Selected Title s i n This Serie s
Volume 11 Jun e Barrow-Gree n Poincare an d th e thre e bod y proble m 1997
10 Joh n Stillwel l Sources o f hyperbolic geometr y 1996
9 Bruc e C . Berndt an d Robert A . Ranki n Ramanujan: Letter s an d commentar y 1995
8 Kare n Hunger Parshal l and David E. Rowe The emergenc e o f the American mathematica l researc h community , 1876-1900 : J . J. Sylvester , Feli x Klein , an d E . H . Moor e 1994
7 Hen k J. M . Bo s Lectures i n the histor y o f mathematic s 1993
6 Smilk a Zdravkovsk a and Peter L . Duren, Editor s Golden years o f Mosco w mathematic s 1993
5 Georg e W. Macke y The scop e and histor y o f commutative an d noncommutativ e harmoni c analysi s 1992
4 Charle s W. McArthu r Operations analysi s i n the U.S . Army Eight h Ai r Forc e i n World Wa r I I 1990
3 Pete r L . Duren, editor, e t al . A century o f mathematics i n America , par t II I 1989
2 Pete r L . Duren, editor, et al . A century o f mathematics i n America, par t I I 1989
1 Pete r L . Duren, editor, et al . A century o f mathematics i n America , par t I 1988 This page intentionally left blank Poincare and the Three Body Problem This page intentionally left blank https://doi.org/10.1090/hmath/011
History o f Mathematic s Volume 1 1
Poincare and the Three Body Problem June Barrow-Green
American Mathematical Societ y London Mathematica l Societ y Editorial Boar d American Mathematica l Societ y Londo n Mathematica l Societ y George E . Andrew s Davi d Fowle r Bruce Chandle r Jerem y J . Gray . Chairma n Paul R . Halmos , Chairma n S . J . Patterso n George B . Seligma n
1991 Mathematics Subject Classification. Primar y 01 ; Secondary 70 .
Photographs o n th e cove r ar e Henr i Poincar e (inset ) an d Osca r II , Kin g o f Sweden an d Norway (background) .
A lis t o f photograp h an d figure credit s i s included a t th e beginnin g o f thi s volume .
Library o f Congres s Cataloging-in-Publicatio n Dat a Barrow-Green, June , 1953 - Poincare an d th e thre e bod y proble m / Jun e Barrow-Green . p. cm . — (Histor y o f mathematics, ISS N 0899-2428 ; v. 11 ) Includes bibliographica l reference s (p . - ) and index . ISBN 0-8218-0367- 0 (acid-fre e paper ) 1. Three-body problem . 2 . Hamiltonian systems . 3 . Poincare , Henri , 1854-1912—Contribu - tions i n dynamics. I . Title. II . Series . QA852.B37 199 6 515'.352- Copying an d reprinting . Individua l reader s o f this publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying, or multiple reproduction o f any material i n this publicatio n (including abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Requests fo r suc h permissio n shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematical Society , P . O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o be mad e b y e-mai l t o reprint-permissionQams.org . © 199 7 by th e America n Mathematica l Society . Al l right s reserved . Printed i n the Unite d State s o f America . Reprinted wit h correction s 1997 . The America n Mathematica l Societ y retain s al l right s except thos e granted t o th e Unite d State s Government . @ Th e pape r use d i n this boo k i s acid-free an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . The Londo n Mathematica l Societ y i s incorporated unde r Roya l Charte r and i s registered wit h th e Charit y Commissioners . 10 9 8 7 6 5 4 3 0 2 01 00 9 9 For m y Mother an d Siste r and i n memory o f my Fathe r an d Brothe r with love and thanks This page intentionally left blank Contents Acknowledgements x i Photograph an d Figur e Credit s xii i Chapter 1 . Introductio n 1 Chapter 2 . Historica l Backgroun d 7 2.1. Introductio n 7 2.2. Mathematica l descriptio n o f the three bod y proble m 8 2.3. Histor y o f the three body proble m 1 4 Chapter 3 . Poincare' s Wor k befor e 188 9 2 9 3.1. Introductio n 2 9 3.2. Th e qualitativ e theor y o f differential equation s 2 9 3.3. Celestia l mechanic s an d the three body proble m 4 1 3.4. Othe r paper s 4 4 Chapter 4 . Osca r IPs 60t h Birthda y Competitio n 4 9 4.1. Introductio n 4 9 4.2. Organisatio n o f the competitio n 5 1 4.3. Kronecker' s criticis m 5 9 4.4. Th e entrie s i n the competitio n 6 1 4.5. Judgin g th e entrie s 6 3 4.6. Th e announcemen t o f the resul t 6 5 4.7. Discover y o f the erro r 6 7 4.8. Publicatio n o f the winnin g entrie s 6 9 Chapter 5 . Poincare' s Memoi r o n the Three Bod y Proble m 7 1 5.1. Introductio n 7 1 5.2. Table s o f contents 7 2 5.3. Poincare' s introduction s 7 3 5.4. Genera l propertie s o f differential equation s 7 5 5.5. Theor y o f invariant integral s 8 3 5.6. Theor y o f periodic solution s 9 1 5.7. Stud y o f the cas e with tw o degrees o f freedom 10 4 5.8. Stud y o f asymptotic surface s 10 8 5.9. Furthe r result s 12 2 5.10. Attempt s a t generalisatio n 13 0 Chapter 6 . Receptio n o f Poincare's Memoi r 13 3 6.1. Introductio n 13 3 x CONTENT S 6.2. Th e view s o f the priz e commissio n 6.3. Gylde n 6.4. Minkowsk i 6.5. Hil l 6.6. Whittake r 6.7. Othe r commentator s Chapter 7 . Poincare' s Relate d Wor k afte r 188 9 7.1. Introductio n 7.2. "Le s Methodes Nouvelle s d e la Mecaniqu e Celeste " 7.3. Th e three body proble m an d celestia l mechanic s 7.4. Genera l dynamic s an d "Th e Last Geometri c Theore m Chapter 8 . Associate d Mathematica l Activit y 8.1. Introductio n 8.2. Stabilit y 8.3. Singularitie s an d regularisatio n 8.4. Numerica l investigation s int o periodic solution s Chapter 9 . Hadamar d an d Birkhof f 9.1. Introductio n 9.2. Hadamar d an d geodesie s 9.3. Birkhof f an d dynamica l system s Chapter 10 . Epilogu e 10.1. Introductio n 10.2. Mors e 10.3. KA M theor y Appendix 1 . A letter fro m Gost a Mittag-Leffle r to Sony a Kovalevskay a Appendix 2 . Announcemen t o f the Osca r Competitio n Appendix 3 . Entrie s receive d i n the Osca r Competitio n Appendix 4 . Repor t o f the Priz e Commissio n Appendix 5 . Titl e Page s an d Table s o f Content s 5.1. Poincare' s Unpublishe d Memoi r 5.2. Poincare' s Publishe d Memoi r Appendix 6 . Theorem s i n [PI ] no t include d i n [P2 ] References Acknowledgements This boo k derive s fro m th e Ph D thesi s I prepare d a t th e Ope n Universit y between 1989-1993 , and I am very grateful t o Jeremy Gray, who suggested the topic and patientl y supervise d th e wor k involved . Hi s help , enthusiasm , an d kindnes s were unfailing an d hi s scholarship a n inspiration . Several peopl e bot h i n thi s countr y an d abroa d hav e gon e ou t o f thei r wa y on m y behal f an d I exten d thank s t o the m all . I particularl y wis h t o than k th e Institut Mittag-Leffle r (show n below ) fo r allowin g m e t o us e thei r archives , an d whose staf f provide d m e wit h ever y possibl e assistanc e durin g th e tim e I spen t there; Jespe r Lutzen , Roge r Cooke , an d Sergue i Demido v fo r helpin g t o mak e m y visit to the Institut Mittag-Leffle r s o rewarding and enjoyable ; an d Stee n Norgaar d for providin g insigh t int o Scandinavian culture , and whos e skill s o f translation an d sense o f humour wer e a constant sourc e o f delight . I als o wis h t o expres s thank s t o m y man y friend s an d colleague s a t th e Ope n University wh o helpe d m e i n man y an d varie d ways . I n particula r I wis h t o thank Dere k Richards and Paul Dando fo r usefu l an d stimulating discussion s abou t Poincare's dynamics ; Joh n Fauve l fo r sharin g s o generously hi s knowledge an d ex - pertise; and Mario n Hall , Merrian Lancaster , an d Angel a Redgewell , wh o provide d essential encouragemen t bot h o n the tenni s cour t an d off . Finally, immeasurable thanks go to Rory Collins, without whom my work would not have begun, and Ray Weedon, Chris Glen, and Marlese von Broembsen, withou t whom it would not have continued. Thei r support a t differen t time s and in differen t places meant mor e than word s can say . xi This page intentionally left blank Photograph and Figure Credits The AMS gratefully acknowledges the kindness of these individuals, institutions, and publishers in granting the following permissions. June Barrow-Green Title page on cover and p. 240; from Henri Poincar´e, Sur le Probl`eme des Trois Corps et les Equations´ de la Dynamique, unpublished memoir; Courtesy of June Barrow-Green, private collection. Table of contents on pp. 241–242; from Henri Poincar´e, Sur le Probl`eme des Trois Corps et les Equations´ de la Dynamique, unpublished memoir; Courtesy of June Barrow-Green, private collection. Figure 6.i on cover and p. 247; from Henri Poincar´e, Sur le Probl`eme des Trois Corps et les Equations´ de la Dynamique, un- published memoir, p. 43; Courtesy of June Barrow-Green, pri- vate collection. Institut Mittag-Leffler Photograph of Jacques Hadamard on p. 200; from Acta Mathematica 1882-1912, Table Generale des Tomes 1-35, p. 144; Courtesy of the Institut Mittag-Leffler. Photograph of Charles Hermite on p. 54; from Acta Mathematica 1882-1912, Table Generale des Tomes 1-35, p. 145; Courtesy of the Institut Mittag-Leffler. Photograph of George Hill on p. 144; from Acta Mathematica 1882- 1912, Table Generale des Tomes 1-35, p. 146; Courtesy of the Institut Mittag-Leffler. Photograph of Gosta Mittag-Leffler on p. 52; from Acta Mathematica 1882-1912, Table Generale des Tomes 1-35, p. 160; Courtesy of the Institut Mittag-Lemer. Photograph of Edvard Phragmen on p. 64; from Acta Mathematica 1882-1912, Table Generale des Tomes 1-35, p. 163; Courtesy of the Institut Mittag-Leffler. Photograph of Henri Poincar´e on cover and p. xvi; from Acta Mathematica 1882–1912, Table G´en´erale des Tomes 1–35, p. 164; Courtesy of the Institut Mittag-Leffler. xiii xiv PHOTOGRAPH AND FIGURE CREDITS Photograph of Karl Weierstrass on p. 56; from Acta Mathematica 1882-1912, Table Generate des Tomes 1-35, p. 176; Courtesy of the Institut Mittag-Leffler. Letter from Poincare to Mittag-Leffler on pp. 120-121; Courtesy of Institut Mittag-Leffler Archives. PrixPrix OscarOscar IIII on on pp. pp 234–235;. 236-237 from; fro Actam Act Mathematicaa Mathematic11a (1888),11 (1888), pp.pp. 401–402;401-402; CourtesyCourtesy ofof thethe InstitutInstitut Mittag-Leffler.Mittag-Leffler. TitleTitle pagepage onon p.p. 243; 245 from; from Henri Henr Poincar´i Poincaree, Sur, Sur le Probl`le Problemeeme des des TroisTrois CorpsCorps etet leslesEquations ´Equations dede la la Dynamique Dynamique,,ActaMathemat- Acta Mathematica, ica,13 (1890)13 (1890);; Courtes Courtesyy of th ofe Institu the Institutt Mittag-Leffler Mittag-Leffler.. TableTable o off Content Contentss o onn pp pp.. 246-247 244–245;; fro fromm Henr Henrii Poincare Poincar´, Sure, Sur le Pro- lebleme Probl` deseme Trois des Corps Trois Corpset les Equations et les Equations´ de la Dynamique, de la Dynamique Acta Math, - Actaematica Mathematica,, 13 (1890); 13Courtes(1890);y of Courtesy the Institu oftthe Mittag-Leffler Institut Mittag-. Leffler. Figures 5.5.i and 5.5.ii on p. 90; from Henri Poincare, Sur le Probleme des Trois Corps et les Equations de la Dynamique, Acta Mathemat- ica, 13 (1890); Courtesy of the Institut Mittag-Leffler. Figures 5.7.i and 5.7.ii on p. 106; from Henri Poincare, Sur le Pro- bleme des Trois Corps et les Equations de la Dynamique, Acta Math- ematica, 13 (1890); Courtesy of the Institut Mittag-Leffler. Figure 5.8.i on p. Ill; from Henri Poincare, Sur le Probleme des Trois Corps et les Equations de la Dynamique, Acta Mathematica, 13 (1890); Courtesy of the Institut Mittag-Leffler. Figure 5.8.ii on p. 112; from Henri Poincare, Sur le Probleme des Trois Corps et les Equations de la Dynamique, Acta Mathematica, 13 (1890); Courtesy of the Institut Mittag-Leffler. Figure 5.8.hi on p. 115; from Henri Poincare, Sur le Probleme des Trois Corps et les Equations de la Dynamique, Acta Mathematica, 13 (1890); Courtesy of the Institut Mittag-Leffler. Figure 5.9.i on p. 123; from Henri Poincare, Sur le Probleme des Trois Corps et les Equations de la Dynamique, Acta Mathematica, 13 (1890); Courtesy of the Institut Mittag-Leffler. Librairie Scientifique et Technique Albert Blanchard Figure 2.3.iv on p. 25; from Henri Poincare, Les methodes nouvelles de la mechanique celeste I, reprinted by Albert Blanchard, 1987, p. 109; Courtesy of Albert Blanchard, Librarie Scientifique et Technique. The Johns Hopkins University Press Figure 2.3.hi on p. 25; from Collected mathematical works of G. W. Hill, American Journal of Mathematics 1 (1878), p. 335; Courtesy of The Johns Hopkins University Press. PHOTOGRAPH AND FIGURE CREDITS xv London Mathematical Society FigureFigure 10.iiilO.iii onon covercover andand p p.. 224;226; fromfrom V.V. I.I. Arnold,Arnold, SmallSmall de-denom- nominatorsinators and andproblems problems of stability of stability of motion of motion in classical in classical and andcelestial celestialmechanics, mechanics Russian, Mathematica Russian Mathematicall Surveys, 1 Surveys,8 (1963)18, p.