This page intentionally left blank · 2000 Mathematics Subject Classification. Primary 00B15; Secondary 14Hxx, 53Axx, 40Axx. Library of Congress Cataloging-in-Publication Data The

America n Mathematica l Societ y

Colloquiu m Publication s Volum e 1

Th e Bosto n Colloquiu m Linear Systems of Curves on Algebraic Surfaces

Henr y Seel y Whit e

Forms of Non-Euclidean Space

Frederic k Shenston e Wood s

Selected Topics in the Theory of Divergent Series and of Continued Fractions

Edwar d Bur r Va n Vlec k

America n Mathematica l Societ y Providence , Rhod e Islan d

This page intentionally left blank

America n Mathematica l Societ y

Colloquiu m Publication s Volum e 1

Th e Bosto n Colloquiu m Linear Systems of Curves on Algebraic Surfaces

Henr y Seel y Whit e

Forms of Non-Euclidean Space

Frederic k Shenston e Wood s

Selected Topics in the Theory of Divergent Series and of Continued Fractions

Edwar d Bur r Va n Vlec k

America n Mathematica l Societ y Providence , Rhod e Islan d

http://dx.doi.org/10.1090/coll/001

2000 Mathematics Subject Classification. P r i m a r y 00B15 ; Secondar y 14Hxx , 53Axx , 40Axx.

Library o f Congres s Cataloging-in-Publicat io n D a t a

The Bosto n Colloquium : Lecture s o n mathematics . p. cm . — (America n Mathematica l Societ y Colloquiu m publications , ISS N 0065-925 8 ; v. 1 )

New York , Publishe d fo r th e Societ y b y th e Macmilla n co. , 1905 . ISBN 978-0-8218-4588- 2 (alk . paper ) 1. Curve s o n surfaces . 2 . Geometry , Non-Euclidean . 3 . Continue d fractions . 4 . Divergen t

series. I . Va n Vleck , Edwar d Burr , 1863-1943 . II . Title . III . Colloquiu m publication s (Amer -ican Mathematica l Society ) ; v. 1 .

QA1 .A5225 vol . 1 0501954 3

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10 98 7 6 5 4 3 2 1 1 3 1 2 1 1 1 0 09 0 8

TO

PROFESSOR JOH N MONRO E VA N VLECK , LL.D .

THESE LECTURE S AR E AFFECTIONATEL Y INSCRIBE D B Y

HIS FORME R PUPIL S

HENRY SEEL Y WHIT E

EDWARD BUR R VA N VLEC K

FREDERICK SHENSTON E WOOD S

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PREFACE.

F O R a numbe r o f year s th e America n Mathematica l Societ y ha s held a Colloquiu m i n connectio n wit h it s Summe r Meetin g a t in -tervals of two or three years . I n th e circular sen t ou t prio r t o the first Colloquium , i n 1896 , th e purpos e an d th e pla n o f th e under -taking wer e describe d a s follows: 1 " Th e object s no w attaine d by th e Summe r Meetin g ar e two-fold : a n opportunit y i s offere d for presentin g befor e discriminatin g an d intereste d auditor s th e results o f research i n specia l fields, an d persona l acquaintanc e an d mutual helpfulnes s ar e promote d amon g th e member s i n attend -ance. Thes e tw o ar e th e prim e object s o f suc h a gathering . I t is believe d howeve r tha t a thir d n o less desirabl e resul t lie s within reach. Fro m th e concise , unrelate d paper s presente d a t an y meeting onl y fe w deriv e substantia l benefit . Th e min d o f th e hearer i s too unprepared , th e impressio n i s o f too shor t duratio n to produce accurat e knowledge o f eithe r th e content or th e method . . . . Positiv e an d exac t knowledge , scientific knowledge , is rarel y increased i n thes e shor t an d stimulatin g sessions .

" On th e othe r hand , th e courses o f lecture s i n ou r bes t univer -sities, eve n wit h topic s changin g a t interval s o f a fe w weeks , d o give exac t knowledg e an d furnis h a substantia l basi s fo r readin g and investigation . . . .

1 Of. Bull Am. Math. Soc, ser . 2 , vol. 3 (1896), p. 49.

vii

viii PREFACE.

