (AMGC, 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 ¢ and ¢’ 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 AdB is equal to a, because ¢ is on the circle; and the lines oA, ga are parallel, and likewise $B, $b, because ¢ is at infinity. Therefore Z agb= 2 A¢B=a, or any two lines through ¢ may be regarded as intersecting at any angle. Hence every circle in the plane passes through ¢, and similarly through ¢’. Conversely, a conic through ¢ and ¢’ is a circle. The orthoptic locus of a curve of the nth class is of the degree n(n—1), since its intersection with the line Infinity consists of @ and ¢’ taken $n(n— 1) times. From the equation e+ y=(et ty) (a—-ty)=0 in rectangular coordinates it seems at first that @ and @’ are indeterminate, because x (or y) may have any direction. But the angles tan-'+7 are indeterminate. — Dr TAYLOR, THE GEOMETRY OF KEPLER AND NEWTON. 219 tan (0 + a) = tan 0 tan a(1 + tan? 6) =0, The equation reduces to and when tan?@=—1, then @ is of the form a +78 with 8 infinite. Page 210 ab Fuclide incepti, etc.] Newton has in mind the words of Descartes in La (Géométrie, “commencée & 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 Xvi. 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 XX1. 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. 1. 4—5) on Inventio Orbiwm. 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 lemma Neutonianum, as we learn from the preface to Simson’s Sectiones Conic. Newton himself, with 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 mazximis et minimis, sur les sections coniques. Dans le questions de maxima et de minima,” and that, “On y retrouve tout ce que les méthodes analytiques d’aujourd’hui nous apprennent sur ce sujet.” This astonishing statement is a too brief summary of the words of Montucla on lib. v. and lib. y1., ‘‘Ils traitent l’un et l'autre un des sujets les plus difficiles de la géométrie, savoir les questions de cinquiéme Apollonius examine particuliérement quelles sont les plus grandes et les moindres lignes qu’on puisse tirer de chaque point donné 4 leur cireconférence. On y retrouve tout ce que nos méthodes analytiques d’aujourd’hui nous apprennent sur ce sujet.’ Chasles goes on to speak of normals as the subject of lib. v. 28—2 IX. Sur les Groupes Continus. Par H. Porncare. [Received 25 September, 1899.] I. INTRODUCTION. La théorie des groupes continus, ce titre immortel de gloire du regretté Sophus Lie, repose sur trois théoremes fondamentaux. Le premier théortme de Lie nous apprend comment dans tout groupe continu il y a des substitutions infinitésimales et comment ce groupe peut étre formé a I’aide des opérateurs Tf = OG) Considérons 7 opérateurs de cette forme (1) Aa Fong hs (Oped zis 5 Al Gays et convenons de poser : X;X;- X,X; — (X;X;). D’aprés le second théoréme de Lie si les symboles (X;X;) sont liés aux opérateurs X; par des relations linéaires de la forme: (2) (X;Xz) = =cusXs, ou les ¢ sont des constantes, les r opérateurs (1) donneront naissance 4 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. Cest le troisitme théortme de Lie qui attirera surtout notre attention. Quelles sont les conditions pour qu’on puisse former un groupe de structure donnée, c’est-a- dire pour trouver r opérateurs X,, Xo, ...... , X, satisfaisant A des relations de la forme (2) dont les coéfficients c¢ sont donnés? On voit tout de suite que les coéfficients ¢ ne peuvent étre choisis arbitrairement. On doit d’abord avoir (3) Cis = — Ciks- SS ees ee bo e M. H. POINCARE, SUR LES GROUPES CONTINUS. 2 Ensuite d’aprés la définition méme du symbole (X;X;) on a identiquement (4) ((XaXy) Xe) + (Xo e) Xa) + (XX a) Xr) = 0, d’ou résultent entre les c certaines relations connues sous le nom ddentités de Jacobi. Une condition nécessaire pour que l’on puisse former un groupe de structure donnée, cest done que les coéfficients c satisfassent & ces identités de Jacobi auxquelles il convient d’adjoindre les relations (3). Le troisitme théoreme de Lie nous enseigne que cette condition est suftisante. Pour la démonstration de ce théoréme, nous devons distinguer deux familles de groupes. Les groupes de la 1° famille sont ceux qui ne contiennent aucune substitution permutable a toutes les substitutions du groupe. Les groupes de la 2° famille sont ceux qui contiennent des substitutions permutables & toutes les substitutions du groupe. En ce qui concerne les groupes de la 1°? famille, la démonstration de Lie, fondée sur la considération du groupe adjoint, ne laisse rien & désirer par sa simplicité. En ce qui concerne les groupes de la 2° famille, Lie a donné une démonstration entiérement différente, beaucoup moins simple, mais qui permet cependant de former les opérateurs X;(b) par lintégration d’équations différentielles ordinaires. Dans une note récemment insérée dans les Comptes-Rendus de VAcadénue des Sciences de Paris, j'ai donné une démonstration nouvelle du 3° théoreme 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. sdchsischen (resellschaft der Wissenschaften 1891 et dans le tome 41 des Mathematische Annalen donné du théoréme en question une démonstration entiérement différente de celle de Lie. Cette démonstration présente la plus grande analogie avee 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 ramene le probléme a l’intégration d’équations ditférentielles ordinaires. Je suis arrivé comme Lie lui-méme & des équations différentielles ordinaires qui méme sont susceptibles d’étre complétement imtégrées. 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 quil y ett quelque intérét a la développer. bo bo bo M. H. POINCARE, SUR LES GROUPES CONTINUS. Je raméne en effet la formation d’un groupe de structure donnée, 4 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 1% famille; mais méme dans ce cas, il peut y avoir intérét & les connaitre, car elles sont dune forme différente et me sen déduisent pas immédiatement. De plus elles sont applicables aux groupes de la 2° famille et dans ce cas elles nous fournissent une solution du probleme plus simple que celle de Lie. Il. DEFINITION DES OPERATEURS. Soit f une fonction quelconque de n variables a,, 2, ..., @p. Soit X un opérateur qui change / en dpe of, ect CS) 7a) Ge be ae ot les (X;) sont n fonctions données des n variables a, a, ..., @, de sorte que: X(N=3(X) S. Soient Y, Z, etc. d'autres opérateurs analogues de telle fagon que: 7 =, Te df | Vea) df PP)=2VN Ts SA)=2AG or les (Y;), les (Z;), ... étant d’autres fonctions de a, a, ..., Zp. Dans ces conditions: X(f)=X(X(P)], XV (PN=XVP/)], LY PI=A[AV (Pf), XVZ(f)=X[V2(F)],--, seront des combinaisons linéaires des dérivées partielles des divers ordres de la fonction Ff, wmultipliges par des fonctions données des 2. Ainsi se trouveront définis de nouveaux opérateurs X*, XY, X*Y, XVZ,..., qui sont des combinaisons des opérateurs simples X, Y, Z, .... On voit que ces produits symboliques obéissent & la loi associative mais n’obéissent pas en général a la loi commutative de sorte que XY ne doit pas étre confondu avec VX. Ces opérateurs sont ainsi symboliquement représentés par des mondmes; mais on peut définir des opérateurs qui seront symboliquement représentés par des polynédmes tels que: 1+aX, aX+bY, aX?+2bXV+cY> ...... 5 en convenant d’écrire par exemple: (1+aX)(f)=f+aX(f); @X+bY)(f)=aX(f)+ bY (f) -ccccccee ae M. H. POINCARE, SUR LES GROUPES CONTINUS. 223 On voit que les polynodmes opérateurs ainsi définis obéissent & la fois A la loi associative et & la loi distributive; de sorte qu’on aura: (aX +bY)(cX + dY) =acX*+adX V+ be VX + bd ¥?, et en particulier: (X+ YP=X?4+XV+ VX + V2, expression qu'il ne faut pas confondre avec X?+2XY+ Y*. On peut aussi introduire des opérateurs qui seront représentés symboliquement par des séries infinies. Je citerai par exemple l’opérateur : fea(Z+V)(f) +e (X + Y(f) 408 (K+ YR (f) teens, que je représenteral symboliquement par: em = > l-a(X+Y) (7), peers l-a(X+Y)’ ou plus simplement par et lopérateur: A. a = = a Ls ae “s ft HX Atq V+ % (f) + sinjelelere ) que je représenterai par e**(f) ou simplement par e™. On peut se demander si l’emploi de ces opérateurs représentés par des séries est légitime et si la convergence des opérations est assurée. ‘ s ott cette convergence est certaine. C’est ainsi que Lie a démontré que Il y a des cas o q gas OPS ii Gre ie 090%) Sta)) ou les «’; sont définis par les équations différentielles : da; 7 F op Iie sag NA BY NOL, 2) +++) Ln); et par les conditions initiales: , Ly = 2; pour ¢=0. 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 : (e%etXeZ) (e-ZebX el) = eV elatv) Xe, Il y a aussi un cas ov ils obdissent A la loi commutative. Soient p(X) = DanX™, (X)= 2d, X", deux séries symboliques dépendant d’un seul opérateur élémentaire X. On a alors 6(X) [v (X) (P= v (X) [6 (4) (PII. Les deux produits symboliques ¢(X) W(X) et W(X) d(X) sont en effet des sommes de monémes dont tous les facteurs sont égaux A WX. Si tous les facteurs sont 224 M. H. POINCARE, 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 3 xAmyn exe¥ = > —_, m! n} avec yuyn evex — >, min! ni avec pe ey es yy p! III. Cancun pes PoLtyNoOMES SYMBOLIQUES. Solent A YA 7, 2 UG), coe. , n opérateurs élémentaires. Par leurs combinaisons on pourra former d’autres opérateurs représentés symboliquement par des monédmes ou des polyndmes. Deux mondmes seront dits équipollents lors quils ne différeront que par Jlordre de leurs facteurs; il en sera de méme de deux polynédmes qui seront des sommes de mondmes équipollents chacun a chacun. Nous appellerons polynéme régulier tout polyn6me qui peut étre regardé comme une somme de puissances de la forme: (aX +BY+yZ+...)?. Il résulte de cette définition: 1°. Que si un polynéme régulier contient parmi ses termes un certain monéme, tous les mondmes équipollents figureront dans ce polyndme avec le méme coéfficient. Cette condition est d’ailleurs suftisante pour que le polynéme soit régulier. 2°, Que parmi les polynédmes équipollents & un polyndme donné il y a un polynéme régulier et un seul. Le polynéme XY-—YX jouit de la méme propriété que les opérateurs élémentaires, c’est-a-dire que (XA V— YX) (f) est comme X(f), Y(f) ete. une combinaison linéaire des dérivées du premier ordre seulement de la fonction f multiplies par des fonctions données des a;. Nous supposerons que les opérateurs élémentaires et leurs combinaisons linéaires sont seuls & jouir de cette propriété. (Si cela n’avait pas lieu, nous introduirions parmi les opérateurs élémentaires tous ceux qui en jouiraient.) Nous devrons done avoir des relations de la forme: @, Re Oy): M. H. POINCARE, SUR LES GROUPES CONTINUS. 225 ou (XY) est une combinaison linéaire des opérateurs élémentaires: nous reconnaissons la la relation de Lie dite relation de structure: X,;X, = XX; => Scie X «. Cela posé, deux polynémes seront équivalents lorsqu’on pourra les réduire Pun A autre en tenant compte des relations (1). Par exemple le produit (2) P[XY-YVX-(XY)]Q (ot le premier et le dernier facteurs P et @ sont deux mondmes quelconques) est équivalent 4 zero; et il en est de méme des produits analogues et de leurs combi- naisons linéaires. Les produits de la forme (2) sont ce que j’appellerai des produits trindmes. La différence de deux mondmes qui ne different que par V’ordre de deux facteurs consécutifs est équivalente A un polynoéme de degré moindre. Soient en effet X et YV ces deux facteurs consécutifs. Nos deux monémes s’écriront LPIA) IBV), P et Q étant deux monémes quelconques, et leur différence P[XY— YX]Q sera équivalente a P(XY)Q, dont le degré est d'une unité plus petit, puisque (XY) est du 1 depreyeX Va aXe du 2° degré. Soient maintenant M et M’ deux mondmes équipollents quelconques, c’est-d-dire ne différant que par l’ordre des termes. On pourra trouver une suite de mondmes TEES, 1M, is. ss, SMe dont le premier et le dernier sont les deux mondmes donnés et qui seront tels que chacun d’eux ne différe du précédent que par l’ordre de deux facteurs consécutifs. La différence M— ’ qui est la somme des différences TE Vin he M,—M’ sera x done encore équivalente & un polynéme de degré moindre. Plus généralement, la différence de deux polynomes équipollents est équivalente a un polyndme de degré moindre. Je dis maintenant qu’un polyndme quelconque est toujours équivalent & un polynéme régqulier. Soit en effet P, un polynéme quelconque de degré n; il sera équipollent A un polynome régulier P’,; on aura alors Péquivalence : Dee) Etna eal oo ot P,. est un polyndme de degré n—1 qui sera & son tour équipollent 4 un polynéme régulier P’,_,, dou Péquivalence : Pe Pa es Vor, X Vili. 29 226 M. H. POINCARE, SUR LES GROUPES CONTINUS. et ainsi de suite; on finira par arriver a un polynéme de degré zéro, de sorte que nous pouvons écrire |’équivalence : Y ere) eit A) colon ON copes trees > dont le second membre est un polynéme régulier. On a done un moyen de réduire un polyndme quelconque a un polyndme régulier en se servant des relations (1). I] reste & rechercher si cette réduction ne peut se faire que d'une seule maniere. Le probléme peut encore se présenter sous la forme suivante; un polynodme régulier peut-il étre équivalent & zéro? Ou bien encore peut-on trouver une somme de produits trindmes de la forme @) P[XY-Yx-(XY]@ qui soit un polynédme régulier non identiquement nul? Toutes les sommes de pareils produits sont en effet équivalentes a zéro. Le degré dun produit trindme sera égal a 2 plus la somme des degrés des polyndmes P et @. Si je considére ensuite une somme S de produits (2), ce que jappellerai le degré de cette somme S, ce sera le plus élevé des degrés des produits qui y figurent, quand méme les termes du degré le plus élevé de ces différents produits se détruiraient mutuellement. Le produit trindme (2) peut étre considéré comme la somme de deux produits, le produit binédme (2 bis) P[XY— YX]Q, ou je distinguerai le mondme positif PXYQ et le mondme négatif — PYXQ; et le produit —P(XY)Q, que jappellerai le produit complémentaire. Soit done S une somme quelconque de produits trindmes de degré p ou de degré inférieur; je pourrai écrire: =8,—T,+Spi-— Tpit ..-.- +8,.—T,, ou S; est une somme de produits bindmes de degré k. (2 ter) PX aveXG| iO) tandis que — 7; est la somme des produits complémentaires correspondants : —~P(XY)Q. Il sagit de savor si la somme S peut étre un polynédme régulier sans étre identiquement nulle. J’observe d’abord que si S est un polynéme régulier, il doit en étre de méme de S,; car S, représente l'ensemble des termes de degré p dans 8; tandis que (Ses — Ty), (Sp-2— Tyr), ---, (S2— 73), — 7, veprésentent respectivement l'ensemble des termes de degré p—1, p—2, ..., 2, 1. M. H. POINCARE, SUR LES GROUPES CONTINUS. 227 On voit immédiatement que S, est équipollent a zéro; comme zéro est un polynéme régulier, et que deux polynémes réguliers ne peuvent étre équipollents sans étre identiques, il faut que S, soit identiquement nul. Soit en particulier p=3, S;= > [XY — YX] Z—- >Z [xX _ YX], le signe = signifie que lon 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: T,=>%(XVY)Z— XZ (XY), puis S,= > [(XV)Z—-Z(XY)I, 1h ECan Al, S=8,;—7,4+8,-—T,= = [XY- YX -(XY)] Z-SZ[XV—-YVX-(XY)] +2 [(XV) Z—-Z(XY) - (XY) Z)}. Il est aisé de vérifier que S, et S,—7', sont identiquement nuls, de sorte que S se réduit A — 7). Or 7, =((X¥) 2] + (FZ) X1+ (2X) VN] est un polyndme du 1” degré, car [(YY)Z] comme (XY) lui-inéme est un polynédme du 1° degré. Or dans un polynéme du 1° degré, chaque terme ne contenant qu’un seul facteur, on n’a pas & se préoccuper de lordre des facteurs. Tout polyndme du 1° degré est done un polyndme régulier. Si done le polyndme 7, n’est pas identiquement nul, la somme S sera égale & un polyndme régulier qui ne sera pas identiquement nul. Done pour quun polynéme puisse étre réduit d’une seule maniére & un polynéme régulier il faut qu’on ait les identités suivantes: (3) ((X¥) 2] +[(V2) X] + (ZX) Y]=0. On reconnait 1a les identités de Jacob’ qui jouent un si grand role dans la théorie de Lie. (Si dailleurs 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 done en réduire le nombre.) Les identités (3) sont done la condition nécessaire pour que la réduction d’un polyndme & un polynéme régulier ne puisse se faire que d’une seule maniére. Il me reste & montrer que cette condition est suffisante. 29—2 228 M. H. POINCARE, SUR LES GROUPES CONTINUS. Je dirai pour abréger une somme réguli#re pour désigner une somme de _ produits trindmes qui est un polynéme régulier. Soit alors S=S,— 7+ Spa— Tpit -: une somme de produits trinémes; les deux premiers termes S, — 7’, représentent la somme des produits trinémes du degré le plus élevé, c’est ce que j’appellerai la téfe de la somme S. J’ai distingué plus haut dans un produit trinédme trois parties que j’ai appelées le monéme positif, le mondme négatif et le produit complémentaire. Je dirai qu’une somme de produits trindmes forme une chaine si le monome négatif de chaque produit est égal et de signe contraire au monéme positif du produit suivant. Le monéme positif du premier produit et le monéme négatif du dernier seront alors les mondémes extrémes de la chaine. Il résulte de cette définition que tous les monémes positifs d’une méme chaine ne different que par lordre de leurs facteurs. Une chaine sera jfermée si les deux monomes extremes sont égaux et de signe contraire. Si S,—TZ, est une chaine fermée de produits trindmes (S, représentant la somme des produits bindmes et — TZ, celle des produits complémentaires), il est clair que S, est identiquement nul puisque les monédmes positifs et négatifs se détruisent deux a deux. Nous avons vu que si S est une somme réguliére, S, est identiquement nul, d’ot il résulte que la téte d'une somme réguliere S se compose toujours d’une ou plusieurs chaines fermées. Si deux chaines ont mémes mondmes extrémes, leur différence est une chaine fermée. Nous nous servirons de cette remarque pour montrer qu'une chaine fermée peut toujours de plusieurs maniéres se décomposer en deux ou plusieurs chaines fermées. Une chaine fermée quelconque peut de plusieurs maniéres étre regardée comme la différence de deux chaines C et C’ ayant mémes mondmes extrémes; soit alors C” une troisitme chaine ayant mémes monédmes extrémes. La chaine fermée C—C” se trouve ainsi décomposée en deux autres chaines fermées C—C” et 0” —C". Il s'agit de montrer que toute somme réguliere est identiquement nulle et en ettet quand cela aura été démontré, il sera évident qu’un polynéme régulier dont tous les coéfficients ne seront pas nuls ne pourra étre équivalente & zéro, puisque tout polynéme régulier équivalent a zéro est par définition une somme réguliére. Supposons que le théoréme 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 régulitre de degré p est identiquement nulle, il en sera de méme de toutes les sommes régulieres de degré p qui ont méme téte. La différence de ces deux sommes serait en effet une somme réguliére de degré p—1 qui serait identiquement nulle d’aprés notre hypothése. a EEE: M. H. POINCARE, SUR LES GROUPES CONTINUS. 229 Il me suffira done de former toutes les chaines fermées de degré p et de montrer que chacune d’elles peut étre regardée comme la téte d’une somme réguliére identique- ment nulle. Toute somme réguhére S d’ordre p a en effet pour téte une de ces chaines fermées, par exemple 8’; si done je montre que Tune des sommes réguliéres dont la téte est S’ est identiquement nulle, il en sera de méme de toutes les autres et en particulier de 8S. Pour établir ce point, je vais décomposer la chaine fermée envisagée en plusieurs chaines fermées composantes. Il me suffira de démontrer la proposition pour chacune des composantes. J’appellerai chaine simple de la 1%? sorte toute chaine ov le premier facteur de tous les mondmes soit positifs soit négatifs sera partout le méme. J’appellerai chaine simple de la 2° sorte toute chaine ow le dernier facteur de tous les monémes sera partout le méme. Une chaine simple peut d’ailleurs étre ouverte ou fermée. Tl est évident que toute chaine fermée peut étre regardée comme la somme d’un certain nombre de chaines simples, alternativement de la 1° et de la 2¢ sorte. Soit done S une chaine femméeiC; 10), <4. Ge edeswehatnes simples de la 1 sorte, Cam OS x. C2 :des chaines simples de la 2° sorte, on aura: S = Cy O40, + O44 20. +e Ch +C'n, le monéme négatif extréme de chaque chaine étant bien entendu egal et de signe contraire au mondme positif extreme de la chaine suivante, et le mondme négatif extréme de 0’, égal et de signe contraire au mondme positif extréme de C,. Soit XY le premier facteur de tous les monémes de ©,, Z le dernier facteur de tous les monédmes de Co Yawie premier facteur de tous les monodmes de C,, T le dernier facteur de tous les mondémes de C’, (je wexelus pas le cas ot deux des opérateurs X, Y, Z, 7 seralent identiques). Soit alors C” une chaine simple de la 2° sorte ayant son mondme positif extréme égal et de signe contraire au mondme négatif extréme de C’,; dont tous les monomes ont pour dernier facteur 7’; et dont le mondme négatif extréme a pour premier facteur XY. Soit C’”” une chaine simple de la 1%° sorte dont tous les monémes ont pour premier facteur X et dont les mondmes extrémes sont respectivement égaux et de signe contraire au mondme négatif extréme de 0” et au mondme positif extréme de C,. La chaine fermée § se trouvera décomposée en deux chaines fermées composantes, & savoir: Ss’ =(O" 4 C) He GH. ae GE af (Ge a oe”), S=— CO" £ O04 O54, Cre On OM. 230 M. H. POINCARE, SUR LES GROUPES CONTINUS. S’ ne contient que quatre chaines simples; car (C’’+(C,) et (C’,+C”) sont des chaines simples; S” contient deux chaines simples de moins que S. En poursuivant on finira par décomposer S en chaines fermées composantes, formées seulement de quatre chaines simples. Il nous suffit done d’envisager les chatnes fermées formées de quatre chaines simples comme par exemple S’. Les monémes positifs extrémes des quatre chaines simples qui forment S’ ont respectivement pour premier et dernier facteurs : pour C’”+C,, Xetiel: »? Ce ? xX et Z, es Ce: Y et Z, yy Gat OF Y et Tf. Soient M,, M’,, M., M’, ces quatre mondmes. Tous ces mondmes sont équipollents entre eux et équipollents & un certain monéme que jappellerai XY PZT. Nous allons alors construire une série de chaines simples, comprises dans le tableau suivant, ot dans la premiere colonne se trouve la lettre qui désigne la chaine, dans le seconde le monédme extréme positif, dans la troisitme le monédme extréme négatif ; je fais figurer dans le méme tableau les quatre chaines simples qui forment S’ et je pose pour abréger: C= XVPZL: Ol KVP, =) XPT Os YXePZr: Nom de la chaine | Monéme positif | Monéme négatif || Nom de la chaine | Monéme positif Monodme négatif aera CY + C, M, —M, 1DY, M, — Q, C’, M’, _ M, D, M’, =O C, M, —M’, E, ah —Q' 0", M’. —M, ja | f =) D, M, — Q: | E, Q. rr Q's D, mM’, Fai Q E’, Q’ pz Q: On peut supposer que tous les monédmes de la chaine PD, ont pour premier et dernier facteurs X et 7’; de sorte que D, est & la fois une chaine simple de 1° sorte et une chaine simple de 2° sorte. Il en est de méme des autres chaines D. On peut supposer de plus que les chaines # se réduisent & un seul produit trindme de maniére que par exemple : E, = XYP [ZT — TZ —(2ZT)). La chaine fermée : S’=(0" +0,)+01,4+6.4+C%. M. H. POINCARE, SUR LES GROUPES CONTINUS. 231 \ peut étre décomposée en cinq chaines fermées composantes, & savoir : U,=C0" +¢,+ D',-—£#,-D,, (Of = C1i+ D, — EF’, — D',, U, = C,+ D, a E, = D,, U4 = 60-2622 SE a— D, Vi Eye 4 Het Eis I] s‘agit done de montrer que chacune de ces cinq chaines fermées est la téte dune somme réguliere identiquement nulle. Pour les quatre premieres, qui sont des chaines simples fermées, le théoréme est évident. On l’a supposé démontré, en effet, pour les chaines fermées d’ordre inférieur a p. Or il est clair que lon a par exemple: Ul = XGHE H étant une chaine fermée d’ordre p—1. Quant a V, ce sera la téte de la somme réguliére [XY — VX —(XY)] PZT + VXP [ZT — TZ -(ZT)| -—[XY — YX —(XY)] PTZ —XYP [ZT —TZ -(ZT)| + (XY) P [ZT —TZ—-(ZT)| — [XY — VX —(XY)] P (27), qui est identiquement nulle. Il reste a@ envisager ce qui se passe quand deux des opérateurs X, Y, Z, 7’ sont identiques, par exemple XY = Y, ou Y=Z. Nous devons alors distinguer le cas ou les divers mondmes positifs ou négatifs de notre chaine contiennent deux facteurs identiques, l'un jouant le réle de X et lautre le réle de Y (ou lun le réle de X et Vautre celui de Z); il ny a alors rien a changer aA l’analyse qui précede. Et d’autre part le cas ot ces mondmes ne contiennent qu'un seul facteur X. Le premier cas pourra seul se présenter si l’on suppose X=Z, ou X=TZ7, et sil 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: OOF — PZ On — eee Zs La définition des diverses chaines demeurera d’ailleurs la méme et on constatera immédiatement que la chaine V est identiquement nulle. Le théoreme est done démontré pour les sommes d’ordre p, sil lest pour les sommes ordre moindre. La démonstration précédente n’est toutefois pas applicable au cas de p=3; car la 232 M. H. POINCARE, SUR LES GROUPES CONTINUS. chaine V n’existe que sil y a au moins quatre facteurs. Mais la seule chaine fermée du 3° ordre est la chaine S,;—T7, envisagée plus haut et nous avons vu quelle est la téte d'une somme réguliére qui est identiquement nulle si les identités (3) ont leu. Le théortme est done établi dans toute sa généralité. Toute somme réguliére est identiquement nulle. Done un polynéme régulier qui n’est pas identiquement nul ne peut pas s’annuler en vertu des relations (1). Done en résumé, Si les identités (3) ont lieu, les relations (1) permettent d'une maniére, et d'une seule, de réduire un polynéme quelconque a un polyndme régulier. TV. PROBLEME DE CAMPBELL. Soient LGA. OS Seno. 6 r opérateurs élémentaires; supposons qu ils soient liés par les relations (1) XaXp— XpXq = (XaX,), (X,X,) étant une combinaison linéaire des X;; supposons de plus qu’on ait les identités (3) ((XaXy) Xe) + ((XpXe) Xa) + (XX a) Xz) = 0. D’aprés le deuxiéme théoréme de Lie, ces opérateurs donnent naissance & un “groupe P continu,” qui admet 7 transformations infinitésimales indépendantes. Ces transformations infinitésimales changent / en t+ eX, Ga: e étant une constante infiniment petite. Soit T =t,X,+bX.+...+6,X,, une combinaison linéaire de ces opérateurs. Les t sont des coéfficients constants quel- conques. La transformation finie la plus générale du groupe s’exprimera par le symbole: cua): Soient maintenant T=tX,+hXot... +tX,, V=4,X, + Xet.-.. $Updr, deux combinaisons linéaires des X. Comme les transformations e? forment un groupe, le roduit a ever devra également faire partie du groupe, de sorte que nous devrons avoir: (4) eVeT=eW, M. H. POINCARE, SUR LES GROUPES CONTINUS. 233 ou W =w,X, + wXo+... + w,X, est une autre combinaison linéaire des X. Les coéfficients w sont évidemment des fonctions des v et des f. Développons le produit : mn f= SS V m!n! VrTn : Le terme général fal est un polynéme d’ordre m+n. Par les relations (1) on m! n! peut le réduire A un polynéme régulier, et cette réduction ne peut se faire que d'une seule maniere. Nous pouvons done écrire: Sao a= =>, W (p,m, 0), ou W(p, m,n) est un polyndme régulier et homogéne d’ordre p(p = m+n); on a done: eVeT= Sa meaeps m™, n). Si nous réunissons les termes de méme degré et que nous posions Wo = 2m.aW (p, m, 1); il viendra : eS BYE = PM ire Le second théor’me de Lie, que je viens de rappeler, nous apprend que le second membre doit étre de la forme e”, et par conséquent que: WP = Pp 1 - C’est 1A une proposition dont la simplicité serait imattendue, si l’on ne connaissait (5) W, pas la théorie des groupes. Si on pouvait la démontrer directement on aurait, comme l’a remarqué Campbell, une nouvelle démonstration du second théoréme de Lie. Mais il y a plus; on aurait aussi une nouvelle démonstration du troisiéme théoreme de Lie. Les égalités (1) nous font connaitre des relations entre les opérateurs ¢lémentaires et les combinaisons XY— YX; ce sont ces relations qui constituent la structwre du groupe. Cette structure est done entitrement définie quand on connait les r* coéfficients c¢ des r° fonctions linéaires (XY). Mais ces r* constantes c ne sont pas toutes indépendantes; tous les coéfficients de (XX) doivent étre nuls; les coéfficients de (YX) sont égaux et de signe contraire a iViots XoVallile 30 234 M. H. POINCARE, SUR LES GROUPES CONTINUS. ceux de (XY). Enfin les constantes ¢ doivent étre choisies de telle fagon que les identités (3) soient satisfaites. J’adjoins donc aux identités (3) les identités suivantes qui sont evidentes : (3 bis) (XX)=0, (X Y)=—(YX). Le 3° théoreme de Lie nous apprend qu’on peut toujours trouver un groupe de structure donnée; pourvu que les coéfficients c qui définissent cette structure satisfassent aux identités (3) et (3 bis), c’est-a-dire aux identités de Jacobi. Mais supposons inversement qu’on ait démontré directement Jlidentité (5) et par conséquent la formule (4). Les coéfticients w seront donnés en fonctions de v et de t; et je puis écrire: (6) We= Px (Vi, ti). Pour former les fonctions ¢,, il suftit de savoir former le polyndme W,, par consé- quent de savoir former les polyndmes W(p, m, n): c’est-a-dire de savoir réduire un polyndme queleonque en polyndme régulier; pour cela il suffit de connaitre les co- éfficients c. Soit evel — ei: )weWel—¢2 . Seleli—ed. O= su, X4, Z= AX, Y= DyyX;. Le caracttre associatif de nos opérateurs nous montre que l’on a: oil ee d’ot les relations suivantes : We= Oe (i, ti); Ye = he (Li, Ua). (7) = PE(Wi, Ui)= HEC, Yi)- Regardons dans les équations (6) les ¢ comme des constantes; ces équations (6) définiront une transformation qui transforme 2, 2%, ..., ¥- EN W;, We, -.-, 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: Ld ab -oonigetaas Cpa GEE ZC du, dt; ’ ou dans ¢,(v;, t;) on annule les ¢ apres la différentiation. Les r substitutions infinitésimales Y;(j) sont linéairement indépendantes. Et en effet, pour qu’elles ne le fussent pas, il faudrait que le déterminant fonctionnel des go, par rapport aux ¢ fat nul, quels que soient les v quand les ¢ s’annulent. Or cela n'a pas leu car ce déterminant devient égal & 1 quand les v sannulent. Ayant ainsi défini les opérateurs élémentaires X;(/), leurs combinaisons 7’ = >t; X;(f), e”. ete. se trouvent définis eux-mémes. M. H. POINCARE, SUR LES GROUPES CONTINUS. 235 Ces opérateurs étant associatifs, on aura eX (f)=e7e™(f), cest-a-dire, en négligeant les quantités du 3° ordre par rapport aux ¢ et aux x: rary op MUaur D’autre part, d’aprés la maniére dont ont été formées les fonctions bz, On vérifie que Y= yer U + 3 (TU)= Dt X; + TG + 3 > (t; Up — tyuz) (X;X;,), et la comparaison de ces deux identités donne: X;X, — XX; =(X; Xx); ot les coéfficients des fonctions linéaires (X;X;) sont bien les 73 coéfticients ¢ donnés. Le groupe ainsi formé a donc bien la structure donnée et le troisiéme théoréme de Lie est démontré. C'est au fond la démonstration de Schur. Ce que j’appellerai le probleme de Campbell consiste donc & démontrer directe- ment la formule (5), ce qui démontre A la fois le second et le troisitme théor&me de Lie. V. Le Sympore ¢ (6). Considérons 7 opérateurs élémentaires Ca Ce Baan) Goa et une de leurs combinaisons linéaires : REX eX, <2. tN Soit ensuite V un autre opérateur élémentaire qui pourra étre ou ne pas étre une combinaison linéaire des opérateurs X, Supposons que les opérateurs V et X soient liés par des relations de la forme : VA XV b 5X by Ret Orplee (C1, eo r), on aura alors: VT -TV = Su,X;, ou UE = Db;.zt;. Je poserai VT—-TV=6(T). Done @(T) est comme 7 une combinaison linéaire des X; et les coéfficients de 0(T) se déduisent de ceux de 7 par une substitution linéaire, 30—2 236 M. H. POINCARE, SUR LES GROUPES CONTINUS. Je poseral A[(L)\=@(L), 0 [6 (Ly]=0™ (7), de sorte que 6”(7') sera comme 7 une combinaison linéaire des X, les coéfficients de oe (7) se déduisant de ceux de 7 en répétant m fois cette méme substitution linéaire. Si maintenant $ (9) = ge est un polynéme ou une série ordonnée suivant les puissances croissantes de @, j’écrirai: $(8)(T) X90" ( Ty au lieu de Considérons |’équation, dite caractémistique : b, — 6, lis Mires by, (1) es Bese 4 =(?). ty wre Si cette équation a toutes ses racines distinctes et si ces racines sont 6,, 0, ..., %,, il existe » combinaisons linéaires des X;, A savoir: (2) VY,= Xan Xi, telles que: VY; — YipV = OY x. Si alors on a: : T=st,X;==t;Yp, on aura: Si nous posons: $ (0) (L)=3hiX;, nous yvoyons d’abord que les coéfficients h; sont des fonctions linéaires des ¢; ce sont d’autre part des fonctions des b; étudions ces fonctions. Si #(@) est un polynome entier d’ordre p en @, les A; seront des polyndmes entiers d’ordre p par rapport aux b. Si done @(@) est une série ordonnée suivant les puissances de 6, les h; se présenteront sous la forme de séries ordonnées suivant les puissances des b. Nous allons voir bientét quelles sont les conditions de conver- gence de ces séries. Des équations (2) on tire en effet: Xi==hx Yi, dou: ° ty, = DBti, (8) (LT) == (Ox) C0. Xi, Wk, el POINCARE, SUR LES GROUPES CONTINUS. 237 dou enfin : hy = St (9x) « dix Bix. Pour déterminer les produits 4,8), faisons 1 $ @ =e , & étant une constante quelconque. On a alors: 1 —— 1 =>, lily = FSM. hex; = H, ou bn Bix i Syd Ik h;=> EGR On tire de lA (E-0)(H)=T, ce qui peut s‘écrire: Eh; — Tbyhy => tj. De ces équations on peut tirer les h en fonctions des ¢; on trouve: ay Bs 3) = Se ee) ou Py est un polynéme entier par rapport aux b et A &; quant a F(&) cest le premier membre de l’équation (1) ot @ a été remplacé par &. Le second membre de l’équation (3) est une fraction rationnelle en £; décomposons la en éléments simples; il viendra: eae Sb aT FP’ (Ox) (& — Ox) ou P;* est ce que devient Pj; quand on y remplace & par 6. On a donc: _ Pi On Bix SACP: d’ot enfin pour une fonction ¢(@) quelconque: ty Pishb (Ox) Xi 4 0)(T) == 7 47 : () g(@(T)= 8 Es On voit que les h; s’expriment rationnellement en fonctions des b, des 6, et des (0). La formule (4) peut se mettre sous une autre forme; nous pouvons écrire: lyin dE (E) St Pi X; (4 bis) $ (0)(T) = a= Jaa! Fe” Vintégrale étant prise dans le plan des & le long d’un cercle de rayon assez petit pour que la fonction $(&) soit holomorphe a lintérieur; nous le supposerons de plus assez grand pour que les points 0,, 6,, ..., 6, soient A l’intérieur du cercle. Cela nous améne 238 M. H. POINCARE, SUR LES GROUPES CONTINUS. 4 supposer en méme temps que le rayon de convergence de la série $(&) est plus grand que le plus grand module des quantités @,, @2, ..., 6,. On a alors pour tous les points du contour d’intégration : |E|> 6, |, &—|>| 6], seeeee > E|>|6,|, dou il résulte que la fonction rationnelle Py. F(&) est développable suivant les puissances croissantes des b. Il en est done de méme des A,. Nous avons dit plus haut que les h; sont développables en séries procédant suivant les puissances des b; et d’aprés ce qui précéde, il suffit, pour que ces séries convergent, que le rayon de convergence de la série $(&) soit plus grand que la plus grande des quantités Wea IGANG Gascon 5 Cie Si done $(£) est une fonction entiére, les h; seront des fonctions entieres des b. Quw’arrive-t-il maintenant si |’équation caractémstique F(6)=0 a des racines multiples? Il est aisé de s’en rendre compte en partant du cas général et en passant a la limite. Je suppose par exemple que 6, soit une racine triple. Alors F(&) contient le facteur (£—6,). Si je décompose le second membre de (3) en éléments simples, trois de ces éléments deviendront infinis pour &= @,. Soient A, A, A," g—0, " €—ay * E-4) ces trois éléments simples. Alors il faudra dans la formule (4) remplacer le terme : sy GPO) Xi ee de) (qui n’aurait plus de sens dans le cas d'une racine multiple) par les trois termes suivants: 2A," Xip (A,) — (1!) 5A, Xi! (A) + (2 !) AB Xi" (A). On opérerait de méme pour les autres racines multiples. Done les h;, dans le cas des racines multiples, sont des fonctions rationnelles des b, des 6, des $(0,) et de leurs dérivés $'(O;), pb’ (Ax), -.