(1963), 100; Cour- p.tes 100;y of th Courtesye Londo ofn Mathematica the London Mathematicall Society. Society. The Royal Swedish Academy of Sciences Photograph of Hugo Gylden on p. 140; from Yearbook of the Royal Swedish Academy, 1939; Courtesy of The Royal Swedish Academy of Sciences. Springer-Verlag Berlin Figure 5.5.hi on p. 91; from June Barrow-Green, Oscar IPs prize competition and the error in Poincare's memoir on the three body problem, Archive for the History of Exact Sciences, 48, No. 2 (1994) p. 124; Courtesy of Springer-Verlag. The Swedish National Art Museums Portrait of King Oscar II by Oskar Bjork on cover and p. 50; Courtesy of The Royal Palace Library. TheThe followingfollowing figuresfigures andand photographsphotographs areare inin thethe publicpublic domain.domain. FigurFiguree 8.4.i 8.4.iii on on p p.. 195 195;; fromfrom SirSir GeorgeGeorgeDarwin, Darwin,Periodic Periodic Orbits Orbits,, ScientifiScientificc Paper Paperss IVIV, p,. 12 p.12.. Figure 9.2.i on p, 205; from Jacques Hadamard, Œuvres de FigurJacquese 9.2. Hadamardi on p, 206,; vol. from 2, Jacque 1968, pp.s Hadamard 741. , (Euvres de Jacques Hadamard, vol. 2, 1968, pp. 741. Figure 9.2.ii on p, 205; from Jacques Hadamard, Œuvres de Jacques Hadamard, vol. 2, 1968, pp. 744. Figure 9.2.ii on p, 206; from Jacques Hadamard, (Euvres de Jacques Hadamard,Figure 9.2.iii vol. on2, 1968 p, 207;, pp. from744. Jacques Hadamard, Œuvres de Jacques Hadamard, vol. 2, 1968, pp. 758. FigurPhotographe 9.2.hi on of p Alexander, 208; from LiapunovJacques Hadamard on p. 178., (Euvres de Jacques Hadamard, vol. 2, 1968, pp. 758. Photograph of Karl Sundman on p. 188. The AmericanPhotograp Mathematicalh of Alexande Societyr Liapuno holdsv o copyrightn p. 178. to the following. PhotograpFigure 10.iih of onKar p.l 222;Sundma fromn Raoulon p. 188 Bott,. Marston Morse and his mathematical works, Bull. Amer. Math. Soc. 3 (1980), p. 915. The AmericaPhotographn Mathematica of Georgel Societ Birkhoffy hold ons copyrigh p. 210. t to the following. Figure 10.ii on p. 224; from Raoul Bott, Marston Morse and his mathematical works, Bull. Amer. Math. Soc. 3 (1980), p. 915. Photograph of George Birkhoff on p. 212. \ . • ~ rA Henri Poincar e This page intentionally left blank APPENDIX 1 A lette r fro m Gost a Mittag-Leffle r to Sony a Kovalevskay a Helsingfors, 7.6.188 4 I agree with Weierstrass, i f none o f the answer s o n the se t questio n ar e worth y of th e prize , the n th e meda l mus t b e awarde d t o th e mathematicia n wh o withi n recent year s ha s mad e th e bes t discoverie s i n highe r analysis . Bu t I canno t agre e with the view that i t will not further th e progress o f science to propose specific priz e questions, i n particular i f they ar e stated reasonably . Wha t abou t th e significanc e for the development o f the theory o f linear differentia l equation s resulting fro m th e last priz e question fro m th e Frenc h Academy . Thi s questio n provide d th e startin g point fo r Poincare' s work . Furthermore , ther e doe s no t exis t a priz e exclusivel y intended fo r pure analysis, hardly a prize exclusively intended fo r pure mathematics, apart fro m Steiner' s priz e in Berlin. Th e shortcoming o f the Steine r priz e i s that i t is awarded too often, ever y year. Bu t we should not award our prize more frequentl y than every fourth year. Malmste n and the King want the prize jury to be appointe d by the Kin g and to consis t o f 1. Th e mai n edito r o f Acta Mathematica 2. A German o r Austrian mathematicia n - = Weierstras s 3. A French o r Belgia n mathematicia n - = Hermit e 4. A n Englis h o r America n mathematicia n - = Cayley ? o r Sylveste r 5. A Russian o r Italian mathematicia n - = = the firs t tim e Briosch i o r Tschebychef, th e secon d tim e Mr s Kovalevskaya . After eac h priz e givin g tw o o f the priz e judges shoul d leav e the jury an d ne w ones shoul d b e appointe d b y Kin g Osca r a s lon g a s h e i s aliv e - h e mus t b e abl e to appoin t [substitutes] fo r bot h th e leavin g members . Afte r Kin g Oscar' s death , the thre e remainin g mus t appoin t tw o ne w member s bu t alway s i n suc h a wa y as t o fi t th e categorie s mentione d above . Imagin e i f on e ha d t o awar d a priz e to th e bes t mathematica l wor k whic h ha s appeare d durin g th e las t fou r years . Then th e nationa l difference s woul d certainl y sho w u p an d th e differen t view s o n what constitutes the essential substance o f mathematics would be clearly expressed . Cayley and Briosch i migh t wan t t o make the award to Go d know s which master o f calculating and Tschebyche f migh t op t fo r Go d knows what od d ideas . I t i s quite a different matte r whe n on e has to judge answer s to a specifi c question . I n thi s cas e one i s forced t o stic k to muc h mor e objectiv e criteria . 227 228 APPENDIX 1 And finally on e mor e reason . Kin g Osca r i s convince d tha t w e shoul d onl y announce specifi c priz e questions and I doubt tha t i t wil l be possibl e to change hi s mind unless I propose that the prize should be used to honour Swedis h or Norwegian papers - lik e th e anatomi c priz e yo u mentione d whic h i s probabl y restricte d t o Russian work s - but I d o not wan t thi s a t all . Th e competitio n woul d no t ge t th e international scientifi c reputatio n whic h I ha d imagined . I thin k tha t honourin g only the works published i n Acta would be more in the interests o f the journal tha n in the interest s o f science . An d thi s I do not wan t either . Th e interest s o f scienc e must com e first. I f later I can do something fo r Act a at th e sam e time then that i s another matte r an d I will do it wit h al l my heart . APPENDIX 2 Announcement o f the Osca r Competitio n Nature 30.7.1885 THE HIGHER MATHEMATICS Prof. G . Mittag-Leffler , principa l edito r o f the Acta Mathematics forward s u s the followin g communication , whic h wil l shortly appea r i n that journal: - His Majest y Osca r II , wishin g t o giv e a fres h proo f o f hi s interes t i n th e ad - vancement o f mathematical science, an interest already manifested b y his graciously encouraging th e publicatio n o f the journal Acta Mathematical whic h i s placed un - der hi s augus t protection , ha s resolve d t o awar d a prize, o n Januar y 21 , 1889, th e sixtieth anniversary o f his birthday, to an important discover y i n the field o f higher mathematical analysis . Thi s prize will consist o f a gold medal o f the eighteenth siz e bearing his Majesty's imag e and having a value o f a thousand francs , togethe r wit h a sum o f two thousand five hundred crown s ( 1 crown = abou t 1 franc 4 - centimes). His Majesty ha s been pleased to entrust th e task o f carrying out hi s intention s to a commissio n o f thre e members , Mr . Kar l Weierstras s i n Berlin , Mr . Charle s Hermite i n Paris , an d th e chie f edito r o f thi s journal , Mr . Gost a Mittag-Leffle r in Stockholm . Th e commissioner s havin g presente d a report o n thei r wor k t o hi s Majesty, h e has graciously signifie d hi s approval o f the followin g final proposition s of theirs. Having take n int o consideratio n th e question s whic h fro m differen t point s o f view equall y engag e th e attentio n o f analysts , an d th e solutio n o f whic h woul d be o f the greates t interes t fo r th e progres s o f science , the commissio n respectfull y proposes to his Majesty to award the prize to the best memoir on one of the followin g subjects:- (1) A syste m bein g give n o f a numbe r whateve r o f particle s attractin g on e another mutuall y accordin g t o Newton' s law , i t i s proposed , o n th e assumptio n that ther e neve r take s plac e a n impac t o f two particles t o expan d th e coordinate s of eac h particl e i n a serie s proceedin g accordin g t o som e know n function s o f tim e and convergin g uniforml y fo r an y spac e o f time. It seem s that thi s problem , th e solutio n o f which wil l considerably enlarg e ou r knowledge wit h regar d t o th e syste m o f th e universe , migh t b e solve d b y mean s of th e analytica l resource s a t ou r presen t disposition ; thi s ma y a t leas t b e fairl y supposed, becaus e shortl y befor e hi s deat h Lejeune-Dirichle t communicate d t o a friend o f his, a mathematician, tha t h e had discovere d a method o f integrating th e differential equation s o f mechanics , an d tha t h e ha d succeeded , b y applyin g thi s method, to demonstrate the stability o f our planetary system in an absolutely stric t 229 230 APPENDIX 2 manner. Unfortunatel y w e know nothing about this method except that the starting point fo r it s discover y seem s t o hav e bee n th e theor y o f infinitel y smal l oscilla - tions.334 I t may , however , b e suppose d almos t wit h certaint y tha t thi s metho d was no t base d o n lon g an d complicate d calculation s bu t o n th e developmen t o f a simple fundamenta l idea , whic h on e ma y reasonabl y hop e t o find agai n b y mean s of earnest an d perseverin g study . However, i n case no one should succee d in solving the proposed proble m withi n the period o f the competition, th e prize might b e awarded to a work in which som e other proble m o f mechanic s i s treate d i n th e indicate d manne r an d completel y solved. (2) Mr . Fuch s ha s demonstrate d i n severa l o f hi s memoirs 335 tha t ther e exis t uniform functions o f two variables which, by their mode of generation, are connected with th e ultra-elliptica l functions , bu t ar e mor e genera l tha n these , an d whic h would probabl y acquir e grea t importanc e fo r analysis , i f their theor y wer e furthe r developed. It i s proposed to obtain i n an explicit for m thos e functions whos e existence ha s been proved b y Mr. Fuchs , in a sufficiently genera l case, so as to allo w o f an insigh t into and stud y o f their mos t essentia l properties . (3) A study of the functions define d b y a sufficiently genera l differential equatio n of th e first order , th e first membe r o f whic h i s a rationa l integra l functio n wit h respect t o the variable , the function , an d it s first differentia l coefficient . Mr. Brio t an d Mr . Bouque t hav e opene d th e wa y fo r suc h a stud y b y thei r memoir o n this subjec t (Journal d e l'Ecole pol y technique, cahie r 36 , pp. 133-198) . But mathematician s acquainte d wit h th e result s attaine d b y thes e author s kno w also that thei r wor k ha s no t b y an y mean s exhauste d th e difficul t an d importan t subject whic h they have first treated. I t seem s probable that, i f fresh inquirie s were to be undertaken i n the same direction, they might lea d to theorems o f high interest for analysis . (4) I t i s well known ho w much ligh t ha s bee n throw n o n the genera l theor y o f algebraic equation s b y the stud y o f the specia l function s t o whic h th e divisio n o f the circl e into equal parts an d the divisio n o f the argument o f the elliptic function s by a whol e numbe r lea d up . Tha t remarkabl e transcendan t whic h i s obtained b y expressing th e modul e o f a n ellipti c functio n b y th e quotien t o f the period s lead s likewise to the modulary equations, that have been the origin of entirely new notions and highl y importan t results , a s the solutio n o f equations i n the fifth degree . Bu t 334See p . 3 5 o f th e Panegyri c o n Lejeune-Dirichle t b y Kummer , "Abhandlunge n de r K . Akademie de r Wissenschafte n z u Berlin, " 1860 . 335 These memoir s ar e t o b e foun d i n (1 ) "Nachrichte n vo n de r K . Gesellschaf t de r Wis - senschaften z u Gottingen, " February , 1880 , p . 170 ; (2 ) Borchardt' s "Journal, " Bd . 