" To exten d th e tim e o f a lecture t o tw o hours , and to multipl y this tim e b y three or by six, would b e practicable within th e limit s of on e week . A n exper t lecture r coul d present , i n si x two-hou r lectures, a moderatel y extensiv e chapte r i n som e on e branc h o f mathematics. Wit h som e ne w matter , much tha t i s ol d coul d b e mingled, includin g fo r exampl e digest s o f recen t o r to o muc h neglected publications. Ther e would b e time for som e elementar y details as well as for more profound discussions . I n short, lectures could b e mad e profitable t o al l wh o hav e a general knowledg e o f the highe r mathematics. "

As a forerunne r o f th e Colloqui a her e outline d ma y b e men -tioned th e Evansto n Colloquiu m o f 1893 , whic h followe d th e Congress o f Mathematic s hel d i n connectio n wit h th e World' s Fair i n Chicago , Professo r Klein , o f Gottingen , bein g th e sol e speaker. Bu t wherea s tha t Colloquiu m covered , i n a descriptiv e manner, a variet y o f topics , — i t comprise d twelv e lectures, — the Colloqui a o f the Societ y have bee n characterized b y clos e con-tact wit h th e actual analytica l developmen t o f th e topi c treated .

The followin g Colloqui a hav e bee n hel d :

I. T H E BUFFAL O COLLOQUIUM , 1896 .

(a) Professo r MAXIM E BOCHER , o f Harvar d Universit y : " Lin-ear Differential Equations , an d Thei r Applications. " This Colloquiu m ha s not been published , but several paper s

appeared a t abou t th e tim e o f th e Colloquium , i n whic h th e author deal t wit h topic s treate d i n th e lectures. *

(b) Professo r JAME S PIERPONT , o f Yal e University : "Galois' s Theory o f Equations. " This Colloquiu m wa s publishe d i n th e Annals of Mathe-

matics, ser . 2 , vols . 1 and 2 (1900) .

*Two of these papers were : " Regular Point s o f Linea r Differentia l Equa -tions of the Second Order"; Harvar d University, 1896 ; "Notes on Some Points in the Theory of Linear Differential Equations, " Annate of Math., vol. 12 , 1898.

PREFACE. ix

I I . T H E CAMBRIDG E COLLOQUIUM , 1898 .

(a) Professo r W I L L I A M F . OSGOOD , o f Harvar d University : " Selecte d Topic s i n th e Theor y o f Functions. " This colloquiu m wa s publishe d i n th e Bulletin of the

Amer. Math. Soc. 9 ser. 2 , vol . 5 (1898) , p . 59 .

(b) Professo r ARTHU R G . WEBSTER, o f Clar k University : " Th e Partial Differentia l Equation s o f Wav e Propagation. "

I I I . T H E ITHAC A COLLOQUIUM , 1901 .

(a) Professo r OSCA R BOLZA , of th e University o f Chicag o : " Th e Simplest Typ e o f Problem s i n th e Calculu s o f Variations. " Published i n amplifie d for m unde r th e t i t le : Lectures on

the Calculus of Variations, Chicago , 1904 .

(6) Professo r ERNES T W. BROWN , of Haverfor d Colleg e : « Mod-ern Method s o f Treatin g Dynamica l Problems , and i n Par -ticular th e Proble m o f Three Bodies. "

I V . T H E BOSTO N COLLOQUIUM , 1903 .

(a) Professo r H E N R Y S . W H I T E , o f Northwester n University : three lecture s o n " Linear System s o f Curve s o n Algebrai c Surfaces/'

(6) Professo r FREDERIC K S . WOODS, of the Massachusetts Institut e of Technology: thre e lecture s on " Forms of Non-Euclidea n Space."

(c) Professo r EDWAR D B . V A N V L E C K , o f Wesleyan Universit y ; six lectures on " Selected Topic s i n th e Theory o f Divergen t Series an d Continue d Fractions. "

This colloquiu m i s her e publishe d i n full .

At Commencement , 1903 , Professo r Joh n Monro e Va n Vleck , M.A., LL.D. , complete d hi s fiftieth yea r o f servic e a t Wesleya n University, an d retire d shortl y afte r fro m th e chair of Mathematic s

X PREFACE.

and Astronomy . Al l thre e o f th e speaker s a t th e Bosto n Collo -quium wer e forme r student s o f his , on e o f the m bein g hi s SOD and colleagu e in th e departmen t of mathematics . I t i s fitting tha t this volume of lectures held at tha t Colloquium be inscribed t o him.

THOMAS S . F I S K E ,

W I L L I A M F . OSGOOD ,

Committee on Publication.

CONTENTS.

PAGE.