--+ ; on pousse jusqu’a op) (6) si 6, est une racine multiple d’ordre p+ 1. \ Remarquons que je n’aurais pu faire ce raisonnement par passage a la limite, si je m’étais restreint dés le début en supposant que V est une combinaison linéaire des M. H. POINCARE, SUR LES GROUPES CONTINUS, 239 X, et que les X sont liés par les relations (1) et (3) du N° IV. (relations de struc- ture et identités de Jacobi). Alors en effet les cas ov l'équation caractéristique a des racines multiples ne pour- raient plus étre regardés comme des cas particuliers de ceux ot toutes les racines sont distinctes. On aurait pu, il est vrai, démontrer directement la formule (4 bis) et se servir de cette formule: mais Jai préféré ne pas m’imposer au début cette hypothése restrictive, quitte & Vintroduire dans la suite du caleul, de facon & avoir le droit de raisonner par passage A la limite. Quoi qu'il en soit, le cas le plus intéressant au point de vue des applications a la théorie des groupes, cest celui ot cette hypothése restrictive est satisfaite. Sup- posons done que V soit une combinaison linéaire des X: V=uX,+0,Xo+...4+0,X>. Supposons de plus que les XY soient liées par les relations (1) du Ne précédent X;X;— X;X;= dcy,X,, et que les constantes c¢ satisfont A des relations telles que les identités (3) du N° pré- cédent aient lieu, On aura alors : 0 (T) = Leijg dit; Xs, dot: bi-k = C).5-40, + Ca.5-K7Vo +... aCe Uys Les résultats, démontrés dans le cas genéral, seront évidemment encore vrais dans ce cas particulier ; si done on pose : b (0) (T)= 3 h,X;, les h; seront des fonctions lingaires des ¢, et des fonctions rationnelles des v, des @,, des (0) et de quelques unes de leurs dérivées, Les #; sont les racines d’une équation algébrique dont le premier membre est un polynéme_ entier homogéne de degré r par rapport aux v et A linconnue 6. De plus les h; ne dépendent que linéairement des $(,) et de leurs dérivées, Si ¢(E) est une fonction entire de E, les h; sont des fonctions entidres des v. Dans tous les cas, le symbole $()(Z’) se trouve entidrement défini. Je terminerai par deux remarques : 1°. Si x (&) est le produit des deux fonctions $ (£) et W(£), on aura: (9) [¥ (8) (T)] = (8) [ (0) (1)] = y (6) (7). 2°, Si on a: $ (9) (T) = U, on aura : 1 q oO \Y=F. Cette derniére égalité n’a de sens que si $(&) ne s’annule pas pour &=0, de telle fagon que a soit développable suivant les pulssances de @. 240 M. H. POINCARE, SUR LES GROUPES CONTINUS. VI. ForRMULES FONDAMENTALES. Considérons |’expression (1) Coane, V et T ayant méme signification que dans le § précédent, tandis que @ et 8 sont des constantes trés petites. Développons cette expression en négligeant les termes du 3° ordre par rapport a a et a @; il viendra: (1 eV E*2 ) (1 + = (1 +aV4"0), ou 1467+ = _ap(vT—TV), ou avec la méme approximation : e8T—ag (VT-TV). On aura donc, toujours avec cette approximation : (2) e- 2 eT eeV— 80 oh U=T— af (T), ou encore avec la méme approximation : (2 bis) e-o¥ BT eeV— BU ot U=e (L). Je me propose maintenant de démontrer que la formule (2 bis) est vraie quelque loin que l’on pousse l’approximation; et dabord qu’elle est vraie quand on néglige le earré de 8 et qu’on pousse l’approximation par rapport & a aussi loin que l’on veut. Supposons done quon pousse lapproximation jusqu’aux termes en 8 et jusqu’aux termes en a” inclusivement. Dans |’expression (1) nous remplacerons e®? par 1+ 7, erY et e*¥ par les m+1 premiers termes de leurs développements; en effectuant le produit (et néglgeant dans ce produit @#”*) nous obtiendrons un polynéme symbolique que nous pourrons rendre régulier par les procédés du N° III. Soit p (a, 8)==ATl, le polynéme régulier ainsi obtenu; [I est un mondme symbolique, et A son coéfficient qui est un polynome entier en a et B. Nous avons alors : (3) d (a de da, B) = e—(atda) V e8T platda) V — p—da-V ri (a, B) ete V. En effectuant le produit du 3° membre de cette double égalité, et néglhgeant le earré de la différentielle da, on obtiendra un polynéme régulier de méme forme dont les coéfficients sont eux-mémes des polynédmes du 1* degré par rapport & da d’une part, par rapport aux coéfficients A d’autre part. Telle est la forme du polynéme $(a+da, 8). aooooooooorrreeerrrrr_ M. H. POINCARE, SUR LES GROUPES CONTINUS. 241 D’autre part on a: (3 bis) $ (a+ da, 8)— > (a, B)=das 11, Cette égalité, rapprochée de la remarque que nous venons de faire, montre que Oe da est une combinaison linéaire des coéfficients A. Done ces coéfficients A, considérés comme fonctions de a, satisfont 4 des équations linéaires 4 coéfficients constants. De plus pour «=0 ils doivent se réduire aux coéfficients de e®” Ces conditions suffisent pour les déterminer. Or je dis que lon peut y satisfaire en faisant (conformément a la formule 2 bis): icy 3))— 16> Ue l(a) En effet cette formule nous donne: (a + da, B)=&U", U' =¢e-@tdajo(f), et il s'agit de vérifier que: e-da.V BU pla. V — oBU' Or la formule (2 bis) démontrée quand on néglige d'une part le carré de ~, d’autre part le carré de a, peut s'appliquer ici puisque nous négligeons le carré de 8 et celui de da. Nous avons done eda. V eBU gia. V— BU" [J = ¢-da.9(TJ), dow: U" = e-aa. 8 [e-a8 (T')] = «eta 0 (7) = U’. On a done bien: ¢(a+da, 8) = ea V e8U eda. V = o8U", CG, @p 1k, 1D) La formule (2 bis) satisfait done A 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 a 8 que par rapport a a. Nous avons: d (a, B) =e" eh Fert ; dot: (a, B+ dB) =e-2V B+ 4B) TeV — (e-aV e8T e2V) (e—-2V ei. TeV), ou (a, B+d8)=$(4% B)d(a, dp). Comme nous négligeons le carré de df, je puis écrire: ¢(a, dB)=e8-", U=e“(T); Vou. XVIII. 31 242 M. H. POINCARE, SUR LES GROUPES CONTINUS. dou: (4) (4, B+dB)=$(a, B) et?" Cette formule (4) représente sous forme condensée des équations différentielles de méme forme que les équations (3 bis), auxquelles doivent satisfaire les coéfficients A de $(a, 8)=SA.TI. Cest ainsi que la formule (4) représentait sous forme condensée les équations (3. bis). On peut satisfaire & ces équations par la formule (2 bis); cette formule donne en effet: b (a, B+dB) =elb+40)U = BU ot8.U — (a, B) e@8-U, Les équations différentielles ne comportant comme les équations (8 bis) qu'une seule solution, la formule (2 bis) se trouve vérifiée dans tous les cas, Cette formule (2 bis) nest 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 connaitre un mode de raisonnement applicable a des questions analogues; je veux parler de celui ot s‘introduisent les équations différentielles (3 bis) ou les équations analogues. Il importe avant d’aller plus lom de préciser la portée de la démonstration que nous venons de donner. Pour quelle soit valable, il faut que tout polyndme puisse étre réduit d’une maniére et d’une seule a étre régulier. Or, d’aprés le N° IIL, cela a lieu dans deux cas. 1°. Si V et 7 sont des combinaisons linéaires des opérateurs X, Vi peng, Sik, et si ces opérateurs sont liés par des relations XX, — XpXi= Dine Xs, les constantes c satisfaisant aux identités (Xa (X,X.)) Us (X, (X,Xq)) + (X. (X,X)) a 0; si en d’autres termes les opérateurs X définissent un groupe de Lie et si e*’, e8? sont deux transformations quelconques de ce groupe : Dans ce premier cas la formule (2 bis) est toujours vraie. 2°. Elle sera done vraie en particulier si on suppose que Py xX, Xs, POH A sont 7+ 1 opérateurs liés par les relations (5) 2 Va — ak Pee M. H. POINCARE, SUR LES GROUPES CONTINUS. 243 et (6) X,X;,— X;,X; = 0. Ces relations entrainent en effet l’identité (V(X; Xx)) + (Xi (XEV)) + (Xe (VX;)) = 0, en désignant suivant la coutume par (VX,;) et (X;X;) les seconds membres des relations (5) et (6). On aura done dans cette hypothese : (2 bis) ine ee eae Teen ea Gans (CT): On aura de méme en permutant V et 7: (2 ter) GTEC GEES GS NY = GEA). e-® étant un symbole analogue a e~* et défini de la maniére suivante: le symbole m est formé avec 7 comme le symbole 6 avec V; on a donc, si VY est un opérateur quelconque : 4(Y)=2Y=YVP.- On aura donc: n(V)=TV—VT=-6(f), et en vertu des relations (6) 7(X)=0; (V)=0; ™(V)=0, ei (V)—V —Bn(V)=V +80 (7). La formule (2 ter) devient ainsi: (2 quater) e~ BT ecV BT — gaV top0(7), Si lon suppose maintenant que les relations (5) subsistent, mais que les relations (6) naient plus leu, les formules (2 bis) et (2 quater) cesseront d’étre vraies quels que soient a et BP. Cependant supposons que l’on regarde les opérateurs X comme trés petits et qu’on en néglige les carrés; & ce degré d’approximation, les relations (6) dont les premiers membres sont du 2° ordre par rapport aux X se trouvent satisfaites d’elles-mémes. Les relations (2 bis) et (2 quater) sont done vraies, si l’on néglige les carrés des X, ou, ce qui revient au méme, si l’on néglige le carré de 7, ou encore si on néglige le carré de 8 (puisque 7’ ne figure qu’affecté du facteur £). Si done V et les X sont r+1 opérateurs liés par les relations (5), les relations (2 bis) et (2 quater) ont liew aux quantités pres de Vordre de 8°. Au méme degré d’approximation la formule (2 quater) peut s’écrire : erV + ape (7) = eV = BTeV fk et! BT, ou encore: etV +268(T) = erV — eT eo V 4. er¥ BT 31—2 244 M. H. POINCARE, SUR LES GROUPES CONTINUS. ou en vertu de la relation (2 bis): eV +ap0(T) — ery — et e8U 4 er eft: U = es (L): ou, toujours en négligeant le carré de f: er V+ape (7) = eV (1 pas BU+ BT) = etV eB (T- U). Si nous posons: +a0(T)=W; T—U=Y; il vient : as ; l-—e-* (7) ecVteW — etVeb¥; Y= __(P). a@ Soit W= Sw; X; une combinaison linéaire queleonque des X;; peut-on déterminer les coéfficients ¢ de la combinaison 7’=Xt;X; de telle fagon que lon ait +20(T)=W? Cela est évidemment toujours possible si le déterminant des coéfficients bj, 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 ot ce déterminant nest pas nul, de faire varier les coéfficients b d’une maniére continue de fagon que ce déterminant devienne de plus en plus petit et de passer a la limite, pour démontrer que la formule (7) est encore vraie quel que soit W. Si enfin V, au lieu d’étre 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’étre par suite de lintroduction de nouvelles relations entre nos opérateurs. Remarquons que ce raisonnement par passage A la limite n’aurait pas été possible, si nous nous étions restreints dés le début en supposant que V et 7 sont des com- binaisons des opérateurs X, que les X définissent un groupe de Lie, que e*” et e®? sont deux substitutions finies de ce groupe de Lie. Dans ce cas en effet le déterminant des bz, aurait été constamment nul. La formule (7) peut sétablir directement: En effet en négligeant le carré de 8 on a: y @V+e8Wy exV+BW — n! n-1 ts +B%—- (V"2W+V2WV + Vow? SE oa a a Pee Or on trouve aisément n-1 n—2 pe 1 n! n-1 n—2 V2 W+V"2WV+...4 WV = Ga SL W- Ce igh 20(W) n! Fst aay a Ven Sn a—1 ( W), M. H. POINCARE, SUR LES GROUPES CONTINUS. 245 dot: qn —_ = etV+BW — eoV 4 BS =|: = pie “oe V P(— 0)? a) |. ss ( 6 0 aV+BW — paV s | (Vr? (— 28)? > eV: 5 (= 20)?7 eran —e¥+ ps | | er |1+e3 oF |, ou V+ew V 4 Len er + = e* al +BY) =eVehF ; Y= ad = (W). CLG TEND: VII. FORMATION DES SUBSTITUTIONS INFINITESIMALES D’UN GROUPE DE STRUCTURE DONNEE. Soient done X,, Xz, ..., X, 7 opérateurs élémentaires liés par les relations (1) XX, — X~Xi=(K:Xe) = Veins Xs, les ¢ étant des constantes telles que les identités de Jacobi du N° III. aient lieu. Soient T= >4;X;, U=wX;, V= dX; W= dw; X; diverses combinaisons linéaires de ces opérateurs. Considérons le produit eal BT effectuons le produit qui sera une série de polyndmes symboliques; réduisons chacun de ces polyndmes & des polynédmes réguliers en nous servant des relations (1); je me propose d’étudier la nouvelle série ainsi obtenue que j’appelle ¢(a, 8); le raisonnement sera le méme que dans le N° précédent, mais je le développerai un peu plus. Tous les termes de cette série (4,8) sont des polynomes réguliers; et les co- éfficients de ces polynomes se présentent eux-mémes sous la forme de séries développées suivant les puissances de a et de 8. Je puis ordonner (4, 8) suivant les puissances croissantes de 8, en groupant tous les termes qui contiennent en facteur une méme puissance de 8. J’obtiens ainsi: (a, B)=6+8o.4+ Bdo4+... D’autre part j’al: $ (a, B+ dB) = eVerTe.T — (a, B)e?-? = (a, B)(1 + df. 7), ou: ou: (3) mom = Pm el ces conditions jointes a (4) dy = er suffisent pour déterminer ¢. 246 M. H. POINCARE, SUR LES GROUPES CONTINUS. Or on y satisfait de la maniére suivante. Faisons : o(a,8)=e", (a, 8+d8)=e"ta"; soit 7 un symbole qui soit a W ce que @ est a V. Il s’agit de satisfaire 4 l’équation (2) ou ce qui revient au méme & $ (4,8 + dB) = (a, B) e®-?, on doit done avoir: eW+dW — Weds. 7. Or en vertu de la formule (7) du N° précédent, on satisfera 4 cette condition si l’on a: i e” (dW). n (5) dp.T= Cette formule (5) représente symboliquement un systeme d’équations différentielles auxquelles doivent satisfaire les coéfticients w;. En vertu de la formule (4 bis) du N° V,, ces équations peuvent s’écrire : Bee 1 rd&1—e$s=r b t£:dB = a > dw;P;; (5 bis) B al E PE) es (@=1, 2,..., 7) Si lon a: WX; — X;W = dex. :,.weXs, F(&) est le déterminant dont |’élément est (pour la 7 ligne et la s* colonne) — (C1. 4.8W1 + Co, i,gWe +... + Crier), sauf les éléments de la diagonale principale (¢=s) qui sont égaux a = (C14. 1Wr + C2, i,5We Hoe + Cri iWr) +E; les Pj sont les mineurs de ce déterminant. L’intégrale du second membre de (5 bis) est prise dans le plan des &, 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 é€quations (5 bis); la condition (4) le sera également si les valeurs initiales des w pour 8=0 sont Wj = Vie Les équations (5 bis) admettant toujours une solution telle que pour B=0, on ait w;=v;, et dautre part les conditions (2) et (4) suffisant pour déterminer ¢, on aura: d (a, B) = e”, w= Xw;X;, les w étant des fonctions de 8 définies par les équations (5 bis) et les conditions initiales W; = Vj. La série ¢(a, 8) nest done autre chose qu'une exponentielle dont l’exposant est M. H. POINCARE, SUR LES GROUPES CONTINUS. 247 une combinaison linéaire des X;; c'est le théoreme que j’ai annoncé au N° IV.; et comme dautre part ce théoreme a été établi en s'appuyant simplement sur les relations (1) et en en faisant des combinaisons purement formelles, le probleme de Campbell est résolu et le troisieme théoréme 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 (6 bis), Soient &, &, ...... , & les p racines distinctes de l’équation F'(&)=0; ce sont des fonctions algébriques des w, puisque #'(&) est un polynéme entier par rapport a & et aux w. Les ae, seront donnés par des équations linéaires dont les seconds membres dB seront des constantes ; tandis que les coéfficients des premiers membres seront des fonctions rationnelles des w, des & et des e-**; ces coéticients ne dépendront d’ailleurs que liné- airement des exponentielles e~**; ce seront des fonctions symétriques des racines. ; F : dw; Résolvons ces équations par rapport aux —~, nous trouverons : q I PI dB : dw; (6) West set Phe + A, jty, les coéfficients A étant rationnels par rapport aux w, aux & et aux e—%, Le probleme qui se pose a propos du troisieme théoreme de Lie est ainsi com- pletement résolu. I] s'agit de trouver 7 opérateurs XG (GB ZENG Ey. acec86 : [EGS satisfaisant aux relations (1); on y satisfait en faisant OG Ae Ag oe A ‘1 hw, 2 hy, Les équations (5 bis) peuvent se mettre sous plusieurs autres formes. Soit SS Cputa e—Oee On aura (puisque les P;; sont les mineurs du déterminant F’): EP; = Lyi Pr =i() pour 727 et EP — hei Pic =F pour 7=y. Nos équations ee IP». (5 bis) 448 = 1 | l—et 5 du;Pi Seay eae ae (9 248 M. H. POINCARE, SUR LES GROUPES CONTINUS. donnent : dB >itibs = a foes Re 5 Spar HN Dp dou S.dw:P.: , 3 dB Sitsbis = 5 — ze —e-*) sie st , Je gael 1 = : La deuxiéme intégrale étant nulle, nous pouvons écrire tout simplement: (Ster) tbr= a T= Jaga —e hs (k= 29 n): D’autre part l’équation (5) peut s’écrire: dW a dp ta) (7) dot dw; _ 1 Edé Li Py de ova (ie Fee ce qui donne: 6) | eee Oe Jaa Jae) FOF dey; Cette derniére intégrale doit étre prise le long d’un contour enveloppant toutes les racines de F(£)=0, mais n’enveloppant pas les points £=2%rV—-1 (k=+1, +2, ... ad inf). VII. ForMuULES DE VERIFICATION. Soit eV +8V — eVeF, V = >0, Xj, 8V = 60, X;, Y= LyX; ; on aura en vertu de la formule (7) du N° VI. mee (posant : 0(T)=VT—TV comme dans le N° V.). Soit maintenant e-VeTe¥ = eU, on aura par la formule (2 bis) du N° VI. Ujex CL): Wii let POINCARE, SUR LES GROUPES CONTINUS. 249 Soit e-\V+8V) eTeV+5V — QU’, on aura: U’ =e 9+ (7), ot 6+ 66 est un symbole qui est & V+6V ce que 6 est & V. On aura d’autre part: eu —\em teaver evel —emucler. d’ou en négligeant le carré de Y qui est infiniment petit: eU' = eV — VeV + eUY = cU+UY-YU. Dot U’—U=UY-YU. Si je conviens de poser: e +80) _ 9-9 — § (¢-), il viendra: U’ — U =8(e~*) (TP). Nous arrivons ainsi a la formule symbolique suivante: @ se) r=ler| Ger) ]- |S“ er) tran Pour mieux expliquer le sens de cette formule rappelons que nous avons trouvé plus haut: 2) $(0)(L)= = — | deb) SX, ott les h; sont des fonctions rationnelles des ¢, des v et des & données par les équations: (8) Ehi— Vee =ti; du = kr + Co. bie H vee + Cree Ur- Alors on aura: Se-8 (T') = [adee-#38h.X,, 1 2a —1 ou les 6h; sont les accroissements que subissent les fonctions h; quand les variables v; subissent les accroissements 6v;. Si alors les h’; sont ce que deviennent les h; quand on y remplace les ¢ par les dvz, la formule (1) pourra prendre la forme l—-e= é Dans le 1* membre le signe = se rapporte aux r valeurs de l’indice 7; dans le @ebis), 20/1 3x, | dée-tSh, = 3 (X,X,— X,X,) Jag hy / déeh;. 2" membre aux r(r—1) arrangements des deux indices i et k (arrangement 7, k étant regardé comme différent de l’arrangement hk, 7). Cette formule nous fait connaitre un certain nombre de relations auxquelles doivent satisfaire les expressions X;X;,—X,X; ou (X;X;). Ces relations sont curieuses: mais Wor, 2 WIUUL Bo 250 M. H. POINCARE, SUR LES GROUPES CONTINUS. la plupart ont déja été démontrées par Killing et il semble que les autres pourraient se démontrer facilement par les procédés de Killing. Je n’y insiste done que comme sur un procédé de vérification. Les deux membres de cette équation sont d'une forme particuliére. Le premier membre est linéaire a la fois par rapport aux symboles X;, par rapport aux ¢;, aux 6, aux exponentielles e-® (les 6; étant les racines de l’équation F=0). Les coéfficients de cette fonction linéaire sont eux-mémes des fonctions rationnelles des v et des @;. Le second membre est linéaire A la fois par rapport aux symboles (Y;X;), par rapport aux ¢;, aux vg, aux exponentielles e~® et e-%-* (0; et O étant deux racines de F=0). Les coéfficients de cette fonction linéaire sont encore rationnels par rapport aux v et aux 6;. Les @; étant les racines de léquation F=0 sont des fonctions algébriques des ». 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~* et il y ena autant que léquation #=0 a de racines distinctes. Il y a ensuite les exponentielles e~®**) qui peuvent étre distinctes des précédentes, mais qui peuvent également ne pas en étre toutes distinctes si l’une des racines de l’équation F=0 est constamment égale a la somme de deux autres racines. Supposons qu'il y ait g exponentielles et soient CHUSECMES OS ces exponentielles. Les deux membres de léquation (1 bis) seront alors des fonctions linéaires des produits de la forme (4) tm Ovne, ou m et h peuvent prendre les valeurs 1, 2,..., 7, et ot mw peut prendre les valeurs il, 7S coon CE Les coéfficients de ces produits sont des fonctions algébriques des v, ne dépendant ni des ¢, ni des 6v. Pour que Jidentité puisse avoir lieu, il faut que lon puisse égaler dans les deux membres de (1 bis) les coéfficients dun méme produit (4). Nous aurons ainsi un certain nombre de relations linéaires entre les symboles X; d'une part, les symboles (X;X;,) d’autre part; les coéfficients de ces relations linéaires sont des fonctions algébriques des v. Ces relations linéaires doivent étre identiques aux relations de structure ou en étre des conséquences. J’examinerai seulement le cas particulier ot F(&)=0 a toutes ses racines distinctes. Je puis alors supposer que les opérateurs élémentaires X; ont été choisis de telle sorte que: V2 é& = AV = 0; X;, @; étant Pune de ces racines. M. H. POINCARE, SUR LES GROUPES CONTINUS. 251 Egalons alors dans ’équation (1 bis) les coéfficients de tmév,; il vient: @h; vy, At» = (XmX;) Se Jager 2a /—1 Le premier membre ne dépend que des exponentielles e-%, mais le second membre outre l'exponentielle e~’ contient encore e-%~, Egalons les coéfficients de em Si 6,46, n'est pas égal & une racine de F=0, cette exponentielle ne figurera pas dans le 1% membre; nous aurons done (Xm X),) (0) On reconnait lA l'un des théorémes de Killing. Si au contraire O,+ 6m est racine de F=0. l'exponentielle pourra figurer dans le 1" membre et (X,,X,) pourra ne pas étre nul. Je mniinsisterai pas sur les autres verifications, ni sur le cas ot les racines ne sont pas distinctes et ot on retrouverait les autres théorémes de Killing. Je me bornerai a faire remarquer que la verification de la formule (1 bis) n’est 1 1 pas immeédiate, et qu'il faut pour la faire avoir recours aux identités de Jacobi et aux théorémes que Killing en a déduits, IX. INTEGRATION DES EQUATIONS DIFFERENTIELLES ET FORMATION DES SUBSTITUTIONS FINIES DES GROUPES. Soit (1) eVtadv = eV et V=30,X;; dV = Xdy;. XxX; dA sda. X. On aura en vertu de la formule (GO) Glu INP Wal WO (2) dA ————s (aia g Cette formule, identique sauf les notations A la formule (5) du Ne VIL, comprend, sous la forme symbolique, r systemes d’équations différentielles; ainsi que je Tai déja fait remarquer au N° VII, Annulons tous les da, sauf da; égalons ensuite les coéfficients de X,, XG ENG dans la formule (2). Nous aurons équations différentielles qui définiront du, du, dv, da,’ da,’ da. en fonctions des v. Ce sont 14 comme nous Yavons vu au N° VIL, les équations diffé- rentielles qui définissent une des substitutions infinitésimales du groupe, si l’on prend les vy comme variables indépendantes. 252 M. H. POINCARE, SUR LES GROUPES CONTINUS. En donnant 2 Vindice f& les valeurs 1, 2,..., 7, on obtiendra r systemes 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 1° famille (vide supra N° I.), & des équations linéaires, puisque c’est la un résultat bien connu obtenu par Lie. Voici le changement de variables qu’il faudrait faire pour retrouver ces équations ; soit : U=Su;,X;; e~YeVeV=eh; L=31;X;; on aura: (3) L=e-*(U). Cette équation symbolique (3) nous apprend que les /; sont des fonctions des v et des u, linéaires par rapport aux wu, et nous permet de former ces fonctions. Si alors on pose: e-V-aV eUeV+aV — gL+aL, on aura: eltdL — g-dApl gia, ou, puisque A est infiniment petit: (4) dL=LdA-—dA.L. Cette formule (4) représente symboliquement 7 systémes d’équations différentielles qui ne sont autre chose que ce que deviennent les 7 systemes d’équations différentielles représentées symboliquement par la formule (2) quand on prend les 7; pour variables nouvelles. Celui de ces systémes que |’on obtient en annulant tous les da sauf da s’écrit: (4 bis) a = LX, — X;,L. Ces équations sont linéaires et A coéfficients constants et sintégrent immédiate- ment; ce sont celles auxquelles Lie arrive par la considération du groupe adjoint. I] 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 prés le cas des groupes de la 2° famille. Nous pourrons alors choisir les opérateurs élémentaires X; de telle maniére qu’on en puisse distinguer de deux classes. Ceux de la 2% classe seront permutables & tous les opérateurs, ce seront les X”;; quant & ceux de la 1% classe que j’appellerai les .X’;, ils seront caracterisés par la propriété suivante: aucune combinaison linéaire des X’; ne sera permutable & tous les opérateurs, Pour mettre en évidence cette distinction, j’écrirai quand il y aura lieu: LX; = DX’ + Do" X"; VH= IX; VW = WX; VaV'+ Vv". i> a) ————— rt” eee M. H. POINCARE, SUR LES GROUPES CONTINUS. 253 Les v’; seront ainsi les coéfiicients des X’; et les v@eceux kdesiiX'%.. Tea lettres Un Wiel: OF. Uf; “; L’, L: ete. auront une signification analogue. Il est clair qu’on aura: [ef YO — Vy ut fits PRPS Pee = 0, dou 6(T)= VI-TV=V'T'-T'D’. Jintroduis alors un symbole nouveau ; soit: VL = TEV = SN kee DED Gabe je poserai : O (T) = =n';X';; 0" (T)= ZA X”,, et je définis $(0’) & Vaide de 6’ comme Jai défini $(@) & Vaide de 6. On a alors : O(X")=0; O[6’(T)]=0; $(6)(2")=0; et on trouve aisément : $O(D)= 6) (P)= 40) (2) + 0" [PO SO) 2) +802". Remarquons que les expressions : A(T), @(T), 0’ (), dépendent des v’ et des # mais sont indépendantes des v” et des ¢” ; et il en est de méme de $(6).(Z) si $(0) est nul. Les J; étant linéaires par rapport aux wu, je puis écrire: l; -35t UZ. dl; : : : : Les Tu, Sont des fonctions des ». Voyons combien de ces fonctions sont indépen- k dantes les unes des autres. Je dis d’abord que ces fonctions ne dépendent que des 2’. Nous avons en effet (e? étant une substitution quelconque du groupe): CIEE = GVieV’ gaV gletiiaiame dou el — eV —V" eUeV'+V" — eV" e-V' eU eV’ ev” = en eUeV | ce qui montre que Z ne dépend que de V’, mais pas de V”. : : ; Gs Je dis maintenant que le nombre des fonctions dy indépendantes les unes des autres est précisément celui des variables vy’. En d'autres termes, si l’on pose : eL=eVeUeV, el, = e- VieUeN, 254 M. H. POINCARE, SUR LES GROUPES CONTINUS. Videntité Z=Z, si elle a lieu quel que soit U entraine lidentité V’=V,. Si en effet L=T,, on aura quel que soit U: elie VeVeVe hh =e, ce qui montre que ee" est permutable a toutes les substitutions du groupe. C'est done une substitution qui ne dépend que des X”; de sorte que je puis écrire: eVe-M = eh", W’” étant une combinaison linéaire des X”;; on en tire: eV = eF" eerie” dou V =V,+ W", V' = We ‘ ye = Vv", at Ww”. Done V=V4. (CG) Shae LD) : : dl : Nous pourrons prendre comme variables les Tie les v”, au lieu des v' et des v”. dl nes een: : ay . : Les = sont définis par les équations (4 bis), qui étant par rapport a ces variables du des équations linéaires & coéfticients constants sintégrent immédiatement. ; di Les équations (4 bis) nous font done connaitre les = et par conséquent les v’ en du fonctions de la variable «;. Pour obtenir les v”, revenons aux équations (2); si nous posons: 1-—e%=60+ 6 (8), elles peuvent s’écrire : dA’ =dV’' +0’ W(@") (dV), dA” =dV" + 0"wW (0) (dV’). On a dA’ = Sda',.X’,; dA” = Xda",X”;. Si on annule tous les da’ et tous les da” sauf da”,, nos équations donnent simplement : v';=const.; v”;=const.(tZk); v”,=a",+ const. Si on annule tous les da’ et tous les da” sauf da’, les équations deviennent X’,.da’,=adV' +O (0’)(dV), 0 =dV" + 6’ (0) (dV). M. H. POINCARE, SUR LES GROUPES CONTINUS. 255 La premiére de ces équations, équivalente aux équations + bis, est susceptible comme nous l’avons vu d’étre ramenée & la forme dun systéme d’équations linéaires a co- éfficients constants. Lintégration est immédiate et nous donne les y’ la variable a’,. en fonctions de La seconde équation est équivalente A un systéme d’équations de la forme: dv”; + dv’, F, + dv’, Fi, + ... +dvF,=0, les F étant des fonctions données des v’. En remplacant les »’ par leurs valeurs en fonctions de a’,, elle prend la forme: dv”, 3 co) (a’,) da ie = 0 et sintégre immédiatement par quadrature. X. Contact Transformations and Optics. By Professor E. O. Lovert. [Received 15 September 1899.] ‘‘Ayant vu combien les idées de Galois se sont peu & peu montrées fécondes dans tant de branches de Tanalyse, de la géométrie et méme de la mécanique, il est bien permis d’espérer que leur puissance se manifestera également en physique mathématique. Que nous représentent en effet les phénoménes naturels, si ce n’est une succession de transformations infinitésimales, dont les lois de l’univers sont les invariants?*’—Sorxts Liz*. Ir 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 paper + 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 o' transformations in m variables a, %, ..., Un: Di ON (Gis eae Gy, 1) neta ON Greets, inne seo, NCO aN) (RG sadn Cerone (1) where the functions X,, ..., X, are regular analytic functions of a, ..., #, 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 P= fo A T = %, La =Ly, ---, Ln =%y- Then for a value of a, say 6t, 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= 6¢ has the form Oe em a (Coy Span, Le) Cle ban Yq = Xy + Ey (a), -.., Gn) OE+..., | TA (RAE IE (mp sees in) OFF... | Under this infinitely small transformation (2) a, increments .., @, Yeceive the infinitesimal CAS eisaay COUN = (Ste hoog, cosh UPN Salle ecu —onesnonoscecs soc (33) * Le Centenaire de VEcole Normale, p. 489.—Paris, Math.-Phys. Classe, Bd. 48, 1896, pp. 181—133. Hachette et Ci*, 1895. + Geometrie der Beriihrungstransformationen, dargestellt + “Infinitesimale Beriihrungstransformationen der Op- von Sophus Lie und Georg Scheffers, Bd. 1, Leipzig, tik,” Ber. ii. d. Verh. d. k. stichs. Ges. d. Wiss. zu Leipzig, Teubner, 1896. See in particular, pp. 97, 100—103. Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. PAG Such a transformation is called an infinitesimal transformation. The expression apap Uf= pee 24 2 2a ce ee ere (4) is adopted as its symbol, since Uf.é¢ is the increment assigned to any function SF (@, ++.» &,) 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 oe! 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 a, ..., @, gives a system of the form 7 / / A U ~ GS NG acy 645 OD) GhSOR(GR, ooom: Cin Ds coon RSAC CAS soon Bin Wood)» 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 J, is the transformation of the group corresponding to the parameter value a, the inverse 7, of TY, is also found in the group. Further the transformation 7,5, corresponding to the parameter value a+ 6a, is the transformation of the group differing infinitesimally from 7,. The product Ta4s. Zu? which, by the assumed group property, belongs to the group, differs then infinitesimally from the transformation 7,77; 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 infini- tesimal transformation assigns the infinitesimal imcrements SU SSA, sdban a) Clap Bone Olen snl Che, cobs. Map)! ClthceoonappooenasnOdnac (6) to the variables 2, ..., 2,, on neglecting infinitely small quantities of a higher order; if t be interpreted as the time, 2, ..., %, as point-coordinates in a space of x dimen- sions, 6¢ as a time imcrement, and 62,, ..., dv, as the corresponding increments of @,, «++, @, then the equations (6) determine a stationary flow in space of n dimensions. After an interval of time ¢ the point (a, ..., #,) will have assumed the new position (aj, ..., @); the latter position will be obtained by integrating the simultaneous system da Bs da, ‘s dx, x eee) AG! ee) ea) Wore xvas 33 ~~) 58 Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. b with the initial conditions that «a, ..., 2,’ shall reduce respectively to a, ..., 2 for t=0. The n integral equations may be taken in the form U; (Gq, oes Dy) = U; (%, ---, Ln); EGS. soon He) = BCs Boog 2). BRSOOOOOOOOOOOOOOUOOOOOOOOC OG eo ir votecucueceaeereskeceeee(ONE UO, (ac see ny) Ue @, seen aa) Un (@y, tees an) = Un Gr seey Ty) + t; the form of these equations shows that when resolved with respect to a, ..., @,' they represent a group with the parameter ¢; accordingly the given infinitesimal transforma- tion is said to generate a one-parameter continuous group. 4. Consider in particular the case where the preceding transformations are contact transformations. The equations a =X, (a, SOO) Zn,2Z,Pry S500 /Da)k seenilr Nin (Gas OOO) Ln, Z, Pris son /is)) 2=Z(m, se+> Dn, 2, Pis — pi =P; (Gi, --- 5 Fn, 2) Pry ==» Pn)» (CS cacin (0) codnodceocobaco das guSodoodSbaooadsescca0Gntc (9) are said to define a contact transformation when they give rise to a differential relation of the form dz = = pi der =) (Gp oson nn “A Hog econ LO) CE — = pide) SESE eaoe (10); 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 f= Clie, ascy Day £5 Drs s--y' Pn) OF Ao ans see (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 det dGSe-2n= > (orgie) Gece dE =p Ge da) ene: (12). t=1 t=) Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 259 The left-hand member of this equation is a series of ascending integral positive powers of dt; thus the function p must be an ascending series in integral positive powers of 6¢; as the term of zero degree in the left-hand series is dz— Xp;dz;, p must therefore have the form SIEGE dcr mph bellcn Rear a ame omen neeapee? (13). Inserting this value of p and equating the coefficients of corresponding powers of dt we have dt — Spidé; = X7;da; = a (dz “= Xpida;) or d(€— =piéi) + BEdp; — Vajda; =o (dz — Updaz)......ccccecceeceveee (15). This linear and homogeneous condition in dz, dz;, dp; must be true for all values of these differentials; hence, writing Gap eye = (2a Semana ees h Da) eee saan aes ovens ne (16) for convenience, we have Ornar Cylon Ono 0)». (5: 3) joancoopodcossonaspan|ccoo) (17) Eliminating o and solving (16) for € we find E; = Oe c= LpiQy,; — OF i Or = piQz seen eee ec eeeeeeeeee (18). The infinitesimal transformation is therefore completely determined, £, &;, 7; being given by an arbitrary function 0. 5. Let the preceding results be now applied to the infinitesimal contact transfor- mation defined by the characteristic function OQ =V1i+ pet pet... + pre The formulae (18) show that the coordinates of a surface element, by which we mean the ensemble of a poimt anda plane through it, receive the infinitesimal increments Pi —1 62; = == sd ot, oz = ———~— §f, JOR SW! oon dooocadbc bddcoapod |; 1 Virose rasa op (19) This infinitesimal transformation generates a one-parameter group of contact trans- formations, namely the group of dilatations, whose finite equations are found by integrating the simultaneous system V1 + = pi" da =...= V1 + Spi” dz, = V1 + Spi dz= dp, = = dpm Pr Pn i saa Ct c where ¢ is an arbitrary constant. 260 Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. These transformations are obviously characterized geometrically by the property of changing the surface-element (2, ~.., Zn, 2, Pi, +--+») Pn) imto the surface-element (a, ..., @', 2, py’, ..., Pn’) In such a manner that the point of the second lies on the normal to the first and at a constant distance ¢ 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 pomted out for ordimary 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 P, of an isotropic n+ 1-dimensional elastic medium; in an interval of time the motion will have advanced to all pomts P of a sphere whose center is at P, and whose radius is ¢, say, in precisely the same manner as the dilatation (21) would change the surface-elements of the poimt P, mto 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 4, will have advanced to spheres of equal radu f 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 P, in the total time elapsed. But in exactly the same manner the dilatation , Dib pr i ty —— 2. , (=D, sinie(sieie (=1, ese WM) wccvee 22 J+ Spe essen! a ) +++++(22) carries every point P of the sphere about P, into a sphere of radius ¢, about P as center, so that the sphere of center P, will be changed by the dilatation (22) imto the sphere of center P, and radius t,+%, that is imto the sphere into which the point P, 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 @ have in general an envelope ®, and hence the passage from ¢@ to ® is a contact transformation. 7. Let the characteristic function be an arbitrary function of bh, 5007 fry SENy (OE SINI(( Seen consy #7) acdacos sbancocoonepee coor napeagnas=ce (23); the infinitesimal transformation defined by [I is represented by the equations 6a; = I1,,6t, 62= =p: T,,— I, dp;=0, ...... (GSAl yess Mge) actue Sosaseeeeee (24). Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 261 By integrating the simultaneous system da, Oram dz’ _ dp a dpn’ Il apie lee cg ONT oO we have the corresponding one-parameter group of contact transformations Py oe 2 —2-+ (Spl, — egy — peer as (CSA) hactten esc (26). Let t have for the moment a fixed value; the corresponding contact transformation of the group changes the point (a, ..., #,, Z) into a surface whose equation in current coordinates (a, ..., @, 2) is obtained by eliminating the p; from the first n+1 equations; this elimination yields the equation , , , LT —-X Ly — Xo @3 — 3 In — Ln Z2—Z 5 oy ( t 2 > 3) elslely t ) t = 0 sec er ec eeceee (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 7 equations Cin CA Seca NG Pag Cay, Boba an Oy CRO) sony 1a Os soacoseqmentacsoc (28), where r may have all values from 1 to n+1; in the last case the transformations it existent will be point transformations, since the n+1 relations will give the n+1 quantities «;, 2’, as functions of the n+ 1 quantities x;, 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 dz — Spi da! —p(dz— pide) =i), Sueoss (Gr oonah il) ebedecsbonecone (29) 1 1 where the z’, #;, p; are functions of the 2n+1 variables z, «;, p; to be determined. This equation shows that there ought to exist at least one relation between the variables 2’, 2, 2, % containing z and z*. Taking the general case of r different relations ex- pressed by (28), the equation (29) ought to be a consequence of dor 0) dos— Ole ORO ee ee ee ee (30) ; that is, it ought to be possible to find 7 coefficients X,, ..., %, such that the identity n n a dz — Xp; dx; — p (dz — Xp;dzx;) = Xr;do; at 1 2! exists. This demands the following equations : , Co; r Co; 1=),; —s p; = — Din; — 1 oz Pj 1 * Oa; aarti My oseraeriaa ace (31) ; = Sy, Oe: ese: = Ratiag tpi oeted * Oa’ * See Goursat, Lecons sur les Equations aux dérivées partielles du premier ordre, Paris, Hermann, 1891, p. 258, 262 Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. the 2n+2+,r equations (30) and (31) in general determine the 2n+2+7 functions 2’, a, pi, 3, p as functions of 2, x;, pi. Eliminating p we can write the followmg n+7+1 equations for 2’, 2, dj, Ss Sg a j=l... 1, i=1 5 qs2n1 08 (32) Sn =a) o,= 0, on — 0 | i=1 02 resolving these for 2’, z/, Xj, the remaining functions p;, p are found by substituting the values of the former in the remaining equations of the system (30) and (81). 