89 , p . 25 1 (a translatio n o f thi s memoi r i s to b e foun d i n th e "Bulletin " o f Mr . Darboux , 2m e serie , t.iv) ; (3) "Nachrichte n vo n de r K . Gesellschaf t de r Wissenschafte n z u Gottingen, " June , 1880 , p . 44 5 (translated int o Frenc h i n the "bulletin " o f Mr . Darboux , 2m e serie , t.iv) ; (4 ) Borchardt' s "jour - nal," Bd . 90 , p . 7 1 (als o i n th e "Bulleti n o f Mr . Darboux , 2m e serie , t.iv) ; (5 ) "Abhandlunge n der K . Gesellschaf t de r Wissenschafte n z u Gottingen," 188 1 ("Bulletin " o f Mr. Darboux , t.v) ; (6 ) "Sitzungsberichte de r K . Akademi e de r Wissenschafte n z u Berlin" 1883 , i, p. 507 ; (7 ) The memoi r of Mr . Fuch s publishe d i n Borchardt' s "Journal, " Bd . 76 , p . 177 , ha s als o som e bearing s o n th e memoirs quoted . APPENDIX 2 231 this transcendan t i s bu t th e firs t term , a particula r case an d tha t th e simples t one o f a n infinit e serie s o f ne w function s introduce d int o scienc e b y Mr . Poincar e under th e nam e o f "fonction s fuchsiennes, " an d successfully applie d b y hi m to th e integration o f linea r differentia l equation s o f an y order . Thes e functions , whic h accordingly hav e a role o f manifes t importanc e i n analysis , hav e no t a s ye t bee n considered fro m a n algebraica l poin t o f vie w a s the transcendan t o f the theor y o f elliptic function s o f which they ar e the generalisation . It i s propose d t o fil l u p thi s ga p an d t o arriv e a t ne w equation s analogou s to th e modular y equation s b y studying , thoug h i t wer e onl y i n a particula r case , the formatio n an d properties o f the algebrai c relations that connec t tw o "fonction s fuchsiennes" whe n they hav e a group i n common . In cas e non e o f th e memoir s tendere d fo r competitio n o n an y o f th e subject s proposed abov e shoul d b e deeme d worth y o f the prize , thi s ma y b e adjudge d t o a memoir sen t i n fo r competitio n tha t contain s a complet e solutio n o f an importan t question o f the theory o f functions othe r than thos e propose d b y the Commission . The memoir s offere d fo r competitio n shoul d b e furnishe d wit h a n epigrap h and, besides , with the author' s nam e an d plac e o f residence i n a seale d cover , an d directed t o the chie f editor o f the Acta Mathematica before Jun e 1 , 1888. The memoir to which his Majesty shal l be pleased to award the prize as well as that o r thos e memoir s whic h ma y b e considere d b y the Commissio n worth y o f a n honorary mention , wil l be inserte d i n the Acta Mathematica, no r ca n an y o f the m be previously published . The memoir s ma y b e written i n an y languag e tha t th e autho r chooses , but a s the members o f the Commissio n belon g to three differen t nation s the author ough t to subjoi n a Frenc h translatio n t o hi s origina l memoir , i n cas e i t i s not writte n i n French. I f such a translation i s not subjoined the author must allo w the Commissio n to hav e one made fo r thei r ow n use. THE EDITOR S O F ACT A MATHEMATIC A This page intentionally left blank APPENDIX 3 Entries receive d i n the Osca r Competitio n The followin g page s contai n th e announcemen t o f th e entrie s receive d i n th e Oscar Competitio n (Acta, 1 1 (1888) , 401-402) . Th e title s o f the entrie s receive d for the competitio n ar e listed i n the orde r i n which they wer e received . 233 234 APPENDIX 3 Prix Osca r I I Memoires presente s a u concours. Le concour s pou r l e pri x fond e pa r S . M . l e ro i OSCA R I I a ete clos l e i r jui n d e cett e annce . Nou s mentionnon s ci-aprc s e t dans I'ordr e oil il s son t parvenus , le s mcmoire s destine s a u concour s qu i on t etc adresses a u Redacteu r e n che f d e c e journal, a Stockholm : i. Memoire sur Vequation trinome de degre impair x m + or = r . Epigraphe: Le s troi s noinbre s harmonique s elemcntaire s sont 2 , 3 e t 5- 2. Nuova Tear la del Massimi e Minimi degli Integrali defmiti. Epigraphe: Opinionu m comnient a dele t dies ; natura e judici a coufirinat. (Cic. Nat . D. ) 3. Allgemeine Entwicklung der Functionen. Epigraphe: Sic h selbs t z u lobe n is t ei n Fehlcr , Doch jcde r thut\s , de r etwa s Gute s thut . (Wcstdstlieher Diva n vo n Gothe. ) L'auteur y a join t un e traduction franchise : Developpement general des fonctions avec 1 epigraphe : T u n e fai s pa s bien e n t e louan t toi-inein e Mais t u t e loue s to i racme e n faisan t bien . (D apres Goethe. ) 4. Les Fonctions Pseudo- et Ilyper-BernouUiennes et tears premieres applications. — Contributio n elementair e a 1'integratio n de s equation s dif - ferent i el les. Epigraphe: Venien t qu i sin e offensa , sin e gratia , judieent . (Senequc.) 5. liber die Bewegnngen in einem System von Massepunlden mit Kraft en der Form 7 • /• Kpigraphe: ' :\~h>r)± o tiiyo^ 7^ dhj/islac iif>. (Euripides.) Acta nutthrmutira. 11 . Imprh,: * l e 1 7 A«HI1 . 1>HK . ' , \ APPENDIX 3 235 402 Pri x Osca r II . — Meinoire s presente r a u concour.-' . 6. Integration des equations slmidtanees aux derivees partieiles du premier ordre d'an nombre quelconque de fonctlons de plusleurs variables in depend antes. Kpigraphe: Accip e jussi s carmina cept a tuis . 7. liber die Integration der Differentialgleichun gen, ivelche die Be- tvegungen elnes Systems von Puncten best tinmen. Kpigraphe: Nu r schrittweis e gelang t ma n zui u Ziel . Avec un e traductio n francaise , intitulee : Sur Vintegration des equations different lelles qui determinant les mouve- ments d'an systeme de points mater lets, et portan t 1 cpigraphe: Pou r parvenir a u souiuiet, i l fau t marche r pa s a pas. 8. Sur les Integrates de fonctlons a multiplicateurs et leur application au dereloppement des fonctlons abellennes en series trlgonometriques. Kpigraphe: Nou s devon s 1 uniqu e scienc e Que 1 homm e puiss e conqueri r Aux chercheur s don t l a patienc e Kn a laiss c le s fruit s umrir . (Sully-Prudhomme, L c IJonheur. ) Avec u n Supplement . 9. Sur le Probleme des trols Corps et les liquations de la Dynamique. Kpigraphe: Nunqua m praescripto s transibun t sider a fines . 10. Sur le Probleme des trols Corps. Kpigraphe: - - — — — — Ooeluinqu c tuer i Jussit e t erecto s a d sider a tollor e vultus . (Ovide.) 11. Vber die Bewegung der Hlmmelskorper em wlderstehenden Mitt el. Kpigraphe: Pe r asper a a d astra . 12. Recherehes sur la for mule sommatolrc d'Euler. Kpigraphe: Utina m n e nimi s crraveriiu . Juin 1888 . MITTAG-LEFFLER. This page intentionally left blank APPENDIX 4 Report o f the Priz e Commissio n Poincare (Euvres XI, 286-289 French translation sen t to Poincare afte r th e announcement o f the competitio n result. Traduction Proces-verbal dress e pa r devan t S . M . l e Ro i a u palai s d e Stockholm , l e 2 0 Janvier 1889 , en presence de S. Exc. M . le Comte Ehrensvard, Ministr e des Affaire s Etrangeres, M . G . Wennerberg , Ministr e de s Culte s e t d e l'lnstructio n Publique , M. R . O . Schjott , Ministr e Norvegie n e t d e M . G . Mittag-Leffler , professeu r a l'universite d e Stockholm . §1. L a commission , nomme e pa r S . M . l e Roi, e n date d u 2 5 Novembre 1884 , pour examine r de s memoires, ayan t concour u pou r l e prix en mathematiques offer t par S a Majeste , e t compos e d e M . Car l Weierstrass , professeu r a l'universit e d e Berlin, M . Charles Hermite, professeu r a la Sorbonne a Paris, e t M . Gosta Mittag - Leffler, professeu r a l'universite d e Stockholm, ayant termine ses travaux, l e rapport de l a commission fu t soumi s a u Roi . II ressor t d e c e rappor t qu e l a commissio n a et e d e l'opinio n unanime , qu e le memoir e qu i es t intitul e Sur le probleme des trois corps et les equations de la dynamique ave c l a devis e "Nunqua m praescripto s transibun t sider a fines" , es t l'oeuvre profond e e t original e d'u n geni e mathematiqu e don t l a plac e es t marqu e parmi le s grand s geometre s d u siecle . Le s plu s impor t antes e t le s plu s difficile s questions, comm e l a stabilit e d u system e d u monde , l'expressio n analytiqu e de s coordonnees de s planete s pa r de s serie s d e sinu s e t d e cosinu s de s multiple s d u temps, pui s l'etud e o n n e peut plu s remarquable, de s mouvements asymptotiques , la decouverte d e formes d e mouvement o u les distances de s corps restant comprise s entre de s limite s fixes, o n n e peu t cependan t exprime r leur s coordonnee s pa r de s series trigonometriques , d'autre s sujet s encor e qu e nou s n'indiquon s point , son t traites par de s methodes qu i ouvrent, i l n'est qu e juste d e l e dire, une epoque nou - velle dans l a mecanique celeste . Le s notions analytique s inconnue s d e Lagrange e t de Laplace , qu i n'on t et e acquise s qu e d e notr e temps , on t u n rol e essentie l dan s ces question s s i difficile s o u l e talen t d e I'auteu r s e montr e dan s tou t so n eclat . Une foi s d e plu s s e trouv e ainsi s confirm e cett e observatio n qu e le s plu s grand s progres e n astronomie, e n physique e t le s decouvertes qu i etenden t l e domaine de s mathematiques abstractes , s e produisent simultanement , comm e si elles etaient ap - pelees a s e seconder e n concourant a un mem e but, e t qu e l a commission d e mem e a ete unanime dan s l'opinion , qu e I'auteur d u memoir e qu i porte pou r titre Sur les 237 238 APPENDIX 4 integrates des fonctions a multiplicateurs et leur application au developpement des fonctions abeliennes en series trigonometriques, e t a pour devis e "Nous devons T unique scienc e Que rhomme puiss e conqueri r Aux chercheur s don t l a patienc e En a laisse les fruits murir." 336 a montre un talent mathematique de premier ordre, et que son memoire est extreme- ment dign e d e 1'attention des geometres . §2. S . M. le Roi daigna decerner l e prix offert pa r S a Majeste e t compose d'un e medaille sen or evaluee a environs 1,00 0 franc s ains i que la somme 2,500 couronnes a I'auteur d e memoire mun i d e I'epigraphe "Nunqua m praescripto s transibunt sider a fines" e t un exemplaire de la medaille a l'effigie d e Sa Majeste et portant l'inscriptio n "in sui memoriam" a I'auteur d e memoire portant I'epigraphe : "Nous devons l'unique scienc e .. . . " §3. S . M . l e Ro i ayan t e n suit e ouver t le s bulletin s accompagnan t le s di t memoires, i l a et e constat e qu e l e bulleti n a I'epigraphe : "Nunqua m praescrip - tos transibun t sider a fines " portai t l e no m "M . H . Poincare , Paris" , e t celu i a I'epigraphe: "Nous devon s l'unique scienc e ... " le nom d e "Pau l Appell , Paris" . Ainsi passe : A u Chateau d e Stockholm l e 20 Janvier 1889 . Oscar Alb. Ehrensvar d G . Wennerber g P. O . Schjot t G . Mittag-Leffle r Otto Printzskol d Sully-Prudhomme, L e Bonheur . APPENDIX 5 Title Page s an d Table s o f Content s The followin g page s contai n th e titl e pag e an d th e tabl e o f content s fo r [PI ] and [P2] , the unpublishe d an d th e publishe d version s o f Poincare's memoi r o n th e three body problem . 239 240 APPENDI X 5 5.1. Poincare' s Unpublishe d Memoi r SUR L E PROBLEME DE S TROIS CORP S ET LE S EQUATIONS D E L A DYNAMIQU E PAR H. POINCAR E M £ M 0 I R E COURONN & DU PRI X D E S . M . L E KO I OSCA R I I AVEC DE S NOTE S PAR L'AUTEUR . 5.1. POINCARE' S UNPUBLISHE D MEMOI R 241 TABLE DE S MATIERES . Pages. Introduction 5 Premiere partie . Generalites. Chapitre I . Notation s e t definition s 9 Chapitre II . Th6ori e de s invariant s int^graux . § 1 . Propriete s diverge s dp s Equation s d e l a dynamiqu e 1 4 § 2 . Definitio n de s invariant s inte'grau x 2 L § 3 . Transformation s de s invariant s inte'grau x 2 5 § 4 . Usag e de s invariant s integrau x 3 1 Chapitre III . Th^ori e de s solution s periodiques . § 1 . Existenc e de s solution s peYiodiqucs.,. . 