PREFACE vi i

LINEAR SYSTEM S O F CURVE S O N ALGEBRAI C SURFACE S

B Y HENR Y S . WHIT E

Cremona transformation s an d uh e geometr y o n a curve . Linea r series of point set s 1

Rational surface s define d 4 Linear systems o f curve s on an y algebrai c surface . . . . 5 Enriques's theorem on the two definition s o f linearity . . . 7 Hyperelliptic plan e curves , two kinds of linear systems . . . 1 2 Surfaces whos e plan e section s ar e hyperelliptic , Castelnuovo' s

theorem o n their rationality 1 4 Picard's exac t linea r differentials o n a surface; thos e o f first kin d

exist onl y o n singular surfaces 1 8 Poincar6's and Berry' s special quarti c surfaces . . . . 2 5 Humbert's hyperelliptic surface s o f sixth orde r . . . 2 7

FORMS O F NON-EUCLIDEA N SPAC E

B Y FREDERIC K S . WOOD S

1. Th e Firs t Two Hypothese s 3 2 2. Definition s 3 4 3. Th e Thir d Hypothesi s 3 7 4. Th e Line-Elemen t 3 9 5. Geometr y in a Restricted Portio n o f Space . . 4 5 6. Th e Fourt h an d Fifth Hypothese s 5 1 7. Th e Extende d Coordinat e Syste m 5 2 8. Th e Auxiliar y Spac e 2 5 7

xi

xi i CONTENTS.

9. Form s of Space which allow Fre e Motio n as a Whole . . 5 8 Spaces of Zero Curvature 5 9 Spaces of Constant Negativ e Curvatur e . . . . 5 9 Spaces of Constant Positiv e Curvatur e 6 0

10. Form s of Space which d o not allow Fre e Motio n a s a Whol e . 6 1 Spaces of Zero Curvature 6 3 Spaces of Constan t Positiv e Curvatur e 6 5 Clifford's Surfac e o f Zero Curvature 6 9 Spaces o f Constant Negativ e Curvature . . . . 7 1

SELECTED TOPIC S I N TH E THEOR Y O P DIVERGEN T SERIE S AN D O F

CONTINUED FRACTION S

B Y EDWAR D B . V A N VLEC K

PART I . DIVERGEN T SERIES . PAGE.

Introduction 7 5 Lecture 1 . Asymptoti c Convergenc e 7 7

" 2 . Th e Applicatio n o f Integral s t o Divergen t Series . 9 2 " 3 . O n the Determination o f the Singularities o f Func-

tions defined b y Power Series . . . . 10 7 " 4 . O n Series of Polynomials and of Rational Fractions. 12 0

PART II . ALGEBRAI C CONTINUE D FRACTIONS .

Lecture 5 . Pade' s Table of Approximants and its Applications. 13 4 11 6 . Th e Generalizatio n o f the Continue d Fractio n . 15 4

Bibliography 16 7

DIVEKGENT SEEIE S AN D CONTINUE D FRACTIONS . 16 7

BIBLIOGRAPHY O F MEMOIR S RELATIN G T O ALGEBRAI C

CONTINUED FRACTIONS .

In th e followin g bibliograph y onl y work s i n Latin , Italian , French, German, and Englis h ar e included. I n Woljjing's Mathe-matischer Buclierschatz (heading Kettenbruche) several dissertations , etc., ar e mentione d whic h ma y possibl y relat e t o algebraic con -tinued fractions bu t whic h ar e no t accessible t o th e writer . The y are therefor e no t include d here . Th e write r woul d b e gla d t o have hi s attentio n calle d t o an y noteworth y omission s i n th e bibliography.

In man y case s i t ha s been extremel y difficul t t o dra w th e lin e between inclusion an d exclusion, especially unde r divisions v i - ix .

Any classification o f the materia l whic h ma y be adopted wil l b e open t o objections, bu t even a n imperfect classificatio n wil l prob -ably ad d greatl y t o th e usefulnes s o f th e bibliography . Sinc e much o f the work relatin g to algebraic continued fractions appears , elsewhere unde r other headings , i t i s believed tha t suc h a bibliog -raphy as i s here given ma y b e of service .

For a brie f resum 6 of th e theor y o f algebrai c continue d frac -tions th e reade r is referred to Osgood's section o f the Encyklopadie der Math. Wissenschaft, n B l, § § 38-39 .

I. O N TH E DERIVATIO N O F CONTINUE D FRACTION S FRO M POWE R SERIES. GENERA L THEORY .

A. Early Works.

1. Euler . (a ) Introducti o i n analysi n infinitorum , vol . 1 , chap. 18, 1748.

(b) De transformation e serieru m in fractione s continuas. Opus -cula analytica, vol . 2 , pp. 138-177, 1785.

2. Lambert , (a) Verwandlung der Briiche. Beytrag e zum Gebrauche der Mathematik und deren Anwendung, vol. 2V p. 54 ff., p. 161, 1770.

(b) M6moire su r quelque s propri6t6s remarquable s des quantit y transcendentes circulaire s e t logarithmiques . Histoir e d e l'Acad. roy. des sciences et belles-lettres a Berlin, 1768.