2°. In the second place two transformations S and 7 are commutative when the symbolic equation ST= TS obtains. Consider the contact transformation S and the point transformation 7. That the point P is changed into the point P, by the transformation 7 is expressed by the symbolic equation (P) T= (Py). In the same manner, that S transforms P into the surface = is expressed by the equation (P)S=(2). Then if (P) ST =(P) TS, we have also Cys) ie That is, if S transforms the point P into the surface =, and 7’ changes the point P into the point P,, the latter is changed by S into the surface into which the surface = is changed by 7. 3°. In the third place let S be a contact transformation of an n+ 1-dimensional space commutative with all translations 7 of that space. If S changes a definite point P into the surface =, 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 P,; then by the second paragraph above, the point P, is changed by S into the surface =, into which = is changed by the last-named trans- lation; 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 NOM (NEE ig ton Gee C Paty et) | 0) Gobpanqnqaoutdedeoadkace (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. Pror. 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 ¢ and thus operating on a point (a,...,@,, 2) with all the transformations of the group (26), the point is changed successively into similar surfaces and similarly placed. The point P, is changed by the transformation whose parameter is ¢#, into the surface =. Operating on all the points P of = with the transformation whose parameter is ¢,, these points P will be changed into congruent surfaces that are similar and similarly placed to =. These latter surfaces have an outer envelope, a surface >, into which the surface © is changed by the second transformation. The successive application or product of the two transformations is equivalent to the transformation whose parameter is 4+; the latter transformation carries the point P, directly into the new surface %,, and this surface must then be a similar and similarly placed surface to ¥. The preceding geometrical operations and their results suggest the phenomena of wave-motion in an elastic n+1-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 P, gives rise to a series of waves similar and similarly placed with the common center of similarity P,; 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 o’ contact transformations (26) form a group. 9. The group (26) may be generalized and specialized. 1°. Much more general wave-motions may be designed by using in a_ similar manner the most general infinitesimal contact transformation defined by the characteristic function O(Gip coon #57 fo coon JO7))5 a simple geometric construction shows that the normal velocity of the wave is given by the expression Q)/V1 + Spe2. 2°. The case applying to the optics of a double refracting crystal is given by the particular form Q= af a + S aep?, (Gea ceo 0) agen nase toon tetecaEeeenenane (34). 1 Observing that Vi HIG Orem soc rects ceo seraclosiesjsisbincaeisese cies (35), we have D2 ROI ea (CO at ad So Hae ee eee oe Sader (36); 264 Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. hence the finite equations of the group of contact transformations generated by the infinitesimal transformation (34) are Ci =ZzeHASP OT, 2 =SH—APFODE, Py =HPNp ocersoecesveosereoaes (37); eliminating ¢ by means of the first x +1 equations, we have the ellipsoid (x, =&)? | (a — a)? (an =n), (2 — 2)? AY 2Q). (ab? (at? et (at)? =i (ade = dg fdssiisedeeete (38) ; 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 trans- formations *. 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 22— 2+ + Me (x" — a2) = 0, a ’ i i ere PUN a (39), (22 +3 aj/x; — hk? (224+ 3 2;?) (24+ 3 22)=0 | a ne 1 where & 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 Ril Rays it yee teehee eRe (40) are the infinitesimal rotations of n+ 1-dimensional space written in the symbolic form (4), the expression = e/ es es eee eee eee (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 ENE BF CLF DTS RS: Se RRR re RES 1 2 (42) * “ Beitriige zur allgemeinen Transformationstheorie,” Leipziger Berichte, pp. 495—498. Pror. 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 Xe. Xe eae iy talaays | ov (s. pe Go =) Pe Pecans oct. a3 (44), where ee 07 2 (2 Seay : ®, Zama tes, = | ad (ro) Bee (45), 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, a, ..., Gan)) aK changed into a circle whose points are at the same distance from the origin as the point (2, %, ..., %) itself. Further the radii vectores of (Z, %, ..., &) and (CAB neon aa) make an angle with each other whose cosine is k. 11. The particular transformation of the above group, namely that corresponding to &=0 and accordingly defined by the two equations n n Z2— 24> (4,2 — i) XO) Cause eee le Le Ne (46), 2 1 was first studied as a contact transformation by Goursat, in three dimensions +. If in equations (AG) zee a” he regarded as constants and Zien wosa © Gn was current coordinates, these equations define a certain circle C in n+1-dimensional space, the locus of (z, x, 703),)) | Lhat is) the equations make a circle @ correspond to every point (2’, a’, ..., ,), and similarly, since the equations are symmetrical in both sets of variables, to every point (2, %, ..., @) there corresponds a circle © in the current coordinates (z’, a’, ..., Zn). When the point (z, a, ..., I) describes a surface Sbhe circles C’ relative to the several points of =} form a congruence. The focal surface of this congruence is the surface >’ into which = is transformed. 3’ is also the locus of _ the points (z’, 2,', ..., a,’) such that the corresponding circles C’ are tangent to SN. 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. Vote XeVGk 34 266 Pror. 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’, 2, p/ as functions of z, x;, p; may be obtained without difficulty by the method of § 7. If this trans- formation be combined with those of the one-parameter group (37) we shall have o? 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 transfor- mations 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 in 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,” Comptes 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, ¢ the time, z, y, z the coordinates of the point of the medium which produces the light, a, b, ¢ the reciprocals of the principal indices of refraction, 2, 8, y three arbitrary constants, and X, gw, v three other arbitrary Pror. LOVETT, CONTACT TRANSFORMATIONS AND OPTICS. 267 constants entering only by their mutual ratios, Lévy finds the following 4. * solutions of the proposed problem : CD r, Ou ae u poe “gs ew ae aa & Ga) Ov ite Ou OW gt oa te q ew 7,0°w ow ow Xr Ou we iCal a ay EM aaa soak ferent odie wm fpacbestseeBuconngecta-os = att et ae bam poe Gets ) +h B-a a aay ag Bees One oo Geb) ota-nte dent - Be 18 (Set aye) to On) Sabet s BOO ay Ear eeGeiefe nS te ods © fear Gt) ate- ooo, Pro Sgeo (Bel) hon diste-m 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 no 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 applica- tions 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 Pror. 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 limes 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 Mathematische Annalen which has led him to a generalized form* of the theorem of Malus. Lately this manner of changing straight lines into spheres by contact transforma- tions 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 sphere- coordinates, 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 iiber. Sind bei einer solchen Refraction + Hausdorff, ‘‘ Infinitesimale Abbildungen der Optik,” die beiden in Betracht kommenden Pseudokugeln (d. h. Leipziger Berichte, 1896, pp. 79—130. Wellenflachen) wesentlich 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 Klem’s Vorlesungen iiber das Icosaéder. 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), (i) the tetrahedral group of order 12, and (iii) the icosahedral group of order 60. They are defined abstractly by the relations :— @) ales, Beas (Uap, oo wells (Gi) eels — a2 — sl CAB) (mM w= B=1 (AByoSL 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 sub- group 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 repre- sented 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 (1), (ii) and (i), defined above, are representative. 1. Let G@ be a group of even order N, which contains no operations of even order 270 Pror. 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 If A,A, were an operation of order two, 1, A,, As, and A,A,, would constitute a subgroup of G of order four. No such subgroup can exist, and therefore A,A, is an operation of odd order. The m operations Zl vlen, Jalil. sooo 5 eledal an. 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 AAS GABA SE Absit srl Acs: and since An ASAT pA =A An. A, transforms every operation of G, of odd order, into its inverse. Hence A,A,.A,A,=A,A, = A,A,.A,A,; 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, 7, ..., and if A is an operation of order two such that ASA = S82) AA a5 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 A,A, 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=2"m, 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 CARAT and IBY BS ceses * This first case is considered in my Theory of Groups of Finite Order, pp. 143 and 230. Pror. 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, wu, the subgroup generated by A and B would be a dihedral subgroup of order 2u; and in this sub- group 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 there- fore 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 (7 odd). The two together will generate a self-conjugate subgroup H’ of order 2”. If n’ is less than n, there must be one or more conjugate sets of operations of order two not contained in H’. Let 17 (ORO acy 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 2”. 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 2”. 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 2” subgroups of order m which have no common operations except identity. With the case at present under con- sideration may be combined that im which G has a self-conjugate subgroup of order 2”, 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 2m, where m is equal to or is a factor of 2"—1. They have a self-conjugate subgroup of order 2”, and 2” 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 @ of order two form a single conjugate set, while G@ contains more than one subgroup of order 2”. If H and H’ are two subgroups of @ of order 2”, and if J is the subgroup common to H and 4H’, then since H and H’ are Abelian (their operations being all of order two) every operation of J 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 2”. Hence Z must consist of the identical operation only; or in other words, no two subgroups of order 2” have common operations other * Theory of Groups, p. 228. 272 Pror. 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 2” contained in G must be of the form 2" +1. If K is the greatest subgroup of G which contains a subgroup H, of order 2”, self-conjugately; then A must be a subgroup of the nature of those considered in the preceding section, and its order must be 2", where » 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 2”—1 operations of order two in K therefore form a single conjugate set; and hence » must be equal to 2"—1. The order of G@ is therefore given by N = (2% + 1) 2” (2"— 1). That G must be a simple group is almost obvious. A self-conjugate subgroup of even order must contain all the 2"%+1 subgroups of order 2", 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 2” must be contained self-conjugately in one of order 2"(2"—1). Hence a self-conjugate subgroup of even order necessarily coincides with G. If on the other hand G@ had a self-conjugate subgroup J of odd order r, Z would by the first section be Abelian and every operation of G of order two would transform every operation of J 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 J, 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. 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 J, be a sub- group of G of order 2m, and J, the Abelian subgroup of order m, contained in J;. Every subgroup A of J, is contained self-conjugately in J,; 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 J, must be the greatest subgroup that contains K self-conjugately; as otherwise 2m, would not be the greatest possible order for the subgroups of this type contained in G. Let p* be the highest power of a prime p which divides m,; and let K be a subgroup of J, of order pt. If p* is not the highest power of p which divides NV, then K would be contained self-conjugatelyt+ in some subgroup of G@ of order ps. This has been proved impossible. Hence m, and N/m, are relatively prime. Again no two subgroups conjugate to J, can contain a common operation other than identity; for if they did 4 would not be the greatest subgroup of its type contained in G. If J, and the subgroups conjugate to it do not exhaust all subgroups of @ of order 2u (wu odd), let 2, of order 2m, (m, odd) be chosen among the remaining subgroups of G of * Theory of Groups, p. 98. + Ibid. p. 65. Pror. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. 273 this type so that m, is as great as possible; and let J, be the Abelian subgroup of J, of order m,. Then J, has no operation other than identity in common with J, or with any subgroup conjugate to J,; also no two subgroups conjugate to J, have a common operation other than identity, and m, and N/m, are relatively prime. All these statements may be proved exactly as in the former case. If the subgroups of G of order 2 (« odd) are still not exhausted, a subgroup J, of order 2m, containing an Abelian subgroup J; of order m, may be chosen in the same way as before; and the process may be continued till all subgroups of @ of the type in question are exhausted. Now J, is one of N/2m, conjugate subgroups and each contains m,—1 operations which enter into no other subgroup conjugate to J, or to J, or J;.... | Hence the subgroups conjugate to Jy, J., Js, ... contain N N N Im, (m, = 1) + 2m (mz — 1) + 2m, (ms = 1) ate scl distinct operations other than identity. If Z; actually existed, this number would be equal to or greater than NV, which is impossible. Hence there can at most be only two sets of conjugate subgroups such as J, and J,. It was shewn in section 1 that each of the m,—1 operations of J, other than identity can be represented in m, distinct ways as the product of two operations of order two. Similarly each of the m,—1 operations other than identity of J., if it exists, can be represented as the product of two operations of order two in m, 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 (2% +1) (2"—1) operations of order two, and any one of these is permutable with exactly 2"—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 (2° + 1) (2” — 1) 2% (2" —1) = Nk (2"—1). On the other hand as shewn above this number is T Ve Sm - 1)+ A (m, — 1) N or 2 (m,— 1) according as J, actually exists or does not. Hence if J, does not exist m, = 2k (27—1)+1; and at the same time m, is a factor of (27% + 1)(2"— 1). Vor, VII 35 274 Pror. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. These conditions are obviously inconsistent. Hence J, does exist, and M+ mM, = 2 {k(2"—1)+ 1}. It follows that, m, and m, being positive numbers of which m, is the greater, m >2k+1—k. On the other hand, since no two operations of order two contained in J, are permutable, while @ contains only 2"%&+1 subgroups of order 2”, m,<2°k+1. Hence there must be an integer /, less than k, such that m, = 2" +1 —1, and My = 2"k+1+4+1—2k. Now m, and m, are relatively prime factors of (2° +1) (2"—1). Hence (27k + 1)? — 2k (2 + 1) + 2hl —P < (2"k + 1) (2"— 1), and @ fortiori since I is less than k, and 2"k+1 is positive, 2k&+1—2k< 2-1, i.e. kel. The group @ can therefore only exist if & is unity, and this necessarily involves that l is zero. Hence N= (2"+1) 27(2"-1), m=2"+1, m=2"—1, and these are the only values of NV, m,, and m, consistent with the existence of a group @ having the required property. Since G@ is simple, it can be represented as a substitution group of degree 2"+1. The subgroup of degree 2", which leaves one symbol unchanged, has a self-conjugate Abelian subgroup of order 2”, and 2” 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 @ can be represented as a triply-transitive group of degree 2”+1. The Abelian subgroup of order 2”—1 which transforms a subgroup of degree 2” is shewn in an appended note to be cyclical. Assuming for the present this result, the subgroups of G@ of order 2”(2"—1) are doubly-transitive groups of known type. Now @ contains just 2"—1 operations of order two which transform each operation of a cyclical subgroup of degree 2"—1 into its inverse. Since each of these leaves only one symbol unchanged, each must interchange the two symbols left unaltered Pror. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. 275 by the cyclical subgroup of order 2”—1. But there are only just exactly 2"-1 substitutions of order two in the 2”+4+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 mn is known*. In fact the system of congruences (mod. 2), where a, 8, y, 6 are roots of the congruence A1=1, (mod. 2), such that ad— By+=0, (mod. 2), actually define such a group; and the permutations of the 2”+1 symbols CA UE ON Mastodon AP oer where X is a primitive root of w"=1, (mod. 2), which are effected by the above system of congruences, actually represent it as a triply- transitive group of degree 2”+4 1. The set of groups thus arrived at are the analogues of group (111) 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 2” 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 p? operations of H are concerned the two isomorphisms, being permutable, must have the form (CARA A.) (Ala tAl at Airc (Amant As.\. and (Cel AS) (AAS A ee (Ale Aley eA): * 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 Pror. BURNSIDE, ON A CLASS OF GROUPS OF FINITE ORDER. being the p? operations. Moreover any cycle of an isomorphism generated by these two has the form CAR s Aleta Sty» ceeeee Ay+(p—) a, tt(pony)> 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 one of the cycles must give the identical operation. Hence Jeleviley Sosos Ylo Sil Jiliegalseocee Jil Sails Jaliedéles Gosade Ane eee eee eee ee and therefore on multiplication AP = ls or rar —oe The supposition made therefore leads to a contradiction. Hence if H admits a group of isomorphisms of order p™”, 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 p™ is the highest power of p which divides 2"—1, the subgroup of order p™ in the Abelian group of order 2”—1, considered above, is cyclical. Hence the Abelian group is itself cyclical. * Theory of Groups, p. 73. XIL On Green's Function for a Circular Disc, with applications to Electro- static Problems. By E. W. Hopson, 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 dise 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 co- ordinates, 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 circle in A and B; the coordinates of P are then taken to be pale ae. 6 which is the angle APB, and ¢@ 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 @ shall be restricted to have values between —7m and 7, a discontinuity in the value of @ arising as we pass through the circle, so that at points within the circumference of the circle, @ is equal to 7, on the upper side of the circle, and to —7 on the lower side of the circle, the value of @ being zero at all points in the plane of the circle which are outside its circumference. As * Theorie der Elektricitiéts- und Wiirme-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, @ is positive and increases from 0 to 7, whereas as P moves from an infinite distance along a line below the plane of the Fie. 1. circle up to a point within the circumference, 6 is negative, and changes from 0 to —a. The coordinate ¢ is restricted to have values between 0 and 27, and the co- ordinate p may have any value from —x2 to +%, 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 & 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 Ecos, Esing, z will be a system of rectangular coordinates, which can of course be expressed in terms of p, 6, ¢. Let the lengths PA, PB be denoted by 7, r’ respec- tively, then r/r’ =log p; we have Qrr’ cos = 7? + 7? — 4a? = 2rr’ cosh p — 4a?, F 2a? hence hh cosh p — cos @ Again, z.2a=rr' sin 6, ansaa hence ee cosh p — cos @ also since r+ 7? = 2a? + 2CP*, we have CP? =rr cos 6+, = WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 279 whence we find CP po ae = azisinhzon uoree "= (cosh p— cos OF thus &, z are expressed in terms of p, @ by means of the formulae _ __asinh p peal GSU Ojos ~ coshp—cos@’ ~~ cosh p—cos@° E 2. To express the reciprocal of the distance D between two points (p, 6, ¢) and (po, 9, Po), we substitute for & z and &,, z, im the expression pa ena + B+ BP BEE, 008 ($ - 4) their values in terms of p, @ and p., @; we then find 1 1 (cosh p — cos 0)* (cosh py — cos 0) IDS BNO {cosh a — cos (8 — 8,)}# ; where cosh « denotes the expression cosh p cosh p, — sinh p sinh p, cos (6 — ¢,). If we suppose the expression {cosh a—cos(@—6@,);*? is expanded in cosines of multiples of : ; sayreoe fim cos mr @—6,, the coefficient of cos m(@— @,) is as) (Ge acna dyy which is equal*. to Bue Qm—+4(cosha) when Q,-; denotes the zonal harmonic of the second kind, of degree m — ; thus a= = (cosh p — cos @)} (cosh p, — cos A)? = 2,4 (cosh a) cos m (A — @,), where the factor 2 is omitted in the first term, for which m=0. The series in this expres- sion for 1/D may be summed, by substituting for Q,-;(cosha) the expression 1 i eau : Y; V2/, (cosh u— cosh ar Mie) cs pol) we find Ue te, (cosh p — cos 6)! (cosh p, — cos 4,)* Ir =e {1 + 23e—” cos m (6 — 0,)} du D™ wraNv2 gf h J _ (cosh u— cosh a)! ‘ y ; and thus we have the formula 1 “Aa le 1 sinh == = (cosh p — cos 0)? (cosh py — cos A)? ———— du, D wav? (Coppi cs Ok fe I, Vcosh uw — cosh a cosh wu — cos (6 — 0,) where @ is given by cosh a = cosh p cosh p, — sinh psinh p, cos (¢ — ¢,). * See page 521 of my memoir ‘On a type of spherical harmonics of unrestricted degree, order, and argument,” Phil. Trans. Vol. cyxxxvu. (1896) A. 280 Dr HOBSON, ON GREEN’S FUNCTION FOR A CIRCULAR DISC, — GREEN’S FUNCTION FOR THE CIRCULAR DIsc. 4 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 @ lies between —w and 7; for the second space we shall suppose that @ lies between 7 and 37, 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 @ increasing continuously through the value m7, and becoming greater than 7 in the second space. In order that a point P starting from a position P,(p), , ¢), say on the positive side of the disc, may after passing through the disc get back to the original position P,, 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,, @, ¢) where @, is between —7 and 7, is the point (p), @.+27, ¢) in the second space, whereas the point (p), @+47, ¢o) is regarded as identical with the poimt (p, @, ¢.). 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 po, ®, gd, be the coordinates of a point P in the first space, on the positive side of the disc, thus 0<6,<7; taking the expression for the reciprocal of the distance of a point Q (p, 0, ¢) from P, given in the last article, we have, since ee ik cecal sinh x Jigl sala Seo . 1 sinh 5 coshu—cos(@—@,) 2 cosh 5 u — COs (6 — 6) 2 Pe a 4 fete 2 (0-6) 1 l = l sinh 3 u —. = =—>— (cosh p— cos @)' (cosh p, — cos a | ne (]/ ¢ /9 PQ” 24/2:ra a Vcosh w— cosh @ oosh 2 ras cos 5 (6 — 6.) need 1 53 l sinh 5 u aa a AN aah ee i. te 4 ; +5 ome (cosh p — cos 6)! (cosh p, — cos 8,) i inne du; sh u — cosh a cosh 5 u— cos 5 (0 0, — 27) we thus see that 1/PQ is expressed as the sum of two functions, the first of which involves the coordinates 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. xxvm. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS, 281 coordinates p,, 0,+27, , of the point P’ in the second space, which corresponds to P. If @ moves up to and ultimately coincides with P, we have cosha=1; 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, %, bo) given by Ween ye aE (cosh p — cos 6)! (cosh py — cos 0,)4 Ta eel - 1 sinh 5 | VET =hG le ee ees a ee me coon cos (610) the above equation may be written 1 PQ- W (po, a, go) + W (pos A+ 27, dy). It is clear that the function W is uniform in our double space as it is unaltered by increasing @ by 4s; it will now be shewn that it is a potential function. We may express W in the form 1 a 1 m W= : cosh p — cos 6)? (cosh p, — cos 6,)2 | a {1 2>e—™ cos—(6 — 4, \ du, 2 es a y Be ) Ja Vcosh u — cosh a id me 3 , i which may be written in the form W= = (cosh p — cos 6) (cosh p, — cos A)! {Qs (cosh a) + 2 5S Qm 1 (cosh @) cos = (0— a} : since the formula en (n+4) uU V2 Qn (cosh ae ih (cosh u — cosh a)? du, holds for all values of n such that the real part of n+4 is positive (loc. cit. p. 519). Now (cosh p — cos 6)! (cosh Po — COS )4 cos s (8 — A) Qs 4 (cosh a) is a potential form whatever s may be, and thus W is a potential function, and is expressible in the form W= es (cosh p — cos )} (cosh p, — cos 6,)! |@ (cosh a) + 2Q, (cosh a) cos : (9-6) + 2; (cosh a) cos (0 — @,) + a ; the value of W(p,, @,+ 27, dy) being = (cosh p — cos 6)! (cosh Po — cos 0,)# 04 (cosh a) — 2Q, (cosh a) cos : (0—6,) + 2Q_; (cosh a) cos (8 — 0,) — +} ; the two expressions added together give the expansion of 1/D obtained in Art, 2. Vou. XVIII. 36 282 Dr HOBSON, ON GREEN’S FUNCTION FOR A CIRCULAR DISC, ae Sn : ; : 1 4. To evaluate the definite integral in the expression for W, write cosh ;u=<, cosh - =o, Cos : (@—6,)=7, then sinh u (i pill. sinaee du =2 | lei Ja Vcosh u — cosh a aoa x eos (6 — 6) Jo Vat— o?(@—7) 42 @ ee z) A Gti? NZ) ayia where the inverse circular function has its numerically least value; we thus obtain the expression 1 (cosh p—cos 0)! (cosh py — cos)! [a | sy Ved 6 h i t] WaT {cosh a — cos (6 — 0,)}* 7s eo Eee 3 ( 2 which may also be written in the form SNe | aah th Men had p i 1] W = PO E on [eos 5 (@ —8,) sech 3% | nusatseanns te ected (1). This expression W has the following properties:—it is, together with its differential coefficients, finite and continuous for all values of p, @, @ 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 : : : 7 . to P, the inverse circular function approaches —>, and the function does not become 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(p., %, ¢)) in the first space on the positive side of the disc (0<@,<7), we take the function W(p,, , do)—W(p., 2r—@, do) which is the potential for the double space due to the point P and its image P’(p,, 27—@, do), 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 6)! (cosh py — cos 6)! maN2 {cosh a — cos (6 — 6,)|4 1 (cosh p—cos 8)! (cosh py + cos 0)! [ar . _, il 1 t] Sane (eclukaasiecs @OERCTE zt sin {- cos 5 (8+ 6) sech 5 a} |, E + sin {oos XG — 6,) sech : a | { SERA] WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 283 which is the same thing as U= £6 E - * sin {eos : (@ — 6,) sech : a | - PO E = * sin . cos 5 (6 + @,) sech : a Secret anceeet ( where P’ is the optical image of P in the disc. On putting in this expression (2), for U, the values 0=7, @=-—7, and remembering that over the disc PQ=P’Q, we verify at once that U vanishes on both surfaces of the disc. If @ coincides with the point (po, —%, $) the function U remains finite. The Green’s function Gpg 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 Ga — U, hence the PQ required value of G ) is Gpg= a Eee a sin 105 = *(0- @,) sech oa a} |+p : ab + - sin- |- cos ; (@+ @,)sech : al | = Fo'= L cos? feos (@—86,) cane 9% Po-= —cos |eos 5 =(0+86 ») ech 5 54 eiey seas als (3), the numerically smallest values, as on 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 Poin’. 6. If we suppose a thin conducting disc to be placed in the position of the funda- mental circle of the coordinate system, to be connected to earth, and influenced by a charge q at the point P(p, 0, ¢)) on the positive side, the potential of the system at any point Q is qU where U is given by (2), and the potential of the charge on the dise is —q.@pe. We shall now throw these potentials into a more geometrical form. We have 1 cos 5 (6 — 6) ) sin— = Joos 5 (@—- @,)sech 5 a =tan— yas | » | a/ cosh: 5 a— cos? 5 (@—@,) Nicos Can cy) = tan Cam f now take an auxiliary point Z, of which the coordinates are p,, 07, qd, the upper or lower sign being taken according as @ is positive or negative (-7<@<7). Thus L and Q are always on opposite sides of the disc; using the formulae of Art. 1, we find — 2a* cos 0 — 2a? cos 0 gee ae Z P= (CH= —— = Core cosh p, + cos 6 ’ a Ge cosh p—cos 0’ Ley 1 + cos (4 — 0) i cosh p — cos 6)2 cosh a — cos(@—@,)) cosh Po +cos Of ’ PQ” 36—2 284 Dr HOBSON, ON GREEN’S FUNCTION FOR A CIRCULAR DISC, hence ? sin7 jeos : (@—8@,) sech 2 a} =+tan™ (Fo tis) Fie. 2. in order to determine the sign on the right-hand side, we observe that the inverse sine is positive unless @ lies between —(7—@,) and —7, that is unless Q lies within the sphere passing through P and the rim of the disc, and is on the negative side of the dise; thus the sign on the right-hand side is to be taken positive unless Q lies within this spherical segment. Similarly we find ; 1 1 = P'L /@-Ce eh ee 2 me = 9ay—l v sin | cos 5 (0 +8) sech 5a} + tan Gaza a) where the negative sign is to be taken unless Q is on the positive side of the disc 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, ¢) a — CQ) q a Ph {POG OL? — =| ~2P@ E +7 ig Cr 2) em when the ambiguous signs are assigned in accordance with the above rules. The auxiliary point Z may be found from the following construction : Draw a spherical bowl through the rim of the disc on the opposite side to that on which Q lies, and equal to a similar bowl which passes through Q; draw a plane PA’B’ through P and the axis, cutting the rim in 4d’, B’; this plane intersects the bowl in a circle; on this circle Z lies, and is found by taking it so as to satisfy the relation TAS GBI = AY) PB. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 285 In the case in which the influencing point is on the axis of the disc, we have p, =0, hence a=p, and the auxiliary point Z 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 dise to Q; the formulae for the potential then become V=39 E + = sin“ jeos : (@— 6@,) sech : Pt | - PO E + E sin - cos : (@ + @,) sech ; P| ee Seite CO Ge (EL fat COs ae IPO E t= tan ep oa 2P'Q 1 Te ban ul \PQ res) soon (8) 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 —g¢.@pg given by (3), let 6=0, p=0, then a=po, 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 9 == “50 cos} (cos 5 sech : pa) : therefore the whole charge on the dise is . —1 —q-— cos (cos which is equivalent to when Z is a point in the plane of the disc which lies on the bisector of the angle APB. This expression may be interpreted thus :— Fic. 3. Let PL be the bisector of the angle APB, draw the chord NZM perpendicular to AB; the total induced charge is 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 brasil where @, Tv is the angle subtended at P by a diameter of the disc. When P is in the plane of the disc, the angle VPM becomes the angle between the tangents from P to the circular boundary of the disc. 8. The surface density at any point of the dise is given by the formula Leah paar ov’ when dy is an element of normal and is given by + adé ”~ cosh p — cos 0° We thus find for the density p, at the point (p, 7, @) on the positive side of the disc, Die. ; 1 q z 2 += sin“ (sin 78, sech >t 7 2 4 Po de PO ty q 1 coshp +1 did Oe ; Tao! Tce, > ei es rE, a/ cosh? 5a sin’ 56, this expression can be put into a more geometrical form by introducing the auxiliary point Z (p,, 9-7, ¢,) of Art. 6. The point Z is now in the plane of the disc, and external to the disc; denoting this position of Z by Z,, its coordinates are p,, 0, ¢,. We have Rae (aoe 1 ye. /PL, - /@=CR sins (sin 2% sech 54) =a helriee \PQ CL2—a:) 5 Sane é a Bey se) fe ae tales which is equal to 3 7 tan Sat a—C@ ) ; on reducing the second term in the expression for p, remembering that asin 0, z= ——— cosh py — cos 6,’ ey (Es Wie JEG IED; a? — CQ’ and thus the expression for the density at any point Q on the positive side of the dise is given by we find that it becomes nese pai PN (J th hee eG: = Po 97° PQ Ie’ PIPL, @—0@ =" \Pr,V a— OG ’), PN 9 PN(PQ /CLi—@ /PQ nee a where PN is the perpendicular from P to the plane of the disc, and Z, is a point on AB produced, such that AL, : BL,=AP : BP. WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 287 The value p, of the density at the point (p, —7, @) on the negative side of the disc is found in a similar manner to be eg ENCE CL? — a? _ as Wess PQ Oke ere ah) Pa ~ art IPG PL. a? — C2 UPT; CQ Thus the densities at corresponding points on opposite faces of the disc satisfy the relation +t eo foe Oe aeons When P is on the axis of the disc, Z, is at infinity, and the formulae (7), (8) become Gq) URINE sag Te PQ ~ tan ( PQ )t po Ie PQ 20° PP Wag. BO VAQ. BQ = Gf PN PQ 2s PQ p= — ce ae aes Can aT ast (9). The expressions (7), (8), (8) agree with those obtained by another method by Lord Kelvin*. When P is in the plane of the disc it coincides with Z£,; in this case we find that the density on either side of the disc is given by a yey) ee CP? — a p=—5 PON Geos irre eeeeeeeeoceensnnneeee (10). 9. If the influencing point P is on the axis at @,, we find from (5) the following expressions for the potential at points on the axis:—On the positive side of the disc ¢ PQ7 te PQ O- %)~ a5 py (9+) when @>6,, eile q PQ' 27 159 (6- 8) — On PQ? t )» when 0<4@,. On the negative side of the axis Qt = "9 8- 0) + 5" pO im PO (9+96,), when @+ 4, is positive, PQt i Poe 0) — PO (0+ 6), when 6+, is negative. If we denote by 2 the distance of P from the disc, and by z the absolute value of the distance from the dise of a point @ on the negative side of the disc, the potential at @ is given by the expression q q ( —; 20) q hetpnbs 2 Z ao ae cot = cot Diet CObm |) Z2+2 w(z+%) 7 iy a 1 (2)—2) (0 a coe a * See his papers on “ Electrostatics and Magnetism,” p. 190. 288 Dr HOBSON, ON GREEN’S FUNCTION FOR A CIRCULAR DISC, if z, be given as a multiple of a, say z,=na, the expression g / z ( Zz oe aetna) (Org toot n) Tas) (hi Ee al ae n) might be used to tabulate the values of the potential at points on the negative side of the axis. When z=0, z= 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 disc; 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. 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 1 yemains finite, say equal PQ THE ELECTRIFICATION INDUCED ON A DISC PLACED IN ANY FIELD OF FORCE. to —A. We can easily shew that ‘ 1 1 Ge a 2a sin? = = = — — YY sin (cos 3 @ sech 3 p) 3 — sin (5 =) : where 7,, 7 are the greatest and least distances of the point (6, ¢,p) from the circular rim of the disc. We thus find for the potential of the electricity on the disc, the € < sin , which is the potential of an insulated disc elec- ui 1 trified freely to potential A. well-known expression 11. To find the potential due to the charge on the dise when placed in a field of force of potential jaw, when a 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(p., 0, 0), P’(—po, 0, 0) on the axis of x; the potential of the charge induced by these on the disc is at any point (p, 0, ¢) 1 a il 2 C08 5 (] 2q 1 be COS 5 (] mee pg 7 he Pore ; cosh 3% cosh 5 a’ where cosh a = cosh p cosh p, — sinh p sinh p, cos ¢, cosh @’ = cosh p cosh p, + sinh p sinh pcos ¢ ; WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 289 now let p, 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, 1 1 aie | ag! = ) cos cosh 5 p 3 008 5 @sinh 5 p cos == (Ap tS=) 4cos— SS 0 OeQqQQQYQYQYYe ; Y 1 D2 1 1 2 1 ene Oe | cosh gP a/ cosh? : p- cos" 5 6 cosh? QP | seal 5 cos Lg egiey * cos x sinh + pcos 1 x 2 2 2 2 2 +4 (ap - Ge) {cos Sa ees 7 ee SS hee si cosh 5P a/ cosh? 5P— cos" 5 r) cosh? 5 p asinhp, — 2a : : : 2q now CP=———_°’ _=~" hence if q be made indefinitely great so that Cp = we coshp,—1 py find for the required potential Lame 5 1 1 cos 5 @ sinh p —— p+ cos (cos 5 @sech 5 p)—a cos ¢. i. = ae is | r v2 cosh? = p Veosh p — cos | 2a _ Vcosh p— cos 0 a sinh p now =—_—————-, © = 00s co) +1, cosh p — cos 8’ V2 cosh : p hence we find that the potential due to the induced electricity, in a field of force of potential yz, is 2 : 2a 2a V(r, + 7? — 4a?) == pa 4sin — SAT ama | NOC DUO IECCCEEA RCo 11). cae | M+ 1% (7 + 72)? J oS) 12. In order to find an expression for the potential of the induced electricity on the disc, when it is placed in a given field of force, we apply the well-known theorem that if o 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 @, the potential function at Q which has values given at every point of the conductor is [VoedS, the integration being taken over the whole surface of the conductor. Suppose V(p, ¢) to be the given potential function at the element p, @ whose area we denote by dS, on either side of the disc; the potential function at the point (py, Oy, ¢.) external to the disc which on the dise takes the values V(p,$) is then, using the expressions found in Art, 8, 1 (1+ cosh p) cos = 8, ——____ 1 2 Zz l—cos@ Dea ———————— SSE tan ee a V 4 ds, 27° Ra my rail ™ Re Vcosh a+ cos 6, oy | cosh? 5 %4— sin? 5 0, Vout. XVIII. 