4 8 § 2 . Exposant s caracte'ristique s 5 8 § 3 . Solution s periodique s de s Equation s de l a dynamiqu e 6 5 § 4 . Calcu l de s exposant s caracferistique s 8 0 § 5 . Solution s asymptotique s 8 8 Deuxi&me partie . Equations d e l a dynamiqu e e t probiem e de s n corps . Chapitre I . Etud e d u ca s of t i l n' y a qu e deu x degr£ s d e liberte . § 1 . Representation s geometrique s diverse s 9 7 § 2 . Equatio n de 3 surface s asymptotique s 11 2 § 3 . Constructio n de s surface s asymptotique s (premier e approximation ) 12 2 § 4 . Constructio n exact e de s surface s asymptotique s 13 5 § 5 . Solution s periodique s d u 2 me genr e 14 4 242 APPENDIX 5 Pages. Chapitre II . R6sum £ g£n£ra l de s r£sultats . § 1 . R&ultat s positif s 15 3 § 2 . R&mltat s n<5gatif s 15 5 Chapitre III . Tentative s d e generalisatio n 15 8 Notes. A. Su r l a divergenc e de s serie s d e M . Lindsted t 16 3 B. Nouve l expose * de s resultat s 17 4 C. Su r le s invariant s intdgrau x 18 3 D. Su r le s Equation s lin^aire s a coeflficient s p^riodique s 18 8 E. Su r l e calcu l de s limite s 19 3 F. Su r le s surface s asymptotique s 21 9 GL Su r l a non-existenc e de s integrate s uniforme s 24 3 H. Su r le s exposant s caracteVistique s 24 9 I. Su r le s solution s asymptotique s 25 1 5.2. POINCARE' S PUBLISHE D MEMOI R 24 3 5.2. Poincare' s Publishe d Memoi r SUR L E PROBLEME DE S TROIS CORP S ET LE S EQUATIONS D E L A DYNAMIQU E PAE H. POINCARE ; & PABIB . MAMOIRE COURONN S DU PRI X D E S . M . L E EO I OSCA R I I LE 2 1 JANVIE R 1S80 . 244 APPENDIX 5 TABLE DE S MATIERES . Pages. Introduction 5 Premi&re partie . General ites. Chapitre I . Propri£t6 s g£n6rale e de s Equations difterentielles . $ 1 , Notation s c t definition s 8 § 2 . Calcu l dc s liraitc s 1 9 § 3 . Application s d u calcu l dc s limite s au x Equation s au x ddrivdcs partielle s 2 6 § 4 . Integratio n de s Equation s lindaire s k coefficient s periodique s 4 1 Chapitre II . Thgori e de s invariants int£graux . $ 5 . Propric'tc s divcrsc s dc s Equation s d e l a dynamiqu e 4 G S 6 . Definitio n dc s invariant s intdgrau x 5 2 £ 7 . Transformation de s invariant s intdgrau x 0 2 § 8 . Usag e de s invariant s intdgrau x C 7 Chapitre III . Th^ori e de s solutions periodiques , £ 9 . Kxistcnc e dc s solution s pdriodiquc s 8 8 $ 10 . Kxponant s caracterisfique s 9 7 S 11 . Solution s periodiques dc s Equation s d e l a dynamiqu e 10 3 S 12 . Calcu l de s exposant s caractdristique s 12 2 S 13 . iSolution s asymptotique s 13 6 S 14 . Solution s asymptotique s de s Equation s d e l a dynamiqu e 14 0 Deuxi&me partie - Equations d e l a dynamique e t problem e de s n corps . Chapitre I . Etud e de s cas oil il n' y a qu e deux degr£ s d e liberty. 3 15 . Representation s gdomdtrique s divcrses . 1(3( 3 5.2. POINCARE' S PUBLISHE D MEMOI R 245 Chapitre II . Etude s de s surface s asymptotiques . S 1G . Expose * d u proMeni c 1 SI >* 17 . Prcmier o approximatio n 1 Hi S 18 . Deuxftm e approximatio n 10 7 S It) . Troisicm o approximatio n 21 9 Chapitre III . Rdsultat s divers . $ 20 . Solution s pcViodique s d u 2 mp genr e 22 S S 21 . Divergenc e de s serie s d o M . Lindsted t 2i 0 § 22 . Nonexistenc e do s intdgrale s uniforme s 25 0 Chapitre IV . Tentative s d e generalisation . ?? 23. Problem o de s n corp s 2G 0 This page intentionally left blank APPENDIX 6 Theorems i n [PI ] no t include d i n [P2 ] In th e chapte r o n invarian t integral s i n [PI ] Poincar e include d tw o extension s to Theorem III, as well as a further theorem , Theore m IV . These theorems ar e no t included i n [P2 ] because either they becam e redundant a s a result o f the discover y of the erro r o r they ha d n o particular relevanc e to an y othe r par t o f the memoir . FIRST EXTENSIO N T O THEORE M III . Suppose that A nBn coincides partly with AQBQ and partly with the extension of AQBO such that AQBQ is an nth order invari- ant curve. Suppose also that the distance between AQ and B p is a small quantity of qth order, where p is prime to n, then the distance from AQ to its nth consequent An is a very small quantity of qth order. This i s the cas e describe d b y Figure 6.i where n is taken to b e 5 . Combining thi s generalisatio n wit h th e Corollary , Poincar e deduce d a furthe r generalisation i n which h e gave the conditions unde r whic h h e claimed that th e se t of curve s forme d b y A 0B0 an d it s successiv e consequent s woul d for m a "closed " invariant curv e o f firs t order . B y invokin g th e Corollar y h e agai n reiterate d th e error h e had mad e earlier . Poincare preface d th e secon d extensio n t o Theore m II I [PI ] b y sayin g that h e did no t expec t t o us e i t i n wha t followed , althoug h h e late r cite d i t twice , bot h references becomin g redundant i n the revision . SECOND EXTENSIO N T O THEORE M III . A curve without being rigorously in- variant may be invariant up to a very small quantity of pth order. If the distance FIGURE 6. i 247 248 APPENDIX 6 between a curve C, which is not rigorously invariant, and an arbitrary point of its nth consequent is a very small quantity ofpth order, then such a curve is called nth order semi-invariant up to very small quantities of pth order. If a semi-invariant curve is quasi-closed such that the distance between the points of closure A and B is very small of qth order, the distance from the point A to its nth consequent A n will be very small of order at least q providing 2q < p, and of order p — q providing 2q> p> q. THEOREM IV . Consider a transverse section S which is simply connected. Let a point on S be determined by a particular system of coordinates (to be defined) which is analogous to polar coordinates. Let O be an arbitrary point on S at which infinitely many branches of a curve meet, in the same way that radius vectors meet at the pole in polar coordinates. Suppose that O is the only common point of any two branches of the curve and that an arbitrary branch is defined by the angle q between its tangent at O and a fixed line passing through O. Consider a second system of closed concentric curves containing the point 0. Furthermore, suppose that any curve of the second system has one and only one point in common with any curve of the first system. Consider a fixed branch of the first system B 0 and let P be the point where it cuts a moving curve of the second system. Let r be the length of the arc of the curve B 0 between 0 and P. The moving curve can then be defined by r. Finally suppose that through an arbitrary point P of S there passes one and only one branch of the first system. The coordinates r and q can then be used to define the position of P on S. Let a be a simply connected area of S limited by a closed curve k. Let a n be the nth consequent limited by the closed curve k n. If the two areas a and a nhave a part in common and 0 belongs to this communal part, if the points of k have same coordinate q as their n consequents, if the curve k meets each of the branches of the first system at one point (such that when one crosses the closed curve k, q varies between 0 and 2n), if there is a positive invariant integral, then two at least of the points k coincide with their nth consequents. Poincare als o gave a second, mor e succinct, statement o f the theorem whic h di d not involv e the coordinate system defined above . Le t k be a closed curve on a simply connected transvers e sectio n S with nth consequen t k n. I f eac h o f the point s o f k can be joined to its nth consequent b y arcs of curves on S in such a way that n o two of these arc s hav e a poin t i n common , and , moreover , ther e i s a positiv e invarian t integral, tw o at leas t o f the point s o f k will coincide with thei r consequents . There appear s t o b e n o clea r reaso n wh y Poincar e include d thi s Theore m i n [PI], a s h e mad e n o us e o f i t there . Hi s onl y referenc e t o i t wa s a n expressio n of regre t tha t h e di d no t hav e th e opportunit y t o sho w ho w i t coul d b e applie d in th e stud y o f th e spatia l distributio n o f close d trajectories . I t ma y hav e bee n because h e wa s writin g t o a deadlin e tha t h e decide d i t wa s altogethe r easie r t o keep i t i n his competition entry , o r i t ma y hav e been simpl y that h e was not goo d at organisin g his material. Alternatively , i t may have been because he decided tha t having establishe d th e resul t i t mad e sens e t o publis h i t s o that i t wa s availabl e should h e ever nee d it , althoug h i t doe s not appea r tha t h e ever mad e us e o f it. References Items liste d includ e al l published work s referred t o i n the text an d som e other s found usefu l i n its preparation . Abbreviations Acta Act a Mathematic a BAAS Britis h Associatio n fo r the Advancemen t o f Scienc e Cahiers Cahier s d e Seminair e d'Histoir e de s Mathematique s Comptes Rendus Compte s Rendus Hebdomadaires de s Seances de TAcademie de s Sciences / M-L Institu t Mittag-Leffle r R. C . Archibal d [1936] Unpublishe d Letter s o f James Josep h Sylveste r an d othe r ne w Infor - mation concernin g hi s Lif e an d Work , Osiris 1 , 85-154 . V. I . Arnol d [1963] Proo f o f a theore m o f A . N . Kolmogoro v o n th e invarianc e o f quasi - periodic motion s unde r smal l perturbation s o f the Hamiltonian , Rus- sian Mathematical Surveys 18 , No. 5 , 9-36 . [1963a] Smal l denominator s an d problem s o f stabilit y o f motio n i n classica l and celestia l mechanics , Russian Mathematical Surveys 18 , No . 6 , 85-192. [1988] Dynamical Systems III, Springer-Verlag , Berlin . H. F . Bake r [1914] Henr i Poincare , Proceedings of the Royal Society A 91, vi-xvi. [1916] O n certain linear differential equation s o f astronomical interest, Philo- sophical Transactions of the Royal Society A 216 , 129-186 . I. Bendixso n [1901] Su r les courbes defmies par des equations differentielles, Acta 24 , 1-88 . K-R. Bierman n [1988] Die Mathematik und ihre Dozenten an der Berliner Universitat 1810- 1933, 2n d edition , Akademie-Verlag , Berlin . G. Birkhof f [1950] George David Birkhoff Collected Mathematical Papers, 3 volumes , American Mathematica l Society , Ne w York. Republishe d Dover , Ne w York, 1968 . 249 250 REFERENCES [1912] Quelque s theoremes sur l e mouvement de s systemes dynamiques, Bul- letin de la Societe Mathematique de France 40, 305-32 3 = Collected Mathematical Papers I , 654-672 . [1913] Proo f o f Poincare's geometri c theorem, Transactions of the American Mathematical Society 14 , 14-2 2 = Collected Mathematical Papers I , 673-681. [1915] Th e restricted problem o f three bodies, Rendiconti del Circolo Matem- aticodi Palermo 39 , 265-33 4 = Collected Mathematical Papers I , 682 - 751. [1917] Dynamica l system s wit h tw o degree s o f freedom , Transactions of the American Mathematical Society 18 , 199-30 0 = Collected Mathematical Papers II, 1-102 . [1920] Recen t advance s in dynamics, Science 51 No. 1307 , January 16 , 51-55 = Collected Mathematical Papers II, 106-110 . [1920a] Surfac e transformation s an d thei r dynamica l applications , Acta 43 , 1-119 = Collected Mathematical Papers II, 111-229 . [1922] Celestia l Mechanics, Bulletin of the National Research Council 4 (Par t 1), 1-2 2 = Collected Mathematical Papers II, 252-266 . [1925] A n extensio n o f Poincare's las t geometri c theorem , Acta 47 , 297-31 1 = Collected Mathematical Papers II, 230-251 . [1927] Dynamical Systems, America n Mathematica l Society , Providence , RI . Reprinted 196 6 with additiona l reference s b y Jurgen Moser . [1927a] Stabilit y an d the equations o f dynamics, American Journal of Mathe- matics 49 , 1-3 8 = Collected Mathematical Papers II, 295-332 . [1927b] O n th e periodi c motion s o f dynamica l systems , Acta 50 , 459-37 9 = Collected Mathematical Papers II, 333-353 . [1928] A remark o n the dynamical rol e o f Poincare's las t geometri c theorem , Acta Litterarum ac Scientiarum 4 , 6-1 1 = Collected Mathematical Pa- pers II, 354-359 . [1931] Un e generalisatio n a n dimensions d u dernie r theorem e d e geometri e de Poincare, Comptes Rendus 192 , 196-19 8 = Collected Mathematical Papers II, 395-397 . [1931a] Proo f o f the ergodi c theorem, Proceedings of the National Academy of Science, USA 17 , 656-66 0 = Collected Mathematical Papers II , 404- 408. [1935] Su r l e problem e restrein t de s troi s corp s (Premie r memoire) , Annali della R. Scuola Normale Superiore Pisa (2) 4 , 267-30 6 = Collected Mathematical Papers II , 466-505 . [1935a] Nouvelle s recherche s su r le s systeme s dynamiques , Memoriae Pontif- ical Academia della Scienze Novi Lyncaei (3) 1 , 85-21 6 = Collected Mathematical Papers II , 530-661 . [1936] Su r l e problem e restrein t de s troi s corp s (Secon d memoire) , Annali della R. Scuola Normale Superiore Pisa (2) 5, 1-4 2 = Collected Math- ematical Papers II, 668-709 . [1942] Wha t i s th e ergodi c theorem ? American Mathematical Monthly 49 , 222-226 = Collected Mathematical Papers II, 713-717 . REFERENCES 251 G. Bisconcin i [1906] Su r l e probleme de s troi s corps . Trajectoire s l e lon g desquelle s deu x ail moin s de s troi s corp s s e choquent . Condition s qu i entrainen t u n choc, Acta 30 , 49-92 . K. Bohli n [1887] Ube r di e Bedeutun g de s Princip s de r lebendige n Kraf t fu r di e Frag e von de r Stabilita t dynamische r Systeme , Acta 10 , 109-130 . [1888] Uebe r ein e neue Annaherungsmethode i n der Storungstheorie , Bihang till Svenska Vetenskap Akademiens handlingar 14 , No . 