168 THE BOSTO N COLLOQUIUM .

3. Trembley . Recherche s su r le s fraction s continues . Mem . d e l'Acad. roy . d e Berlin, 1794 , pp . 109-142 .

4. Kausler . (a ) Expositi o method i serie s quascunqu e data s i n frac -tiones continua s convertendi . M6m . d e l'Acad . imp . de s sci -ences de St . P6tersbourg , vol . 1 , pp . 156-174 , 1802 .

(b) D e insign i us u fractionum continuaru m i n calcul o integrale . Ibid., vol . 1 , pp . 181-194 , 1803 .

5. Viscovatov . (a ) D e la mSthode g6neral e pour reduire toutes sortes des quantity en fractions continues . Ibid. , vol . 1 , pp. 226-247, 1805.

(b) Essa i d'un e mSthod e g6n6ral e pour reduir e toute s sorte s d e series e n fraction s continues . Nov a Act a Acad . Scient . imp . Petropolitanae, vol . 15 , pp . 181-191 , 1802 .

6. Bret . Th6ori e generat e de s fraction s continues . Gergonne' s Annales d e Math. , vol . 9 , pp . 45-49 , 1818 . Unimportant .

7. Scubert . D e transformatio n serie i i n fractione m continuam . M6m. d e l'Acad . imp . de s science s d e St . P6tersbourg , vol . 7 , pp. 139-158 , 1820 .

8. Stern , (a ) Zu r Theorie der Kettenbriich e un d ihr e Anwendung . Jour, fu r Math. , vol . 10 , pp . 241-265 , 1833 .

(b) Zu r Theorie der Kettenbriiche . Jour , fu r Math. , vol . 18 , pp. 69-74, 1838 .

9. Heilermann . (a ) Uebe r di e Verwandlun g de r Reihe n i n Ketten -briiche. Jour , fu r Math. , vol . 33 , pp. 174-188 , 184 6 ; als o vol . 46, pp . 88-95 , 1853 .

(b) Zusammenhan g unte r den Coefficiente n zweie r gleiche n Ket -tenbriiche vo n verschiedene r Form . Zeitschrif t fu r Math, un d Phys., vol . 5 , pp . 362-363 , 1860 . Unimportant .

10. Hankel . Uebe r di e Transformatio n vo n Eeihe n i n Kettenbriiche . Berichte der Sachische n Gesellschaff c de r Wissenschaft z u Leip -zig, vol . 14 , pp . 17-22 , 1862 .

11. Muir . (a ) O n the transformatio n o f Gauss 1 hypergeometric serie s into a continue d fraction . Proc . o f th e Londo n Math . Soc , vol. 7 , pp . 112-118 , 1876 .

(6) Ne w genera l formula e fo r the transformation o f infinit e serie s into continue d fractions . Trans , o f th e R . Soc . o f Edinburgh , vol. 27 , pp . 467-471 , 1876 .

The genera l formula e i n these memoirs , which Muir suppose d to be new, ha d bee n previousl y give n b y Heilermann i n 9(a) .

12. Heine . Handbuc h de r Kugelfunction , 2 te Auflage, 187 8 ; chap . 5 , Die Kettenbriiche , pp . 260-297 .

This gives a good idea o f th e stat e o f th e theory u p to 1878 .

DIVERGENT SERIE S AN D CONTINUE D FRACTIONS . 16 9

B. The Modern Theory. The beginning s o f thi s theor y ar e t o b e foun d i n Nos . 11 0

and 111 . 13. Frobenius . Uebe r Eelationen zwisehe n den Naherungsbriichen von

Potenzreihen. Jour , fu r Math., vol . 90 , pp . 1-17 , 1881 . This fundamental memoi r mark s an importan t advance . Se e

16(a). 14. Stieltjes . Su r l a reductio n e n fractio n continu e d'un e s6ri e pro -

c£dant suivan t le s puissance s descendante s d'un e variable . Ann. d e Toulouse, vo l 3 , H , pp . 1-17 , 1889 .

15. Pincherle . Su r un e applicatio n d e l a th6ori e de s fractions contin -ues alg6briques. Comp . Rend., vol . 108 , p . 888 , 1889 .

16. Pade . (a ) Sur la representation approche e d'un e fonctio n pa r des fractions rationnelles . Thesis , publishe d i n th e Ann . d e l 7Ec. Nor., ser . 3 , vol . 9 , supplement , pp . 1-93 , 1892 .