37 290 Dr HOBSON, ON GREEN’S FUNCTION FOR A CIRCULAR DISC, the value of the required function, R denoting the distance PQ. We now introduce new coordinates r, 7, @ instead of p, 6, ¢, these being given by c= J/r+a?sinncosd, y=J/r+a?sinnsing, z=rcosn; to express 7 in terms of p, 0, we have 2 e+y= (7+ a) sin?n = ( =a a?) sin? 7, cos* CQ2— a 2 hence cos‘ + od cos? n =S¥ where CQ?=a°+ y°+ 2°; hence we have 2 VCQ?—a? ~ V(CQ@ — a?) + 4022 cos? 7 = — Oa Da? Ree. cosh p—cos 0’ 2a? cos 8 N gg ee ee ie cosh p — cos 6’ and it is easily found that V(CQ? — a®)?+ 4a%z? ae ay) I —cos 6 1M cosh p—cos 6” and therefore cos m=, / Scola cosh p,— cos 0, hence we have Also as P is on the plane of the disc (r=0), we have CP=asinyn, hence e&= ee , ay, 7) from which we find 1+ cosh p=2sec?y. Remembering that in del V1+ cosh p Veosh p, — cos 0, R av2 cosh a + cos 8, : 1— cos 0, @ COS 7H COS Hy oe V cook a+cos 0, R ' and also hp) cos £8 hh tise ere (1 + cos p) C08 5 Mi ree + cos pices tet ND} : Qe R° Vcosh py — cos 0, R* cosy’ a V2 cos m aR cos 7 cos ” ac ah a/ cosh? 5 asin? 58, then since dS=a?sinyncosndndd, we have for the potential function at an external point 7%, , $, which has the value V(y, ¢) at the point 7, @ of the disc, the expression ¥ Zz folie { @COSN COS], _ @ COS 7) COS \) é eral Res Vm, #44 ae Sarr) a tan ee dnd eecces (12); here the coordinates of the external point at which the potential is found are the elliptic coordinates given by Z=7,Ccosm, 2=Vre+a?sinycosd, y=Vr2+a?sin m sin do, the coordinate 7) 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. 11. 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 funda- mental circle for its rim. If the angle of the bowl is 8, we must suppose that in the first space @ has values from @—2z7, 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, @ increases from 8 up to 8+27, when the positive side of the bowl has again been reached. If the convexity of the bowl is upwards, @ is less than 7; if down- wards, 8 is greater than 7. The image of a point P(p,, @, ¢$,) in the first space and above the bowl is the point P’(p,, 28—@,, ¢) in the second space, and below the bowl. The expression 1g ele ely i! 1 UG [ rio sin {eos 5 (@- @,) sech 3 at | cosh p, — cos 6, 1 a a eae 1 5 1 fi = hes a= 008 (28—6,) PO E +7 sin {cos 5 (0+ 8 28) sech >t| Sade (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 Gp, is therefore given by the formula Cope = cos— {eos : (@—8@,) sech 5 a} + V aS or 8.) Fy E +> sin Joos (6 + 0,— 28) sech 3 a\| S900 (14). By introducing an auxiliary point LZ whose coordinates are p,, 9+, qo, 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 disc. 37—2 XIII. Demonstration of Green’s Formula for Electric Density near the Vertex of a Right Cone. By H. M. Macponatp, M.A., Fellow of Clare College. [Received 13 October 1899.] In a footnote in his Hssay on Electricity Green makes the following statement*: “Since this was written, I have obtained formule 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 r”*; r being the distance Op, and the exponent n very nearly such as would satisfy the simple equation (4n+2)8=37; where 28 is the angle at the summit of the cone. If 28 exceeds 7, this summit is directed outwards, and when the excess is not very considerable, n will be given as above: but 2¢ still increasing, until it becomes 27—2y, the angle 2y at the summit of the cone which is now directed outwards, being very small, n will be given by 2n log = 1. 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 nm in terms of « for which the harmonic P”() vanishes. * Green, Essay on Electricity and Magnetism, 1828; + I have been unable to obtain Mehler’s paper con- Mathematical Papers, p. 67. taining the results for the cone and have had to rely on + Green’s statement is quoted and applied by Max- Heine’s account of it, Theorie der Kugelfunctionen, Vol. 11. 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, ete. 293 § 1. Green's case. With the usual notation, the expression V,— Ar” P,(y) 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 P,,(«) must uot vanish for any value of @ between a and 7; 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 semi- vertical angle is a Hence n must be such that P, (w) does not vanish for a value of @ which is greater than a; now the kth zero of P,(m) considered as a function of n diminishes as @ increases*, therefore n must be the least zero of P,, (cosa). 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 V,, is V,— Ar”P, (u), where 7 is the least zero of P,(cosa) and A is a constant depending on the form and size of the surface. Hence+ the density of the distribution im the neighbourhood of the vertex of the cone varies as r®, where r is the distance from the vertex and n is given by n=«x,/a« where a is the least zero of J,(#), when @ is small, by (4n+2)a=37, when a is nearly 7/2, and by 2n eg sae when a is nearly 7 and t—a=y. Thus Green's results 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 6=a, where r, 0, ¢ are polar coordinates, and let there be a charge g at the poimt r=7", @=0. The conditions to be satisfied by the potential are V=0, when r=a and a>6@>0, V =0, when r= 0) and a>6>0, V=0, when 0=a anda>r>b, a Viz ovale | oV) and = a Ge Gp Ton (1 = p’) Bill + 4p =0 throughout the space. Put r=ae™, then the equation to be satisfied by V becomes Gy Oe 1G \\ RECNG ore Dina tae) ga + Aaa p=0; * Macdonald, “On the zeros of the harmonic P,,”(u) considered as a function of z,” Proc. Lond. Math. Soc. 1899. + Loc. cit. 294 Mr MACDONALD, DEMONSTRATION OF GREEN’S FORMULA A and, writing V = Ue?, U has to satisfy the equation 5A Ul Uineed ee) ser apa te men et 1-H Go| + trae p=0, with the same boundary conditions as V. Assume oa 5 » TS We sin : a Ao where No= log Fs this satisfies the first two boundary conditions and will be the solution required if W,, can be determined to satisfy the conditions W,»,=0, when 6=a2 and a>r>b, mmr -3 + 47a’e ~p=0, and also Pa) OW. men? 1 : hl ee a ) Ved = [2 fom" - (Cadre) em that is r) OWm mr 1 Sra? (As) Sein a=) at (= +3) Wat ze | pe 2sin r, dy = 0. Assuming VWe= DAtm ln (1), all the conditions are satisfied if this summation extends to all the values of m which make P,(cosa) vanish and A,,, is determined so that LOE ein ne a UL > = —- =—— —— ZAnm {(n4+5) SRE j Pa@) Se 2 sin dx, that is, if ( A\ ys ania i Ne =F ff -» . mmr Ann | nts) +5 _(Palw)itdu=S™ | | pe 2 Pa (w)sin So dnd where »,=cosa. Now u 2 = ih bo OPn (p) =I [ew du=- 5" | on On pie tt therefore 2 2 0 5A 1 =e OP 0Pn_ _ 87a ik hi nes (u) ne oa No ~ bo 0 Xo / 1 m7 Ae {(n 3 5) ae ie | Making p vanish except at the point r=7r’, 0=0, where q=— 27pa° edn’ dy’, In+1 on op FOR ELECTRIC DENSITY NEAR THE VERTEX OF A RIGHT CONE. 295 the expression for V becomes mr . mmr se (2n+ 1) sm—— sin —— P, (un) Va neo ee EEE! ANy 1 ; oP, oP. IN ne |) 1 =& 2 n n ( , ) Di re Cais i seo) anes effecting the summation with respect to m, this becomes At’ cosh (2 + Z Ny —A+NX’) — cosh ( n + u (Ay — A — 2’) V. 2ge 2 s! 2 2) P ue sinh (» at 5) Ao (1 — p42) OPn OP n n(H) or Dy one ais when A>2X, and AtX" cosh (» aE =| (XA) — Xr’ + X) — cosh (» sy 5) Osi — 70) 2) 2 / 2 Pa age s / ee: h( Ag ae ee ke ee. sin nv 5) 0 Mo Ar Oe when X<2. Making »,=0 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 gq at the point r’=ae-* on the axis is given by “x +4) (A-A’) ( (A 2ge 2 — e-(MtH(A-N) __ p—(m+4) (AtN) Vases sy $$ Pur) a 1 2) ot n OP ela On Op when A>2, and by MeN } rn 2¢0e o- (n+) (A-A’) _ e— (m+) (A+A’) Faces > —___—— Pu) a a 2) JPSOR= eo) On Op when X<2X’, that is by ete 2 x ( yr ie Te) Je. (“) Sot yp nt qent when 7’ >r, and by y’n pry n TP, (“) =_— 9 5 Sait eae n+ ; fa V q (a a} Gas) OP n OP n BAT Op when r>r’. To obtain the potential in Mehler's case when the cone extends to infinity put @=o and then YatgS 2 lS nn a ee 2) OP ny OPn ? on Om 296 Mr MACDONALD, DEMONSTRATION OF GREEN’S FORMULA when 7’ >r, and nr 12 (uw) se Sb rs » OP OP’ Po) On Om when + >7’, where the summations extend to all the positive values of n which make P,, (cos a) vanish, When a=7/2 OE Os £5 On Cu : an and V= 2q> pn IE, (#), when 7’ >r, where the summation extends to all the positive odd integers, that is Vs g = q = a7 -\ Vr +r2—2rr'cosO V22+ 7 + Ir’ cos O 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 & be the angle in the segment of the circle whose revolution describes the spindle, € the angle in = 105 any other segment of a circle on the same chord, n=log—, where 7,, 2 (71>) are 2 the distances of a point on a segment from the extremities of the chord; then putting q=-—V,r' and observing that the cone of angle & in the dielectric inverts into the spindle the generating segment of which contains an angle &, the potential at any point due to the spindle when charged to potential V, is given by eh P,, (cos €) OPn aE | ; (Ls) E Om M=Ho V=V,+2V, V2 (cosh n — cos €) = where ,=cos &, and the summation extends to all the positive values of » which make P,,(cos &) vanish. The case of the sphere is that when £=7/2. 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 V, if d Me e— (n+h) 7 — Bap? (2 (cosh 9 — cos &,)}? = ———a5 ; sin &, a FOR ELECTRIC DENSITY NEAR THE VERTEX OF A RIGHT CONE. 297 and near one of the conical points this becomes Va yr 1 Dire gee eile where 7’ is the length of the axis of the spindle, 7 the distance of the point on the surface from the conical point and n is the least zero of P,(cos&). Now when &, equal i WD, 7 : to m—y, is nearly 7, aa er ae and the values of n which occur are k+m, where as k is any positive integer and 2n,log-=1*; on substitution and summation, the ex- Vomr’ pression for the density at any point becomes — * Loe, cit., Proc. Lond, Math, Soc. 1899. WO, SS\A00L 38 XIV. On the Effects of Dilution, Temperature, and other circumstances, on the Absorption Spectra of Solutions of Didymium and Erbium Salts. By G. D. Liverne, 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 didy- mium and erbium in various degrees of dilution. Since then most of the observations have been repeated with improved apparatus, whereby several anomalies in the photo- graphs 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 preliminary 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 spectro- scope, 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 spectro- scope 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 2427, 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 half- prisms, all of calcite, the object glasses of the telescopes were quartz lenses of 18% inches focal length for the sodium yellow light. The photographic plate was of course Pror. LIVEING, ON THE EFFECTS OF DILUTION, TEMPERATURE, erc. 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 Jens 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 hight 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 cireular 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 fillmg the tube, and to allow of expansion of the liquid when it was heated. Tubes of four lengths in geometrical progression, namely of 38mm., 76mm., 152°5 mm., and 305 mm., and a cell with quartz faces having an interval of 67mm. between them, were used to hold the solutions; and for a few observations a cell of only 5mm. 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 tem- perature of the room; the trough was then heated by one or more gas lamps until the water boiled, the gas lamps 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 Pror. 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 con- sequent 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 prism, 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 2370, 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 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 sensi- tiveness of the photographic 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 same 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 com- pared 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 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 spectrum. 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. lanthanum, but it had not been found possible to get it, or the erbium, so free from The didymium was spectroscopically free from yttrium *. there is no method at present known for separating the various metals of which ordinary erbium is Indeed for advantage in doing so; though for a quantitative estimation of the concentration of No attempt was made to separate the neodymium from the praseodymium, and supposed to be a mixture. my purpose there would be no 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 4629 grams of anhydrous chloride+. These each contain 1862 gram-molecules of the salt * Lanthanum and yttrium cannot be recognised by any absorption bands, but when induction sparks are taken of three sets in the green and citron of which the brightest begin at 15599 and 45380 respectively, and the third at from solutions of their salts, each gives a very character- istic channelled spectrum, by which it is easily recognised in a solution containing one per cent., or even less, of the salt. The yttrium channellings are in the orange, the brightest of those of lanthanum in the citron and green, and both fade towards the red. Thalén in his paper (1874) on the Spectra of Yttrium and Erbium, and of Didymium and Lanthanum, gives the wave-lengths of the sharp, more refrangible edges of the yttrium channellings, one set beginning at \6131 and the other at \5970°5. He does not give those due to lanthanum. These I find to consist \5173. There is another weaker set in the orange beginning at 15865, and two sets in the indigo beginning at \4419 and 44370 respectively. My measures were not made with any large dispersion and the last figure of the measured wave- length may not be quite correct, but near enough for recog- nition of the channellings which are easily seen with a small spectroscope, especially the two first mentioned. + The (crystalline) didymium chloride in this solution was dissolved in just about twice its weight of water; the equivalent solution of nitrate had still less water. 302 Pror. 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 concen- trated 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 7266 grams. These each contain 2°67 gram- molecules 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 2679. A weak band at about A 623. A rather weak band at about 596. The strong group extending from about 2590 to 2570, 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 528 to X 520. A less strong group of two diffuse bands with the centre about 2510. ABSORPTION SPECTRA OF SOLUTIONS OF DIDYMIUM AND ERBIUM SALTS. 303 A well marked triplet at about 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 4462, and extending nearly down to the most refrangible band of the triplet above mentioned. A very broad band with its centre about 444. A very weak band i * NAS: A strong, narrow, sharply-defined band at about \ 427. A very weak diffuse band with its centre about 418. A still weaker one with its centre about 415. Another weak diffuse band at about 2» 406. A very broad strong band with its centre about 403. A very weak diffuse band at about 391. A diffuse band at about 380. Another, wider, at about 375. A weaker band at about 364. Four, nearly equally distributed between 2358 and 2350, which in all but the weakest solutions run into one broad band extending beyond the above-mentioned limits. A weak diffuse band at about 338. And a broad diffuse band with its centre about 2» 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 retam so much the same 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 hydro- chloric 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 ) 653. A group of four, of which the more refrangible two are much stronger than the others, lying between 536 and 2 549. 304 Pror. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE A weak band at about 2 527. A very strong one at about 4523. A weaker one at about 520. A rather broad band, strongest on its more refrangible side and fading towards the less refrangible, with its strongest part at about 491. A strong band at about 2 488. A weaker one at about 486. A broad but weak band with its centre about 472. A sharp but weak band at about 2 467. A broad, diffuse band with centre about 454, reaching almost up to a stronger, and narrower, band at about 2449. These two are merged into one with concentrated solutions. A weak band at about 2 441. A narrow one at about A 422. A weak one at about 418. A broad band, fading on its less refrangible side, and extending from about » 415 nearly down to the band at 418. A pair of nearly equal bands, rather strong, at about \ 404 and » 407. A very faint but broad band extending from about 1396 to » 402. A well-marked, rather narrow band at about 379, 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 2377, in solu- tions a little stronger. A weak diffuse band with centre about » 367. A strong band at about 2365, accompanied by One rather less strong at about 2363, which become merged together when the solution is rather stronger. A band rather weaker than the last at about 2357, and A broad weaker band with centre at about 4353, which soon merges in the former when the solution is a little increased in strength. All these bands more refrangible than 2404, 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 2390: see plate 3, at the end of the volume. With the strongest solution in a thickness of 38mm. 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 2 375, or thereabouts, it seems to cut off all the light, but the diffuse edge extends with the strongest didymium chloride as low as 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 didymium 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 1350 to 2600. 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 2380. Plate 9 is a reproduction of these photographs. The triple band at about » 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 2355, a thickness of 152 mm. of a very dilute solution of didymium Vor XVilnL, 39 306 Pror. LIVEING, EFFECTS OF DILUTION, TEMPERATURE, ETC. ON THE chloride transmits a sensible amount of light as high as 315 (the highest part of the spectrum included in my photographs) but with a gradually fading intensity from 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 1360. Didy- mium 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 con- centrated 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, zine, 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 4355. 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 thick- nesses, 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 solu- tion 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 145. The extent, down the spectrum, of the absorption now in question, is increased, as might be ex- pected, 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 con- centrated 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 didymium 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 mm., 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 alter- nately, 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 mm. 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 », 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 perceive, 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 cor- responding 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 im the green appears with the 39—2 308 Pror. 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 67 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 2427, 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 67 mm. of solution containmg 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 2333 seems to be absorbed, while with the chloride light gets through beyond 2315; and the strongest solution of the nitrate in a thickness of 38 mm. does not entirely cut off the light below 2360, 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 con- centration. 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 ditfuse- ness 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 room, 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 hght through to affect the plate. The creeping down with the higher temperature of a diffuse absorption from the most 310 Pror. 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 photographic 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 sup- position 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 THE 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 some 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 con- centrated 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 2451. 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 Pror. LIVEING, EFFECTS 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 com- paring 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 2679 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 mid- way 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 direc- tion, 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 mm. 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. Vou. XVIII. 40 314 Pror. 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 molecules 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 com- pounds 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 com- bination. 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 im 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 distri- bution of energy in the field, the ions are charged with electricity. That is not neces- sary 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 redistribu- tion 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 mani- festation 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 circum- stances 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. y, Pp y [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. Some 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.” Pror. 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 interest- ing 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, ete. 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 nm 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 §. 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 Fie. 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 Pror. 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 imsurmountable—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: Fie. 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 m consequence of dust particles which invariably deposit on the glass surfaces, in spite of the greatest possible pre- cautions, it is practically impossible to imsure a perfect contact, or even constancy m 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 =¢, the thickness of each element of the echelon; @, 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 mA = pt—ace. t Now ac = —— cos(a+ 8), cos a * 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. Pror. MICHELSON, THE ECHELON SPECTROSCOPRE. 319 or, since @ is always very small, t i ac =~ (cosa—@sina)=t¢(1—@ tana), COs a and hence NUN) (AMOI NS Ole meeneseeee atc cr set oeces coe ee te (1). Fic. 3. To find the angle corresponding to a given value dy, differentiate for X and we find d@_1/,_ du ah ts (m a Putting in this expression the approximate value of m=(u-1)s, : dé oe dy ual we have ia [ur] foo! Ba EGOBOTD ca CaSO Rad ON ae erase (11). 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=dd/dA for the limit. For this limiting value the angle @ will be 2X/ns, where » is the number of elements: whence ns=the effective diameter of the observing telescope. Substituting these values we find xr TS bat nislolejeielv\via[eis\slels\aisiaisialels(e]siuialelalele/eteislcisiisisisicteleverale (IIT) To obtain the angular distance between the spectra, differentiate (I) for m; we find d@ X 320 Pror. MICHELSON, THE ECHELON SPECTROSCOPE. or putting dm = unity, The quantity d\/A=Z£ corresponding to this is found by substituting this value of d@ in (II), whence 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 A/bt the two contiguous spectra (of the same line) will 1 17000° to examine lines whose breadth is greater than the fourteenth part of the distance between the D lines. It is evidently advantageous to make ¢ as small as possible. overlap. As a particular case, let us take ¢=7 mm, H= It will be impossible Now the resolving power, which may be defined by = is proportional to the product nt. 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 bemg 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 peda, —s/2 2a where p= = 0. Hence As Pror. MICHELSON, THE ECHELON SPECTROSCOPE. 321 This expression vanishes for @=+2/s which is also the value of d@,, the distance between the spectra. Hence in general there are two spectra visible as indicated in Fig. 4. e7 7 0 7 em Fic. 4. By shghtly 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 Gi LL CA ysey | ae ar Fe Piz: Sane le Gas: and formula (IV) becomes wae dm ys" Repeating the same operations as in the former case, we find fet Xr ~ net’ Xr 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 multiplied by this factor. But as there is now a surface water-glass which reflects the light, the loss due to this reflection will be Wot, XOVWIURE 4] 322 Pror. 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 3°55 times less than before. For many radiations the absorption due to thicknesses of the order of 50 em. 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 re- fleeting 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. aS 3) eee Fic. 5. Fia. 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 Pror. MICHELSON, THE ECHELON SPECTROSCOPE. 323 r/na. For instance, for 20 plates the average error may be a whole wave-length if the inclination @ is ;4. As there is always a more or less perfect compensation of the errors, the number of plates, or the inclination, may be correspondingly greater. Accord- ingly 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 indepen- dently 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). Set Ses ee | Fie. 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. Fie. 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 Minimal Surfaces. By H. W. Ricumonp, 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. PA. 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 Surfaces 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 my work; but the appearance of Mr Bromwich’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, d(p, l, m, n)=0, (where $ is a homogeneous function of p, 1, m, n, but not necessarily algebraic), ex- presses the condition that the plane [RSET PAR OA YO Saoaccocosssnoosse8e esascHessbas35000007 (aby, 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 $(p, 1, m, n)=0 will always be regarded as defining a dependent variable p as a function of three independent variables /, m, n;— (D=IP(H Ws, TW) bococeosadvoocsdacopboosbodoeapendobodgeTaDd @)5 but the function is of necessity homogeneous and of the first degree. The coordi- nates of the point of contact of the plane (1) with the surface enveloped by it are Di, vena nme EP al Yam! By TTT Teenie eeetetessseneeesaces e= so that 2, y, z are expressed as homogeneous functions of J, m, n, of degree zero, 1.e. as functions of the ratios 7:m:n. It is therefore possible to eliminate J, m, n from equations (3) and so to obtain a relation in «, y, z, alone, the equation of the surface in point coordinates. The condition that the surface should be minimal is established without difficulty, viz. 0? dm? dn? EPP CEG Ee ee OR Nie ay (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), 7s a minimal surface. When the condition (4) is satisfied, I shall say that p has a minimal value, or is a minimal function of 1, m, n. It is of importance to observe that, in what precedes, the condition P+? +r2=l, is not imposed: provided only that p is of the first degree in J, 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 (u) + F, (uw), lL+im r+n l—im r+n where b= Oh = — r—n tl—iwm r—n Ut+im and r=(P+ m+ n°). These quantities w and w, 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 v,, the former a function of w and the latter of w, we reach the solution of Monge (D. §§ 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 +f (m)] — @— mm) f(w) — (+ im) fi): 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[uy’ (uv) + wx (tH)] — 2 [XK (W) + yr (UH) ween eee ee cence ee eee (5); and this is the value which will be used in the following applications. From it I derive, by differentiation with respect to J, m, n, the expressions , in 2 uM () 1 9 ” z=’ (u)+ 3% (1 — wu) x” (uw) + x0" (4a) + 3 Uy (1 — 7) yr” (tH): er, iy. as ae 1a Sa y=X (u) +5 tu (1+) x” (u) — iy (4) — 3g mal + Uy?) ¥1"" (4); z=—x (u)t ux’ (vu) + wy” (uw) — x4 (ta) + tay’ (ta) + 2D” (tm). It will be seen that the two forms are in agreement if FwM=uxy (uw); A 4) = tH 4_1 (%4). 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 y=, (which in the case of a real surface implies that y is a real function), the surface has 20a as a plane of symmetry and cuts it orthogonally: moreover, if we fix directions by Euler’s two angles, @ the colatitude and ¢ the longitude, (so that l:m:n:7::sin @cos¢:sin@sing: cos@: 1, and u=e't cot 6 ee bot 6) © 2 > ty a o 2 ? the functions y and xy, are determined by the equation x (cot : é) =x (cot 5 6) =~ 5 cosee 6. | pd®, the quantity p being the length of the perpendicular from the origin on any tangent of the given plane curve, laid in the plane 20s, and @ the inclination of that per- pendicular 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 «Oy; let p, denote the perpendicular from O on the enveloping line, and ¢, the inclination of that perpendicular to Oz. In a complete circuit of the contour, the enveloping line will turn through some multiple of two right angles, and return to its original position; p, is therefore a periodic function of ¢),—the period being a multiple of m,—and may be expanded in a Fourier’s series even when p, or its differential coefficients have discontinuities: thus Po =X (a; sin kg, + by cos ky). In the case of an oval curve or a closed convex polygon the period of p, is 27; k will then receive only integer values. In a cardioid the period is 87, and 3h will always be an even integer, etc., etc. The minimal surface sought will be represented by the tangential equation p= > {(é — cos @) cot} 6+ (k + cos @) tant 5 a} (a, sin kh + b;, cosk) + 2h. For this typical term may be obtained trom the general formulae (5) by making x (Ww) = K (uk —u-*), yy (th) = K (mt — w*); K and K, being constants suitably chosen; and we may deduce z= X(k—k*) (cot! : @ — tan ; @) (ax sin kh + by cos kd) ; 1 : so that, when 0=57, z vanishes and p has the correct value. Interesting special cases arise when the given plane curve is an epicycloid or hypo- cycloid; for the series for p, then reduces to a single term po = A cos kd, and the required surface is obtained by making in (5) x (u)=B(uF—u*), x1 (ty) = B(w* — u-*). 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 lmear, the sum of any two of its solutions is itself a solution: if then p, and p, be two minimal functions of 1, m, n, p,+p. is also a minimal function, Stating this theorem in geometrical language, we enunciate the note- worthy 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 J, 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 &+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, parabolas. Following his example, I call a surface of class &+1, which has the plane infinity as a k-fold plane, a k-fold para- boloid; and the family of such surfaces, (the value of & not being specified), Multiple Paraboloids. - 7. The tangential equation of a k-fold paraboloid will be written as p=V +U, U and V being rational integral homogeneous functions of J, m, n, of degree k and k+1 respectively. If for the moment partial differentiations with regard to J, m, n, be indicated by suffixes 1, 2, 3, respectively, the condition (4) that the surface should be minimal gives us the identity V (Uy + Ug + Uys) — U (Vin + Viz + Vis) + 2 (U,V + U,V, + UV) = 2V (U2 + U2+ U3) =U; and so proves that (U2 +U2+U02) +U is a rational integral function of J, m, n:—a result possible only if U be the product of factors which are powers either of P+m+ nn’, or of linear functions such as al +bm-+en, in which e+b6+c=0. Mr RICHMOND, ON MINIMAL SURFACES. 329 But it will appear further that ?+m?+n* cannot be a factor of U; for if in the above identity we substitute U= (2+ m2? 4+ v7 = rT, and take account of the fact that V and 7 are homogeneous functions of degree k+1 and /—2s respectively, we find that (2s? + 3s) VI? +7? is identically equal to an integral function of /, 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 +bm-+en, 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 system of values of the ratios 1: m:n; and, if the corresponding real direction be taken as the z-axis in a new coordinate- system, is reduced to the form C(l + im), 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 l, 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+im)'. That such values actually exist is shewn by the formulae (5), in which if we make xu) =A (= Uys yn (en) =A (ny; we obtain a value of p of the kind sought, viz. p=—A {(n—hr) (nr + (n+ hr) (n— rh} = (U4 im The numerator of this fraction, when the special value 1 = 2% (% — 1) has been assigned to A, will be denoted by p,x(n); thus 2* (hk —1) x(n) +(n — kr) (n+r)F + (n+ kr)(n—r)F= 0. The function p,z(n) is real and may be expanded in powers of n and 7°; or, by rearrangement of the terms, in powers of n and (/?+m)?: moreover on account of the value given to A the coefficient of the highest power of » in the latter form is unity: we might in fact write p(n) = nF + yn (2 +m) + mn (PF + m?P + ..., Vou. XVIII. 42 330 Mr RICHMOND, ON MINIMAL SURFACES. mm, M,... being real numerical constants. The corresponding minimal value of p is of the form p=Ex(n) + (1+ im) = nk = (1+ im) + W+ (1+ im), W denoting some rational integral homogeneous function of J, m,n (with complex numerical coetticients), of degree k. It will be seen that of integer values of # the value &=1 alone fails to give a function yz(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 /+7%m 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+im)y is minimal, it will be convenient to write for a time fz=lt+im, g=l—im, and to use f, g, 7, as Independent variables in place of 1, m, n. The differential equation of a minimal surface is now Spi ysem s ant ’ ofog 0, and is to be satisfied by p= Vi= fF Substituting and multiplying by /*, we find that og tft an integral function = 0, and deduce that the part of V that does not contain f must consist of a single term, Cink, C being a constant. It follows that by subtracting a numerical multiple of the fore- g A g p going particular solution we obtain a new minimal function p,, viz. Pi = {V = Chi (n)} =f", 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 minimal value of p which has (L+im)* for its denominator is pH=al+ Bm+ yn+ VC,u, (n)+(L+ im)’; (s=2, 3, 4, ... k); the quantities ¢, 8, y, Cs, Cs, ... C, bemg 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 (OS WSs Gas) in the differential equation we find on multiplying by /f” that, if S do not contain f as a factor, must be divisible by 7, and infer that the terms in V that are independent of # must be equal to those in S multiplied by n** and a constant. If h be equal to unity we must therefore have V=A.n?.S+terms divisible by /; but substitution in the differential equation proves that A must vanish: if on the other hand / be greater than unity, we may. by subtracting a properly chosen multiple of Bn (n) =f, obtain a new minimal function whose denominator does not contain so high a power of f as f". It follows that the most general rational minimal function with denominator (L+im)e 8 may be obtained by adding to a value with denominator S the terms DC, fe (rn) = (U+im) : (6 =2) 3) 4, ... h): C,, C;, ... Cy, being complex constants. The factors of S may now be subjected to the same treatment; that is to say, first reduced to the form /-+7¢m 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 /, m, n, may therefore be resolved into the sum of a number of terms each separately capable of being reduced by a real trans- formation 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 (L+ im)? = 2n? + 3n (7 + m’), and leads to a surface, 2 (a + ty = 18 (@ + ty) 2 + 27 (a: — ry), 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 RICHMOND, 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. uy, (n) {(U + im + (L— im} = (2 + m2 ; j, b= 23 w aks Every real minimal value of p which is a rational function of 1, m,n, may be expressed as the sum of a finite number of real fractions, each separately reducible by real trans- Jormation of axes to one of the forms just quoted. Terms such as al + Bm+yn may also be present, but are ignored since a change of origin will remove them. If we again introduce Euler’s angles @ and @, as in § 4, the surface corresponding to the above value is pHA. pe (n).