5 ; Zu r Frag e der Convergen z de r Reihenentwickelungen i n der Storungstheorie , As- tronomische Nachrichten 121 , 17-24 . L. Boltzman n [1871] Uebe r die Druckkrafte, welch e auf Ringe wirksam sind, di e in bewegte Fliissigkeit tauchen , Journal fur die Reine und Angewandte Mathe- matik 73 , 111-134 . R. Bot t [1980] Marsto n Mors e and hi s mathematical works , Bulletin of the American Mathematical Society 3 , 907-944. O. R . M . Brende l [1889] O n th e proble m o f thre e bodie s accordin g t o Gylden' s theory , The Observatory (1989 ) 399-403. C. Brio t an d J , Bouque t [1854] Recherche s su r le s proprietes de s fonction s definie s pa r de s equation s differentielles, Comptes Rendus 39 , 368-371 . [1856] Etud e de s fonctions d'un e variabl e imaginaire; Recherche s su r le s pro- prietes de s fonctions definie s pa r de s equations differen t ielles, Journal de VEcole Polytechnique 21 , 85-132 ; 133-198 . D. Brouwe r an d G . M . Clemenc e [1961] Methods of Celestial Mechanics, Academi c Press , Ne w York . E. W . Brow n [1892] Poincare' s Mecanique Celeste , Bulletin of the New York Mathematical Society I , 206-214 . [1896] An Introductory Treatise on the Lunar Theory, Cambridg e Universit y Press, Londo n an d Ne w York. Reprinte d b y Dover , Ne w York, 1960 . [1916] Biographica l Memoi r o f Georg e Willia m Hill , Biographical Memoirs, National Academy of Sciences 8 , 275-306 . [1916a] Th e scientifi c wor k o f Si r Georg e Darwin , G. H. Darwin, Scientific Papers V , Cambridg e Universit y Press , xxxiv-lv . H. Brun s [1887] Ube r di e Integrale de s Vielkorper-Problems, Acta 11 , 25-96. 252 REFERENCES S. G . Brus h [1966] Kinetic Theory, 2 volumes, Pergamon Press , Oxford . [1980] Poincar e an d cosmi c evolution, Physics Today 33, 42-49. G. Canto r [1872] Ube r di e Ausdehnun g eine s Satze s au s de r Theori e de r trigono - metrischen Reihen , Mathematische Annalen 5 , 123-132 . C. Caratheodor y [1919] Ube r de r Wiederkehrsatz vo n Poincare, Sitzungsberichte der Preussis- chen Akademie der Wissenschaften, 580-584 . E. Carta n [1922] Legons sur les invariants integraux, A . Hermann, Paris . M. L . Cartwrigh t [1965] Jacque s Hadamard , Journal of the London Mathematical Society 40 , 722-748. A.-L. Cauch y [1842] Memoir e su r u n theorem e fondamenta l dan s l e calcu l integrate , Comptes Rendus 14 , 1020-1026 . A. Ca y ley [1862] Repor t o n the progres s o f the solutio n o f certai n specia l problem s i n dynamics, BAAS Report 1862, 184-252 = The Collected Mathematical Papers of Arthur Cayley IV, 513-593 . L. Cesar i [1959] Asymptotic behaviour and stability problems in ordinary differential equations, Springer-Verlag , Berlin . R. P . Cesc o [1961] Som e Theorems an d Result s i n the Three Bod y Problem, Proceedings of the International Meeting on Problems of Astrometry and Celestial Mechanics, La Plata, Argentina, 81-92 . J. Chaz y [1920] Su r les singularities impossible du probleme des n corps, Comptes Ren- dus 170, 575-577 . [1922] Su r Failur e d u mouvemen t dan s l e probleme de s trois corps , Annales Scientifiques de VEcole Normale Superieure 3 9 (Serie s 3), 29-130 . [1952] L'integratio n d u problem e de s trois corp s par Sundman , e t se s conse- quences, Bulletin Astronomique 16 , 175-190 . T. M . Cherr y [1924] O n Poincare' s theore m o f "Th e non-existenc e o f unifor m integral s o f dynamical equations", Proceedings of the Cambridge Philosophical So- ciety 22 , 287-294 . REFERENCES 253 [1928] O n periodic solutions o f Hamiltonian system s o f differential equations , Philosophical Transactions of the Royal Society A 227 , 137-221 . A. C . Clairau t [1747] D u Systeme du Monde dans les Principes de la Gravitation Universelle , Memoires de VAcademic royale des Sciences de Paris, 329-364 . [1749] D u Systeme du Monde dans les Principes de la Gravitation Universelle , Memoires de VAcademie royale des Sciences de Paris, 549 . R. Cook e [1984] The Mathematics of Sonya Kovalevskaya, Springer-Verlag , Ne w York. G. Darbou x [1914] Elog e historique d'Henr i Poincar e l u dans l a seance publique annuell e du 1 5 decembr e 1913 , Revue Scientifique 2 5 Jul y 1914 , 97-11 0 = Poincare (Euvres II, vii-lxxi . Sir Franci s Darwi n [1916] Memoi r o f Si r Georg e Darwin , G. H. Darwin, Scientific Papers V , ix-xxxii. Sir Georg e Darwi n [1911-1916] Scientific Papers by Sir George Howard Darwin, 5 volumes, Cambridg e University Press . [1887] O n figure s o f equilibriu m o f rotatin g masse s o f fluid , Philosophical Transactions of the Royal Society 178A , 379-42 8 = Scientific Papers III, 135-185 . [1897] Periodi c Orbit s Acta 21 , 99-242 an d (wit h omissio n o f certain table s of results) Mathematischen Annalen 51 , 523-58 3 = Scientific Papers IV, 1-113 . [1900] Presentatio n o f th e meda l o f th e Roya l Astronomica l Societ y t o M . Henri Poincare , Monthly Notices of the Royal Astronomical Society 60, 406-41 5 = Scientific Papers IV, 510-519 . [1909] O n certai n familie s o f periodi c orbits , Monthly Notices of the Royal Astronomical Society 70 , 108-14 3 = Scientific Papers IV , 140-181 . J. W . Daube n [1979] Georg Cantor. His mathematics and philosophy of the infinite, Har - vard Universit y Press , Cambridge , Massachusetts . T. d e Donde r [1901] Etud e su r le s invariants integraux , Rendiconti del Circolo Matematico di Palermo 15 , 66-131 . C. E . Delauna y [1846] Nouvell e theorie analytique du mouvement d e la lune, Comptes Rendus 23, 968-970 . [1860] Theori e d u mouvemen t d e l a lun e I , Memoire de VAcademie des Sci- ences 28, 1-883 . 254 REFERENCES [1867] Theori e d u mouvemen t d e la lune II, Memoire de VAcademic des Sci- ences 29, 1-931 . L. Dell'Agli o an d G . Israe l [1989] L a theori e d e l a stabilite e t l'analys e qualitativ e de s equations differ - ent ielles ordinaires dans le s mathematiques Italiennes : l e point d e vue de Tullio Levi-Civita, Cahiers 10 , 283-321 . F. N . Diac u [1993] Painleve' s conjecture , The Mathematical Intelligencer 1 5 (2) , 6-12 . Y. Doma r [1982] O n the foundatio n o f Acta Mathematica , Acta 148 , 3-8 . G. Duffin g [1918] Erzwungene Schwingungen bei veranderlicher Eigenfrequenz und ihre technische Bedeutung, Braunschweig . P. Duhe m [1962] The Aim and Structure of Physical Theory, Atheneum , Ne w York . Translated b y Philip P. Weiner fro m La Theorie Physique: Son Objet, Sa Structure, Marce l Rivier e & Cie , Paris , 2n d edition , 1914 . Firs t edition publishe d 1906 . A. Einstei n [1917] O n th e quantisatio n conditio n o f Sommerfel d an d Epstein , Deutsche Physikalische Gesellschaft Berlin Verhardlungen 19 , No . 9/10 , trans - lated b y C . Jaffe , Joint Institute for Laboratory Astrophysics Report 116 (1980) , University o f Colorado . I. Ekelan d [1988] Mathematics and the Unexpected, Th e Universit y o f Chicag o Press , Chicago. L. Eule r [1744] Theoria motuum planetarum et cometarum = Opera (2)28, 105-251 . [1748] Recherches due la questions des inegalities du mouvement de Saturne et de Jupiter, sujet propose pour le prix de Vannee 1748 pa VAcademie Roy ale des Sciences de Paris = Opera (2) 25, 45-157. [1753] Theoria motus lunae = Opera (2) 23, 64-336. [1772] Theoria motuum lunae, nova methodo pertractata = Opera (2 ) 22 , 1-411. E. Fiirstena u [1860] Darstellung der reellen Wurzeln algebraischer Gleichungen durch De- terminanten der Coeffizienten, Marburg . A. Gautie r [1817] Essai historique sur le probleme des trois corps ou dissertation sur la theorie des mouvements de la lune et des planetes, Courcier , Paris . REFERENCES 255 J. Gerve r [1991] Th e existenc e o f pseudocollisions i n the plane , Journal of Differential Equations 89 , 1-68 . G. E . O . Giacagli a [1972] Perturbation Methods in Non-Linear Systems, Georg e Allen & Unwin Ltd., London . C. Gilai n [1991] L a theorie qualitativ e d e Poincare e t l e probleme d e l'integration de s equations differentielles , La France Mathematique, ed . H . Gispert , Cahiers d'histoir e e t d e philosophie de s science s 34, 215-242 . D. L . Gorof f [1993] Henr i Poincar e an d th e birt h o f chao s theory : A n introductio n t o the Englis h translatio n o f Le s Methode s Nouvelle s d e l a Mecaniqu e Celeste, Ne w Method s o f Celestia l Mechanics , America n Institut e o f Physics Press, History o f Modern Physics and Astronomy 13 , 11-1107. I. Grattan-Guinnes s [1971] Material s fo r the history o f mathematics in the Institut Mittag-Lefner , Isis 62, 363-374 . J. J . Gra y [1984] Puch s and the theory o f differential equations , Bulletin of the American Mathematical Society 1 0 (1) , 1-26 . [1992] Poincare , topologica l dynamics , an d th e stabilit y o f the sola r system , An investigation of difficult things. Essays on Newton and the History of Exact Sciences (ed . P . M. Harman an d E. E . Shapiro), Cambridg e University Press , 503-524 . J. Guckenheime r an d P . Holme s [1983] Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vec- tor Fields, Springer-Verlag , Ne w York . H. Gylde n [1887] Untersuchunge n iibe r di e Convergen z de r Reigen , welch e zur Darstel - lung de r Coordinate n de r Planete n angewende t werden , Acta 9 , 185 - 294. [1891] Nouvelle s recherche s su r le s serie s employee s dan s le s theorie s de s planetes, Acta 15 , 65-189. [1893] Nouvelle s recherche s su r le s serie s employee s dan s le s theorie s de s planetes, Acta 17 , 1-168 . J. Hadamar d [1968] (Euvres de Jacques Hadamard, 4 volumes, Editions du Centre National de la Recherche Scientifique , Paris . [1897] Su r certaine s propriete s de s trajectoire s e n dynamique , Journal de Mathematiques (5 ) 3 , 331-38 7 = (Euvres 4, 1749-1805 . 256 REFERENCES [1898] Le s surfaces a courbures opposee s e t leur s ligne s geodesiques , Journal de Mathematiques (5 ) 4 , 27-73 = (Euvres 2, 729-780 . [1901] Notice sur les traveaux scientifiques de Jacques Hadamard, Gauthie r Villars, Paris. Reprinte d Hermann , Paris , 1912 . [1912] L'ceuvr e d'Henr i Poincare , Revue de Metaphysique et de Morale 21, 617-658. [1913] Henr i Poincare et l e probleme de s trois corps, Revue du Mois 16 , 385- 419 = Henr i Poincare : L'ceuvr e scientifique . L'ceuvr e philosophique , Nouvelle Collection Scientifique, Paris , 1914 , 51-114 = (Euvres 4, 2007- 2041. [1915] Su r un memoire de Sundman, Bulletin des sciences mathematiques 39 , 249-264 = (Euvres 4, 1897-1912 . [1921] L'ceuvr e mathematique d e H. Poincare, Acta 38 , 203-287= (Euvres 4, 1921-2005 = (Euvres de Henri Poincare 11 , 152-242. [1922] Th e early scientific wor k o f Henri Poincare, Rice Institute Pamphlet 9 , 111-183. [1933] Th e late r scientifi c wor k o f Henr i Poincare , Rice Institute Pamphlet 20, 1-86 . Y. Hagihar a [1970-1976] Celestial Mechanics 1-2 , MI T Press, Cambridge, Massachusetts, 1970 - 1972; 3-5, Japa n Societ y fo r th e Promotio n o f Science , Tokyo , 1974 - 1976. M. Ham y [1892] Le s Met nodes Nouvelles de la Mecanique celeste I, Revue Generale des Sciences pures et appliquees, 649. [1896] Le s Me t nodes Nouvelle s d e l a Mecaniqu e celest e II , Revue Generale des Sciences pures et appliquees, 39-40 . [1900] Le s Me t nodes Nouvelle s d e l a Mecaniqu e celest e III , Revue Generale des Sciences pures et appliquees, 254-255. T. Hawkin s [1992] Jacob i an d th e birt h o f Lie's theory o f groups, Archive for History of Exact Sciences 4 2 (3) , 187-278 . M. Heno n [1965] Exploratio n numeriqu e d u problem e restreint , Annales d'Astro- physique 28 , 499-511, 992-1007 . C. Hermit e [1877] Su r quelques applications de s fonctions elliptiques , (Euvres de Charles Hermite III , 266-418 . (Th e compilatio n o f the serie s o f articles whic h appeared i n the Comptes Rendus fro m 187 7 onwards.) [1889] Discour s d'Hermite dans la seance publique de l'Academie des Sciences en 1889 , (Euvres de Charles Hermite IV , 567-575 . G. W . Hil l [1905] The Collected Mathematical Works of George William Hill, 4 volumes, Carnegie Institution o f Washington . REFERENCES 257 [1876] Demonstratio n o f th e Differentia l Equation s Employe d b y Delauna y in the Luna r Theory , The Analyst 3 , 65-7 0 = Collected Mathematical Works I, No . 