This very fundamenta l memoi r is the best one to read for the purpose o f learnin g th e element s o f th e theor y o f algebrai c continued fractions . Th e sam e poin t o f vie w i s taken a s b y Frobenius i n (13 ) an d i s mor e completel y developed . Th e thesis was preceded b y the two followin g preliminar y note s :

(a') Su r l a representatio n approche e d'une fonctio n pa r de s fractions rationnelles . Comp . Rend , vol . I l l , p . 674 , 1890 .

(a") Su r le s fraction s continue s r^guliere s relative s a e'. Comp. Rend , vol . 112 , p . 712 , 1891 .

(b) Recherche s nouvelle s su r l a distributio n de s fraction s rationnelles approch6e s d'un e fonction . Ann . d e PEc . Nor. , ser. 3 , vol . 19 , pp. 153-189 , 1902 .

(c) Aper§ u su r les d^veloppement s r^cent s d e l a th^ori e de s fractions continues . Compt e rendu du deuxieme Congr& s inter-national de s math£maticiens ten u a Paris , pp . 257-264,1900 .

Only a restricted portion o f th e field i s here reviewed, an d i n this portion the important work o f Pincherle i s overlooked .

17. Pade . (a ) Su r le s serie s enti&re s convergente s o u divergente s e t les fraction s continue s rationelles . Act a Math. , vol . 18 , pp . 97-111, 1894 .

(a') Su r l a possibilit 6 de dMni r un e fonctio n pa r un e s6ri e enti&re divergente. Comp . Rend. , vol . 116 , p . 686 , 1893 .

See also No. 26a , 76 . II. O N CONVERGENCE .

(For a resume of the criteria for the convergenc e of continue d fractions with real element s see PRINGSHEIM' S repor t in the En-cyklopadie der mathematwehen Wissemchaften, I A 3 , p . 126 , ff.)


18. Riemann . Sull o svolgiment o de l quozient e d i du e seri e ipergeo -metriche in frazione continua infinita, 1863 . Gesammelt e math-ematische Werke , pp . 400-406 .

18. bis. Worpitzky . Untersuchun g iibe r di e Entwickelun g de r mono -dromen und monogenen Functionen durch Kettenbriiche . Pro -gramm, Friedrich s Gymnasium un d Realschule , Berlin , 1865 .

This progra m and the two followin g memoir s o f Thom6 wer e published before Riemann' s posthumou s fragment .

19. Thome , (a ) Uebe r di e Kettenbruchentwickelung de r Gauss'sche n Function F(a, 1 , y , x). Jour , fu r Math. , vol . 66 , pp . 322-336 , 1866.

(b) Uebe r di e Kettenbruchentwickelun g de s Gauss'sche n Quo -tienten

FJja, / ? + ! , y+1, x) F(a, ft, y, x) '

Ibid., vol . 67 , pp . 299-309 , 1867 . 20. Laguerre . Su r 1'integrate

J% x

Bull, d e l a Soc . Math , d e France , vol . 7 , pp . 72-81 , 1879 , o r Oeuvres, vol . 1 , p . 428 .

Historically a n importan t memoi r because o f its developmen t of th e connectio n betwee n a divergen t powe r serie s an d con -vergent continue d fraction . Se e the first footnote in lecture 4 ; also No. 102 , p . 30 .

21. Halphen . (a ) Su r l a convergenc e d'un e fractio n continu e alge -brique. Comp . Rend. , vol . 10 0 (1885) , pp . 1451-1454 .

(b) Sam e subject . Ibid. , vol . 10 6 (1888) , pp . 1326-1329 . (c) Trait 6 de s fonction s elliptiques . Chap . 14 . Fraction s con -

tinues e t integrate s pseudo-elliptiques . 22. Pincherle . Alcun i teorem i sull e frazion i continue . Att i dell e R .

Accad. de i Lincei , ser . 4 , vol . 5 V pp . 640-643 , 1889 . The tes t for convergenc e give n her e i s include d i n a mor e

general criterio n give n late r by Pringsheim , No . 29 . 23. Pincherle . Su r les fractions continue s algebriques . Ann . d e l'Ec .

Nor., ser . 3 , vol . 6 , pp . 145-152 , 1889 . An incomplet e resul t i s her e obtained . Se e No . 32 c for th e

complete theorem . 24. Pade . Su r la convergence des fractions continue s simples . Comp .

Rend., vol. 112 , p. 988,1891 . Als o found in §§ 45-47 of No. 16a .


25. Banning . Uebe r Kugel - un d Cylinderfunktione n un d dere n Ket -tenbruchentwickelung. Dissertation , Bonn , 1894 , pp . 1-33 .