\(U+im) + (l—im)} = (P+ m2); (ESP BY GK ook) =B.coskd. Sin — cos 6) (cot 5 a) —( + cos 6) es tan A )| l > (q@zcce 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 number of standard surfaces (defined above) with their centres co- inciding but with 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 Quartic Surfaces which admit of Integrals of the first kind of Total Differentials. By Arraur Berry, M.A., Fellow of King's College, Cambridge. [Received 15 November 1899. | CONTENTS. Introduction. Analysis of the fundamental differential equation, Be Se) Integration of the differential equation, leading to five possible surfaces. Tabular statement of results. Birational transformation of the surfaces into cones. Numerical genus of surfaces which admit of integrals of the first kind. Geometrical characteristics of the five surfaces. VR SP) SP SL) S23 STs SF2 melee) eat of §$ 1. Iyrropucrion. 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 indepen- 5 5 dent variables. Given an algebraic equation, J (#, y, 2)=0, between three non-homogeneous variables, we may study either double integrals of the t pe R(x, y, 2) dxdy, where J ) g YI ¥ y R is rational, or single integrals of total differentials of the type | (Pax + Qay), where P, Q are rational functions of «, y, z, which satisfy in virtue of f=0 the condition of integrability ar _ 0g Oy ox’ Such integrals of total differentials were introduced into mathematical science by Picard about fifteen years ago*, and have been the subject of several memoirs by himt+. 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). driicke,” Math. Ann. t. 29 (1887). + The most important appeared in Liouville, ser. 1v. | Note sur le mémoire de M. Picard “ Sur les intégrales t. 1 (1885), and ser. ry. t. 5 (1889). There have also been de différentielles totales algébriques de premiére espéce,” a series of notes in the Comptes Rendus. Bull. des Sciences Math. ser. 1. t. x. (1886): Coll. Math. + Comptes Rendus, t. 99 (29 Dec. 1884). Papers, t. x11. no. 852. § “‘Ueber die totalen algebraischen 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 uumber 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 may 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 obtamed 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 problem 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. Poinearé’s results have been adopted by Picard, who has given a proof in outlineS. 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 Picard) that if a surface of order », of which the equation in homogeneous point coordinates is f(a, y, z,w)=0, admits of an integral of the first kind, then f satisfies the partial differential equation ay - of of of Oem spre ie gente ay Orta Ue hte aaa (1), Comptes Rendus, t. 99 (29 Dec. 1884). Picard et Simart, pp. 135, 136. Ib., Chapter V. * “On the deficiency of certain surfaces,”’ Math. Ann. t. mi. (1871): Coll. Math. Papers, t. vii. no. 524. + Picard et Simart, pp. 113, 119, 120. Sr ++ OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 335 where 6,, @,, 0;, ; are quantics of order n —3, which satisfy the equation 00, i 08, a 06; % 06, Ox Oy 02 dw These equations being satisfied, the differential dz, dy, dz of 2 5 ow ,, Os, 9; 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 @ are linear and the equation (1) becomes a familiar partial differential equation. If we write 0; =aae+ by + OS + daw, (2 =, 2 eh, 2h then, in accordance with the usual elementary theory, the integration of the equation (1) depends upon the roots of the algebraic equation Aor bie on (ie WIV iascadttin ingeaccosebascere (3) (ls, b,— 8, on ds | (ls, lids. C3 — 8, ds Gis Brg ae Cho | If the roots of this equation are all distinct we can at once obtain three inde- pendent integrals of the auxiliary system Se eRe ee tee (4), GO VAT Ve: 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 @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 form 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,-tuply as a factor of each first minor of A, p,-tuply as a factor of each * “Bemerkungen zur Integration eines Systems linearer Differentialgleichungen mit constanten Coefficienten,” Mathematische Werke, 1. pp. 75—6. 336 Mr BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS second minor, and so on, and if p—p,=2, pi—po=a@, ..., and B, B’...,y, y ... are the numbers corresponding similarly to the other factors, s—b, s—c,... of A, then (s—a)t, (s—a)*..., (s—b)®, (s—b¥ ..., 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 dw a doe dt = dt a ah Le dt Os, 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 («— a)? being of the form dx - aE SS i Abie =nootopaanooebdendonasabad (5). da, dt diy —— = 2, + Ty, «<0 dt Ie 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) (1) (ii) (iii) (iv) (v) (1) (al) (iil) (III) Two pairs of roots of A=0 equal: All the roots of A=0 equal: (II) Three roots of A=0 equal: (IV) One pair of roots of A=0 equal: (V) All the roots of A=0 distinct: (s—a)}, (s—a)’, (s—a), (s—a/y, (s—a)’, (s—a), (s—a), (s—a), (s—a), (s—a), (s—a), (s—a). (s—a)*, (s—b), (s— a), (s—a), (s— 6), (s—a), (s—a), (s—a), (s—)). G) (s—a), (s—b), (i) (s—a), (s—b), (s—d), (iil) (s—a), (s—a), (s—b), (s—b). (i) (s—a), (s—6), (s—o), Gi) (s—a), (s—a), (s—b), (s—c). (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=0, Case II. = —3a#0, and we may evidently take a=1, b=—3, OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 337 Case III. b=—a+#0, and we may take a=1, b=—1, Case IV. 6+c=-— 2a, Case V. a+b+c+d=0. 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 2, y, z, w, and the auxiliary equations will be expressed in the usual Lagrangean form, the variable t used by Weierstrass being omitted. I. (Gi). The auxiliary equations are: da_dy_dz_ dw O: mater y Tae: three integrals of which are: z=const., y?—2ze=const. y+ 3a°w — 3dxyz=const., so that the general integral of the equation (1) is fH=O(@, y—-22n, y+ 3x°w — d3xyz), where ¢ is an arbitrary function. The only quartic of this form is a sum of terms a, w(y?—22x), (y?— 22a), v(y?+ 3a°w — 3xyz), so that w occurs linearly or not at all, and the surface is therefore a cone or rational. I. (i). The auxiliary equations are: dx _dy_dz_ dw Ohana 7 wo. three integrals of which are: x=const., y=const., 2°— 2yw=const., so that S=O(@, y, 2 2yw). Vou. XVIII. 43 338 Mr BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS The general quartic of this type is (22 — Qyw)? + 2 (2? — 2yw) (w, YP AC, YY HO reeceeceereereneenes (6), where (z, y)” denotes an arbitrary quantic of order n. J. (ii). The auxiliary equations are: de _dy_de_dw ON BPO Be leading to «=const., z=const., yz — #w=const., and S=O(a, 2, yZ— aw). The general quartic of this type is (yz — aw)? + 2 (yz — ww) (@, ZP+(@, ZH Oo eee eee eee eee eee ees (7). I. (iv). The auxiliary equations are: dx dy dz_dw Ore a Oren On leading to the cone F=o@, 2, w)=0. II. (i). The auxiliary equations are: dx dy dz dw a 2ty yt2 —Bw’ of which one integral only, viz. 2*w=const., is algebraic, the other two being logarithmic. Thus the only possible form of f is ¢(a*w), leading to a set of planes. II. (ii). The auxiliary equations are: dz dy dz dw zZ ety 2 —3w’ three integrals of which are: y/x —log «= const., z/z=const., «w= const., so that the only algebraic form of f is $(z/x, zw), and the quartic is the degenerate surface @ 2) w=0: II. (ii). The auxiliary equations are: dx _dy_dz_ dw ZY 1 2 V=3wr three integrals of which are: y/x =const., z/r=const., «w= const.. which lead as before to a degenerate surface (a, y, zPw=0. OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 339 TE Gi) se Ehre auxiliary equations are: dx _ dy whdzen dw 2 e@+y —Z) 2=w’ three integrals of which are: y/« —log «= const., w/z+ log z=const., z#=const., so that the only possible quartic is the degenerate surface irs =). III. @i). The auxiliary equations are : dx _ dy _dz_ dw Z ety —z —w’ three integrals of which are: y/« —log «=const., za = const., z/w =const., leading to f=d(za, z/w), which gives a cone. III. (iii). The auxiliary equations are: dz _dy dz _ dw & Y -2Z —w’ leading to yx =const., #z=const., mw = const., whence f=¢(a#z, ww, y/x), so that the quartic is quadratic in a, y and in Zz, w, viz. of the form IV. (i). The auxiliary equations are: dz_ dy _dz_ dw ? ax xtay bz cw where 2a+b+c¢=0. If a#0, three integrals are: ay/x —logz=const., 2%/x”=const., w/a? = const., of which the first is essentially logarithmic, so that we have S= (24/2, w/a’), leading to a cone. If a=0, so that c=—b+#0, three integrals are: x=const., «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: de _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=0, b=—-c #0. (8) b=0, c=—2a#0. (y) a#0, b#0, c#0. In sUB-CASE (a), three integrals are: #=const., y=const., 2w = const., so that f=¢(a, y, zw), and the quartic is ZI S AI Ge, Op) 4>\ G2, 2) == OscosesenasvaosaeodesnooaoEoscoonobas (9). In suB-CASE (8), three integrals are: z=const., y/z=const., zw =const., so that S=o (eu, y/a, 2). The only possible quartic terms are: 2w (x, yy, 2, so that the surface degenerates. In SUB-CASE (vy) it is a little simpler to work directly with the corresponding partial differential equation a a O\ a (ax a, t YY 7) + bz ag +cw 5) f= 0, and to consider the possible terms in /. 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 2‘, w*, («, y)* cannot exist. If a term of the type (a, yz exists, then 3a+b=0, whence a=c, contrary to hypothesis; similarly no term of the type («, y)’w can exist. Similarly no terms of the types (2, y2, (#, y)'w* can exist, as their existence would involve b=c. Any term of the type (a, y)*zw satisfies the equation. If a term of the type (a, y)2 exists, then a+3b=0, whence c=5b, so that the equation is — 3 (afx + yfy) + 2fz + 5th = 0, ee OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 341 of which the general integral is f=¢o (2a, 22w, y/zx), leading to 2 (0, y)+2w(@, y=, a degenerate surface. Similar reasoning shews that no term of the type (#, y)w* can exist. If a term (x, y)2*w, or («, y) zw* exists, then a=b or a=c, contrary to hypothesis; and if a term z*w or zw* exists, then also a=b or a=c. Thus the only possible terms are of the type (#, y)? 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: (2) d=0, a+b+c=0, (8) a+#0; b+ 0; ¢c-0;, dO! SUB-CASE (a). Proceeding as in IV. (ii) (y) we see that the terms 2*, y‘, 2* cannot exist in f, but a term w* may exist. If a term a*y exists, then b=—3a, c=2a, so that the differential equation reduces to the ~ Bufy + 2efs=0, whence S=o(ey, 2/x, w), so that the only possible terms are ay, ayzw, w. The quartic is therefore rational, since y only occurs linearly, if at all. If a term 27 exists, then a+b=0, and therefore ¢=0, contrary to hypo- thesis. For the same reasons no terms of the types (y, 2)', (z, x), (z, y)! can exist. If a term a*yz exists, then 24+b+c=0, whence a=0, contrary to hypo- thesis. Thus no term of the type (#, y, 2)‘ can exist, so that f contains w as a factor and is degenerate. SuB-casE (8). Under the conditions assumed it is evident that no terms such as «4, or #yz can exist; and there cannot be more than one term belonging to a group of the type (2, y)*. Let the term z*y exist, then b=—3a, c+d=2a, and no other term involving 342 Mr BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS # can exist. If a*z* exists also, then a+c=0, d=3a. The differential equation is now If x — By fy — 22+ Buf» = 0, whence S=o (ay, aw, x2), so that the most general form of the surface is (CA0h URES ORI, CEM) p=" Saspasroosdoncosbaaoced (10). Let the terms a*y, wxz* co-exist, then b=—8a, a=—3c, d=—Tc; and the differential equation is = 3h a 9ufy ot 7 —Twfw = 9, whence f= (ay, x2’, zw), and the only possible quartic terms are ay, xz*, zyzw, so that the surface degenerates. If the terms a*y, wyzw 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 ay exists. If no term cubic in any one variable exists, then the possible terms to be considered are of the two types a*y?, xyzw. If only one or no term of the former type exists the surface degenerates; if terms such as a*y*, 2° co-exist, then b=c; if ay", zw? 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 prima fucie 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 6;, 02, 3, @,, we get the following table: OT OE a OF THE FIRST KIND OF TOTAL DIFFERENTIALS, 343 Surface 6, 0, 05 64 | Reference letter | vy? + 2ry (z, wP+(z, w)t=0 & -y 0 0 A u [as | | 2 (a, y)?+2w (x, y)2+w (2, y)e=0 a | Yi Pz ee B (aw — yz) + 2 (ew — yz) (z, wP + (Zz, wt =0 z | w | 0 0 C 2 (2ew — P+ 2(2aw — y*)(z, wP+(z, w)t=0 y 2 || 0 0 | D i aay? + b2*w + cayzw + daw + eyz? = 0 3a —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 Comptes Rendus for 2 Sept. 1899. § 5. BrraTIONAL TRANSFORMATIONS OF THE SURFACES INTO CONES. Corresponding to each of these surfaces we can at once construct a total difterential by the formula already given (§ 1); and since the 6’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 non- singular cubie 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 trans- formations which effect this object are as follows: (A) We write the equation in the form {ey + (2, w)}?={(z, w)}P?—(z, wy) and choose as a new variable w one of the factors of the right-hand side, so that the latter becomes w(z, w)® The quadric transformation : Bry sziwa=cw—(@, wy): y2: 7 > yw) , i} fi GO) BAe Sin AC, Oe B Oe Boy 8 Oe * Loe. cit. > 344 Mr BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS then leads to Fit Te iran) each ronince Saoeeondcoricochaeodann Coasts (B) Choosing as a new variable y one of the factors of (a, y);, we can write the equation : 2(a, y)P+z2w (a, y)P + wy (a2, yy =0. The quadric transformation : then leads to OP Gis OP Ye ARE (las OL)YS SEES (Cis 07) = On Ssnanamoeedooseeocodvac (C) The coordinates can evidently be chosen so that the point e=y=2z=0 lies on the surface; z is then a factor of (z, w)4, which may accordingly be written 2z(z, w)’. The quadric transformation : J C2 y 322 wW=2 (a +4) saya ye": we ° , eiy i 2 Ww =aw—yzi yz: 2: Zw then leads to GEV AEN Ey OP, Gi) ==10) gnasaoosboossscooobeendanchoses (D) Changing the variables as in (C) and employing the quadric transformation: Was Biy:z:w=_5(e2+y)sy'w : Zw’: w [9 wily 2's w= 2aw— yf: yz 22: ew | i we get EP aS aA, QP aE (2g DIR SO ebssscosoundondsosoocsnese4c (E) The cubo-quartic transformation : By 2 Wi Yeerrne Ww Nance Aue ey ie iw =2w : ow: cyz: cyw ; converts the surface into Cass enciy eo Hoje ab mn seanthy == (0); .onoeysoosesonanage70ce 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 quartic 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 rela- tively to the coordinate planes. Hence we see that a quartic of any of the five types can be birationally transformed—if necessary wa a cubic cone—into a quartic belonging OF THE FIRST KIND OF TOTAL DIFFERENTIALS. 345 to any one of the other types. But im order that two such quartics with given co- efficients 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 cor- responding cubic cone might become rational or degenerate, in which cases no integrals of the first kind could exist. § 6. NuMeErIcAL GENUS OF SURFACES WHICH ADMIT OF INTEGRALS OF THE FIRST KIND. 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; conse- quently 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 met 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 hyper- elliptic 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=0, x=z=w=0; 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 “ Selbstberiihrungspunct ” ; 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 + Liouville, sér. 1v. t. 9 (1893). “On the deficiency of certain surfaces,’ quoted before; t{ Crelle, t. 64 (1864). Picard et Simart, ch. vit. § iv; Castelnuovo & Enriques: § According to Picard and Simart this is the name “Sur quelques récents résultats dans la théorie des surfaces given by ‘les géométres anglais,’ but I have not been able algébriques,” Math. Ann. t. xuvut. (1897). to find any such authority for it. Vout. XVIII. 44 346 Mr BERRY, ON QUARTIC SURFACES WHICH ADMIT OF INTEGRALS the surface can be transformed by linear substitution (# =«+iy, y’=x—vty) into the general quartic surface of revolution. The birational transformation employed 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 eleventh+ 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 some- times 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=w=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 2=w=0 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 coefticients. 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 im 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. Math. Papers, Bologna, ser. 11. t. vir. (1868). t. vy. 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 x =ny’, y= 2’ (where X, # are connected by a cubic equation), corresponds in the ”, Y, 2, W space the variable part of the curve of intersection of the quadrics 2*=)aw, w*=pyz; but these have in common the fixed straight line z=w=0, so that their residual curve of intersection is a twisted cubic. 44__2 XVII. An Electromagnetic Illustration of the Theory of Selective Absorption of Light by a Gas. By Professor Horace Lamp, 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 absorp- tion 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 mechanical 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. Mag., Feb., 1881, and April, 1899; Prof. Love, Proc. Lond, Math. Soc., t. xxx., p. 308; G. W. Walker, Quart. Journ. Math., June, 1899, Pror. 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 10%. 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 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 4772. 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 sodium- light would contain something like 5 x 10° molecules, it is evident that a gaseous medium of the constitution here postulated would be practically impenetrable to radia- tions 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 problem 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 formule 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, 850 Pror. 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 aay esine Ge: 1 Bute es worst} 6 on(8) = (- Fae) Wild. «GrtDlt a 2Casa 2Canensa dae ON ECOS Ge died oes le) ee la eo aay | 4. ¥n(O)=(— sag) gp =e Se $e Se) op © These may be taken as the two standard solutions of the differential equation 2(n+1)dF oe — =P |) ccd sccceahous Sam catee res eRe 4)*; ie a al © the solution ~,(¢) being that which is finite for €=0. In the representation of waves divergent from the origin we require the combination Jn (E) = (- 92) oF = Vn OH) ss 8 a he ee ee (5). The functions W,(f), Ya(S), fr() all satisfy formule of reduction of the types AG ae alrna (Gy cone ccccarcoes sees Sree (6), Glen (0) tet les (6) ain 3 (O)ice ccc aa eeteceteeas eee (7), from which (4) can be verified. We have also the formula vn’ (£) Vn (£)— Vn (0) Vn’ (0) = aa ged hotebincatey eee (8)t. 1. The equations to be satisfied in a medium whose electric and magnetic per- meabilities are K and w may be written, as in Prof. Love’s paper, Ky _dy dB Ky da dy K, dB de ea ae v= dz dz? eo Steen eee eee eeeeeee (9), iti, AL AY. Appia MA Reiman e. dy eae epee RE GER oe’ de dy ee ee cece ecccnce (10), where (X, Y, Z) is the electric force, (a, 8, y) the magnetic force, and c denotes the wave-velocity in the aether. Assuming a time-factor e', we find (V2+h)X=0, (+h) V=0, (VFR) Z=H0 vccccccceccsccceeee (11), : CG aaa Z, with Gdn * dy 1 dg Oren sncncn (12), where | Fd (Trl (eeepc cs COeuOROR ACE Peace oGecdson aura. (13). * See Hydrodynamics, §§ 267, 305. + See Lord Rayleigh’s Sound, § 327. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 351 When values of X, Y, Z satisfying these equations have been found, the corre- sponding values of a, 8, y are given by (10). Or, we may reverse the procedure, determining the general values of a, 6, y 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 = {hry (hr) + (n + 1) vn (hr)} S rT, — nhrvr (hr) xr” Th, = {hrvy’ (hr) + (n + 1) vr, (hr)} i rT, — nhrwy' (hr) yr® Th, Sere (14), Z = {hrvpy (hr) +(n + 1) Wn (hr)} S rT, —nhryp,! (ir) zr” a where 7, is a spherical surface-harmonic of order n*. These make ey oe (hr) yy dz a)" PAllin, K | pa— (ir) (2 Bo EVO, csnsesiesnsete (15). ick Arn (hr) int =y5) | a It follows that MCAS) EET ASAT O(a eI) IES (iA) Gal Eo onoandeswondobomencebon6dt (16), Gal 36 ]/s) 447 Veppoodepnasogaesb00dDe oagodecaduaKdacecoooxoadouees (17); also that Z—2zY = thrwp,! (hr) + (n+ 1) wn (hr)} ly bps 2+) rT,, S&e., &e (18) ¥y L n 2 \ n J) dz dy n> eee cee eeeeee A yy — 28 = aif Wn (hr) (e" T — nar TE) EOE ANCE G Rae MRO SGN SEER ES eee eee (19). In the solutions of the second type we have a= {hrvy’ (hr) +(n +1) vn (hr)} a Uy, — nhrvp,! (hr) ar? Un, B= fhryy’ (hr) + (n+ 1) Vn (hr)} 5 Of, = Toran (Wor) ofr Ulie Ss sennnee (20), = {hry (hr) + (n+ LD) Wn (hr) } cal Un — nhrrvby’ (hr) zr” Uy, v dz * These are equivalent to the forms given in Hydro- formule relating to spherical solid harmonics, such as dynamics, § 305 (6), divided by 2n+1. The proof of the a 7 (Ga—ro-h Gh aie ) equivalence requires the use of (6) and (7), together with ‘ n= ond dx da 72nt1 } © 352 Pror. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY where U,, is a surface-harmonic. From these we deduce ,_ top -\ | ie ee n xA= - Yn (hr) y ae cere 7 y=" » (tr) (2 aX) Ua. ea nae (21). Z= -- n Gin) (© yan)! “U, | Hence ea + YB + 2y = (+ 1) ben (AN) Ty cece see erree renner sense (22) Ca la) aloe Hie Ror et doc dori hbo ACUR SRO EEO SEBAGACOOSOCOOSOneHOCE (23); also yy — 2B = shrva’ (hr) +(n +1) vn (hr)} (v5 2 =) TOs A055, (AES Goon sabar (24), yZ—2zV= — “ys Wh, (hr (s = ar-*U,) PROC ACOICARY. cca RU ER cE econ: (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 Toko ae 2. Let us now suppose that a sphere of radius a, having the origin as centre, whose electric and magnetic coefficients are A and yw, is surrounded by an unlimited medium (the aether) for which A=1 and w=1. The disturbance in this medium may 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 out- wards 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, w=1, and replace h by k, where | GES Co | RPP RRBEE sos SABRA Son aBED bec ROG cE A cddAddos (26). As usual, 27/k is the wave-length of plane waves of the period 27/c. In the corresponding expressions for the divergent waves, we must further replace wr(hr) by fr (kr), where f, is the function defined by (5). This is necessary in order that the formule may represent waves propagated outwards, the complete exponential factor being then e*(t™, It is necessary to have some 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 Zn, U,, simply; those relating to the scattered waves by T7;,, U,‘; and those relating to the inside of the sphere by a Ore 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 lhe (Cae) Mee rs CR) TES = LEG (2) IE soopeoneoocenpnceoeaceEss se (27), Nin (He MOF eye (te) (UL > = jiath. (ia) OS brine ndeonacodceoapanaapeane (28). Again, it is easily seen that the continuity of the tangential components of electric and magnetic force implies the continuity of the vectors (y¥Z—zY, zX —aZ, «Y—yX) and (yy—2B, 24—ay, «B—ya), respectively. Hence from (18), (19), and (24), (25), we have, in addition kann’ (ka) + (n+ 1) Wy, (ka)} Tr + {haf,’ (ka) + (n +1) fr (ka); 7 = fharry (ha) + (+1) Wn (ha)} Dy s.....scccecsecseaccosseees (29), and [hayrn' (ka) +(n +1) Yn (hka)} Un + {hafy’ (ka) + (n+ 1) fy (ka)} Un’ = {hasb, (ha) = (que 1) teen (lie) Ue sada seaetcans daege wees secs (30). Hence T, ef. Ky (ha) kan’ (ka) + (n +1) Wn (ka)} — {hay (ha) + (n +1) Wn (ha)} Wr (ka) TS Ky, (ha) {kay (ka) + (n+ 1) fn (ka)} — {hawrn’ (ha) + (n + 1) Wn (ha)} fr (ka) it tae aed, (31), Ue — HN (ha) {kayn’ (ka) + (m+ 1) Wn (ka)} — fhawpy’ (ha) + (n+ 1) Wn (ha)} Wn (ka) (op BY n (ha) (kafn’ (ka) + (nm + 1) fn (ka)} — Shay’ (ha) + (n+ 1) vn (ha)} fr (ka) 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 yw are not greatly different from unity, so that ha is also small, we should obtain at once approximate expressions equivalent to those given by Prof. Love, viz. TAN Ses (n a Ie 1) (ka) : soe nK +(n+1) {ies Cn emai teen eee teen ee (33), 7 (n+ 1)(w— 1) (kaya ; Un = 77 2 5 Wiringgo0aono 3¢ me+(n+1) ~ {1.3...(2n—1)}?(2n + 1) Un sreeesereereens (34). It is our present object, however, to examine the case where K is large. For simplicity we shall suppose that ~=1, so that K=h2/k*. It will be found that the first factors on the right hand of (33) and (34) must be replaced by (n+1)(K —1), (ha) —... ft RGREaren Va (ia) SRO soe (35), Vout. XVIII. 45 354 Pror. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY and _ han (ha) + ... Wn (ha) + ey oo ese et ee ee respectively, where the terms omitted are of the order /*a* 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 a(ha)=0, or to a root of ar,-.(ha)=0, 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 2-negative, and that the electric vibration is parallel to y, we may write, symbolically, If this be resolved into a series of disturbances of the types (14) and (20) we must have, by (13) and (19), SDI Coe Ral) es Ue) ce ES 1G sos onansoonaponaasopaacsasonsec¢ (39), re (rt -F ly ir (er) eg — ree ence ee eee (40). Now if we put tz=reos@, y=rsn@coso, z=rsmOsin@ ..............0.-+--- (41), we have ikye* = > (2n + 1) (ckr)" Wp (Ar) sin 8 cos @ Pp’ (COS B) «.....-e reece (42)*, where P, (cos @) is the ordinary zonal harmonic. We infer, by comparison with (39), that 2n+1 ae GED (sic) =" sim) @ cos eel?s(COS\@) i n--eeseee= eee eee ae (48). Similarly, we find 2n+1 ae CE: ACSC (aie) "= sins 6) Sin ople nt (COSiO) eas cnesensene scence (44). In particular 11s ey roe 3 2 : STS Se ene (45), oe ; 3z U,=—5sin@sin o=—5- Hee San Sons SE OeNO eee poe ewe sds (46) * Proved most easily by differentiating with respect to cos 6 the known identity elk CoS OS (2n+1) (ikr)" Yp (kr) Pp (cos 8). OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 355 If we substitute the above values of T, and U, in the formule (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 - qr om d \ DS De aye \ a qa a / d apy \ yy? T. \ \ (48) ~ (ery ° Nag mT, —nyr ys Sesser a Sac tath cee ; n—1 Z= a e—ikr (¢ yn ips — nzgrr— iE) qr ei Gs ad —— —ikr [ , yn op: N t= Fpnpnt © (y dz a) oe n—1 ey oar; d F B= pean e7ikr 2 aaa x =) TERR en es tate ican aaa cect (49). qn-l ee d ad nip Y= Fnpn & : (25,-9 a) rT, We notice that X, Y, Z are ultimately of the order 1/r, whilst the radial electric force («aX +yV+2Z)/r is zero to the present order of approximation. It is really of the order 1/r2 The radial magnetic force (wa+y8+2y)/r is accurately zero, If the contour-lines of the harmonic 7', be traced on a sphere of large radius 7, for equal infinitesimal increments of T,,°, 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 7,’ 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 qr Te d F ~ ‘ a Fae El nO, ieee (OL = (kr) da qn + d f As ‘ B= ny enh Ge rT, —nyr? = Ui, ) » odogouRpoancostonecrotasesd (50), we —tkr d nT pn—2 U, . Y= (kr) é AE 7 n — N27 n * Cf. Proc. Lond. Math. Soc., t. xu., p. 194, 356 Pror. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY w4sss1fld d b " X=— herp \y dz wa =) r U), ? ea git a ( ig Saleh antl z dz oe [aa in ? PiatalctcsclelelalulstaiatetelnicieintetsteTetnibinistetefave hy x ey | qn d d k Teale dy =| AT 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 Z| | tr — 62) + m (a2 —yX) + n(BE = AV) dS coe (52), taken over the boundary of the space, J, 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 7, 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 a i Gel) eV Ge Nae) ey (GeV pak) eee (53), taken over the sphere of radius r. Before applying this result, the values of a, 8, y and X, Y, Z must of course be expressed in real form. To take first a solution of the first type, since 77,', as given by (31), will in general be complex, let us write gl ERE RAL Vo BCE anarene nO TeeOdcO neBenca sO Sar (54). Restoring the time-factor in (48) and (49), and taking real parts, we find 1 d d\. : ae aire (y Eee = {®, cos (ot —kr+e€)— >, sin (ot —kr+e)}, &e., We, ......... (55), and yZ—2Y= fee y a= iz -) {@, cos (ot — kr + €)— d, sin (ot —kr+.e)}, &e., We.,...... (56), where e€ may be 0, or +47, or 7, according to the value of n Hence the mean value of the expression (53), per unit time, is found to be jhe. Hy d®,, _ ,2Pn\? aD, oes / i yo) Sark?" =| dz dy / File da Cae Idn Ibn ddn see don doy, 3 CaP sae Gan a q Mae -yE) fas * Phil. Trans., 1884, p. 343. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. B07 which may also be written c (/d®,, d®,\? _ (d®,\) wae, aterm || ae) + Cay) + ae) - 2 ® ddbn\? ~/ddbn dd, ) ae fe) |e (2) — nee} as RRCOOHOT (58). The expression under the integral signs im (58) is equal to the sum of the squares of the tangential components of the vectors (d®,/dx, d®,/dy, d®,/dz) and (dd,/dx, dd,/dy, dd,/dz). Now if S, be a surface-harmonic of order n, we have dS, dS, plr far 2 k 7 I {C i(' > a) + es a ) sin 6d@dw=n(n+ 1) ihe ii 'S,2sin 0dOdw ...... (59)*. Hence (58) may be written c ih eae 2 ; \j2 n(n+1) aaa || (®,72+ $,°) dS, or n(n+1)——— or im ffir. RGkorsoogddeadase (60), where |7,\| denotes the modulus of 7, and dw is an elementary solid angle, viz. dS =rda. In a similar manner, a solution of the second type gives the result n (n+ 1) oem om If 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-harmonics of different orders*. Hence, if = 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 Cc n(n+1 V9 oy eh Wn eee ts Sete Be (62). In the case of plane incident waves the harmonics are tesseral, of rank 1. Writing, for shortness, TEN Be Ue OSU ee eenee nan oreecacce cnc onceteein (63), * Proved easily by partial integration, making use of the differential equation i Gi ja. pals 1 @&S, sin 6 0 (sine rT ) a ametpaan + The integrals which arise from combinations of the two types are of the form [ee @%- 1)S,=0. Zz i) a aH. Las. This involves products of surface-harmonics of orders m—1 and n, and will therefore vanish unless m=n+1. But writing it in the form if IXn (Am _ 4} {= ~ dy we see that it also vanishes unless n=m-+1. vanishes in any case. y sn) 4. a6. att. Las, dz Hence it 358 Pror. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY where the values of B, and C, are as given by (31), (32), and 7,, U, have the forms given in (43), (44), then since Ale 2 ‘ . | | {sin 8 cos wP,, (cos@)}? da =n(n +1). = ee (64)*, the expression (62) reduces to gp 2 (2n+1) {I Bal? + (aN. sk caceas peereeeneee (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 formule this is c/87. Hence, if J denote the ratio which the energy scattered per unit time bears to the energy-flux in the primary waves, we have T= Fe Em) {BaP [Cal ----2--saeeanseeceese renee (66). For example, in the case to which the formulx (33), (34), refer, the constants HK and uw for the sphere being not greatly different from unity, we have 2K-1 2-1 B, => 3 K - 5) hea’, C; => 3 FEED TRUS cece cance eee eee (67), and thence aA (SSN 7 ee = 3 Ta iG) a= =) (ka) teen eee e eee n eee eereeeeneee (68)+. 6. We may proceed to examine more particularly the case where K is a large number, whilst » is (for simplicity) put =1. The types of free vibration which can exist in the absence of extraneous disturbance are found by making 7,=0, U,=0 in (81) and (32). In the first type we have harpy’ (ha) + (m+ 1) Wn (ha) _ 7, kafn' (ka) + (n + 1) fp (ha) a aa (69), where, it is to be remembered, k/h =1/K?. We are specially concerned to find the solutions of this equation for which ka is small. On this hypothesis we have harp,’ (ha) + (mn + 1) Wn (ha) i§es a) UE cp inet emesis (70), nearly. This is satisfied approximately by ha =z, where z is a root of Nal) — Oe eer meRRR eas nvic enol cen'h onus «aw one OSRORe (71), and more exactly by ha= (1 — =z) 7 Sa gOaco OOO TOO ICUCROR ONO SER OF O53 50: (72). * Ferrers, Spherical Harmonics, 1877, p. 86. + This agrees with a result given by Lord Rayleigh, Phil. Mag., 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 Apo 8\03}, CRE), BIOS), cea. cooopsenscousoscencopade (73)*. orresponding to any one of these roots we have a simple-harmonic electric oscillation of C ling t y f tl t have a simple-harmo lect llat ft frequency and wave-length 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 w=1, har, (ha) — kafn’ (ka) * SIS NE) "cee eT eee 76), Faia), 7 fal) Se Ge Mire=st 2) ire ee (77), Yn(ha) fr (ka) This is satisfied approximately by ha=z, where z is a root of AU pen (23) =O) co dseccosvon once csbocnsocboccDDobebodbo0Gd (78), and more accurately by 1 = ha = {2 = a yrt BD wvvvccrvecvvcvevcoccvevecersecseees (79): When n=1, (78) takes the form sinz=0, whence /Afipil, 2), Bh, fico coaaodabanenosvopedondddeonoosoboKonC (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 formule (33) and (34) will no longer apply. If in the accurate formula (31) we make the substitution VAC) = We, Ut) = Ola (HO) oa cancncoeoccacbeseauneasgneccoud (81), we find that it takes the form where g(ha) stands for the expression in the numerator + of (31), and G@(ha) is derived from g(ha) by the substitution of WV, (ka) for Ww, (ka) The modulus of the expression * The lines of electric force in the sphere are for the in Electricity and Magnetism, p. 317. most part closed curves in planes through the axis of the + Which may be regarded as a function of ha since the harmonic 7,. Their forms are given in Phil. Trans., Pt. 11. ratio of k to h is fixed. 1883, p. 532; and in J. J. Thomson’s Recent Researches 360 Pror. 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 harry’ (ha) + (n +1) Yn (ha) _ K ka WV! (ka) + (n + 1) Wn (ka) Wn (ha) 7 WV, (ka) ee ceeceeneee or 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 satistied exactly, we have If the incident waves be plane, the dissipation-ratio (68) takes the form - 2(2n+1)7_2n+1,, = = ae Saag if 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 5 (kay. The wave-length of maximum scattering is of course very sharply defined. If we put [ik (IE) 7) cenesedeeenboosdonodoh cocofcoanconsankesonor (87), where z is a root of (84), and e is a small fraction, I find x a Wn (ha) _ _7.1.3...Q2n —1) = g (ha) = wie (C= G (ha) = “hay AE (Ui ola@S Bemanacnc (88), approximately, whence Ti a ee 89). Te n {ll 73... — I) ae ( 1 ar oe (kaye = iKe 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)*/K of itself. The second type can be treated in a similar manner. Writing (32), with ~=1, in the form US g (ha) To dha) wig Wha) eee escenene (90), the equation G(ha)=0 which determines the wave-lengths of maximum dissipation may be written Vna(ha) Vy (ka) Vathay = Wi (hay (rrreirisesesseeeeeeeeeeee (91). 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 US Ae Up c5 ee: (92), leading to the same formula (86), as before, for the dissipation-ratio when the incident waves are plane. Also, if we write eG ee eeriteiasisissosievidecinciens aceseawessenainays (93), where z is a root of (91), I find n (ha) 1 .3...(2n —1 2 g(ha) = t x oe 1)’ G (ha) = a ) Aes (0G) oEococacnono oc (94), approximately. Hence On _ a . Ge = SS aCe = Di Se i ee (95) 1+ —— ——~. iKe (ka)y" 1 The definition is now less sharp than in the case of (89), in the ratio ha’. 8. It remains to examine what sort of magnitudes must be attributed to the quantities @ and # in order that our results may be comparable with ordinary optical relations. Since ka(=27a/X) must in any case be small, and since ha must in the case of synchronism satisfy (71) or (78) approximately, and must therefore be at least comparable with 7, it follows that if our molecules are to produce selective absorption within the range of the visible spectrum, the dielectric constant AK (=h?