26 , 227-232. [1877] O n the part o f the Motion o f the Lunar Perige e which i s a Function o f the Mea n Motion s o f the Su n an d Moon , John Wilson & Son, Cam- bridge, Massachusetts. Reprinte d i n Acta 8 , 1886 , 1-3 6 = Collected Mathematical Works I, No . 29 , 243-270. [1878] Researche s int o the Luna r Theory , American Journal of Mathematics 1, 5-26 , 129-147 , 245-26 0 = Collected Mathematical Works I , No . 32 , 284-335. [1896] O n th e convergenc e o f serie s use d i n th e subjec t o f perturbations , Bulletin of the American Mathematical Society 2 , 93-9 7 = Collected Mathematical Works IV, No . 59 , 94-98. [1896a] Remark s on the Progress o f Celestial Mechanics since the Middle of the Century, Presidentia l Addres s delivere d before th e America n Mathe - matical Society , 1896 , Bulletin of the American Mathematical Society 2, 132-3 . P. Holme s [1990] Poincare , celestia l mechanics , dynamical-system s theor y an d "chaos" , Physics Reports 19 3 (3) , 137-163. E. Hop f [1930] Zwe i Satz e iibe r de n wahrscheinliche n Verlauf de r Bewegunge n dy - namischer System , Mathematische Annalen 103 , 710-719 . S. S . Houg h [1901] O n certai n discontinuitie s connecte d wit h periodi c orbits , Acta 24 , 257-288 = G. H. Darwin, Scientific Papers IV, 114-139 . C. G . J . Jacob i [1836] Su r l e mouvemen t d'u n poin t e t su r u n ca s particulie r d u problem e des troi s corps , Comptes Rendus 3 , 59-6 1 = Gesammelte Werke IV , 35-38. [1843] Su r I'elimination de s noeuds dans l e probleme des trois corps, Journal filr die Reine und Angewandte Mathematik 26 , 115-13 1 = Gesammelte Werke IV, 295-314 . [1844] Theori a novi multiplicatoris systemati aequationum differentialium vul - garium applicandi , Journal fur due Reine und Angewandte Mathe- matik 27 , 199-268 ; 29 , 213-279 , 333-37 6 = Gesammelte Werke IV , 317-509. [1866] Vorlesungen iiber dynamik, Reime r Publisher , Berlin . Lord Kelvin : se e W . Thomso n M. Klin e [1972] Mathematical Thought from Ancient to Modern Times, Oxfor d Uni - versity Press , Ne w York . 258 REFERENCES G. Koenig s [1895] Applicatio n de s invariant s integrau x a l a reductio n a u typ e canon - ique d'u n system e quelque s d'equation s differentielles , Comptes Ren- dus 121, 875-878. A. H . Koblit z [1983] A Convergence of Lives. Sophia Kovalevskaia: Scientist, Writer, Rev- olutionary, Birkhauser , Boston . A. N . Kolmogoro v [1954] O n conservatio n o f conditionall y periodi c motion s unde r smal l per - turbations o f th e Hamiltonian , Dokl. Akad. Nauk SSSR,98, No . 4 , 527-530. (Russian) . S. Kovalevskay a [1875] Zu r theorie der partiellen differentialgleichungen , Journal fur die reine und angewandte Mathematik 80 , 1-32 . [1978] A Russian Childhood (tr . Beatric e Stillman) , Springer-Verlag , Ne w York. Firs t editio n publishe d i n 188 9 i n Swedis h a s a novel, From Russian Life: the Rajevski Sisters. Firs t published in autobiographica l form i n Russian i n 1890 . L. Kronecke r [1869] Ube r Systeme von Functionen mehrerer Variabeln, Monatsberichte der Koniglich Preussischen Akademie der Wissenschaften zu Berlin 1869, 688-698 = Werke I, 213-226 . [1888] Bemerkunge n iibe r Dirichiet' s ietz e Arbeiten , Sitzungesberichte der Koniglich Preussischen Akademie der Wissenschaften 1888, 439-44 2 = Werke V, 471-476 . J. L . Lagrang e [1772] Essa i su r l e probleme de s trois corps , (Euvres 6, 229-331 . C. Lanczo s [1970] The Variational Principles of Mechanics, 4t h edition , Universit y o f Toronto Press . Republishe d b y Dover, Ne w York, 1986 . P.-S. Laplac e [1799-1825] Traite de Mecanique Celeste, 5 volume s = (Euvres completes 1-5 , Gauthier Villars , Paris , 1891-1904 . Englis h translatio n b y Nathania l Bowditch, 1829-1839 , reprinted b y Chelsea , Ne w York , 1966 . J. L a Sall e an d S . Lefschet z [1961] Stability by Liapunov's Direct Method, Academi c Press , Ne w York . H. Lebesgu e [1902] Integrate , longeur , aire , Annali di Matematica Pura ed Applicata (3 ) 7, 231-259 . REFERENCES 259 T. Levi-Civit a [1901] Sopr a alcun i criteri di instabilita, Annali di Mathematica Pura ed Ap- plicata (3 ) 5 , 221-307. [1903] Su r le s trajectoire s singuliere s d u problem e restrein t de s troi s corps , Comptes Rendus 135 , 82-84 . [1903a] Conditio n du choc dans l e probleme restreint de s trois corps, Comptes Rendus 135 , 221-223. [1903b] Traiettori e singolari ed urti nel problema ristretto dei tre corpi, Annali de Matematica Pura ed Applicata (3 ) 9 , 1-32 . [1906] Su r la resolution qualitative du probleme restreint de s trois corps, Acta 30, 305-327 . [1915] Sull a regolarizzazion e de l problem a pian o de i tr e corpi , Rendiconti delVAccademia Nazionale dei Lincei (5 ) 24, 61-75. [1918] Su r l a regularisation d u problem e de s trois corps , Acta 42 , 92-144 . [1926] Su r les chocs dans le probleme des trois corps, Comptes Rendus du 2me Congres international de mecanique appliquee (Zurich 1926), 96-106 . J. Lev y [1952] Notes , (Euvres de Henri Poincare VII , 623-632 . A. M . Liapuno v [1907] Problem e genera l d e l a stabilite d u mouvemen t (tr . A . Davaux) , An- nals de la Faculte des Sciences de Toulouse (2 ) 9 , 203-474 . Reprinte d with the same pagination, Princeto n University Press, 1947 . Firs t edi - tion publishe d i n Russia n b y th e Mathematica l Societ y o f Kharkow , 1892. Translate d fro m Frenc h int o Englis h b y A . T . Fuller , Interna- tional Journal of Control 5 5 (3) , March 1992 , 531-773. A. Lindsted t [1883] Beitra g zur integration der differentialgleichungen de r storungstheorie , Memoires de VAcademie Imperiale des Sciences de St. Petersbourg (7 ) 31, No . 4 , 1-19 . [1883a] Su r la form e de s expressions de s distances mutuelle s dan s l e probleme des trois corps , Comptes Rendus 97 , 1276-8 , 1253-5. [1884] Su r la determination des distances mutuelles dans le probleme des trois corps, Annales de VEcole Normale (3 ) 1 , 85-102. J. Liouvill e [1838] Not e su r l a theorie d e la variation de s constantes arbitraires , Journal de Mathematiques Pures et Appliquees 3 , 342-349. E. O . Lovet t [1912] Generalisation s o f th e proble m o f severa l bodies , it s inversion , an d an introductor y accoun t o f recen t progres s i n it s solution , Quarterly Journal of Pure & Applied Mathematics 42 , 252-315. 260 REFERENCES L. Lusterni k an d L . Schnirrelman n [1930] Existenc e d e troi s geodesique s fermee s su r tout e surfac e d e genr e 0 , Comptes Rendus 188 , 534-536 . J. Lutze n [1990] Joseph Liouville 1809-1882, Springer-Verlag, Ne w York. W. H . McCre a [1957] Edmun d Taylo r Whittaker , Journal of the London Mathematical So- ciety 32 , 234-256 . S. W. McCuske y [1963] Introduction to Celestial Mechanics, Addison-Wesley , Reading , Mass - achusetts. R. McGehe e [1986] Vo n Zeipel' s theore m o n singularitie s i n celestia l mechanics , Exposi- tions Mathematicae 4 , 335-345. R. S . MacKa y an d J . D . Meis s [1987] Hamiltonian Dynamical Systems, Ada m Hilger , Bristol . W. D . MacMilla n [1913] O n Poincare's correctio n to Bruns' theorem, Bulletin of the American Mathematical Society 15 , 349-355. J. Mawhi n [1994] Th e centennia l legacy o f Poincare an d Lyapuno v i n ordinary differen - tial equations , Supplemento ai Rendiconti del Circolo Matematico di Palermo 34 , Studie s i n the Histor y o f Modern Mathematic s I , 9-46 . C. Marcha l [1990] The Three-Body Problem, Elsevier , Amsterdam . R. Marcolong o [1914] L e recent i ricerch e d i K . Sundma n su l problem a matematic o de i tr e corpi, Giornale de Matematiche Battaglini (3 ) 52 , 171-186 . [1919] / / problema dei tre corpi da Newton (1686) al nostri giorni, Hoepli , Milan. J. Mathe r an d R . McGehe e [1975] Solution s o f the collinear fou r body problem which become unbounde d in finite time , Lecture Notes in Physics 38 , Springer-Verlag , 573-597 . K R . Meye r an d G . R . Hal l [1990] Introduction to Hamiltonian Dynamical Systems and the N-Body Prob- lem, Springer-Verlag , Ne w York . REFERENCES 261 G. Mittag-Leffle r [1902] Un e pag e d e l a vi e d e Weierstrass , Comptes Rendus du deuxieme Congres International des Mathematiciens, tenu a Paris 6-12 aout 1900, Gauthier Villars , Paris, 131-153 . [1912] Zu r Biographi e vo n Weierstrass, Acta 38 , 29-65. M. Mors e [1921] A one-to-one representatio n o f geodesies o n a surfac e o f negative cur - vature, American Journal of Mathematics 43 , 33-51. [1921a] Recurren t geodesie s o n a surfac e o f negativ e curvature , Transactions of the American Mathematical Society 22 , 84-110 . [1934] Calculus of variations in the large, America n Mathematica l Societ y Colloquium Publication s 18 , Providence, RI . [1946] Georg e Davi d Birkhof f an d hi s mathematica l work , Bulletin of the American Mathematical Society 5 2 ( 5 i), 357-391. M. Mors e an d G . Hedlun d [1938] Symboli c Dynamics, American Journal of Mathematics 60 , 815-866 . J. Mose r [1962] O n invarian t curve s o f area-preservin g mapping s o f a n annulus , Nachricht von der Akademie der Wissenschaften, Gottingen II, Math. Phys. Kl , 1-20 . [1973] Stable and Random Motions in Dynamical Systems, Princeto n Univer - sity Pres s an d Universit y o f Toky o Press, Princeton, NJ . [1978] I s the sola r syste m stable ? Mathematical Intelligencer 1 , 65-71 . F. R . Moul t on [1914] An introduction to celestial mechanics, 2n d edition , Macmillan , Ne w York. Reprinte d Dover , Ne w York, 1970 . [1920] Periodic Orbits (i n collaboratio n wit h D . Buchanan , T . Buck , F . L . Griffin, W . R . Longley , W . D . MacMillan) , Carnegi e Institutio n o f Washington, Washington . V. V . Nemytski i an d V . V . Stepano v [1960] Qualitative theory of differential equations, Princeton University Press, Princeton, NJ . P. Nabonnan d [1995] Contributio n a L'histoire d e la theorie des geodesiques ou XIXe siece , Revue d'histoire des mathematiques, 1 , 159-200 . S. Newcom b [1874] O n the Genera l Integrals o f Planetary Motion , Smithsonian Contribu- tion to Knowledge 31 , 1-31 . Sir Isaa c Newto n [1934] Philosophies naturalis principia mathematica I - III, Th e Roya l Soci - ety, London , 168 7 = Mathematical Principles of Natural Philosophy, 262 REFERENCES translated b y Andre w Motte , 1729 . Revise d an d edite d b y Floria n Cajori, Universit y o f California Press , 1934 . P. Painlev e [1896] Su r le s singularites de s equations d e l a dynamique e t su r l e problem e des trois corps , Comptes Rendus 123 , 871-873. [1897] Lecons sur la theorie analytique des equations differentielles professees a Stockholm (Septembre, Octobre, Novembre 1895), A. Hermann, Paris. [1897a] Su r le s integrales premiere s d e la dynamique e t su r l e probleme de s n corps, Comptes Rendus 124 , 173-176 . [1897b] Su r le cas du probleme des trois corps (et des n corps) ou deux des corps se choquent au bout d'un temps fini , Comptes Rendus 125 , 1078-1081. [1898] Su r le s integrale s premiere s d u problem e de s n corps , Bulletin As- tronomique 15 , 81-113. [1900] Su r les integrales uniformes d u probleme des n corps, Comptes Rendus 130, 1699-1701 . [1912] Henr i Poincare , Le Temps, 1 7 July 191 2 = Acta 38 , 399-402 . C. Parik h [1991] The Unreal Life of Oscar Zariski, Academi c Press , London . L. A . Par s [1968] A Treatise on Analytical Dynamics, 2n d edition, Heinemann , London . J. Percho t [1899] H . Poincare . Le s nouvelle s methode s d e la Mecaniqu e celeste , I , Bul- letin des Sciences Mathematiques (2) 23 , 213-242 , 245-260 . E. Phragme n [1889] Poincare'sk a faile t a f tekropparsprobleme t ( = O n som e dynamica l problems whic h ar e relate d t o th e restricte d thre e bod y problem) , K.V.A. Bihang till Handlingar 1 5 (1 ) No . 