26. Stieltjes . (a ) Recherche s su r le s fractions continues . Annale s de Toulouse, vol . 8, J , pp . 1-122, an d vol. 9, A , pp. 1-47. 1894-95 . Published also in vol . 3 2 of th e M6moire s present s k l'Acad . des sciences de l'Institut Nationa l d e France .

A ric h memoir , developin g particularl y th e connectio n

between a n importan t clas s o f continue d fraction s an d the cor -responding integrals .

(a') Su r u n d6veloppemen t e n fractio n continue . Comp . Rend., vol . 99 , p . 508 , 1884 .

(a") Sam e subject. Ibid. , vol . 10 8 (1889) , p . 1297 . (a"') Su r une application de s fractions continues. Ibid. , vol .

118 (1894) , p . 1315 . (aIV) Recherche s su r le s fraction s continues . Ibid. , vol . 11 8

(1894), p . 1401 . Markoff. (b) Not e su r le s fraction s continues . Bull , d e l'Acad .

imp. de s science s d e St . Petersbourg , ser . 5 , vol . 2 , pp . 9-13 , 1895.

This gives a discussion o f th e relation o f hi s wor k t o that o f Stieltjes.

27. H . von Koch, (a ) Su r un th6or6me d e Stieltjes e t su r le s fonction s d&inies par des fractions continues . Bull , d e la Soc . Math , d e France, vol . 23 , pp. 33-40 , 1895 .

(a') Su r la convergenc e de s determinants d'ordre infini e t de s fractions continues . Comp . Rend. , vol . 120 , p . 144 , 1895 .

28. Markoff . Deu x demonstrations de la convergence de certaines frac -tions continues . Act a Math. , vol . 19 , pp . 93-104 , 1895 .

Contained also i n hi s Differenzenrechnun g (deutsch e TJeber-setzung), chap . 7 , § 21-22.

This discusses th e convergenc e o f th e usua l continue d frac -tion fo r

fhf(y)dy

when f(y) > 0 between th e limits o f integration . 29. Pringsheim . Uebe r di e Convergen z unendliche r Kettenbriiche .

Sitzungsberichte der math.-phys . Class e de r k . bayer'sche n Akad. de r Wissenschaften, vol . 28 , pp . 295-324 , 1898 .

The most comprehensive criteri a for convergence ye t obtained are found i n 29 , 31 , and 326 .


30. Bortolotti . Sull a convergenz a dell e frazion i continu e algebriche . Atti dell a B . Accad . de i Lincei , ser . 5 , vol. S v pp . 28-33, 1899 .

31. Va n Vleck. O n the convergenc e o f continue d fraction s wit h com -plex elements . Trans . Amer . Math . Soc , vol . 2 , pp . 215-233 , 1901.

32. Va n Vleck. (a ) O n th e convergenc e o f th e continue d fractio n o f Gauss and other continued fractions . Annal s of Math. , ser . 2 , vol. 3 , pp . 1-18 , 1901 .

(b) O n th e convergenc e an d character o f th e continue d fractio n

axz a 2z a zz

T~+~T+r+ •• • Trans. Amer . Math . Soc , vol . 2 , pp . 476-483 , 1901 .

(c) O n th e convergenc e o f algebrai c continue d fraction s whos e coefficients hav e limitin g values . Ibid. , vol . 5 , pp . 253-262 , 1904.

33. Montessus . (a) Su r le s fractions continue s alg£briques . Bull , d e la Soc . Math , d e France , vol . 30 , pp . 28-36 , 1902 .

The content o f this memoir was discussed in lecture 5 . (b) Sam e title . Comp . Kend., vol . 13 4 (1902), p . 1489 .

See also 37a7, 41 .

III. O N VARIOU S CONTINUE D FRACTION S O P SPECIA L FORM .

A. The Continued Fraction of Gauss.

34. Gauss . Disquisitione s generate s circ a seiiem infinita m

1 + l-y x+ l-2y(y + l) x+ '-'

Deutsche Uebersetzun g vo n Simon , o r Werke, vol . 3 , pp . 134 -138, 1812 .

34, bis. Vorsselma n de Herr . Specime n inaugurat e d e fractionibus con -tinuis. Dissertation , Utrecht , 1833 .

Numerous reference s ar e give n her e t o th e earl y literatur e upon continued fractions .

34, ter. Heine . Auszu g eine s Schreibens uber Kettenbriiche von Herr n E. Hein e a n de n Herausgeber . Jour , fu r Math. , vol . 53 , pp . 284-285, 1857 .

See also 40c, p . 231 . 35. Euler . (a ) Commentati o i n fractione m continua m i n qua illustri s

Lagrange potestate s binomiales expressit . M6moire s de 1T Acad, imp. de s sciences d e St . P6tersbourg , vol . 6 , pp . 3-11 , 1818 .