/k?) must be a very large number. Again, it appears from two distinct lines of argument* that im a gas composed of spherical dielectric molecules the index of refraction (#,) for rays which are not specially absorbed is given by the formula 8 K=1 (ASS i) ra. oe eo ad vende Sacer (96), 4 id where p— Ne 3 Fifth atneio oOraHaOOUSCHCOCOR ORC COD CHE neroan (97), N denoting the number of molecules in unit volume. On our present hypothesis this takes the simpler form * Maxwell, Electricity, § 314; Lord Rayleigh, Phil. Mag., Dec. 1892, and April 1899. Vou. XVIII. AG 362 Pror. LAMB, AN ELECTROMAGNETIC ILLUSTRATION OF THE THEORY Hence if yp, = 10003, we have p=2x10~. This determines the product Na’, 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 N=2 x 10%, we find a=13x10-°cm. Hence if X=6 x 10 cm., we find ka = 14x 105% so that, if ha=7, we must have K =h?/F=5 x 105 In a dense medium composed of the same molecules the formula (98) is replaced by py+2 where the accents refer to the altered circumstances. Comparing, we have , 21 2 eg po Mn ee (100). 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’(=w") for dense media, taken in the bulk, are not incompatible with an enormous value of A 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 assemblage 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 n=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 enfeeblement of light traversing such a medium is Wie NEL TI «eet ta eee (101). * This is Lorentz’ result. Lord Rayleigh’s investigations shew that it will hold approximately even if p’ be not a very small fraction. + Lord Rayleigh, U. c. OF SELECTIVE ABSORPTION OF LIGHT BY A GAS. 363 We may inquire what value of the dissipation-ratio J would make the intensity diminish in the ratio 1/e in the distance of a wave-length. If we write so that J,, denotes the maximum value of the dissipation-ratio for n=1, the requisite value is given by i £2 N 103) mS as == Oy? gOS HOCEC OAC ES RECROOLOCnS CeO TnCner core € Tare On our previous numerical assumptions this is about 4x 10-7. The corresponding value of € in (95) is about 4x10. 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 meet with in the vibrations of the first type, and to the increase of sharpness (in each type) with the index x of the mode considered, it would appear that there is no prima 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. Iv 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 right- angles 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/2m) are given by the equations by gee Lebicrar (ie): Pe a pat ln? ” ae a) and Parag pat l+yn+n’ where @ is the radius of the circle formed by the wire, c the radius of the section, p, the mass per unit of length, # the Young’s modulus and 7 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 forms 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 dis- placement 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 wave- length. Pror. 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 coimcides with the radius of the cylinder, and the displacement w is reckoned positive when it is inwards along this normal; the displacement w is reckoned positive when it is im the sense in which the are is measured, and then the positive sense of the displacement v is determined by the convention that the positive directions of wu, 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 8, about the tangent to the helix, and § is reckoned positive when 8 and w form a right-handed rotation and trans- latory 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 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 27/m 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 m?>1/p?—1/e°, but if m*<1/p?—1/c7 this relation is reversed. * Proc. Lond. Math. Soe., xtx. 1888. 366 Pror. 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 8 and w should be always in the same 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 circular 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 m?=1/p?+1/o°, (b) when m?=1/p?—1/o*, (c) when the helix is very flat or 1/o can be neglected. In ease (a) one of the modes of deformation is equivalent to a rigid body displacement of the helix at mght 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 47. 3. The wire is taken to be of uniform circular section (radius c), and of homo- geneous isotropic material, and in the natural state the line of centres of its sections forms a circular helix of curvature 1/p and tortuosity 1/c¢. The displacement of a point on the central line is specified by components u, v, w along the principal normal, the binormal and the tangent in the senses already defined, but it is necessary to fix the meaning of the angular displacement 8. 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 1/p about the binormal and 1/¢ 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 frame attached to the unstrained wire, 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 La cae) GES fy 8 Gays where ds is the element of are 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 may be resolved into components V, along the principal normal, V, along the binormal, and 7 along the tangent, in the senses in which u, v, w are reckoned positive. The couple may be resolved into two flexural couples G,, Gs and a torsional couple H about the same three lines. The couples are expressible in terms of the displacements by the equations l/ow vw Co (55 are 1 p | 2 ol [Oy GL= ON fie E as & #5) ee Ga) HiGKih a) Oh il (ov u GA leet = = (ze ale erat eee (1), ,| 08 u H= o|2 = +(B+5)| in which A,=+}#7c', is the flexural rigidity, and C,=42#7ci/(1+ 7), 1s the torsional rigidity. Further, the displacements u, w are connected by the relation of inextensibility of the wire ow u SS Moat soboa em etionrees soniyau:s sehadamcnsawolid cabot 2). as p (2) When rotatory inertia is neglected the stress-couples are connected with the stress- resultants and with each other by the three equations of moments = = - +2 -N=0 ine = #5 NE = (0), }oecaconcodecnocccoboonscpoccsanooo (3) aH _G, a, ds) p is The equations of small motion are the three equations of resolution ON Nay. £ 2 Ou One wa ph LS One oN, iN ov as pee leer SISOOOOROOOOOOOOOOCOOON OOOO OOO OOO IG (4), a a. G2 fp Bi acer in which p, is the density of the material of the wire and , =7c’, 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 u=— mp W sin (ms—pt), v=Vcos(ms — pt), w= W cos (ms — pt), 8B = B cos (ms — pt) base (5), 368 . Pror. LOVE, THE PROPAGATION OF WAVES in which uw and w have been adjusted so as to satisfy the equation (2) of inextensibility. Further, we shall take the forms of G,, G., H to be G, = dg, cos (ms — pt), G, = Ag. sin (ms — pt), H =(mp)~ Ag, sin (ms — pt) in which G, and H have been adjusted so that the third of the equations of moments (3) is satisfied identically. We then find by (1) hee (ne + =) V+ (=P = =.) W, p o (on po 2 mp m = === + (mp + a Seite ess aaseinccea tiene (7), ta 1 = Cm (B+ a mp o of which the first and third give a Nei aces 1 7 aes | (1+ Gass) =! Se | Jeldeeosatceee sane (8), and the second is & v2 4W(m—445) Rs date athe oe (9). mp po Buenos 1 1 A : If Tea does not vanish we can solve for V and W and obtain ee AW et ol ce fy H(i =) = wh (1+ Gia) (™ Sage) er eee » «---+( 10): Cares eee eae la), Dea se Gal eo ( Poko pn Cm?p?/ po Is mp OA Mas } The first two of the equations of moments (3) now give us N,=—A (- at mgs) cos (ms — pt), Y edearetetecouenieaee (11). = 1 1 : N,=— n= ae = N, A {(m =5) Gaia o.h sin (ms pt) We eliminate 7 from the equations of motion (4) and obtain GEG, ah iL GE . an ae + o Fay = Pomp? (1 + mp’) (W/p) cos (ms — pt), oN, , N. == = = — p,wp*V cos (ms — pt), or, on substituting for WN, and N,, 2 1 A E & Ga - a + Mpgo ( m — 2 + =| = poop? (1 + mp?) W,| oe eee (12). rd! tee! 2m = A|o.(m— 243) +9. | = puap't OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE. 369 6. We may now substitute for g, and g, in terms of V and W and obtain by elimination of V and W the equation for p? poop? : )( A ) - Bee (2 2 aaa) ( a =) ( A ) (1 + ep ae Cm*p? A oi mp? + Cmep* ue rl ee OC ae 1 ay iN See a eee +(m—— =) =0 horas (13). il : : , : If m?— ——— does not vanish this equation can be written FE o 2 {1 — (1 + wo) a} {1 — (1 + oe) a} (1 — t= te)? = Bea (2 + eo — 1) =O eoreereree (14), by putting ky = A/Cm'p®, «,=1/m'p*, Kk. = 1/m?o* a= A (1—k, — kK) powp?m™ Since A/C=1+ 7, where 7 is the Poisson’s ratio of the material, Ky —Ki = 1K), and this is always positive; so that, if for « in the left-hand member of the above equation (14) we substitute the values ©, 1/(1+«,), 1/1+«,), 0, the expression has the signs +——+, and thus one of the two values of # exceeds 1/(1+«,) and the other is less than 1/(1+x,), both values being positive. It follows that there are two possible velocities for waves of given length, the speed of one exceeding le mp? (m* — 1/p* — i no) 1 + mp? and that of the other being less than A. mip (mt — 1p 1/89) Po® l+y+ mp? ’ these two expressions become the speeds of the corresponding waves round a circular ring by writing n for mp and omitting 1/c. The left-hand member of the equation (14) for # breaks up into factors rational IM Ky, Ky, Ko if [(2 + ko + Hy) (1 — Hy — Hy)? + Bq (2 + ky — 1)? — 4 (1 + He) (1 +) (1 = 4, — 12)! is the square of a rational function of , «, . This is the case when 24+ «,—«,=0, or when 1—x«,—«,=0, or when «,=0, or when 1—«,+«,=0, for in the last case the expression becomes 16 (1 = 4)? [1 = ey) (Ko — 1) — 2 (eo + 1) PP. Of these cases the first cannot happen since «,>,, and the third is the limiting case in which the helix becomes a circle; the two remaining cases will be discussed later. Vou. XVIII. 47 370 Pror. LOVE, THE PROPAGATION OF WAVES 7. With the view of determining the character of the motion corresponding to one value of p? we observe that by the second of (12) combined with the first of (10) A 2 1 XL 2m ( oh f = 1 Vie. a I+ {1 —2x (1 + Gre =i (m - ae =) + 92 i +a u =) (I) Beapoch oe (16), where z is given by (15), and it has been assumed that m*—1/p?—1/c? does not vanish. Hence we find (gece =) 2m — si \™ po +9: om W/E LN 2m 1 WEN wn +G ) [ae Jo 3 Ce Gas) and therefore 1l-—a#(1+«)=—- we la 4 (G- i) z a} | (me — 5-5) mp") & 1 r=; | a fe (A NPS ie 1\? a = {1 +5 Poy Ee = Sof Ay} = aa s) mip*| (m <5 =) godase (17) It follows that in the wave for which «(1+«,)<1 we must have 1Hew eb we 1+ 5p077 (m2- =e =) ZO 1.20 ee a (18). Again, by combining the first of (12) with the second of (10) we find Oy fy eel Serle nt | (1- = wap) tel + He) (l+m)|+g2 Ae) flo(l+<)}=0 03) Hence - 2 1 HR ia Ql ae wa 9s (me = ae #) ; eG z(1+“4)=- ca ; (+6) 9, (m— +5 py te and therefore 2 2h fo 571) cat /(w- 4-4) 1 a) ea ea & 1 mip? (m ae) Pre Sh oe keene 2p 5 | 2 1) 5 +/ pel esl Ne £ --l7 (m— e —+ =) + = (me Sle A + (4- mpi (1 + &) (m-— RE -=) Ree (20). Tt follows that in the wave for which x(1+x,)>1 we must have Vi ipo QUE Sates ee : W (m ia + =) ae (m - = BS (Dieicacids =i neitasis onepsen eemnetine (21). The two inequalities (18) and (21) are not mutually exclusive for all values of 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 2 from (17) and (20) in the form ee Ue eal Cay (on? 1 ) We We Lee ell — — Se — 2 —— == = = is = = (Kp k;) (m p =) ( aE mp? ) | a @ aE mp E ats 2 po V (a p? AF =) | 2 lr peer a =) =e ( 4 =) |} = ACL aR CH as m——)|}, po for Gr o pal Wal 9(V2— We mp2) | m? — — — or 2{V W? (1 + mp°)} (m Sie) 8 A/C-1 TAS — WP RALPE ne / Oya: = = 29 uA E Po Imp? + A/C— 1 (m p> 2) | Bane (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 o 1 20 / / #5008) 0} -& (me — 545), — 7 (m: —=.) |(me- 545), Z p o [A Ne ar placing these values in order of decreasing algebraic magnitude. For shortness we write and then ae 8 according as (m? = 1/p? +1/0*) 5 0. There are three cases depending on the signs of m?—1/p? and of m*—1/p?+1/o*, In any case when we substitute V/W=a, the left-hand member of (22) becomes eee oN AN ee pol 2m? + A/C—1 (m p” 3) (m p* 2 oe)” and when we substitute V/W=£ the left-hand member of (22) becomes gee MOR ie 8 Now in the slower wave we have l-a + <0, which shows that if m*—1/p?+1/c* is positive 0>V/W>a, and if m?—1/p?+1/o" is negative a>V/W>0. In the quicker wave we have 3 (m3 — 1/¢?-+ 1/3) Gr- B) >0. 47—2 372 Pror. LOVE, THE PROPAGATION OF WAVES Thus when m?—1/p?+1/o? and m?— 1/p* are both positive we have V/W>B, when m?—1/p*+1/c? is positive and m*—1/p? is negative V/W>B>0, when m?—1/p?+1/c*? and m*?—1/p? are both negative 0>B8>V/W. When m?—1/p? and m?—1/p?+1/c? are both positive we obtain after substitution in the left-hand member of (22) the signs shown in the table I] Wai—paco, SOR Sena, —oco eS a When m*—1/p? is negative and m?—1/p?+1/o? is positive we obtain VW, = SogeBee0 eo —co +--+ + When m?—1/p? and m?—1/p?+1/c? are both negative we obtain VIW = 02 a0 B —-o By comparison of these results we see that when m?—1/p?+1/o? is positive V/W is positive in the quicker wave and negative in the slower one, but when m?—1/p?+1/o* 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 displace- ments 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 aseca(w/o+v/p), and the displacement parallel to the circular section is aseca(w/p—v/c), and the wave is in a certain sense right-handed or left- handed according as (Wip —V]a)+(Wle+V/p) is positive or negative. We write & for this, and then the values of & in the two waves satisfy the equation 2 [(1/p — e/a)? — (1 + mp’) (E/p + 1/a)"] (m? — 1/p*+ 1/0") ey ENGELS A/C-1 ( ae =| o=5 (E+ 2) |S 00 sae ao iG p° = =0; 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 m very great and substitute for € in the left-hand member the values 0 — pio — 0; the signs are = 3F = 5 OF ELASTIC DISPLACEMENT ALONG A HELICAL WIRE. 373 showing that both values of € 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 B/w have the same sign, and thus the waves are not respectively right- handed and left-handed in regard to 8 and w. 9. We have already noted that in three special cases the equation (13) for p? can be solved rationally in terms of m, Pp, o Taking m?=1/p?+ 1/c? =a~ cos® 4, it is convenient to put ms=6, and then @ 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 from which the are is measured to the point at which the arc is s. In this case equations (12) become _ Powp” ain Va q |b g i + la? = oe and equations (8) and (9) become f A sade Fare ypie le a Cale i Ae _ 2m / p o ‘ (V+ 4W e), sony fee so that mgt gle= 1+ as nant (VW), and thus either V+ Wp/c=0 and p=0, or else V=W ; (1 + m’p’), poop? , ie = ( mm Cp? ) (5 ; “ii and a (1 + m’p?) ANS tem + Omep? +1+m’p? The second kind of motion is an example of the quicker wave, and the speed p is given by 8A (p?+ 0? (A +C) o? + 2Cp? 1 poo pot (A+C)o*+ Cp* (207+ p?)* 2 p= In the displacement for which p=0 the equation V+Wp/o=0 shows that there is no displacement parallel to the axis of the helix; we also have W cosa—V sina=W (cosa +sin a tan a) = W seca, 374 Pror. LOVE, THE PROPAGATION OF WAVES, ec. and thus the displacement along the tangent to the circular section of the cylinder is W sec acos 6; the displacement along the radius vector outwards is —asec?a a~cosa Wsin ms, or W sec asin 0, and thus the displacement is Wseca at right angles to the plane from which @ is measured. The helix is displaced bodily, and there is no deformation. 10. Again, taking m?=1/p?—1/c?, where o® is supposed >p* or a<41m, we find that equations (16) and (19) show that either A (m?—1/p?—1/c°? 1— mp? 92=90, p= poo 1+A/Cmp? 1+ mp?’ gel p : Sq Slee or n=0, p= poo (m? — 1/p? — 1/0°) mp? The motion for which g,=0 is an example of the slower wave, the speed p of this wave is given by ae p? (a? — p*) BS pow o (2a*= pi) (1 + A/C) a = pi)’ and the flexural couple G, in the osculating plane and the displacement v along the binormal both vanish at all points of the helix. The motion for which g,=0 is an example of the quicker wave, the speed p of this wave is given by p op = Pot P 4 p= = and the flexural couple G, about the principal normal, the torsional couple H, and the displacements w 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. Tayutor, 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 poimt 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 ircotes pass through the same line*. the surface. There are sets of three triple tangent planes, no two of which 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 In agreement with a notation adopted in a former paper*. In accordance with this notation, the lines in these three planes are denoted by Arabic numbers as follows :— BC, 1 BO’, 6 TEAOM, I) CA, 2 CA’, 4 CAG 2 Zl J35, -B) ACB, 5 AUG ae mal 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 tangent planes which pass through each line on the surface. i, 2 3 1, 6, 15 1g 1, 16, 19 Se Bae Dales 2, 4, 12 2, 8, 14 2, 20, 23 2, 21, 22 ae ae ory 3, 10, 13 3, 24, 27 3, 25, 26 4, 2, 12 4,5, 6 4, 9, 13 4, 16, 27 4, 17, 26 meee 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 Ts. 5 (ue ee 7, 12, 15 7, 16, 23 7, 17.02 8, 2, 14 8, 6, 10 Baeritee 9 8, 18, 27 8, 19, 26 9,1, 11 9, 4, 13 a 9, 20, 25 9, 21, 24 10, S013 10.6 8 10, 11, 12 10, 16, 21 10, 17, 20 Lis 1} ll, 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 1G CAT 13. 4° 9 13, 14, 15 13, 18, 23 13, 19, 22 14,2, 8 14, 5, 11 14, 13, 15 14, 16, 25 14, 17, 24 ele 6 15,.7, 12 15, 13, 14 15, 20, 27 15, 21, 26 16, 1, 19 16, 4, 27 16,- %e.28 16, 10, 21 16, 14, 25 ilps 17, 4, 26 Vie ote 17, 10, 20 17, 14, 24 TS) ila Je 18, 5, 21 Seas) a 18, 12, 25 18, cB23 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, 10, 16 21, 15, 26 22, 2, 21 22, 6, 25 Boe den ali 22, 11, 27 22, 13, 19 23, 2, 20 23, 6, 24 ony IG 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 bo = 1 a to rs bo on > (or) bo ot 0 _ 194) bo oa — _ bo bo bo SS = a bo oO 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 BiCey CAC At 5 eAl3. -AcBr is in the line of the hinges. Each of the remaining twenty-one straight lines is repre- sented 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 9;, 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 7,12,15;, 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’BC’, 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 8, 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 lime 7,8; cuts BC in 9 and cuts the line XYZ in a point such that the straight line joing it to the point 7, gives the points 8,, 9,. In a similar way 4, and 9; give 13, 6; ” 8: ” 10; 1, ” 9; ” ike 2, ” 8, ” 14, Since 10, 11, 12, and 13, 14, 15 form triangles, 10; and 12; give 11, 11, and 12, give 10, WS, dle Ge Ales, 4 elton eee all3. 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 lmes 16 to 27. One of them 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 17, lies on BC and 17, on O’A’, and that the pairs of lines 7,17;, 7,17, and 14;17;, Vou. XVIII. 48 378 Mr TAYLOR, THE CONSTRUCTION OF A MODEL 14,17, meet on the line XYZ 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 LOT eO 95-20) =p 20 1225) sel 13,18 ,, 23 135 22) ey 9 U2, LO ae 24 9,24 , 24 LD eu peas 0 15,20 , 27 LOM 2 eG 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 limes 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 left- hand figure (triangle ABC) :— 6514s bSloy 2 Satie Os 20s (25s 3 26. 16, aloe Os 10K 2 17, 14, 24; 18, 8, 27. It may be noticed that in the left-hand figure the points 4...15 he 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 pots, 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 all 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 :— Wie oa, 9; LS) 12; 0s. (6) 20025) doye23. oO soe LO: ell) DAR es 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 he 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 Dynamics of a System of Electrons or Ions: and on the Influence of a Magnetic Field 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 =(87r)> IG + 8? ++") dr, W = 2m? | (f2+9?+h) dr, wherein 67 represents an element of volume, (a, 8, y) is the magnetic force which specifies the kinetic disturbance, and (jf, 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 units, dyd8 da dy dB da), d,, Gs dz’ de , ae" dm Fa 9"), or say curl (a, £, ase (y g, h), which restricts (f, g, h) to be a stream vector satisfying the equation of continuity: it also confirms the view that (a, 8, y) is of the nature of a time-fluxion or velocity. It is assumed that (a, 8, y) is itself a stream vector, which must be the case if electric waves are of wholly transverse type. On substituting in these expressions (€, », €), the independent variable or coordinate of position, of which (a, 8, y) is the velocity, so that (4, 8, y)=d/dt(&, », €), the dynamical equations of the free aether can be directly deduced from the Action formula 6 fir- 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. -e ke Mr LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: ere. 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 lv +mg+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 7 may be effected by introducing the auxiliary variable (F, G, H) defined by curl (F, G, H)=(a, B, 4). = (i(jaleh CG: (dF dH dG dF Thus T = (87) a - te) *t(g a =| Be ae ay) 7 oe = (87) fae — BH) 1+ (aH —yF)m+ (QF —a@) n} dS Ae ienidy. dC da dy dp da . rs lb tie pd eel df dg dh Om) | Re Ga tH ae male? ed = Oke | a, iB yf ' Now it follows from the definition of (F, G, H) that , d (dF dG dH\_ (dy dp ie dx = dy a i aca Ge x) Bes ii == tere > with two similar equations. Solutions of these equations can be at once obtained by taking dF/dz+dG/dy+dH/dz to be null: this makes F, G, H the potentials of volume distributions throughout the medium of densities i gs h, together with contributions as yet undetermined from the singular points or electrons. The most general possible solution adds to this one a part (F,, G,, H,) which is the gradient of an arbitrary 382 Mr LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: function of position y: but this part does not affect the value of (a, 8, 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 displace- ment 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 pomt 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 4, 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 point- singularity 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(z, y, 2) 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(#, y, 2), concentrated at the point (2, y, 2). 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 4d/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 ee ce a ee 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 beimg 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 G g, h) the flux of the electrons of type e(é, y, 2) wherever electrons occur. In the transformed expression for 7 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 un- changing 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 Ta 5 [Put Gv + Hw) dr, wherein (F, G, H)={. v, w)rodt: and in these expressions the total electric current (wu, v, w) will consist of a continuous part Ga g, h) which is not electric flow at all, and a discrete electric flux or true current of amount e(#, ¥, 2) 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 ef. Phil. Trans. 1897 A, or ‘Aether and Matter,’ Ch. v1., Camb. Univ. Press, 1900. + It may be formally verified, after the manner of the formula for 7 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 1d rd ; ex ages ice t r F= in which r represents the distance of the point from the element of volume in the integral and from the electron respectively. Thus T= 2 | Gig: + get hyhs) M1 dr,d7, + Lex [Ars dt. + Ley [aera dt. + Lez phar dt. + LLepes (dydig + PYo + 2122) Mos in which each pair of electrons occurs only once in the double summation. Also W= 270? li: + 9? + h’) 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 Liv bol Le? (a? + 7+ 2*): or we may write T,= i m (@ + +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 aether- strain (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, 8, y) 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 other- wise it could not be permanent or intrinsic. If the latter coordinates were adopted AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 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 function- theory*. 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 superfluous, 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 8 |(1— W) dt =0, 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 con- stitution 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 wltra-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 im terms of which 7 and W are expressed are not all independent, for when the distribution of (fg, h) is given that of the electrons is involved. The connexion between them is completely specified by the relation (z dg ay rine as =aG. \det dy * dz * More concretely, the relation curl (a, 8, y)=4m(f,g,h) kind whose velocity is (a, 8, y); that are required to intro- involves f(t mg+nhk)dS=0: now {(lf+mg+mnh) dS isnot 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. Vor. 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 a (df dg dh o=|v (E+ dy Ge qe) a7 30 the variational equation will by Lagrange’s method assume the form 5 (2+ T,- W+ Q)dt=0 in which W 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 must 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 4 [fdr involves 4 [[fifirw dnd, that is S3f fir dn8r, because each pair of elements appear together twice im the double integral of a product, but only once in a double summation, we obtain as the terms involving f in the complete variation déf 8 | at [Rar — Aero? [at [refer + abare dr, leading, through the sista integration by parts, to | F8fdr } = Jae [Pera ae | dt i fofdr+ | dt i Pfayae J 2 i dt | = ofan The coefficient of 6f must vanish in the volume integral, giving dF dv 470° f = — an aa BOSON OOOOOOCOOOOOCOOOOSOOOOOOOM OOOO CCOCG (i). Similar expressions hold for g and h. Again, the terms in the variation involving the electron e at (a, y, z) are 8 [dte(aF + yG42H) + bd {at @+p+e)— 8 |atew, yielding as regards en of the position of this electron [ate ree + Gdy + Hds + adF +98G +23H) + m Jat (@8é + 984 + 282) — fateav in which 8% means the change of the velocity of the electron, so that we have on integration — [ate {az Be eG by + 8) - eae bz) +. > by parts é Féa + Gdy + Hbz t dt d tite ia: S, dv —m [at ules ae ae (Ge ut zz bc) +m | ada + ydy +282 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 IDM GRR CNM GM aia The vanishing of the coefficient of 6x for each element of volume gives # ( DEV dE dG a | ML = e\ — z dt me dx ant! dx rhe dx da Similar expressions hold good for mi and mi. The form of W shows that 47c? is the coetticient of aethereal elasticity corresponding to the type of displacement (f, g, h): the right-hand sides of equations (1) 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 dF dv P’=—-—--—. dt dex 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 or, in terms of physical quantities only, by P=yy- B2 + 4c? f. We do not now go into the case of a magnetically polarized material system, for which in certain connexions* (a, b, c) replaces (a, 8, y) mm 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. Morion IN AN ImpRESSED MAGNETIC FIELD. When the electrons or ions constituting a molecule describe their orbital motions in a uniform magnetic field (a, 8); Yo), 1ts influence is represented by an addition to the vector potential (F, G, H) of the term (yoY — Boz, 2 — Yor, Boe — ayy). * Cf. loc. cit. ante. 49—2 388 Mr LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: Thus 7,+T=13m (a+ y+ 2) +5 Se (Fé + Gy + He) +5 | (RF+ Gg + Hh) dr - a nw +55e| * y cee F geld. |a ¥ z | = TROT % Bo 90 | | ty Bo Yo As the aether is stagnant, so that the position of the element of volume 67 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 (yoy — Bod; oz = Yo, Bot zz Ai). 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 (@r, @,, wz) measured with reference to their instantaneous positions, these quantities being either constant or assigned functions of the time. For the velocity, imstead of (a, y, 2) there must now be substituted, in the formula for 7’— W, (@— yoz+ 2oy, J — 2Z@z+ Taz, Z— LW, + Yor), and for ( iB gd, h) there must be substituted (f—go:+ho,, g—hez+foz, h— fw, + gor), while (#, y, 2) remain unchanged. Referred to these moving axes the kinetic energy, which was, so far as it involves the ion e,(#, %, 4), given by 1 Sey ows j 3 ‘ T,+ T= 5m (é2+ yet 27) +e (Fit, + Gy, + MA) +... where (F,, G,, H,) is the value of the vector potential at the point (m4, %, %), has now additional terms which on neglecting the square of the angular velocity are —m|\e% f% 4\+talu Hh aA \+4(¢60F,4+9,5C,4 49H)), boa) eal aller ae @, Wy Wz) | we @y @: ‘ Cody dr: wherein oF, =8 (= SS i=) Tis Tyo 5 22(@y2a = @r2) oe “ Tie Tie 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 e, the first of these terms is the same as that due to an impressed magnetic field given by 2m (a, [shy %) = = 2 z 1 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. (@z » Wy, 2). Mutruat Forces or 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 C00 1g) (4 L2 + YrYo t+ 2120) + FeV €.V2011»/ds, dsp, where ds,, ds, are elements of their paths described with velocities v,, v. The Weberian theory of moving electric particles involves on the other hand a kinetic energy term heer. 1 (dr,,/dt: 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 (/, m, n) adds to the force acting on an ion of effective mass m and charge e, situated at the point (a, y, z), the term eH (ny — mz, l2—n&, mx —1y), so that its dynamical equations are modified by change of #, ¥, 2 into Z—x«(ny—mz), y—« (l2—ne&), 2—K(me — ly), where « =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 x, and this alteration of the dynamical equations of the molecule will be, to the first order of «, the same as would arise from a rotation of the axes of coordinates to which the system is referred, with angular velocity $« around the axis of the impressed magnetic field. Hence the alteration produced in the orbital motions is simply equi- valent 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 «/27, 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 4« 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 same 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 4« 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 prcportional 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 m, 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 ee : md, m(*—re*) and rar (7) Fes heer LO Se F to m{F—r(@+o’)*} and zal ir? (@+ @’)}. Thus, » being small compared with @, the new forces required will be —2mree’ and = — (ra) ; whereas the force arising from the magnetic field acting on an ion moving with velocity v is 2mvw’ 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. Phil. Mag., 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 con- figuration 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 in 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 con- figuration 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 10° diameters of the molecule in one of the periods concerned,—-the error is in fact of order 10-*. The motional forces between two electrons are of type, as regards one of them, KO Gh hs \ a aa, ~ a,) 2 HL + Yo + 22, Ary, \ ( Ti + au, Ets P 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 mv*/}r=c*e/r%, * 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=10': thus v°=4c"e/mr, so that, taking r to be 10-*, e=10™, we obtain »=10*c; thus the orbital period comes 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 terms in the motional forces between two electrons are of orders ¢,¢4/r, €,@a°/r* as compared with their statical attraction of order c*e,e./r? and the forces arising from the impressed magnetic field H of order eH; the ratios are thus of the order of 10-* to 1 to 3.10-°H. Thus when H exceeds 10%, 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 GyrRosTaTIC TYPE. 5. Sensible damping of the vibrations of the molecule owing to radiation cannot actually come 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 6 |(Z- W)at=0 where T=tim(@’+7+2)+4d«|\ a y 2 Cay ae |o m n thus it comes under the same 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 momenta 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 comes 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 more- over 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 « 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 But if that these periods would not be definite, and instead of a sharp magnetic resolution of each molecule with regard to the magnetic field. were so, optical line there would be only broadening with the same general features of polar- ization. To that extent the phenomenon was in fact anticipated from theory, except as The definite resolution of the what would have been predicted on an adequate theory, and thus furnishes a clue regards its magnitude. lines is however an addition to 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 Each of them will periods of vibrations, slightly modified however by the mutual influence in the molecule, instead of being fused together. therefore preserve its free proximity of the other one. For the case of a molecule containing two identical atoms with their each of these identical periods would be doubled*: thus the series of lines belonging to the atom revolving at a distance large compared own dimensions, 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 Their vibrations relative to a system of axes of reference rotating along with them will thus other with but slight disturbance of the internal constitution of each of them. 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. atom in the molecule, those of a molecule consisting of two such atoms would thus If the lines of a spectral series are assumed to belong to a definite 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 cations 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 lmes of a series were thus two modifi- * In illustration of the way this can come about, consider two parallel cylindrical vortex columns of finite section in steady rotation round each other. Each by itself has a system of free periods for crispations running revolution is different, and each single undisturbed period becomes two adjacent disturbed periods. Analogous con- siderations apply to the interaction of the two atoms of the molecule, rotating round each other. round its section: when one of them is rotating round the other, the velocity of the crispations which travel in the direction of rotation is different from the velocity of those that travel in the opposite direction: thus the period of Vout. XVIII. According however to Smithells, Dawson, and Wilson, Phil. Trans. 1899 A, it is the molecule of sodium that gives out the yellow light, that of sodium chloride not being effective. 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 @ priort be incom- plete: 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 piezvelectric 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 system—nor in a more general kind of action of the same type,—which can introduce electric polarization. en 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 inde- pendent of the velocities or masses of the ions. The fact that the polarization is proportional to the imducing 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-*cm., there will be 10% molecules per unit volume in a solid or liquid, and the aggregate of their intrinsic electric polarities may be as high as 10%.10-*ec electrostatic units, where ec is 3.10-°. Now the moment of polarization per unit volume for an inducing field F is (K —1) F/87; 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 7 the period of a molecular orbit, which is e/nv or 10-%v per cubic centimetre, where v is the velocity in a molecular orbit whose linear dimension J is 10. Thus the superior limit of the magnetization if the molecules were all completely orientated would be of the order 10~*v, which is large enough to include even the case of iron if » 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 diminished 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—2 396 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 pw of a transparent medium with the frequencies (p,, ps, .