13 , 1-33 . E. Picar d [1896] Traite d'analyse III , Gauthie r Villars , Paris. [1902] L'CEuvr e scientifiqu e d e Charles Hermite, Acta 25 , 87-111 . [1913] L e problem e de s troi s corp s a propo s de s recherche s recente s d e M . Sundman, Bulletin des Sciences mathematiques 3 7 (serie s 2), 313-320. H. Poincar e [1915-1956] OBuvres de Henri Poincare, 1 1 volumes, Gauthie r Villars , Paris . [PI] Su r l e probleme de s trois corp s e t le s equations d e la dynamique ave c des note s pa r l'auteu r - memoir e couronn e d u pri x d e S . M . l e Ro i Oscar II . Printed i n 188 9 but no t published . [Pla] A s [PI] but with handwritten correction s and additions by the author . [P2] Su r l e probleme de s trois corps et le s equations d e la dynamique, Acta 13, 1890 , 1-27 0 = (Euvres VII, 262-479 . [MN I ] Les Methodes Nouvelles de la Mecanique Celeste I , Gauthier-Villars, 1892. Republishe d b y Blanchard, Paris , 1987 . REFERENCES 263 Les Methodes Nouvelles de la Mecanique Celeste II , Gauthier-Villars, 1893. Republishe d b y Blanchard, Paris , 1987 . Les Methodes Nouvelles de la Mecanique Celestelll , Gauthier-Villars, 1899. Republishe d b y Blanchard, Paris , 1987 . Lecons de Mecanique Celeste I-III, Gauthier-Villars , 1905-1910 . Sur les proprietes des fonctions definie s par les equations aux differenc e partielles, Premier e These , Gauthier-Villar s = (Euvres I , XLI X -CXXIX. Sur le s courbes definie s pa r un e equation differentielle , Comptes Ren- dus 90, 673-67 5 = (Euvres I, 1-2 . Memoire sur le s courbes definie s pa r un e equation differen t ielle, Jour- nal de Mathematiques (3 ) 7 , 375-42 2 = (Euvres I, 3-44 . Memoire sur le s courbes definie s pa r un e equation differen t ielle, Jour- nal de Mathematiques (3 ) 8 , 251-29 6 = (Euvres I, 44-84 . Sur l'integratio n de s equation s differentielle s pa r le s series , Comptes Rendus 94 , 577-57 8 = (Euvres I, 162-163 . Sur les series trigonometriques, Comptes Rendus 95, 766-768 = (Euvres IV, 585-587 . Sur certaine s solution s particuliere s d u problem e de s troi s corps , Comptes Rendus 97 , 251-25 2 = (Euvres VII, 251-252 . Sur le s serie s trigonometriques , Comptes Rendus 97 , 1471-147 3 = (Euvres IV, 588-590 . Notice su r le s traveaux scientifiques , Gauthier-Villars. Sur certaine s solutions particuliere s d u problem e de s trois corps , Bul- letin Astronomique 1 , 65-7 4 = (Euvres VII, 253-261 . Sur la convergence des series trigonometriques, Bulletin Astronomique 1, 319-327 = (Euvres IV, 591-598 . Sur un e equatio n differen t ielle, Comptes Rendus 98 , 793- 5 = (Euvres VII, 543-545 . Sur le s courbe s definie s pa r le s equation s differentielles , Journal de Mathematiques (4 ) 1 , 167-24 4 = (Euvres I, 90-161 . Sur le s serie s trigonometriques , Comptes Rendus 101 , 1131-113 4 = (Euvres I, 164-166 . Sur l'equilibre d'une mass e fluide animee d'un mouvemen t d e rotation, Acta 7 , 259-38 0 = (Euvres VII, 40-140 . Sur le s courbe s definie s pa r le s equation s differentielles , Journal de Mathematiques (4 ) 2 , 151-21 7 = (Euvres I, 167-222 . Sur le s integrates irreguliere s de s equations lineaires , Acta 8 , 295-34 4 = (Euvres I, 290-332 . Sur un e method e d e M . Lindstedt, Bulletin Astronomique 3 , 57-6 1 = (Euvres VII, 546-550 . Sur un moyen d'augmenter la convergence des series trigonometriques, Bulletin Astronomique 3 , 521- 8 = (Euvres IV, 599-606 . Sur les determinants d'ordre infini, Bulletin de la Societe mathematique de France 14, 77-9 0 = (Euvres V, 95-107 . Sur le s series de M. Lindstedt, Comptes Rendus 108 , 21-2 4 = (Euvres VII, 551-554 . Sur l e problem e de s troi s corps , Bulletin Astronomique 8 , 12-2 4 = (Euvres VII, 480-490 . 264 REFERENCES [1891a] L e problem e de s troi s corps , Revue Generate des Sciences pures et appliquees 2, 1- 5 = (Euvres VIII, 529-537 . [1891b] Su r l e developpement approch e d e la fonction perturbatrice , Comptes Rendus 112 , 269-27 3 = (Euvres VIII , 5-9 . [1892] Su r l'application d e la methode d e M. Lindstedt a u probleme de s trois corps, Comptes Rendus 114 , 1305-130 9 = (Euvres VII, 491-495 . [1896] Su r l a divergence de s serie s d e la mecanique celeste , Comptes Rendus 122, 497-49 9 = (Euvres VII, 558-560 . [1896a] Su r l a divergenc e de s serie s trigonometriques , Comptes Rendus 122 , 557-559 = (Euvres VII, 561-563 . [1896b] Su r l a methode d e Bruns, Comptes Rendus 123 , 1224-122 8 = (Euvres VII, 512-516 . [1896c] Su r les solutions periodiques et le principe de moindre action, Comptes Rendus 123 , 915-918 = (Euvres VII, 224-226 . [1897] Su r les solutions periodiques et le principe de moindre action, Comptes Rendus 124 , 713-71 6 = (Euvres VII, 227-230 . [1898] O n th e stabilit y o f the sola r system , Nature 58 , 183-184 . Originall y published i n Annuaire du Bureau des Longitudes an d Revue Scien- tifique 9 (4) , 609-613 = (Euvres VIII, 538-547 . [1901] Su r l a theori e d e precession , Comptes Rendus 132 , 50-5 5 = (Euvres VIII, 113-117 . [1904] Su r l a methode horistiqu e d e Gylden , Comptes Rendus 138 , 933-93 6 = (Euvres VII, 583-586 . [1904a] Su r l a methode horistique . Observation s su r Particle d e M. Backlund , Bulletin Astronomique 21 , 292-295 = (Euvres VII, 619-621 . [1905] Introduction to The Collected Mathematical Works of George William Hill, Volume 1. Carnegie Institution o f Washington, VII-XVIII . [1905a] Su r la methode horistique de Gylden, Acta 29 , 235-271 = (Euvres VII, 587-618. [1905b] Su r le s ligne s geodesique s de s surface s convexes , Transactions of the American Mathematical Society 6 , 237-27 4 = (Euvres VI, 38-84 . [1912] Su r u n theorem e d e geometrie , Rendiconti Circolo matematico di Palermo 33 , 375-407 = (Euvres VI, 499-538 . A. Ro y [1968] Orbital Motion, Ada m Hilger , Bristol . D. Saar i [1990] A visi t t o th e Newtonia n n-bod y proble m vi a elementar y comple x variables, The American Mathematical Monthly 97 , 105-119 . G. Sarto n [1913] Henr i Poincare , Ciel et Terre. Bulletin de la Societe beige d'Astro- nomic, 1-11 , 37-48. A. Schlisse l [1976] Th e development o f asymptotic solutions o f linear ordinary differentia l equations 1817-1920 , Archive for History of Exact Sciences 16 , 307 - 378. REFERENCES 26 5 E. Schol z [1980] Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincare, Birkhauser, Bosto n an d Basel . K. Schwarzschil d [1898] Ube r ein e Class e periodische r Losunge n de s Dreikorperproblems , Astronomische Nachrichten 147 , 17 , 289. C. L . Siege l an d J . K . Mose r [1971] Lectures on Celestial Mechanics, Springer-Verlag , Berlin . Firs t editio n published a s Vorlesungen uber Himmelsmechanik, Springer , 1956 . V. I . Smirno v [1992] Biograph y o f A . M . Liapunov , International Journal of Control 5 5 (3), 775-784 . Translate d b y J . F . Barrett fro m th e Russia n articl e i n A. M. Lyapunov: Izbrannie Trudie (A . M. Liapunov: Selecte d Works ) (Leningrad: Izda t Akad . Nau k SSR , 1948) , edite d b y V . I . Smirnov , 325-340. D. E . Smit h an d J . Ginsbur g [1934] A History of Mathematics in America before 1900, The Mathematica l Association o f America (Caru s Mathematical Monograp h Numbe r 5) . S. Sternber g [1969] Celestial Mechanics, Parts I and II, W . A . Benjamin , Ne w York . I. Stewar t [1989] Does God Play Dice ? , Blackwell, Oxford . K. F . Sundma n [1907] Recherche s sur le probleme des trois corps, Acta Societatis Scientiarum Fennicae 34 , No . 6 , 1-43 . [1909] Nouvelle s recherche s su r l e problem e de s troi s corps , Acta Societatis Scientiarum Fennicae 35 , No. 9 , 1-27 . [1912] Memoir e su r l e probleme de s trois corps , Acta 36 , 105-179 . V. Szebehel y [1967] Theory of Orbits, Academic Press , Ne w York an d London . W. Thomso n (Lor d Kelvin ) [1891] O n periodic motion o f a finite conservative system, Philosophical Mag- azine 32, 375-383; Instability o f Periodic Motion, Philosophical Maga- zine 32 , 555-56 0 = Mathematical and Physics Papers, IV, Cambridg e University Press , 1910 , 497-512. F. Tisseran d [1887] Su r l a commensurabilit e de s moyens mouvements dan s l e systeme so - laire, Comptes Rendus 104 , 259-265 ; Bulletin Astronomique 4 , 183 - 192. [1889-1896] Traite de Mecamque Celeste I , 1889 ; II , 1891 ; III, 1894 ; IV , 1896 ; Gauthier-Villars, Paris . 266 REFERENCES O. Veble n [1946] Georg e David Birkhoff, Yearbook American Philosophical Society, 279 - 285. E. B . Va n Vlec k [1915] Th e rol e o f the point-se t theor y i n geometr y an d dynamics , Bulletin of the American Mathematical Society 21 , 321-341. H. vo n Zeipe l [1908] Su r le s singularitie s d u problem e de s n corps , Arkiv for Matematik, Astronomi och Fysik 4 (32) , 1-4 . [1921] L'CEuvr e Astronomiqu e d e Henri Poincare , Acta 38 , 309-385. R. S . Westal l [1980] Never at Rest: A Biography of Isaac Newton, Cambridg e Universit y Press. E. T . Whittake r [1899] Repor t o n the progres s o f the solutio n o f the proble m o f three bodies , BAAS Report 1899, 121-159. [1901] O n the solution of dynamical problems in terms of trigonometric series, Proceedings of the London Mathematical Society 34 , 206-221 . [1902] Periodi c orbits, Monthly Notices of the Royal Astronomical Society 62 , 186-193. [1902a] O n periodi c orbit s i n the restricte d proble m o f three bodies , Monthly Notices of the Royal Astronomical Society 62 , 346-352 . [1912] Prinzipie n de r Storungstheori e un d allgemein e Theori e de r Bahnkur - ven in dynamischen Probiemen, Encyklopadie der mathematische Wis- senschaften V I (2) , 512-556. [1917] O n the adelphic integral o f the differential equation s o f dynamics, Pro- ceedings of the Royal Society of Edinburgh 37, 95-116 . [1937] A treatise on the analytical dynamics of particles and rigid bodies, 4th edition, Cambridg e Universit y Press . Firs t editio n publishe d 1904 . E. T . Whittake r an d G . N . Watso n [1927] A course of modern analysis, 4t h edition, Cambridge University Press. First editio n publishe d 1902 . S. Wiggin s [1993] Global Dynamics, Phase Space Transport, Orbits Homoclinic to Reso- nances and Applications, America n Mathematical Society , Providence, RI. A. Wintne r [1941] The Analytical Foundations of Celestial Mechanics, Princeto n Univer - sity Press, Princeton, NJ . Z. Xi a [1992] Th e existenc e o f noncollision singularitie s i n Newtonian systems , An- nals of Mathematics 135 , 411-468 . Index Acta Mathematical 51 , 57 Bohlin's method , 15 9 Adams, Joh n Couc h (1819-1892) , 2 4 Boltzmann, Ludwi g (1844-1906) , 148 , 21 7 [1877], 2 7 [1871], 8 3 Appell, Pau l (1855-1930) , 57 , 62, 6 3 Bonnet, Ossia n (1819-1892) , 4 4 Arnold, V . I. , 15 8 Bott, R . [1963], 22 4 [1980], 22 2 [1988], 15 6 Bouquet, Jean-Claud e (1819-1895) , 31, 44 Asymptotic expansions , 15 9 Brendel, Ott o Rudol f Marti n (1862-1939) , Asymptotic geodesies , 20 6 21 Asymptotic series , 45, 10 3 Brioschi, Francesc o (1824-1897) , 5 3 Asymptotic solutions , 45 , 92, 99-10 2 Briot, C . an d Bouquet , J.-C , 5 9 of Hamiltonian systems , 102-10 4 [1854], 7 6 Asymptotic surface , 10 1 Briot, Charle s A . A. , (1817-1882) , 31 , 44 [1854], 7 6 Backlund, Alber t Victo r (1845-1922) , 16 4 Brown, Ernes t Willia m (1866-1938) , 2 2 Baker, Henr y Frederic k (1866-1956 ) [1892], 16 3 [1916], 15 9 [1916a], 19 3 Bendixson, Iva r (1861-1935) , 20 8 Bruns, Heinric h (1848-1919) , 81, 127, 18 3 [1901], 17 6 [1887], 8 , 155 , 16 4 Bernoulli, Daniel , 1 4 Brush, S . G . Bernoulli, Johann , 14 , 8 2 [1966], 14 8 BirkhofF, Georg e Davi d (1884-1944) , 1 , 168, [1980], 8 6 172, 181 , 191, 199, 209, 210, 211-21 7 minimax method , 21 6 Calcul des limites (se e method o f majorants ) minimum method , 21 5 Cantor, Geor g (1845-1918) , 36 , 59, 125 , 208 restricted thre e bod y problem , 212-21 5 derived set , 3 6 [1912], 22 2 perfect set , 3 6 [1913], 16 9 Caratheodory, Constan t in (1873-1950 ) [1915], 209 , 21 2 [1919], 8 6 [1917], 21 5 Cartan, Eli e (1869-1951 ) [1920], 21 7 [1922], 17 7 [1922], 191 , 214 Cartwright, M . L . [1925], 17 3 [1965], 20 2 [1927], 168 , 173 , 21 6 Cauchy, Augustin-Loui s (1789-1857) , 4 4 [1927a], 21 7 [1842], 7 6 [1927b], 21 7 Cauchy-Kovalevskaya theorem , 8 0 [1928], 17 3 Cayley, Arthu r (1821-1895) , 7 , 53 , 55 , 148 , [1931], 17 4 227 [1935], 21 2 Cesari, L . [1935a], 21 4 [1959], 18 1 [1936], 21 2 Cesco, R . R [1942], 21 7 [1961], 19 1 Bisconcini, Giuli o (b . 