Pade. (b) Su r l a generalisatio n de s dfiveloppement s e n fraction s continues, donne s pa r Gauss e t pa r Enter, d e l a fonctio n (1 + x) m. Comp . Rend. , vol . 129 , p . 753 , 1899 .

(c) Su r l a generalisatio n de s d6veloppements e n fractions contin -ues, donn6 s pa r Lagrange d e l a fonction ( 1 + #) w. Ibid. , vol . 129, p . 875 , 1899 .

(d) Su r V expression g£neral e d e la fraction rationnell e approche e

de (1 + x) m. Ibid. , vol . 132 , p . 754,1901 . See also Nos. 11 , 32a , 65 .

B. The Continued Fractions for e x.

36. Winckler . Uebe r angenahert e Bestimmungen . Wiene r Berichte , Math.-naturw, Classe , vol . 72 , pp. 646-652 , 1875 .

37. Pade . (a ) M6moir e su r les developpements e n fractions continue s de la fonction exponentielle , pouvan t servi r d'introductio n k l a th6orie de s fraction s continue s alg6briques . Ann . d e l'Ec . Nor., Ser . 3 , vol . 16 , pp . 395-426 , 1899 .

(a') Su r l a convergence de s r6duites de la fonction exponentielle , Comp. Rend. , vol . 127 , p . 444 , 1898 .

See also Nos. 16a" , 106 , an d page s 243- 5 of 40c.

C. The Continued Fraction of BesseL

38. Gunther . Bemerkunge n fibe r Cylinder-Functionen . Archi v de r Math, un d Phys. , vol . 56 , pp . 292-297 , 1874 .

39. Graf , (a ) Relation s entre la fonction Bess61ienn e d e l re espfec e e t une fraction continue . Annal i di Mat., ser. 2, vol. 23, pp. 45-65, 1895.

Giving reference s t o earlie r works where th e continued frac -tion of Bessel i s found .

Crelier. (6 ) Su r quelque s propri6t£ s de s fonction s Bess61iennes , tiroes d e l a th6ori e de s fractions continues . Annal i d i Mat. , vol. 24 , pp . 131-163 , 1896 .

See also Nos. 25 , 32a .

D. The Continued Fraction of Heine.

40. Heine , (a ) Uebe r die Reihe

( g « - i ) ( g g - i ) ( r - i ) ( r + 1 - D ( ^ - i ) ( ^ + 1- i ) . , 1 + ( j _ l ) ( j y - l ) * + ( s - l H ^ - l X g v - l X g Y ^ - l ) * + " "


Jour, fu r Math., vol . 32 , pp . 210-212, 1846 . (b) Untersuchun g libe r die (selbe ) Reihe . Ibid. , vol . 34 , pp . 285 -

328, 1847 . (c) Uebe r di e Zahler und Nenner de r Naherungswerthe vo n Ket -

tenbriiche. Ibid. , vol . 57 , pp . 231-247 , 1860 . Christoffel (d) Zu r Abhandlun g "Uebe r Zahle r un d Nenner "

(u. s . w. ) de s vorige n Bandes . Ibid. , vol . 58 , pp . 90-91 , 1861 . 41. Thomae . Beitrag e zu r Theori e de r durc h di e Heine'sch e Reih e

darstellbaren Funktionen . Jour , fu r Math., vol . 70 , 1869 . Se e pp. 278-28 1 where th e convergenc e o f Heine' s continue d frac -tion is proved .

See also 32a . 42. (O n EisetisteM's continue d fractions) .

Heine, (a ) Verwandlun g von Reihen in Kettenbniche. Jour , fu r Math., vol . 32 , pp. 205-209 , 1846 .

See also vol . 34 , p . 296 . Muir. (6 ) O n Eisenstein's continue d fractions . Trans . Roy . Soc .

of Edinburgh , vol . 28 , par t 1 , pp . 135-143 , 1877 . Muir plainl y wa s no t awar e o f th e precedin g memoi r b y

Heine.

E. The Continued Fraction of Stieltjes. (See No. 26.)

43. Borel . Le s serie s d e Stieltjes, Chap . 5 o f hi s Memoir e su r le s seriesdivergentes. Ann . del'Ec . Nor., ser . 3 , vol. 16 , pp. 107 -128 ; an d als o chap . 2 of hi s treatise , Le s Serie s divergentes , pp. 55-86 , 1901 .

44. Pade . Su r la fractio n continue de Stieltjes. Comp . Rend., vol . 132 , p. 911 , 1901.

45. Va n Vleck. O n a n extensio n o f th e 189 4 memoi r o f Stieltjes. Trans. Amer . Math . Soc , vol . 4 , pp . 297-332 , 1903 .