-- Pn)/2~7 of the principal free vibrations of its molecules, which are so great that radiation travels over 10° molecular diameters in one period, is of type YP ieee we+2 ~ pp in which A, is a constant which is a measure of the importance, as regards dispersion, of the free principal period 27/p,. The quantity on the right-hand side of this equation, of form f(p*), 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 «x 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 4«. This will modify the configuration of the vibrating system relative to the circular wave-train passing over it in the same 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 p/27 to p/27+x/47, the latter term 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—d« or p+43« for p according as the train is right-handed or left-handed, thus giving when «* is neglected, pea 0+ a5— a, = K,R +a, yo *), 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 (J, m, n), for which 5) (72, (4) R) x expe ¥7 (Ix + my +nz—Vt), p=2nrV/N, V=cKk', AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 401 this may be expressed in the form aes fo - ee , _n OF NA; My Act (f’, 9, W)=-(Kp ee eee Me so that, when Ky, K., K, 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 20K *x-2 (lay, ma,, nds), Which implies a coefficient of rotation of a plane-polarized wave equal per unit distance to 27°K—A~“(la,, ma,, 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 (a, ds, @s;) 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 @ with that axis is proportional to cos’ @, not to cos @ 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 con- stitutive relation of type 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 has 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 found 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, Journ. de Phys., 1885; Lefebvre, loc. cit., 1892; O. Wiener, Wied. Ann., 1888. Wor, SOVi0Ul 51 402 Mr LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: GENERAL VIBRATING SYSTEM IN WHICH THE PRINCIPAL MopES 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 ito 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 harmonics, 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 cirenlar 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 complete 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 imdependent molecules, the circularly polarized lght 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 &, 7 variables in what follows. For simplicity let us take the axis of z parallel to the impressed magnetic field, and let (XY, 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(&#—Ky)=X, m(y+nr)=Y, me=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 E=a2+vy, n=x—vwY, f AND THE INFLUENCE OF A MAGNETIC FIELD ON OPTICAL PHENOMENA. 403 so that 2a=E+m, 2w=E-—n, (Sty Up eet pied dé da dy’ dn dx dy’ and therefore 2 the equations become F m (E +exKE) =X + cY, m (7 —ixn)= X —cY, mz =Z. If therefore X +cY is a function only of the & coordinates of the electrons, and X —vY a function only of the » 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 = (d ad ad Cae where & 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 Mrs oe PA ae 2k dW pe Oe Ge mi —iY)=-— Etat The solution of the complete system of equations, three for each ion, will in any case involve the expression of &, 7, 2 for each ion as a sum of harmonic terms of the form e”* 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 system of elliptic harmonic motions of definite forms and phases. Their components in the plane &, » will be circular vibrations only when the £ and 7» coordinates vary inde- pendently of each other, that is when dW/dy is a function of the £ coordinates of the ions alone and dW/dé a function of the 7 coordinates alone. This condition can only be satisfied, W being real, when it is a linear function of 2 and of products of the form &,», or &,n5: it may thus be any quadratic function of the coordinates which is invariant in form as regards rotation of the axes of #, y around the axis of z. Under these circumstances the free periods for € coordinates, 7 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 é,. = DA, etPeitiar a, being chosen so that A, is real: thus p= =A, cos(p;t+a,), Yp==A,sin (p,t + a,) 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 Or = =B,eat + 8, * Routh, Essay on Stability, 1887, p. 78; Dynamics, vol. 1., § 319. 51—2 404 Mr LARMOR, ON THE DYNAMICS OF A SYSTEM OF ELECTRONS OR IONS: so that x, = =B,cos(qt+B8,), Yr=—>B,sin (qt + By), 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 beimg invariant in form with respect to rotation of the axes of courdinates 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 On writing fa ht Ep! et they become BPE eae __ kaw “Tiare The form W will be unaltered when it is expressed in terms of €&, 7’, provided it depends only on the mutual configuration of the ions, and « is the same for all of them; hence when x* is negligible compared with unity, (&, 7’, z) are determined by the same equations as would give (& 7, 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, 2) of an ion be resolved into two parts, namely (#, 7%, 4) which are known functions of the time and represent its mean or steady motion, and (a#’, y’, 2’) which are the small disturbance of the steady motion constituting the optical vibrations. When this substi- tution 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 (z’, y’, 2’) 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 harmonic, 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 =—434,, (a, — ay)? + (yr — Ys + (Zr — Ze PB} + UE Bre (Cr ls + Yr Ys + Zr Ze ), =— 42 Ars {(E, ay iE) (ny = Ns) 3° (Ee = eee) =F LE Bre (E, Ns. + Ee ny ar 22,25). Thus in the absence of a magnetic field the vibrations of the a 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 m/k=X and mk/k=X', the equations of motion are of type ou 20 N= N+ 2 =O, sanliesny NE + E+ 2 Bes dn The periods of the right-handed circular vibrations, of type & x e”', period 2zr/p, will be given by the equation | =i Ai'p =2ZAt,, Cr, Cy, Cre era, Ci 0) Cu, = Gir De p — ZA,,, C3; Cus; Con | Oe Cis, = Asp” = dsp = Ae (Cin, tee Cr | in which C,,=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 A’ 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+ 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 + e.g. K, Cy, Cy; --- Cy, Co’, «.. in Phil. Trans. 1897 A, electric field, it is only the vibration polarized so that its pp. 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 Mathematica, t. Xxu. (1888), p. 89; and depends for its interest on the remark- able 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. Ul. 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 con- currently with the real part; (11) and thereby, as would appear, the coefficients in the quadratic function used by M. Poincaré (Acta Math. t. xxt. p. 174) are shewn to be zero. (ili) The theory is put in connection with Kronecker'’s formulae (Werke, Bd. 1. 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 ¢,,, §§ 12, 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. u. In that paper there remains in the resulting formula an integral function of which the existence is proved, for which however no definite expression 1s 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 (n—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. inal Mr BAKER, ON THE THEORY OF FUNCTIONS, ere. 409 Part J. 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 (a, ...,2,) the n points P, with coordinates (2, + d,a, ..., 2 +d,%p), P,, with coordinates (#,+ dpa, ..., 22 +dn2pn), it being supposed that the determinant, M, of n rows and columns, whose (7, s)th element is d,#,, is not zero. At each of the points P,,..., P, we can similarly take n independent consecutive points, those at P, being Py, Py, ..., Pm; 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 P,, namely P,;, is the same as the rth satellite pomt of P;, or P,,, or in other words that the suffixes shall be commutative, we can associate the deter- minant M with the ‘cell’ which is defined by the 2” points Iz P}; MOOS) ae Jer VDOe] ee COOK sens whose suffixes consist of all the combinations of not more than n different numbers from 1, 2, ..., nm. 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 dS,. Similarly if we have in n dimensions a space of (n—r) dimensions, defined suppose by r equations fila, +s-, Zn) =O, ..., JAG tees In) = 0, with a certain number of inequalities, we can associate with every point P of this space (n—r) satellite points, P,, ..., P,_,, also lying in this space, the coordinates of these points being denoted by @, + AyX,, --., Ln + dtp; [py 2 conn (os 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 2”~” points; with each of these cells we can as before associate an element of extent for this space, which we denote by dS,_,; this is defined as the positive square root of the sum of the squares of all the determinants of (x—r) rows and columns which can be formed from the matrix of m columns and (n—7) rows | day, dy, eee, dy @p |, Healt 2, sey (n—7), or, what is the same thing, as the positive square root of the determinant of (n—7) rows and columns which is formed by multiplying this matrix into itself, row into row, in the ordinary way. Vou. 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—7)-fold. We also use @) quantities, called the direction cosines of the normal to the (n—vr)-fold; let the (m—1)-folds A) be always supposed taken in the same order, given by the suffixes; let 6, d, e, ...,h, k be any r of the numbers 1, 2, ..., n, no two of them equal; then the ratio of the Jacobian Ih = Of =» Fr) > I) (xp, ---> Ze) on an) to the positive square root of the sum of the squares of all the possible (") such Jacobians is denoted by x»,a,c,...,4,e, 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 ©) direction cosines |r — 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 —7+1)-fold given by GPs 505 F 2=O, is completely bounded by a closed (non-singular) (n—7)-fold given by (Aa cot Fae (SO, and that throughout the limited portion of the (n—7r+1)-fold we have f,<0; let P be any function of a, ..., 2, for which it is supposed that itself and its first differential coefficients are finite and continuous (and single-valued) throughout the space considered ; then oP oP [Ic WA a (- ID ras: tk Opt +e. + Kbd...h | ASn—r3 = Kea...tk P -ASp_r, 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 (7—1) numbers in their natural ascending order. If we introduce & functions such as P, and make the rule that an interchange 1 8 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 f OP ae... -hkm [fete .— dS, 1 = Dikod...AkLba...Ak@Sn—r> Ox m ee 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, ..., , and the second summation extends to the (n—7r+1) different values from 1, 2, ..., » which m can have so as not to be equal to any one of d,e,..., h,&k; on the right under the integral sign the summation extends to every combination of r different numbers 0b, d, e, ..., h, k from Ts OAS eens 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 H,_, of an (m—2)-fold, which is given by the equations VA Charron 2) i Cescny Cn) Oy to be bounded by a closed (non-singular) (n —3)-fold H,_; given by TAM@Bp oe5 DSO, GACH cn, GDHO, j(Go o055 >) =O We can imagine H,,_, divided into cells in a manner before indicated, the satellite points of P, whose coordinates are (#7, ..., 2,), being denoted by P, whose coordinates are (+ dpa, ..., n+ adptn), k=) 2. 225, @—2); In general the differentials d,#, are arbitrary, save that the determinants of (mn —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 Cyaaky > Un aXa; tee Un ohn all but three, say all but d, a, dpa, dn.x,, are zero; the ratios of these three will then be determined from Of. INOf or ee eae eee Oxy nao + 02, n—2%e + One An—2 Lp =0, Oe gn + Oh af, ie ax, nay + 02, dye + aa, An—s%p, = 0 ; it is clear, in fact, that we can draw on H,_, through every point P a one-fold (or curve) along which all the coordinates except , #,, %, are constant; taking then any point P and taking (n—3) of its satellite points P,, ..., P,_, arbitrarily, we can draw such a curve through P and each of P,, ..., Py_s, and take for the satellite point P,» 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,, ..., Pys, as edges, 4. A set of (n—3) neighbouring points Q,, ..., Qn; in which the curves drawn on H,, through P,, ..., Py, intersect the (n—3)-fold H,_; may then be taken as the satellite points on H,_, of the neighbouring point @ in which the curve through P intersects H,_,; we have thus a possible basis for the division of H,-; into cells, which it will later be convenient to adopt. We assume that the curve on H,-. which is drawn 52—2 412 Mr BAKER, ON THE THEORY OF FUNCTIONS through P intersects the closed H,_, in an even number of points; and to shorten the proof we shall speak only of two, say Q® and Q. Then if the differentials dyp_.a,, dn%, Ant, be always taken in the same direction along this curve Q® PQ, the expression Of; of: a Ons fs = A, Cymay ar a Ona at oie naa @h will have different signs at Q® and Q, and in fact, since f,<0 over H, 4, the expression will be positive at the point, say Q, where the curve through P leaves H,_,, and negative at the point Q, where the curve enters H,_». 5. Considering now any point P of H,_,, and its satellite points (a ar dry, see y En Se dx), k = i 2 aise hs (vn = 2s in regard to which we do not until special mention is made of the fact introduce the con- vention that all but three of the differentials (dno, Bot pal) are zero, we have oS on x ae, dx, SP aao SF om AL, = (i), Of Of = a aya, +...+ ae, Gao, —0; and hence easily find ee aT Se =lEn=ay) SLY; wherein Jr. = gh oi > Oiaa Ons Ea OX,’ Oils and M,, denotes a determinant of (n—2) rows and columns, obtained by taking the deter- minant which remains when im the matrix of nm columns and (n—2) rows CRAs cos CBS |) 5 k=1, 2, ..., (n—2), the rth and sth columns are omitted, and prefixing to this determinant the sign (—1)**7 or (—1)"** according as rs. We require now to make it clear that we can suppose the sign of the ratios €,_, to be the same for all points of the limited H,_,; for this purpose suppose Py, With coordinates (a, + dn 2, -.., Gn + In+2n); and P, , with coordinates (a,+dn2, ..., n+dntn ), OF SEVERAL COMPLEX VARIABLES. 413 to be satellite points of P of which P,_, is on f,=0 but not on fr:=0, and P, is not on either of the two A=0, £=0; then we have Jn nck Sen €xn—2 = 37 ce — 49 ‘ieee rl M ™m : eG Jn Chas Char coo ae Ua hn Bp VE pana ee eae eee Le (aha ee dna pez, _ ; where M, denotes the value obtained by taking the determinant left when in the matrix of m columns and (n—1) rows Ghehs coon Chee \\ 5 [PEL PX, Sao (GS A), the rth column is omitted and prefixing the sign (—1)"*” to this determinant; hence as dn f,=0 we have of: En—2 _ Oly aye Si of, Oa, OL, ~ M, MY,’ Ores 1 a Ge Gis T WhChaGe ao awe _ an fi Sie where WM is as before the determinant Cres coon Ch, || » aM PL oon 1 Thus on the whole we have pa. Jy fr nS drs ze = anf » Ina fr «Una Sa nl Cerrames aan a ae ee We now make the assumptions (1) that for all points P of Hy» the satellite point P, is taken in space on that side of the (n—1)-fold f,=0 for which f,>0, so that d,f, ts constantly positive, (ii) that the satellite point P,_, is taken on f,=0 on that side of f.=0 for which f2>0, so that dr fz is positive, (111) that the satellite points P,, ..., Pn» are constantly taken on f,=0, fo=O in such a relation to P, and P, that M is constantly positive over Hy». Under these assumptions each of the ratios J,,/M,,; maintains a positive sign over 414 Mr BAKER, ON THE THEORY OF FUNCTIONS H., and the direction cosine x,, of the (n—2)-fold A=0, f.=0, which by definition is given by Krs = Jrs =| VSJ*5 i and is therefore equal to Sgn €n_o. My, + | VEN, where sgne,. means +1 or —1 according as €_, is positive or negative, has through- out H,_, the sign of M,,. Thus Keds. = M,, sgn (#2) = M,, sgn (- on fi tea) 5 oe 6. Next we consider any point Q of H,_;, and its satellite points (@ +dya,, ..., Gn +dztn), b=1e 2s 5. (Ue) From the equations Aidit + ... + fidztn = 0, SodeX, + ... + fodeXn = 0, Fedex, + arte + frdyx, = 0, [p= LD oars ee (n— 3), we find as before Jk af = Syst = = r ap ae Ens, SBY; where , ie] B, BH es an 02,” OX,” Oz, | H % *, OL,’ Ox,” Oat, We a a Om? Ox,” Ox, | and M,,. is the value obtained by omitting the rth, sth and &th columns in the matrix of n columns and (n—3) rows | dpa, ---, Uptn|, k=1, 2, ..., (w—8), 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 x rows and columns whose first (n—3) rows are formed by the matrix just described, whose (n—2)th row is Aj, ..., An, whose (n—1)th row is B,, ..., B,, whose nth row is C,, ..., Cy; then the expansion of this determinant is 1ton DS> ABC Meret; r+s$l thus when r, s, ¢ are in ascending order the sign to be prefixed to the determinant OF SEVERAL COMPLEX VARIABLES. 415 by which M,. is formed is (—1)"+*+*+*4, Hence, taking the satellite point Qn», of Q, upon 7,=0 and f,=0, but not upon f,=0, we have Tra ws Jy0 a — Sren 8 Mi eg eens? = Tre Oh ih qr ooo Sr deen Oh ins p Mra dn 2%, + EM pn dn En’ here the numerator is ofr of d oye ax,’ Or,’ of. of s . ax, 5) On, ? An—of 2 of: OF: we Js ? Ghrenyf 3 and therefore, since dp» f,;=0, dnf,=0, is equal to es Ano fs; the denominator can be seen to be exactly equal to M,,; thus we have Irs En—3 = 1,0" af 3) and so obtain Krst = Syst > | NSS at | M rst Ts ase ok - nafs) the element of extent dS,_, being by definition equal to | VSM? 4 |. 7. At this point it is convenient to recall the connection which has been established above (§ 4) between the division of H,_, and H,_, into cells. With that arrangement the J,, and M,,; now arising in the consideration of x, may be regarded as identical with those that arose just previously in considering «,,. We proceed to utilise this. Consider the determinant of n —2 rows and columns Chend@an Chenlity cot Chess wherein (2+d,2,...,7+d,.”) are satellite points of (#), upon H,_,, the columns con- taining @,, @s, # are omitted, and P,,. is a function which is single-valued, continuous and finite upon H,_, and H,_;, and possessed of single-valued finite differential coefficients. The suffix of P,. 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, 8, ¢ are in ascending order, is equal to oP. rst ) oP, rst aa ne + Ga Mr il: (—1)yt#+# kK hey = On the other hand, supposing as in § 3 that all the differentials d,.2,, ..., dno®n except* dp ot, Anois, Un—2% are zero, the determinant is equal to CG 1) see Ano Pret. Myst. Hence finally we can evaluate the integral (OP str OP res OP,, I (wee Gont kn are + Mra ae *) cis taken over H,_,. Suppose H,_, divided into strips as in § 3, and find the contribution of one of these strips. The integral is [ 2 / rs \ OP str OP vig OP rst II en (a,.) ° | Ma Ok, aie Ory + Mrs 0a, |’ which by the identity just found, and because we can suppose sgn (J;/M,s) to be the same over the whole of H,_., is equal to pf sade as we pass along the strip under consideration the determinant M,, is constant; thus the integration along the strip gives he aan (iz) | Melee — PO, where the single integral sign indicates an integration extending to all the strips, and P®,, P®, are the values of Py. at the points Q”, Q° where the curve of integration through the point P, along which only the three coordinates «,, as, a vary, respectively leaves and enters H,_,. We have seen that df s Kret@Sn—3 = Ma sgn oe (eae fi) ; and moreover that d,_,f, is positive at Q” and negative at Q; hence the element cgn (372) Mun [P= Pa) is the sum of the two elements of the integral over H,-; which is expressed by fora PradSn-s which arise corresponding to the cells at Q” and Q. Thus we have proved that the * In § 3 the differentials not zero were denoted by dy» %y, Ayo Xe, Ano The OF SEVERAL COMPLEX VARIABLES. 417 latter integral, over H,_,, is equal to the integral from which we started which is taken over H,_.; and this is what was desired. We can then by a summation infer that i 2 Krst spat WS ,-.= || = Kr, = Se : USn—2, 1, 8,t rT, 8 t Okt which is a convenient way in which to state the result. There are (5) terms in the left- hand integral, and (n — 2) (5) =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, ee ec rs t On, ? ee 0x5 we have = 0) o } as nm necessary conditions that the integral | | Se Krs ING: diSi=s5 taken over a finite portion of an (n—2)-fold may be capable of being represented as an integral over the closed (n —3)-fold bounding this portion. If these conditions are satisfied, functions P,,, satisfying the equations : OP rst Xe nett Ts ; Ox must be found, in order that the expression may be possible; but it is necessary that the functions P,. so found should be finite on the (n—3)-fold (cf § 28). The equations yy xe = Xr _ s Os 0 have been given by Poincaré (Acta Math. 1x. (1887), p. 337) from a somewhat different point of view. We can as an application generalise Cauchy’s theorem to the p-fold integral | a, le (Gap ee abeeae where &, ..., & are complex variables. For example, for n=4, the integral ||P Go &) dédé., Vou. XVIII. 53 418 Mr BAKER, ON THE THEORY OF FUNCTIONS taken over a closed (n—2)-fold, which (see § 9 below) may be interpreted as [Fe E.) (Heys + Uheyy + tas — Kas) TSy_o, is equal to the integral | Cc eae) (2 ny -) Fee) (e He x) r| dS taken over the (n—1)-fold bounded by the (n—2)-fold; and this vanishes identically on account of Gime Omer CMON 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 (of § 9 below) Ky3= Ky, Ky = — Kos, and therefore Ky3 + Uky4 + Ukeg — Ko, = 0. Part IT. The expression of an integral function whose zeros are given. 9. In what follows we consider a space of n dimensions, x being even and equal to 2p. The points of this space being as before given by the n coordinates 2, ..., Zp, we define from these p complex variables by means of the equations Ep — Capen te on, (r= il. 2; Ts) Pp) 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 («,, ..., @,) of itself by the vanishing of an ordinary power series in the quantities £—&, .... &—&,, where — — 7 ,4+ 729. 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 &, ..., E, (§ 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 P(E, ---, E)=utrw=0, where wu, uv are real functions of the n real variables a, ..., #,, which satisfy the equations ou du ou ov Oba Ota) | Ona Omeee, OF SEVERAL COMPLEX VARIABLES. 419 thus if we denote du/dz, and dv/dx, respectively by wu, and v,, the direction cosines of the (n—2)-fold, defined in § 2 ante, are given by Kr, ¢ = (UyUs — Usd,)/h, where fA is the positive square root of SE (py — Ugdp)? = (42 + eee HU P= (OE +o. En); r,s now we have Woyp—1 Vog—a — Ung—1 Voy—1 = — Voy Ung F Vog Use = — Usp Ung + UsyUos— 5 Uoy—1 Vog — Ug Vor—1 = Voy Uog—y — Vos—1 Wor = Uny—1 Uog—1 + Uy Uog, so that Zz __ Uzy Urgs—1 — Wor—1 Uog ae aL uy” sEand + Up? ; Ker—1,28 — — Kor, 034 = ee a ey a Up+... Un? ’ and Rig + Ky Foe. + Kno = L. 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 con- sideration 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 a, ..., %, the coordinates of a point on the (n—2)-fold or on the (n—3)-fold, and by (4, .--, tr) the coordinates of a finite point of space not on either of these, the corresponding complex variables being as before given by Te lee tbe BN OB cso ya Let L,, ..., In and R,, ..., Ry be single-valued functions of 2, ..., a, and of eco en which are continuous and finite, with their differential coefficients, so long as (a, ..., x») is upon the (n—2)-fold or (x —3)-fold under consideration, and the point (t, ..., t,) is in finite space and not upon the (n—2)-fold or (n—3)-fold; further suppose that these functions are such that OL; df; ; i Be (Gas = 2) cal oR, , oR, oR a +3, tot aa =, Consider the n integrals bpm [ela + one + kml) dus Gaia a 420 Mr BAKER, ON THE THEORY OF FUNCTIONS taken over the limited portion of the (n—2)-fold; we have OC, 0b, __ oR, oR OR, _ oR, at, ~ Ot, =-|(cn Oars +... +r Or, — Ky On, see — Ken =) dSn—2, and therefore, adding to the right hand the vanishing quantity oR, , oR, ORn [re (Fe ry nea a) dS,_2, we have Se eS ots Ot, i=1 ie tls = $ (feet aR; oR; (« a ae =) Saas where i does not take the values r, s. Thus by the formula proved in Part I. we have OC, == [3 at, — 2, =/2 KR; SUAS as += where the integral is taken over the (n—3)-fold bounding the limited (n—2)-fold over which the integrals &, ..., & are taken. 11. It follows that if the (n—3)-fold integral vanishes, the expression Edt, + ...+ Grdtn is a perfect differential; on grounds further considered below (§§ 12, 22) we suppose that this (n—38)-fold integral does vanish; we suppose also that L,,=7L,.,, and that the (n—2)-fold is a complex (n—2)-fold; then from the equations Ker—1,23—1 = Ker,osy Kor—1,08 = — Ker, 23-1 it follows that Gy = [tem a AiGor's) I, AP ose ar (Ker, or) Tie +... + (ice. n= UKer, n) Dinah WSs ma i [fem SF UK 27-1, 2) q, SOO Sr (tara, or) Dp Seeetste (Kora, ett UK2y—1,n) Ly} aSp-2, so that oer = KEL and therefore a Otos tor Obra Otosa : which gives ce + iz) for = 0. Thus, under the hypotheses introduced, all the functions &, &, ..., are functions of the complex variables 7,, ..., Tp, and there is a function ®d = [Ga ar f.dt, fora et Endtn) = [Gar aF Edt. + vee + Snap) OF SEVERAL COMPLEX VARIABLES. 421 for which 12. We now give a more special value to the functions Z,, ..., Z,. Let -2 [(a, —t)? +... + (aa — ‘ace n—2 ¢ (x|t) =— and Ay,= Qe (ax|t) -—¢ (x|0) + € 2) a (2|O)—... + S Die (: aie P (el0), m! 0a m being a finite positive integer ow zero, and 0 0 0 (+5) = t Fre Poon Nara When r, =#2+...+¢,7, is sufficiently small in regard to R?, =a,°+...+4,7, we have (—1)"# (-— 1yrn ( a) (m+ 2)! m+1 ‘ eC RRG by t=) a (x|0) + a) m+2 ee @(z|0)+... to o, ypmrl x gmt ~ Rem King (w) + Rum Kins2() +... to 0, where yw denotes (af, +...+@,tn)/Rr and is numerically less than unity, and K,() denotes, save for a factor independent of ~ but depending on n and s, ( ies ae ( es aa Lal 5 dps ye 7 : As we can find a real angle 6=cosy, we have = 1 Le Ei eae oF ee —— @ (alt) — — - a1 Rie (-Re ) } by expanding the bimomials and considering the explicit expression for the coefficient of rt4/R7*k1 it is immediately obvious that this coefficient is not greater than if 6 were zero. Thus when rvr has zero for limiting value as & increases indefinitely. It follows that a value R, can be chosen such that for fixed n, m, r and all values of R>R,, we may have page, esate Reem ! with B a finite constant independent of R. 422 Mr BAKER, ON THE THEORY OF FUNCTIONS It is easily seen that an equation which holds also for negative m provided H,» be then understood to mean g («| t); and thus 2 3 oe eee @ (aaa * aga t + ea) # (ars ons ) ae ar Hence if we put sks. Oleks Ti tLe (5 on a a) ; ‘Maks j0Hma Re =the 3 —— ( Oly Or ' 2 where m is to be kept the same throughout the investigation, we have aL, _ aR, Ote Oars g ORS Ror) _ ; Gre les et) - ey ae che Fe See ( OFsra = OX3y , so that the conditions of § 10 are satisfied. We suppose also the further condition, of § 11, namely that the integrals [2 = (Ky, 32k + UK, & ok) (ee * goer) dSn-s, OX Oxy, / 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 (t, ..., t,) is in finite space and (a,,...,%,) is very distant, the function Hy. is a small quantity of the order of (a°+...+.,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 limit- ation 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 f,,...,€, to extend over the whole infinite extent of the (n—2)-fold, we suppose that they and their differential coefficients in regard to &, ..., tf are convergent. OF SEVERAL COMPLEX VARIABLES. 423 Then we have the result ; the p functions . x “ OH m . 0H, Cora = 4 i = (Kor, os—1 + Uop—1, 2) ( tin 8 =1 \OXo3 4 O2og ) dS», extended over the infinite complex (n—2)-fold, are functions of the complex variables Ti, +++) Tp, and are the partial differential coefficients of a Function 2 =| dr, + €,dt.+...+ GeidTp). 13. We proceed now to put this result into a new form, from which it will appear that the real part of the function ® 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 Grey = 6a, + COs ’ we have, clearly, On 0H, 0H, al Oor—1 = ls 1 Ae Koy, 2 an pet Kor, n Ala ) dja; vy 2 n oH, OH. n OH», ul Cro [- Koy—1, 2 a, + Koy—1,1 aa + Koy, n-1 a ; ) GNSS ; of these the latter, in virtue of the equations Koyr—1, 23-1 = Koy, 08, Kar—1, 99 = — Kar, 28-1) can be written OH oH. 0H, i Sar = | ( Kor, a + Koy, 9 Bo + eee + Mor, n | AS Ox, 0X2 0Ly, We proceed now to shew that in fact pe) Ona = ees WSs bo, = — in. aSna, J Oo, J OX oy 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 6,,., namely in ET 0H m Soy = -| Kor—1, or = > WSn=a + | [+ 21 ar | + Koy, oe ; OTS oy 5 kaa orl k=1 Loe OW ng where & does not assume the value 7; over a finite portion of the (n—2)-fold, bounded by a closed (n—3)-fold, the integral i oH. OD n 0H, |(+s orl aE Koy, ok—-1 Dave, + Kok, ok Aya ) ASj-» Oop orl is by the results of Part I. equal to | Koki, 2k, 2r—1 Jal, m OSs 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 which, since Ky t Wy + 20 + Kya, n= 1, is the result stated, namely that memes Ns bs Hm a oH Beam |( tone Gye Kant gig too Maret gg) Sane Thus also = = bya =- | (2 —1 2s dS_s, r / rl 2r and, if we put (Poincaré, Acta Math. t. xxu. p. 168) V = Anas aS, so that V, which may be regarded as a generalised potential, of the (n — 2)-fold of integration at the point (4, ..., t,), is the real part of ®, or differs therefrom by a constant. 14. Supposing that the integral is taken from the point 4=0=#=...=t,, which is supposed not to be on the (x —2)-fold of integration, we may write ? =[ (fdr, ae dt, + 309F fop-14Tp), 0 = | (Sia Nae eee, a eee) 0 +i | (geste + one, FO dee): 0 of this, in virtue of the results of § 13, bearing in mind that Hy» 4, and therefore also V, vanishes for 4,=0, ..., f:=0, the real part is exactly V; the imaginary part is iQ where OF SEVERAL COMPLEX VARIABLES. 425. Q =) (8, dt, +... + Sndtp) 0 , oa oH 0H, 2 Neches ipa ( CELE cbt My ose) ac n- att 1° er—1 | Ker—1, 1 On, + + Koy, n Oa, yah a TT + dt», (n,1 5 a + Kor, n> Sk Ox, OLy lake ™ =[ [as,.¥ kro (die gy OE , Jo/ 7,8 Oils Ou, the summation extending to all pairs of different numbers 7, s from 1, 2, ..., n; now we have seen (§ 5) that if M,, denote, for r). With these results we combine now the following, which is a particular case of a theorem of Kronecker’s. Let f(m,..., 7) be a single-valued function finite and con- tinuous upon a certain closed (n—1)-fold, whereof «,, ..., «, are the direction cosines; consider the integral 1 f ; 0 =e ; 3 fi) 0 eh Ge anes) \@ + Uk) ee —i =| Q (a|t)+ ... + (Kn + Un) Ge —71 | g(x | o} Ship where (#,...,,) denotes a varying point upon the (n—1)-fold. By Green’s theorem it is immediately clear that this integral is unaltered by any deformation of the (n—1)-fold of integration which does not involve a crossing of the point (4, ..., t,) or of any point where f(m, ..., Tp) ceases to be finite, continuous and single-valued. For the condition for this is simply (Part I. of this paper) (pe tie) SE,» &) (Se -iZ) oC). =0, Gi +254...) p(e|)=0. namely ( ane Oa,2 54—2 428 Mr BAKER, ON THE THEORY OF FUNCTIONS Hence, if f(t, .--, 7) be single-valued, finite and continuous for the whole interior of the (n—1)-fold of integration, the integral is zero when (t,...,f,) is outside, and, when (t,.-.,t,) is imside the (n—1)-fold, it is equal to. f(m%,.-.-, Tp), as. we see by supposing the (n—1)-fold of integration to be deformed to (a, — 6)? +... + (@n—fP=r, and then taking 7 to diminish without limit. Now consider, in the region of convergence of the series ¢, a (multiply-connected) closed region, bounded by (i) part of the (n—1)-fold = surrounding the (n— 2)-fold ¢@=0 which has already been described, (ii) part of a closed (n—1)-fold S described in the region of convergence of ¢, the part being limited by the (closed) (nm —2)-folds in which S is intersected by =; and take f(r)= tog 6, where r is one of 1, 2, ..., p. Then when (4, ..., t)) is interior to the (multiply-con- nected) region above described, we shall have PO=L[PO lea ried (F -1Z) 0 CI0+--f dun where the integral is taken over the two partial (n—1)-folds denoted by (i) and (1). The part (ii) of this integral is finite for all the positions of (4, ..., f,) under con- sideration; consider the limiting value of the part (4) as the (n—1)-fold & is taken nearer and nearer to the (n— 2)-fold 7, namely by the decrease of the quantity denoted above by «. By what has been stated above we may ultimately put Kog— + Uog = — ($s) 7 ee i » f®= “ , G@Sr. = edédS,_.; then, if (t,...,f,) have some fixed position at finite, not infinitely small, distance from the (n—2)-fold J, we obtain, for this part of the integral, of =| do | Fed Sas {3 (gy 2" * (go = isa) eld) --7/% ae ($e) Gea igi) | 9} An now the direction cosines of the (n—2)-fold J are (§ 9) such that Mora Yona = Uog—1 Vora + 4 (Usp Vog — Uog Vor—1) Kor, 98-1 + 1K orl, he : a ‘ ‘ = le (Usp + Wor) (Uos—1 = Wag), = Es oy (bs) ; OF SEVERAL COMPLEX VARIABLES. 429 thus this integral becomes Qqri F ue) [= (Ker, os—1 + UK ora, os) ( ego a) Q(x | t) Sa: Ow O8og—1 Therefore it follows that f(r) differs from this integral by a function which is finite and continuous for all positions of (t,...,t,) in the region of convergence of 4, except actually upon ¢=0. Recalling a previous notation (§ 12) this is equivalent to the fact that 0 Qar or, log $(r) a Cora remains finite and continuous: as (7,,..., Tp) approaches the (n—2)-fold J, so that also 2ar log (7) — = | (fxd, +... + End) remains finite and continuous as (7, ..., 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 @ by which the (n—2)-fold I is defined; describe in this region as before a closed (n—1)-fold S, containing in its interior a portion of the (n—2)-fold Z; about J suppose as before an (n—1)-fold = satisfying the condition that every point of it is at a small distance e from some point of Z. Then the portion of n-fold space imside S and outside = is multiply-connected; but it can be rendered simply-connected by supposing an (n—1)- fold diaphragm P to be drawn, bounded partly by the (n—1)-fold = 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 is single- valued. Hence, if (7,,..., 7») be a point within this space, we have, as explained above, 1 ; 0 S50 Ales, log ¢ (tT) = = | log $ (é) c + tks) = —1 a) @ (a|t)+ | OSna6 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 >, and excluding the (n — 2)-fold in which the diaphragm intersects S; (ii) that over the part of = lying within S, excluding the (n—2)-fold in which P intersects &; (iii) that over the two sides of the (limited) diaphragm P. The part (i) remains finite and continuous for all positions of (7, ..., tT») within the n-fold space under consideration. The part (11) ultimately vanishes when the quantity e diminishes indefinitely; for we have seen that dS, ,=ed@dS,_., and it can be shewn, as by Poincaré (Acta Math. xxu1.), that as the point (&, ..., &’) on a normal of J, at 430 Mr BAKER, ON THE THEORY OF FUNCTIONS a distance e from J, approaches indefinitely near to J, the limit of e log ¢(&), when e and therefore $(£) vanishes, is zero. The part (iii) of the integral is equal to 2 [l(a + ie (Z ie “)p@l)+.. | a8 taken over only one side of the (limited) diaphragm P; for the values of logo at two near points on opposite sides of P differ by 27. Consider now the real part of this integral, namely 2r [ (0p | 1, Of ; | (Geant) by the theorem of Part I. of this paper we can replace this by an (n— 2)-fold integral taken over the (x —2)-fold which forms the boundary of the diaphragm; this (n— 2)- fold lies partly on ¥ and partly on S; the (n—2)-fold is =F { (ee + Ky +... F Kn, 3) ~ (a | t) dSy=35 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 (7,, ..., Tp) 18 within the region considered, log ¢ (7) differs only by a finite and continuous function from a function whose real part is equal to 2Qar =F [9 (lt) dSn-s, where the integral may be supposed to be taken only over the part of J which lies within S; for we have seen (§ 9) that for a complex (n — 2)-fold Kyo + Kg +... + Kn—an = 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—2)-fold, then, under certain conditions of continuity and single-valuedness for the function U, we have (7 oU oU e UdSh-s = | (1 Oat, — Ke Oa, apc 4 USn-a> the first imtegral being taken over the closed (n—2)-fold, and the second over the bounded portion of the (x —1)-fold. We now extend this idea to the (n—2)-fold J, 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 = [Bir d8.-2= {f(e et — GE.) da Ly where the right-hand integral is taken over the infinitely extended (n —1)-fold diaphragm bounded by the complex (n — 2)-fold. And, similarly, we have (§ 14) a == |(« rl OP + Kren | ASp_-» E | i (- eae nh ae itn) (i (Cs foe PHN) 35 = Ky 0a, 00, Xn (— Oat, Ot, nr ( On? arc joape =) An» =F [f(x OH m+ Ve bore OBm . oes Oay . Bsa where we have put CH pn PH Hm A oat 4. of Se OL, OL y= Ox,” thus, taking © to vanish when (&, ..., tr) =(0, ..., 0), we have 9) =|f{(« OH ms fh oo bien os Obs ys Gs O2n, Thus, as has been indicated in connexion with the definition of 0 as an (n—2)-fold integral, © is a generalised solid angle. It is not a single-valued function of (fh, ..., tn); its values at two near points on opposite sides of the (n—1)-fold of integration differ by integral multiples of @; this follows, in a well-known way, from the fact that the value of the integral taken over the closed (n —1)-fold (2, — a)r Jen dk (@n oa tn? = 7 is ultimately o when r diminishes indefinitely. Thus it is obvious that 2" (7+i0) (S)(Cmig coon Ur) = 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. Some 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=1. For this latter case there is no continuous (n—2)-fold of integration; the corresponding thing is a series of discrete points, in general of infinite number. We have in this paper found a formula, OlGis ---5 Tm) = 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—2)-fold constituted when n=2 by any enumerable system of discrete points &, &,..., having infinity as a point of condensa- tion, in regard to which it is assumed that for some positive integer, m, the series Em 4+ eae ee is absolutely convergent; this condition corresponds to the general one that the integral [Bema dS,-. 