1880 ) Characteristic exponents , 39 , 92, 9 4 [1906], 18 6 calculation of , 98-9 9 Bohlin, Kar l (1860-1939) , 7 4 Chazy, Jea n (1882-1955 ) [1887], 87 , 8 8 [1920], 19 2 [1888], 74 , 9 8 [1952], 19 1 267 268 INDEX Cherry, Thoma s MacFarlan e (1898-1966 ) Fuchs, Lazaru s Immanue l (1833-1902) , 44 , [1924], 15 5 46, 59 , 76 , 8 2 [1928], 22 1 Clairaut, Alexis-Claud e (1713-1765) , 1 5 Gamma function , 4 5 Collision, 78 , 137 , 161 , 183 , 18 7 Gautier, A . Consequents (iterates) , 3 3 [1877], 7 Conservation o f angula r momentum , 9 Genus o f a surface , 3 5 Contact transformation , 14 , 17 7 Geodesies o n a conve x surface , 16 7 Convergence, 1 8 Geodesies o n surface s o f negative curvature , concepts of , 155-16 0 204, 21 1 strength of , 4 2 Geodesies o n surface s o f positiv e curvature , Cooke, R. , 55 , 76 , 8 0 201 Cycles without contact , 3 3 Geometric representation , 104 , 10 6 Gerver, J . Darboux, Jea n Gasto n (1842-1917) , 44 , 4 9 [1991], 19 3 Darwin, Si r George Howard (1845-1912) , 28 , Gilain, C . 160, 19 3 [1991], 3 0 [1887], 19 3 Goroff, D . L . [1897], 16 , 19 4 [1993], 152 , 21 9 [1900], 28 , 177 , 19 3 Gray, Jerem y [1909], 19 6 [1984], 7 6 Darwin, Si r Franci s [1992], 30 , 35 , 148 , 177 , 18 1 [1916], 19 3 Guckenheimer, J . an d Holmes , P . de Donder , T . [1983], 22 0 [1901], 17 7 Gylden, Hug o (1841-1896) , 20 , 42 , 66-68 , de Longchamps , Guy , 6 2 139, 140 , 15 9 Delaunay, Charles-Eugen e (1816-1872) , 19 , convergence o f series, 4 2 23, 10 5 horistic function , 21 , 164-16 6 Dell'Aglio, L . an d Israel , G . method o f integration, 15 9 [1989], 18 1 polemic with Mittag-Leffler , 66 , 139 , 141- Derived sets , 3 6 143 Diacu, F . N . theory o f absolute orbits , 2 0 [1993], 19 3 [1887], 139-14 3 Differential equation s [1893], 16 5 qualitative theor y of , 29-4 1 Gylden-Lindstedt equation , 15 9 Dillner, Gora n (1852-1906) , 6 3 Dirichlet, Johan n Pete r Gusta v Lejeune- , Hadamard, Jacque s (1865-1963) , 31 , 41, 44, (1805-1859), 60 , 6 1 149, 199 , 200, 201-209 , 21 1 Domar, Y . geodesies o n surface s o f negativ e curva - [1982], 57 , 5 8 ture, 204 , 21 1 Doubly asymptoti c solutions , 108 , 16 1 geodesies on surfaces o f positive curvature, Doubly asymptoti c trajectory , 11 8 201 Duffing's equation , 9 7 well pose d proble m 199 , 20 8 Duhem, P . [1897], 19 9 [1962], 20 9 [1898], 199 , 22 2 Einstein, Alber t (1879-1955 ) [1912], 19 9 [1917], 22 1 [1913], 19 9 Ekeland, I . [1915], 19 1 [1988], 21 9 [1921], 149 , 19 9 Elimination o f the nodes , 1 0 [1922], 19 9 Equation o f evection , 15 9 [1933], 19 9 Ergodic theorem, 21 7 Hamburger, M. , 8 2 Ergodic theory , 14 9 Hamilton-Jacobi theory , 11 3 Escary, J. , 6 3 Hamy, Mauric e Euler, Leonhar d (1707-1783) , 14-16 , 23 , 8 2 [1892], 16 3 [1896], 16 3 First retur n map , 3 8 [1900], 16 3 Floquet, Gasto n (1847-1920) , 8 1 Hansen, Pete r Andrea s (1795-1874) , 2 3 INDEX 269 Hawkins, T . Kovalevskaya, Sophi a "Sonya " Vasilievn a [1992], 1 4 (1850-1891), 44 , 51 , 53, 55 , 58 , 67 , 80 , Henon, Mauric e 227 [1965], 19 7 Kronecker, Leopol d (1823-1891) , 38 , 43, 58- Hermite, Charle s (1822-1901) , 51 , 53 , 54 , 61, 66 , 68 , 142 , 16 9 55, 57 , 68 , 134 , 14 2 Kronecker's inde x fo r a close d surface , 3 8 opinion o f Poincare's memoir , 13 4 [1888], 13 6 Hessian, 9 6 Kummer, Erns t Eduar d (1810-1893) , 6 0 Heteroclinic solutions , 162 , 16 3 Lagrange, Josep h Loui s (1736-1813), 10 , 15, Hilbert, Davi d (1862-1943) , 7 , 14 3 16, 22 , 8 1 Hill, Georg e Willia m (1838-1914) , 43 , 67 , Lagrangian (o r libration ) points , 1 6 143, 144 , 15 9 Laplace, Pierre-Simo n (1749-1827) , 1 8 curves o f zer o velocity , 24 , 19 4 [1789-1825], 12 9 Hill's equation, 26 , 15 9 Last Geometri c Theorem , 16 9 infinite determinant , 2 6 Last multiplier , 8 5 moons o f maximu m lunation , 2 4 Lebesgue, Henr i (1875-1941) , 8 6 motion o f luna r perigee , 2 3 Lecons de Mecanique Celeste, 15 1 opinion o f Poincare's results , 143-14 6 Legg, Cyrus , 6 3 periodic solution s o f the three bod y prob - Les Methodes Nouvelles de la Mecanique lem, 22 , 25 , 4 3 Celeste, 151-16 4 restricted thre e bod y problem , 1 6 Levi-Civita, Tulli o (1873-1941) , 175 , 181 , [1877], 22-2 8 184 [1878], 22-28 , 87 , 8 8 regularisation o f the thre e bod y problem , [1896], 14 5 184 [1896a], 14 3 stability theory , 181-18 2 Holmes, P . J. , 1 [1900], 18 1 [1990], 22 0 [1900a], 18 1 Homoclinic point , 108 , 11 8 [1900b], 18 1 Homoclinic solutions , 108 , 161 , 16 3 [1901], 18 1 Homoclinic trajectory , 11 8 [1903], 18 4 Hopf, Hein z (1894-1971 ) [1903a], 18 4 [1930], 15 3 [1903b], 184-18 6 Horistic function , 21 , 164-16 6 [1906], 185 , 212 , 21 4 Hough, Sydne y Samue l (1870-1923 ) [1906b], 18 6 [1901], 195 , 19 6 [1918], 19 1 Levy, J . Implicit functio n theorem , 7 8 [1912], 16 1 Infinite determinant , 2 6 Liapunov, Alexande r (1857-1918) , 175 , 177 , Integrals 178, 179-18 1 of area , 9 [1907], 179-181 , 20 3 "Vis Viva", 1 0 Libration Invariant integrals , 39 , 83 , 160 , 177 regions of , 12 4 Libration points , 1 6 Lie, Sophu s (1842-1899 ) Jacobi, Car l Gusta v Jaco b (1804-1851) , 1 6 theory o f contact transformations , 17 7 [1836], 1 2 Limit cycles , 3 3 [1843], 1 0 Lindstedt, Ander s (1854-1939) , 21 , 67 [1844], 8 5 Lindstedt's method , 21 , 36, 40 , 42 , 15 6 [1866], 11 3 Lindstedt's series , 42 , 119 , 126 , 157 , 22 3 Jacobian integral , 12 , 13 , 16 , 19 4 Liouville, Josep h (1809-1882 ) [1883], 8 3 Kelvin, Lord , se e Thomson, Willia m Lovett, Edga r Odel l (1871-1957 ) Koenigs, Gabrie l Xavie r Pau l (1858-1931 ) [1912], 7 , 14 7 [1895], 17 7 Lunar apsides , 1 5 Kolmogorov, Andre i (1903-1987) , 15 8 Lunar theory , 1 5 [1954], 22 3 Lusternik an d Schnirrelman n Kolmogorov-Arnold-Moser (KAM ) theory , [1930], 16 8 158, 22 3 Liitzen, J. , 3 0 270 INDEX Machin, John , 1 5 questions set , 58 , 22 9 MacMillan, W . D . report o f priz e commission , 134 , 237-23 8 [1913], 16 4 winning entries , publicatio n of , 6 9 Marchal, C . Oscar II , King of Sweden and Norway (1829 - [1990], 8 1907), 1 , 49, 50, 51 , 53, 134 , 183 , 22 7 Marcolongo, Robert o (1862-1942 ) [1914], 19 0 Painleve, Pau l (1863-1933) , 49 , 183 , 19 2 [1919], 7 integrals o f the thre e bod y problem , 15 5 Mather, J . an d McGehee , R . singularities o f the n an d three body prob - [1975], 19 3 lems, 138 , 18 3 Mathieu's equation , 2 6 [1896], 18 3 Mawhin, J . [1897], 18 3 [1994], 152 , 20 9 [1897a], 155 , 18 4 McGehee, R . [1897b], 18 4 [1986], 19 2 [1898], 155 , 18 4 Melnikov, V . K. , 22 0 [1900], 15 5 [1963], 22 0 [1912], 16 9 Method o f majorants , 44 , 7 6 Parikh, C . Minimax method , 21 6 [1991], 16 9 Minimum method , 21 5 Perchot, J . Minkowski, Herman n (1864-1909) , 14 3 [1899], 16 3 Mittag-Lemer, Magnu s Gost a (1846-1927) , Perfect set , 3 6 51, 52 , 55 , 57 , 135 , 13 9 Periodic solutions , 24 , 25 , 43 , 75 , 153 , 206 , polemic wit h Gylden , 66 , 139 , 141-14 3 208 [1912], 13 9 denseness of , 125 , 152 , 20 8 Morse, Marston (1892-1977) , 212 , 215 , 222- of Hamiltonian systems , 9 5 223 of the secon d class , 122 , 16 0 Morse trajectories, 22 2 of the secon d species , 131 , 16 1 [1898], 22 3 Perturbation function , 15 4 [1920], 22 2 Phragmen, Lar s Edvar d (1863-1937) , 63 , [1921a], 212 , 22 2 64, 67 , 69 , 74 , 99 , 11 9 [1934], 16 8 Picard, Emil e (1856-1941) , 57 , 6 5 [1938], 22 3 [1896], 15 2 [1946], 21 5 [1902], 14 2 Moser, J . [1913], 19 0 xvi , [1962], 22 5 Poincare, Jule s Henr i (1854-1912) , 57 , [1973], 59 , 15 8 61, 19 6 Moulton, Fores t Ra y (1872-1952) , 1 definition o f restricte d thre e bod y prob - lem, 73-7 4 [1920], 197 Fuchsian functions , 5 9 n bod y problem , 59 , 130 , 19 2 Last Geometri c Theorem , 16 9 singularities 192-19 3 Legons de Mecanique Celeste, 15 1 Nabonnand, P . Les Methodes Nouvelles de la Mecanique [1995], 19 9 Celeste, 151-16 4 Newcomb, Simo n (1835-1909) , 22 , 2 4 Poincare's error, 74 , 89, 101 , 103, 112 , 11 4 [1874], 15 6 Poincare's inde x theorem , 3 2 Newton, Si r Isaa c (1642-1727) , 14 , 1 5 Poincare's opinio n o f Gylden's [1887] , 13 9 Noncollision singularities , 19 2 qualitative theory o f differential equations , Normal set , 22 2 29-41 [1879], 4 4 Oscar competition , 49-7 0 [1881], 29 , 3 0 announcement o f the result , 6 5 [1882], 29 , 3 0 judging th e entries , 63,6 5 [1882a], 176 , 19 0 Kronecker's 188 5 objections , 59-6 1 [1882b], 4 1 list o f entries , 23 3 [1883], 4 3 origins, 5 1 [1884], 3 8 Poincare's intentio n t o enter , 61-6 3 [1884a], 4 3 prize commission , 53 , 5 5 [1884b], 41 , 157 public announcemen t of , 58 , 22 9 [1885], 29 , 34 , 75 , 86, 169 , 177 , 20 3 INDEX 271 [1885a], 4 1 Siegel, C . [1885b], 177 , 19 3 [1971], 19 2 [1886], 29 , 34 , 43, 75 , 83, 92, 94 , 176 , 19 0 Singular points , 31 , 3 8 [1886a], 45 , 56 , 83, 10 3 Singularities an d regularisation , 182-19 3 [1886b], 40 , 42 , 15 6 Small divisors , 17 , 18 , 15 9 [1886c], 4 2 Smirnov, V . I . [1886d], 2 7 [1992], 17 5 [1889], 15 6 Solar syste m [1891], 108 , 145 , 149 , 16 6 stability of , 7 , 34 , 16 6 [1891a], 16 6 Stability, 34 , 166 , 17 7 [1891b], 15 5 coefficients of , 92 , 9 5 [1892], 157 Stability o f equilibrium, 20 3 [1896], 14 5 Steiner's prize , 22 7 [1896a], 14 5 Stewart, I . [1896b], 16 4 [1989], 92 , 21 9 [1896c], 16 0 Stieltjes, Thoma s Ja n (1856-1894) , 55 , 8 1 [1897], 16 0 Stirling's series , 4 5 [1898], 16 6 Stromgren, Svant e Eli s (1870-1947) , 197 [1901], 16 4 Sturm, Jacque s Charle s Frangoi s (1803 - [1904], 16 4 1855), 3 0 [1904a], 16 5 Sundman, Kar l Frithio f (1873-1949) , 7 , 187 , [1905a], 20 4 188, 189-19 2 [1905b], 16 7 [1907], 187 [1912], 16 9 [1909], 18 7 [1915], 21 2 [1912], 187 [1935], 21 2 Surface trajectory , 7 5 [1936], 21 2 Surface withou t contact , 3 8 Poisson stability , 40 , 86 , 87 , 21 1 Sylvester, Jame s Joseph (1814-1897) , 24 , 28, Poisson, Simeon-Deni s (1781-1840) , 3 4 53 Prix Bordin de UAcademic des Sciences, 32 , Symbolic dynamics , 193 , 22 3 199 Szebehely, V . Pusieux, Victo r (1820-1883) , 7 9 [1967], 13 , 153 , 196 , 197 , 21 2 Qualitative theor y o f differentia l equations , Thome, Ludwi g Wilhel m (1841-1910) , 4 6 29-41 Thomson, Willia m (Lor d Kelvin ) (1824-1907 ) Quasi-closed, 8 9 [1891], 14 8 Three bod y problem , 7 , 30 , 37 , 71-131 , 15 2 Recurrence theorem , 86 , 148 , 203 , 21 7 definition, 7 Recurrent motion , 20 9 early attempt s a t solving , 1 5 Restricted thre e bod y problem , 11 , 73-74 , history, 7 212-215 Lagrangian (o r libration ) points , 1 6 definition, 1 1 mathematical description , 8 first formulation , 1 6 particular solutions , 1 6 geometric representation , 104 , 10 6 periodic solution s of , 22 , 24 , 43, 15 2 Jacobian integral , 12 , 19 4 reduction o f order, 9 mass parameter, 8 8 restricted, 11 , 73-74, 212-21 5 Poincare's definition , 73-7 4 restricted, firs t formulation , 1 6 Rotation number , 3 6 restricted, mas s parameter , 8 8 singularities an d regularisation , 182-19 3 Saari, D . stability, 16 0 [1990], 137 , 19 1 Tisserand, Frangoi s Feli x (1845-1896) , 149 , Sarton, Georg e (1884-1956) , 15 1 190 Schering, Erns t Christia n Juliu s (1833 - [1887], 10 5 1897), 5 1 [1896], 149 , 19 0 Schlissel, A. , 4 5 Transfinite set s Schwarz, Herman n Amandu s (1843-1921) , Cantor's theor y of , 20 8 65 Transverse (o r Poincare ) section , 38 , 85, 88, Schwarzschild, Kar l (1843-1921 ) 214, 216 , 22 1 [1898], 15 3 Trigonometric series , 41 , 81-8 2 272 INDEX Tschebychev, Pafnut i (1821-1894) , 53 , 55 , Well pose d problem , 199 , 20 8 227 Whittaker, Edmun d Taylo r (1873-1956) , Two bod y problem , 1 4 146-147 van Vleck , Edwar d (1863-1943 ) [1899], 7 , 23 , 146 , 163 , 16 5 [1915], 8 6 [1937], 7 , 8 , 127 , 17 7 Variation o f parameters , 1 5 Wiggins, S . Variational equations , 7 5 [1993], 22 0 Veblen, O Wintner, Aure l (1903-1958 ) [1946], 21 6 [1931], 15 3 von Koch , Helge , 2 7 [1941], 8 , 15 5 von Zeipel , Hug o (1873-1959 ) [1947], 86 , 18 7 [1908], 19 2 [1921], 14 9 Xia, Zhihon g [1992], 19 2 Weierstrass, Kar l Wilhel m Theodo r (1815 - 1897), 51 , 53, 55 , 56 , 57 , 68 , 126 , 135 , Zermelo, Erns t Friedric h Ferdinan d (1871 - 136, 14 2 1953), 14 8 collisions i n the three bod y problem , 137 - 138 report o n Poincare' s memoir , 134-13 8 Weierstrass's conjecture , 18 7 [1842], 7 6