18 convergent; for instance the points may be those given by a formula a+ 2hkw + 2k'o’, where k, k are integers and o’/# is not real, in which case, as is well known, it is sufficient to take m=1. Taking Q (a\t) = z log [(a, — 4) + (#2 — &)*], = and, as in the general case (§§ 13, 14), V= lela, Sp» — LAs (- 1) fa] m+. ES oa feo alOve ous ; == [pip 9(2|0) +... +7, (ta) e(e\0)), where the summation extends to all the points &=2,+7z,, and a aGiales OH im .0) =| |(a& Oa = dt. i) dSn—a; we find easily aaa 1 | hence as a= 27 for n=2 we have eh ’ aml oman {a4 =), é 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 T.=N+ 20; 1A +... + 20; op Aap, (Ci Ecoony))) where 2» is a constant and 4, ..., Ax», are real variables each between 0 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 m=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 f=; putting y=a+v7b this is then the two-fold given by 2,=a, #,=b. The matrix d,2, d;%,- dt, ,%, |, ELM itu Bribe cel with the help of which the direction cosines may be defined, may be taken to be (OO ce, © |, |o 0 0 de so that «,,=0 except x,=1, and dS,_,=da,dx, As the integral mesos da,dx, =| "ae fe rdr (7@+0+af+aZP? Jo Jp, (D+r7P vanishes when #&, R, are infinite we infer that it is sufficient to take m=0(0, and therefore Hy = 9 (@\t)—E(¢\0), Anis =e (a|t)— e (w|0) + (« 2) e(x|0) ; then (§ 12) we obtain, for Sec OHm .0Hm 5 (like, sO) = f= i | is ( an, v Ou, ) + (13 + 1k14) \ On, a, =) dis; 5 ff . Om _ jem . OH in * 0H, &= ‘| \(«s + UKs) ( der, aa ") + Uk 34 ( O25 aU = dSn_2, the respective values g; =0 j 1 —t,-—t(a@—t) @, — 1X> =—||ded = 4 SE If paiva i = th) Eesti (a4 ita) > eee = : Vou. XVIII. 55 434 Mr BAKER, ON THE THEORY OF FUNCTIONS wherein, in the latter, z,=a and 2,=b, and the integration in regard to 2, 2, is to be taken for each of them over the whole range —% to +. Hence we obtain which, as the general theory requires, is a function of the complex variables (in fact only of 7,=t,+%t,). Thence id = | (Sar, + &dr2) =7 e + log (1 -")| ; 0 y Y and therefore, as a =277 for n=4, @(n, 7) = oP" =( =) er, 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 J, of the zeros of the function, V 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 WV zeros whose centres are at distance at least R from some definite finite pomt of the plane taken as orgin, 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 #& 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 V zeros beginning with log R is at most log (R +.) — log R = log @ ae 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 NV 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 R06, OF SEVERAL COMPLEX VARIABLES. 435 where ( is the area of such a circle, may be finitely greater than zero for all values of R greater than a certain assignable R,. 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 m=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; con- sider the greatest number of zeros of such a function which can lie in the annulus between two such circles of radii r and 7” (7’>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 ym (f = iB where B is some quantity greater than zero. Let k be the number of these regions, which is finite so long as 7 is finite. Then T (7 — r) m1 = kB F 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 peatm? which in turn is less than aM (7? — 7?) 7m B pttm , which, if 7’ =r(1+e), is equal to l ™ (WY ek Sahl B ( +e) ( +€) (= ail 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 7, is less than aM 1 lies m (2 2 B (1 +e) Cade and can be made as small as we please by taking r large enough. This proves the convergence of the series. 22. Pass now to consider an integral function of p complex variables, and consider the (n—2)-fold over which the function vanishes, this being supposed to extend to infinity. Imagine closed (n—1)-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—1)-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 —1)-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 in 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—1)-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 Tie 3) 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 [ DSn_s i 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—1)-folds to be described, with centre at the finite point from which R is measured, and consider the extent of the (n—2)-fold lymg in an annulus bounded by two of these spheres, of radii r and 7, (7,>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 RTL 183, where B is some constant greater than zero, Let & be the number of these regions, which will be finite when 7, is finite. Then @ Nn yn -m—1 z=. . a (7; myn, Z2kB; as the total extent of the (n—2)-fold lying in the annulus is kM, the contribution to the integral GS pe which arises from the annulus is less than kM pntm ~ OF SEVERAL COMPLEX VARIABLES. 437 and therefore less than a M GRE = 7”) Te n B putm ? which, if 7, =r(1+e), is equal to ao M (feel ty ppt Oe esa) 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 (m—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 (7m, ..., Tp) be a power series, the extent of the (n—2)-fold ¢=0 which hes within a closed (n—1)-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 =| Aimi:dSn_; denote by (a, ..., @) as before a point of the (n—2)-fold, and by (4, ..., t,) a finite point not upon the (n—2)-fold of integration; when R*=a,?+...+,? is large, that is, for the very distant elements of the integral, and =4°+...+¢,? is finite, we have pre da Va fe = Rrim Kine (/) AP 60015 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 R, such that the integral ASn—» Rem a taken over the part of the (n—2)-fold of integration, extending to infinity, for which R>R,, shall be less than « We have in § 22 proved that this is so under the hypothesis advanced. 24. The method just applied to the integral [HmesdSn-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—1)-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 p oH, 0H, & Caustic Oils = 3, kA Kr,s,2k) |= v | ¥ OXo.-1 OM /x=1 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 (mn —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 — 1)-fold integrals V=|{(« es zap OB ms ss 9 dso. Lo Ox, a= {(« OAs ne OH mss jas \ 1 0a, 2 Oa. 220 n—1) taken over the infinite (n—1)-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—1)-fold integrals taken over the portion of the (n—1)-fold which lies at infinity can be replaced by an (n — 2)-fold integral taken over a closed (n—2)-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 (4, ..., tn) approaches indefinitely near to the (nm — 2)-fold of integration the integral Qar = |p (alt) dn» becomes infinite like log mod. ¢, where ¢=0 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 dS,_, satisfy the following equations, the origin of reckoning being taken at the point of the (n — 2)-fold, UjX, + UX, +... + UnTn = O, U,X, + VoL +... + Unln = 0; OF SEVERAL COMPLEX VARIABLES. 439 these give (UsV_ — Uys) 73+... (UyU3 — Usd) y+... = ———— , &=— = 5, ERS, HI B00 UyVo — Usd UVe — Ug; whereby all the coordinates are expressed in terms of the (n—2) quantities 2, ..., Zn. Thence, using the equation «,.d@S,-,= M,,, we have Up +t... + Un’ a de = 2 n—2 Uy? + Uy" FRY 35 Olina We can further suppose the axes so chosen that Cy = o00= Vi 0h = oo = On =O so that, for dS,., #,=0 and 2,=0; and dS,_,=da,da,...dz,. Also, the origin being at the point of dS,_, which is to be considered, ay, ...,@, are, for dS,., subject to a condition of the form RO Soa Gh RS Se ir) where a is small and fixed; these coordinates are otherwise unrestricted; we can there- fore put dSn—» = 1" sin” 46, sin” *9,... sin 0,_,.drd@, ... dAn, where the limits are r=0 toa, @,=0 to m, 6,=0 to m,..., O7.—0 to m 6,2,— 9) to 27. The point (4, ...,t,), as it approaches the (nm — 2)-fold, can be taken subject to Bib tee aly tigi — Opty taste teNlins "eos where 2,=0, 2, =0 and (0, 0, 25, ..., @) is any point of dS,_». Then to the integral Qa [ a = le (a\t) dSp_» the contribution arising from dS,_, is 1 aa yn-3 dr Were | sin", ... sin Oy» d6,... Ons, n—-2oa) which is easily seen to be ra rr 3dr il (7? ae ey 2) putting n=2p, r? =z, 2+1=+t, this is ey es Le 2 1 tP- 440 Mr BAKER, ON THE THEORY OF FUNCTIONS of which the infinite part, for diminishing € and fixed a, is exactly loge. But as we approach the (n—2)-fold in the way here taken we can put = hee®, (see § 16); so that the infinite part of log¢ is also loge. Thus in the limit the difference Qr [ ay = lp (it) dS,» — log remains finite, as stated. 26. In this paper we have hitherto supposed the (n—2)-fold of integration to be given @ 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'(1, ..., Tp) is known to exist for all finite values of 7, ..., Tp, and to have no essential singularities for any finite values of 7, ..., Tp, namely can be represented in the neighbourhood of any finite point (7,", ..., Tp) in the form F=, (1 —7:", -.-, Te— Te”) = Gol —1™, «--, Tp — Tp”), where yy), ¢) are ordinary power series (of positive powers) with a presumably limited common region of convergence. If the series w%, ¢ have a common vanishing factor at (4, ..., Tp”), that is, are both divisible by another convergent series which vanishes at (4, ..., Tp), this factor may be supposed divided out (Weierstrass, Werke, 11. (1895), p. 151). There is then a region about (7,", ..., Tp”), within the common region of convergence of y and ¢,, but not necessarily coextensive with it, such that, if (q+, +) Cp + Tp) be any point in this region, and the series yy, $ be written as power series with this point as centre, by putting ™%—7° =ch+uz, the resulting series in %, ..., Up have no common factor vanishing at u,=0, ..., up=O0 (Weierstrass, loc. cit., p. 154). This region we may temporarily call the proper region of (7,°, ..., tp”) for the function F. There may be points within this region at which Yr, ¢p 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 yy, ¢ as originally given have no common factor vanishing at (7, ..., tp”) 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 (7,, ..., 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 K, by an expression, belonging to an interior point 7, =v eer OF SEVERAL COMPLEX VARIABLES. 44] wherein yy, $ have no common factor vanishing at any point of K,; as we pass to a contiguous region A, we need a representation belonging to a point (7,”, ...) interior to K, of the form rev di By considering the equality iow we og fi in the region common to the proper regions) Of (Gases Tp), and (qj) 25 tm"))) we are then able to deduce that all the points for which y,=0 are also points for which vr,=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 ©=0, and the latter by ®=0, the function F can be represented in the form (3) = sy Boe where A is an integral function; and ©, ® 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—1)-fold be taken actually within the region of convergence of a power series in the complex variables &,, ..., &, say $(&, -.., &), where n=2p, the extent of the portion of the (n— 2)-fold given by ¢=0 which lies within the (n—1)-fold is finite. For consider the points of this portion for which &= yp, ..., &)= yp, where yo, ..., Yp are certain definite values; these points are given by the equation in &, #(&, y, ---, Y)=0, wherein &, is capable only of a limited range of values determined by the (n—1)-fold; as this range is included within the region of convergence of the &-power series @(&, yo, ---, Yp), there cannot be an infinite number of values of & within this range for which $(&, y, .--, yp)=0. Thus on the portion of the (n —2)-fold $(&, &, ..., &)=0 lying within the (n—1)-fold there exists only a finite number of values of &, corresponding to given definite values of &, ..., &. Let dS,_. be an element of the (n—2)-fold ¢=0; we have | i= | pass POT: | Anse the integrals being taken over the portion of the (n— 2)-fold which les within the (n-1)-fold; to prove that [ass 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 age, Chis, Ch DO, O5 O , (He, (he, 0, Cir. D5 OW. tim, Che, O , OW , Gee, UW. chim. thes, OU, O 5 O 5 che, Oo for) Wor, XOVOUE 442 Mr BAKER, ON THE THEORY OF FUNCTIONS where, for instance, d,,a,, de,,%, are determined in terms of di»,., by the equation 1 (day1y + Uday 422) + Griid@er41 = 0, and dz,2,, d»,a, are determined in terms of daz,,, by the equation fy (doy; + Udy, L2) + Uy 41a 49 = Oe Then «,dS,-, may (§ 5) be replaced by dx,dx, ... Ain Ay 3 since then the range of values for each of 2, %, ..., ,, for points under consideration, is finite, and, as proved, there is only a finite number of points of the (n— 2)-fold for which Ig, «++» ty have a given value, it follows that the integral dx, ... Atn 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—1)-fold bounded thereby, the equation (42g. Pos + Ky Py + KP + Ky Py + Koy Pog + Po) dS» = + OP. , OPis , OP = [as,.. {1 = ~ an + =) ms { OP» OP. ae OP, se (oa, 02s ae) w (22s 4 Pe Ps) (ss Oa, ]\? = [a8 (4,Q) + 2 Qo + 3Q3 + K,Qs), say; (22s oP aPa) Gea em Ge thus Gling 2 SE ara OX> OL 02, Py OP os ge OP» = On. f0z,. Lotau <2 OP OP., ORs Sg), OD, OX OX; OP a OP» OP Q,; 02, 0X2 OL: OF SEVERAL COMPLEX VARIABLES. 443 30 + oe + B05, OM =i), therefore Oa, ay 0%; Oars which is a necessary condition for the consistence of the four equations just written. It is satisfied for instance by Q=1Q=1 (7 | oj £), Q=10=-1(% +i), aa 1 f being any function of 2, a, 2%, 2,; corresponding to these values the four equations just written are satisfied by P,=P,=0 P.=f, Pua Ps=f Pa=-f But it does not thence follow that [x + Ukyy + Utss — Ko) ifs Sj» SN aeeeaies) Lik P) (es tin) (5 GEN ee is for the first mtegral vanishes for a complex (n— 2)-fold, and the second integral does not necessarily vanish, as we see by taking for instance 1 = (a, —4,)? +(@,-—h) 2(& — 7) (&—t) (4 —hP + (m— bY + (% — 6)? + (4s —t,)?’ f= when we get y Of “Sere lige Cans (diane tae an Gm p (|e), On. oe Gates i) 9D whereby the second integral becomes ise ( Os On $ : GG = Jas ce + ik») lee =e) Q (@ t) + (Ks + Ux,) (= —1 sa) a) (wit) ; which is not always zero. In explanation it may be noticed that on the (n—2)-fold there are points where &,=7,; and for these f is infinite. 25 July 1899. 56—2 - INDEX. Absorption, selective, electromagnetic illustration of, 348 Analytical representation of a monogenic function of one variable, 1; of an integral function of several variables, 418 Arrhenius, 131 Ashworth, 135 Baker, on functions of several variables, 408 Becquerel, 397 Bendixson, 36 BERRY, on quartic surfaces with integrals of the first kind of total differentials, 333 Bigelow, 131 BontzMaNnN and Mace, on Van der Waals’ law, 91 Bonnet, Ossian, 325 Borel, 6 Briggs, 203 Bromwich, 324 Brown, on differential equations of the lunar theory, 94 BURNSIDE, on groups of finite order, 269 Campbell, 221 Cantor, 204 Capstick, 186 Cavalieri, 204 Cayley, 328, 333 Chasles, 219 Clifford, 328 Condenser, oscillatory discharge of, 136 Coradi, 116 Corbino, 398 Cornv, la théorie des ondes lumineuses, xvii Cremona, 346 Cubic surface, model of, 375 Darboux, 9, 324 Dawson, 393 Declination, magnetic, 107 Desargues, 204 Descartes, 197 Didymium salts, absorption spectra of solutions of, 298 Differential equations, Forsyth on the integrals of sys- tems of, 35; of lunar theory, E. W. Brown on, 94 Dise, circular, Green’s function for, 277 Echelon spectroscope, 316 Eckholm, 131 Elastic displacement, waves of, on a helical wire, 364 Electric density near vertex of a right cone, 292 Electromagnetic illustration of selective absorption, 348 Electrons, dynamics of a system of, 380 Electrostatic problems, application of Green’s function to, 277 Equations, differential, Forsyth on, 35 Erbium salts, absorption spectra of solutions of, 298 Ewing, 396 Fabry, 9 FitzGerald, 397 Fontenelle, 197 ForsytH, on the integrals of systems of differential equations, 35 Frisch, 197 Functions, of one variable, representation of, 1; of several variables, 418 Gauss’s formula for solid angle, 425 Genus, numerical, of a surface, 345 Geometry of Kepler and Newton, 197 GLAZEBROOK, on the discharge of an air condenser, 136 446 Goursat, 36 Green’s function for a circular disc, 277; formula for electric density near the vertex of a right cone, 292; formula in integral calculus, 409 Greenwich Observatory, magnetic declination at, 107 Groups, Poincaré on continuous, 220; of finite order, Burnside on, 269; and optics, Lovett on, 256 Hadamard, 9 Heine, 292 Helical wire, waves of elastic displacement on, 364 Henrici, 116 Hill, 97 Hopson, on Green’s function for a circular disc, 277 Horn, 37 Hornstein, 131 Humbert, 345 Integrals of first kind on quartic surfaces, 333 Ions, dynamics of a system of, 380 Jordan, 37 Kelvin, Lord, 137, 277, 287, 291 Kepler, geometry of, 197 Kerr, 406 Klein, 269 Klugh, 116 Konigsberger, 36 Kronecker, 408 Kummer, 345 Lame, electromagnetic illustration of selective absorp- tion of light by a gas, 348; referred to, 365 Lanthanum, 301 Laplace’s equation, with minimal surfaces, 325 Larmor, on dynamics of a system of electrons or ions; influence of a magnetic field on optical phenomena, 380 Levy, 266 Leyden jars, oscillatory discharge of, 136 Lie, 220, 256 Light, selective absorption of, by a gas, 348 LIVEING, on absorption spectra of solutions of didy- mium and erbium salts, 298 Loneg, on the discharge of an air condenser, 136 Lorentz, 362 Loria, 203 Love, referred to, 348; on waves of elastic displacement along a helical wire, 364 Lovert, on contact transformations and optics, 256 Lunar theory, differential equations of, 94 Macaluso, 398 MacCullagh, 380 INDEX. Macponaxp, on Green's formula for electric density near vertex of a cone, 292 Mace, see Boltzmann MacManon, on Partition analysis, 12 Magnetic declination, periodogram of, 107 Magnetic field, influence of on optical phenomena, 380 Maxwell, 137, 143, 292, 361 Mehler, 292 Meteorology, 107 MicHELsoN, on the echelon spectroscope, 316 Minimal surfaces, Richmond on, 324 Mirrac-LEFFLER, representation of a monogenic func- tion, 1 Monge, 325 Moore, 275 Newton, geometry of, 197 Noether, 333 Optical phenomena, influence of a magnetic field on, 380 Optics, and contact transformations, 256 Oscillatory discharge of an air condenser, 136 Painlevé, 7 Partition analysis, 12 Periodogram of magnetic declination, 107 Picard, 36, 333 PoINcaRs, on continuous groups, 220 Poinearé, 36, 333, 408 Poncelet, 204 Poudra, 205 Preston, 391 Quartic surfaces with integrals of the first kind, 333 Rayleigh, Lord, 109, 139, 147, 348 Rede Lecture, xvii RICHMOND, on minimal surfaces, 324 Rotation, magneto-optic, 396 Routh, 403 Runge, 2 Russell, 301 Salmon, 346 Schur, 221 ScuustTEr, on the periodogram of magnetic declination from records of Greenwich Observatory, 107 Smithells, 393 Sommerfeld, 277 Spectra of solutions of didymium and erbium salts, 298 Spectroscope, the echelon, 316 Stokes, 409 Substitutions, infinitesimal, 245 Surfaces, minimal, 324; quartic, with integrals of the first kind, 333; cubic, model of twenty-seven lines upon, 375 INDEX. 447 Taytor, C., on the geometry of Kepler and Newton, Vitellio, 201 197 Voigt, 398 Taytor, 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. vi ul Phil. Soc. Trans. XVIII. Plate T. DIAGRAM OF 26 DAY PERIOD 5 10 15 20 26 Year: 1320 1200 1080 960 720 360 240 120 The points of strongest amplitude are marked thus @ weakest Ax 5) » +F intermediate ,, +5 > 6 ” » ” ” Phil. Soc. Trans. XVIII. Plate II. DIAGRAM OF 27 DAY PERIOD. 1320 1200 1080 960 720° 600 360 240 120 o The points of strongest intensity are marked thus ©® & = “ intermediate ,, — 5 ©) a . p weakest ; ot #2 oe ’ Cambridge Philosophical Transactions, Vol. XVIII, Plate PLATES 3—23 ILLUSTRATING Proressor Livetne’s Paper (pp. 298—315), On the effects of Dilution, Temperature, and. other circumstances, on the Absorption Spectra of solutions of Didymium and Erbium salts. These plates are all reproductions, enlarged to double the size, of photographs of some of the spectra from which the conclusions in the text have been deduced. In the processes of enlargement and reproduction some of the fainter details visible im the original negatives have (perhaps unavoidably) been lost : but they present the salient features of the changes in the spectra produced by the variations of circumstance. The references to these plates in the text applied to the original negatives and were printed before the reproductions were ready. The latter, being positives, are reversed, and in order that the references may be easily intelligible it has been necessary to place the red ends of the spectra on the left hand. The figures at the top of each plate are the approximate wave lengths of the bands in the spectra beneath them, and sufficiently indicate the range of the spectrum photographed. PLATE 3. Absorptions of solutions of didymium chloride in four degrees of dilution in thicknesses inversely as the dilutions. The most concentrated solution contained 140-7 grams per litre, and the absorbent thickness of this solution was 38mm. solution 1/8 strength 1 z " 305 mm. thick 6 strongest solution 38 mm. thick : solution 1/4 strength ‘ 152°5 mm. thick ; solution 1/2 strength 76 mm. thick It will be noticed how very nearly identical these four spectra are. The original photograph shews a number of faint bands which have not come out in the reproduction. They are however as nearly identical in all four spectra as are the stronger bands here reproduced. 2 o. Cambridge Philosophical Transactions, Vol. ROVIAIE Plates A, PLATE 4, Absorptic ms ¢ f solution of didymium sulphate in four degrees of dilution. 8 469 144 ) 403 5 18: tbe The diffuse bands at about 380, 375 and 364, are visible in the lave nearly disappeared in the reproduction quite PLATE 5. Absorptions by solution of erbium nitrate in four deerees of dilution, 566 grams of the salt to the litre. ) SS 149 14] l 404 387 7 bo increased concentration of the with The increasing diffuseness of the bands series; the weak band about 441 when the soluti that about A449 is much broader and the details within it obliterated. seeming to be washed out saturated solution 38 mm. thick half-strength 76 mm. thick quarter-strength 152°5mm. thick one-eighth strength 305 mm. thick original photograph, but the strongest containing one-eighth strength 305 mm. thick quarter-strength 152°5 mm. thick half-strength 76 mm. thick strongest solution 38 mm, thick solution is seen in this m is concentrated while a. =e eS mm ge Cambridge Philosophical Transactions, Vol. XVIII, Plates 6, 7. PLATE 6. Absorptions by solutions of didymium nitrate, concentrated, and extremely dilute. The most concen- trated had 6111 grams of the salt per litr he othe is part of the same solution diluted to 15-5 times its bulk = > ae =) x > 1. 1D gS as = RD & stronger solution 6-7 mm. thick 1/45°5 strength 305 mm. thick bo these two spectra except that the band in the vellow i little difference between 1eS¢ those at A476 and 427 more washed out. broader with the stronger solution, and PLATE 7. chloride of concentrations equivalent to those of the nitrate Absorptions by solutions of didymium used for plate 6: the stronger containing 462°9 crams of the chloride per litre. stronger solution 6-7 mm. thick 1/45°5 strength 305 mm. thick There is no definite difference between these two spectra. a a . . eee dh Bea Ss . ae Es * . > x ; 5 a = va J es =~ == =e ot eg . = - et = =~ - . - = = Sa ee Cambridge Philosophical Transactions, Vol. XVIII. Plate PLATE 8. Absorptions of a solution of erbium nitrate containine 467°6 crams of the salt per litre, and of a solution made by diluting the former to 45°5 times its bulk. 1 104 8 7 3 188 140) stronger solution 6°7 mm. thick 1/45°5 strength 305 mm. thick The bands are more diffuse with the stronger solution, that at about A377 being decidedly broader. the original and is more diffuse with the stronger The band at about A449 is more distinctly seen in solution than with the weaker, PLATE 9. Absorptions by solutions of erbium chloride of concentrations equivalent to those of the nitrate used the stronger solution containing 363°3 grams of the salt per litre. for plate 8; co oD L =) 10> =H io t oO nN 22 =H an S a > Yn Yes aa + SH an 1/45°5 strength 305 mm. thick stronger solution 6°7 mm. thick any difference between these two spectra except that the band about 377 is rather probably to the overlapping end. The fainter There is hardly stronger with the more concentrated than with the dilute solution, owing of the general diffuse absorption of the concentrated chloride at the more refrangible bands which are visible in the original photograph can hardly be traced in the reproduction. Cambrida Philosophi i Transactions, Vol. XVII, Plate PLATE 10. Absorptions by solutions of erbium chloride and equivalent solutions of erbium nitrate, alternately four degrees of concentration, the strongest having 726°6 grams of the anhydrous chloride to the litre, (2 and the equivalent nitrate 935'2 gsrams to the litre. ~ 188 149 141 1 104 strongest solution of Er Cl* 38 mm. thick strongest solution of Er (NO®) 38 mm. thick half-streneth chloride 76 mm, thick half-strength nitrate 76 mm. thick quarter-strength chloride 152-5 mm. thick quarter-strength nitrate 152°5 mm. thick one-eighth-strength chloride 305 mm. thick one-eighth-strength nitrate 305 mm. thick The ecreater diffuseness of the bands with the more concentrated solutions of the nitrate is evident, and so is the extension of the general absorption at the more refrangible end of the spectrum with the most concentrated solution of the chloride, The difference between the absorptions by the chloride and nitrate diminishes with dilution and has almost, or quite, disappeared in the case of the weakest solutions. 10. Cambridge Philosophical Transactions, Vol. X VITI., Plate 11. PLATE 11. Absorptions by didymium chloride and nitrate, alternately, in equivalent solutions of four degrees of concentration, beginning with the strongest solution containing 462°9 grams of the anhydrous chloride to the litre, followed next with the equivalent solution containing 611°1 grams of nitrate to the litre. aS oS} + mre @ o 1p oo a) + ON or) ios 1S no) sa x s+ sts oD strongest solution of DiCl* 38 mm. thick : : strongest solution of 2 Di(NO?#)3 38 mim. thick : E a half-strength chloride + 76 mm. thick , : 9 7 half-strength nitrate : i 76 mm. thick a quarter-strength chloride 5) . : 152-5 mm. thick Ps quarter-strength nitrate 6 ¥ eae 5 152°5 mm. thick F Ay one-eighth-strength chloride 7 ; 305 mm. thick be one-eighth-strength nitrate 8 A 305 mm. thick The extension of the general absorption at the most refrangible end of the spectrum with the concentrated solution of chloride is evident in the uppermost figure. With such strong solutions as were used for these photographs other differences between the absorptions by chloride and nitrate can be seen only in the weaker bands such as those from 433 to A406. These are weakened by diffusion in the ee of the nitrate, but there is very little difference between the absorptions by chloride and nitrate in the most dilute solutions. 2 ew aek '- = De es: 2 ' Lf ( ‘ambridae PLATE 12. Philosophical Transactions, Vol. X VIIL., Absorptions by solutions of hydrochloric acid in alcohol, and in water, compared with the absorption by pure water. The effect of the hydrochloric while diminished concentration ] nas acid the absorption with diminished concentration of the at the more no ettect acid is seen in the aqueous solutions Nos. Oe teh 7, in the case refrangible end of the 1S alcoholic solutions Nos. 1, 2, strongest solution of HCl] in aleohol, 38 mm. thick half-strength, do. 76 mm. thick quarter-strength, do. 152 5 mm, thick pure water 305 mm. thick strongest solution of HC] in water, 38 mm. thick half-strength, do. 76 mm. thick quarter-strength, do. 152°5 mm. thick one-eighth-strength, do. 305 mm. thick visible, and the diminution of Plate 12. 4 2 Cambridge Philosophical Transactions, Vol. AX VITT., Plate 13. PLATE 13. Absorptions by solution of erbium chloride, cold and hot alternately, in two degrees of concentration. {88 149 104 half-strength solution 76 mm. thick at 23° C, 1 half-strength solution ») a 76 mm. thick at 97°C. stronger solution 3 38 mm. thick at 253°C. stronger solution : 38 mm. thick at 99°C The extension of the general absorption at the more refrangible end of the spectrum by a rise of temperature is manifest in these photographs, and so is the greater diffuseness of the bands at about \449 and A488. Cambridge Philosophical Transactions, Vol. X VIIL, PLATE 14. Absorptions by solutions of erbium nitrate, cold and hot alternately, in four degrees of dilution, in thicknesses inversely as the dilutions. The strongest solution had 566 grams of erbium nitrate per litre. D = x = 4 {04 strongest solution 38 mm. thick at 22°C. do. do. at 94° C. solution 1/8 strength ) 305 mm. thick at : do 6 do. at 94°C. a solution 1/2 strength i a vant 76 mm, thick at 23°C, do 5 It will be noticed that the eftect of heating the solution is in general to render the absorption bands more diffuse, and that it is the bands that increase in difftuseness with increasing concentration of the solution which are most affected by the rise of temperature. The original photographs shew several fainter bands which have not come out in the reproduction, and also shew the lighter interspaces between the absorptions in the ultra violet much more distinctly than the reprodu distinct in the spectra of the cold solutions than in those of the hot solutions. ion. Even in the reproduction these lighter interspaces in the ultra violet are more Plate 14. a = cog tae . . . 1 - 7 7 r ae a ee. at a fea Cambridg: Philosophical Transactions, Vol. X VIIT., Plates PLATE 15, Absorptions by solution of didymium sulphate, cold and hot, in two degrees of concentration. The stronger solution was a saturated solution at 20°C. = nN 510 403 S80 1 38 mm. thick, at 23° C. same solution and same 9 : : = thickness, at 90° C. half-strength solution 2 76mm. thick, at 243°C. half-strength solution I 76 mm. thick, at 92° C. The extension of the general absorption at the more refrangible end of the spectrum, and the increased diffuseness of the bands in the blue, by the rise of temperature is plainly seen in these photographs. PLATE 16. Absorptions by solution of erbium chloride, neutral and acid, in two degrees of concentration; the stronger neutral solution haying 726°6 grams of the chloride to the litre, and the acid solution having besides an amount of hydro hlorie acid equivalent to the amount of neutral salt. {S88 149 t1? 104 8 stronger neutral solution 1 ¢ 38 mm. thick stronger acid solution ”» 2 38 mm, thick one-third strength neutral a solution, 152°5 mm. thick one-third streneth acid 4 solution, 152°5 mm. thick The thickness of the absorbent solutions is not proportional to the dilutions, so that the absorptions of figures 3 and 4 are produced by a quantity of salt one-third greater than that which gave figures 1 and 2, which makes the bands of 3 and 4 stronger. The effect of the acid is chiefly to extend the general absorption at the more refrangible end of the spectrum. stronger solution of Di2(SO+4)* 15, 16. Bi need vy , ne ee Seo (sated ‘py! r ’ - A i ' hi (oy ‘ ” ag , ; ic | a Be 7 hw ; \ i t ¥ a A ' a as = 4 ” Sih J 1 il : i ye i : ol - 7 Ss Ff - i _ oo - © wee ) ae a «= Cambridge Philosophical Transactions, Vol. XVIII, Plates 17 PLATE 17. Absorptions by solutions of erbium nitrate, neutral and acid, in two deerees of concentration. The stronger neutral solution had 935:°2 erams of the salt per litre, and the acid solution had in it besides as much nitric acid as was equivalent to the amount of neutral salt, ) 88 449 ] 104 8 stronger solution, neutral 1 38 mm. thick stronger solution, acid 2 20 F 2 38 mm. thick half-strength, neutral 3 76 mm. thick half-strength solution, acid t 76 mm, thick The eftect of the acid in rendering the bands more diffuse is seen in these photographs, and in the extension of the general absorption at the more refrangible end of the second figure, PLATE 18. Absorptions by solutions of didymium chloride, neutral and acid, in two degrees of concentration : the acid solutions containing the same amount of didymium per litre as the neutral solutions but with hydrochloric acid in addition. 10 t44 ) 37 B80 169 103 oe 5 : stronger solution ef DiC], neutral, 38 mm. thick 9 stronger solution, acid 38 mm, thick _ half-streneth, neutral 3 76 mm. thick { half-strength, acid 76 mm. thick The chief eftect of the acid is to extend the ceneral absorption at the more refrangible end of the spectrum, 18. Cambridge Philos: PLATE 19. Absorptions by nearly equivalent solutions of didymium chloride in water, in alcohol, and in alcohol , charged with hydrochloric acid, The acid solution was prepared from the neutral alcoholic solution by passing hydrochloric acid gas into it and was found to be about nine-tenths of the streneth in didymium of the neutral solution. — ti =) oS bs tad on > on = Jo) H N = Te ID 1D — meh =H = 1 neutral aqueous solution 2 neutral alcoholic solution 3 acid alcoholic solution The eeneral absorption at the more refrangible end is extended : little by the alcohol, and still more by the addition of acid. The bands are generally rendered more diffuse by alcohol and a little shifted towards the red end of the spectrum, the shift increasing as the refrangibility decreases. The acid seems to diffuse away the bands in the blue, the strong paw at ibout A520 are just visible in the spectrum of the acid solution considerably shifted towards the red. And the strong group in the yellow is still more shifted, and so spread out that several of the component bands are separated. PLATE 209. Absorptions by equivalent solutions of didymium nitrate in water and in glycerol. 183 169 596 5 103 380 aqueous solution alycerol solution No definite shift of the bands by the elycerol appears in the photograph, but there is an extension of the rene ral absorption at the more refrangible end ot the spectrum, and the bands are rendered More diffuse by the glycerol. cal Transactions, Vol. XVIII, Plates 19, 20) Cambridge Philosophical Transactions, Vol. X VIII., Plates PLATE 21. Absorptions by glass of borax coloured with didymium oxide and by a solution in water of didymium nitrate containing a quantity of didymium equal to that in the glass. 10 183 169 144 ) 103 380 borax glass coloured with didymium aqueous solution of didymium nitrate These photographs are disfigured with horizontal lines due to dust on the slit of the spectroscope. It will be seen that the bands are for the most part shifted by the borax but very unequally so: also that the bands are rendered more dittuse by the borax and some almost diffused away. PLATE 22, Absorptions by equivalent solutions of didymium acetate in acetic acid and of didymium nitrate in water. 596 5 5 } 169 ti4 10; 380 1 didymium acetate dissolved in acetic acid 2 equivalent aqueous solution of nitrate The bands are generally shifted towards the red by the acetic acid, and in the photograph the shift diminishes as the band is less refrangible; but the dispersion of the spectroscope also diminishes as the light is less refrangible; so the apparent diminution of the shift is not altogether real. The acetic acid also increases the diffuseness of the bands, as is very manifest in the case of the band at about A476, and may be traced in others. . Bia ; Cambridge Philosophical Transactions, Vol. XVIIT., Plate 23. PLATE bo we Absorptions by solutions of didymium chloride in water, and of didymium tartrate in water charged with ammonia, 48 169 444 42 403 E aqueous solution of didymium chloride ammoniacal solution of tartrate The tartrate has all its bands more diffuse than the chloride, some of them almost diffused away, and they are shifted towards the red. Cambridge Philosophical Transactions, Vol. XVIII, Plate X a J EZ, Se ee ‘Ail Cam. Phil Trans Vol XVI Plate XXV. “ BINDING 2.2 Fr. jut 19 1968 Q Cambridge Philosophical 41 Society, Cambridge, Engl C1ga Transactions v.18 Physical & Applied Sci. Seriale PLEASE DO NOT REMOVE CARDS OR SLIPS FROM THIS POCKET UNIVERSITY OF TORONTO LIBRARY gior ace Proat hts a3 Crit ied oA 8: Sie tank Be agri nate Masigetoen Sd Per a ee Fi SB dint ~ . eta ku SANK WA AS Ny tal € -. pee Dr hal i Shaye 1 uy a Woaiate gy a8 RAYS sas oh Pen Sh wee Vee VA ari feat i's we