See also Nos. 27 , 102 .

JP. The Continued Fraction for

1 -f - mz -f - m{m + n)^ -f - m{m -j- ri) (m -f- 2n)a 3 - | —

and its special cases. 46. Euler . (a ) D e seriebu s divergentibus . Nov i commentari i Acad .

scientiarum imperiali s Petropolitanse, vol . 5 , pp. 205-237, 1754 -5 ; in particular pp . 22 5 and 232-237 .

(6) D e transformation e serie i divergenti s

1 — mx + m(m + n)x 2 — m(m + n) (m + 2n)x 3 -f • •


in fractione m continuam . Nov a act a Acad , scientiaru m im -perialis Petropolitanae , vol . 2 , pp . 36-45, 1784 .

Gergonne. (c ) Reeherche s su r les fractions continues. Gergonne' s Annales de Math. , vol . 9 , pp . 261-270 , 1818 .

47. Laplace , (a ) Trait 6 d e mecaniqu e celeste . Oeuvres , vol . 4 , pp . 254-257, 1805 .

Jacobi. (b) D e fraction e continu a i n qua m integral e I e dx

evolvere licet . Jour , fu r Math. , vol . 12 , pp . 346-347 , 1834 , o r Werke, vol . 6 , p . 76 .

See also p . 7 9 of No. 20 , an d the first note unde r lecture 2 .

Q. Periodic Continued Fractions, and Continued Fractions Connected with the Theory of Elliptic functions.

48. Abel , (a) Su r V integration d e l a formule differentiell e pdxl\/lt, B et p etan t de s fonctions entieres . Jour , fu r Math. , vol . 1 , pp . 185-221, 1826 , o r Oeuvres, vo l 1 , p . 10 4 flf.

Dobnia. (b) Su r l e d6veloppemen t d e YE e n fractio n continue . Nouvelles Ann . de Math. , ser . 3 , vol . 10 , pp . 134-140 , 1891 .

49. Jacobi . (a) Not e su r un e nouvell e applicatio n d e V analyse de s fonctions elliptique s a l'algebre . Jour , fu r Math. , vol . 7 , pp . 41-43, 1831 , o r Werke, vol . 1 , p . 327 .

Borchardt. (b) Applicatio n de s transcendante s ab^lienne s a l a theorie des fractions continues. Ibid. , vol. 48, pp. 69-104, 1854 .

50. Tchebychef . Su r Integratio n de s diff<§rentielle s qu i contiennen t une racin e carr£e d'un polyndm e d u troisi&me ou d u quatriem e degr6. M6moire s d e PAcad . imp . de s science s d e St . Peters -bourg, ser . 6 , vol . 8 , pp . 203-232 , 1857 .

51. Frobeniu s und Stickelberger. Uebe r die Additio n un d Multiplicatio n der elliptischen Functionen . Jour , fu r Math. , vol. 88 , pp. 146 -184, 1880 .

52. Halphen . Su r les integrales pseudo-elliptiques . Comp . Rend., vol . 106 (1888), pp . 1263-1270 .

53. Bortolotti . Sull e frazion i continu e algebrich e periodiche . Rendi -conti de l Circol o Mat . d i Palermo, vol . 9 , pp . 136-149 , 1895 .

See also Nos. 21 , 26(a) , 40 .

H. Miscellaneous.

54. Euler . (a) Speculatione s supe r formula integral i

/xndx

Ya2 — 2bx+cx 1


ubi simul egregia e observationes circa fractiones continuas occur-rent. Act a Acad, scientiaru m imperiali s Petropolitanae , 1784 , pars posterior, pp . 62-84 , 1782 .

(b) Summati o fractioni s continu e cuju s indice s progressione m arithmeticam constituunt. Opuscul a Analytica, vol . 2 , pp. 217 -239, 1785 .

55. Spitzer . (a ) Darstellun g de s unendlichen Kettenbruch s

1 1 1

x + X+1+X+2+X+3+

in geschlossene r Form , nebs t andere n Bemerkungen . Archi v der Math, un d Phys. , vol . 30 , pp . 81-82 , 1858 .

(b) Darstellun g de s unendlichen Kettenbruch s

- 1 1 1

2x +1 2 s + 3 + 2 s +5 + 2 x + 7 +

in geschlossener Form . Ibid. , vol . 30 , pp . 331-334 , 1858 . (c) Not e iibe r ein e Kettenbriiche . Ibid. , vol . 33 , pp . 418-420 ,

1859. (d) Darstellun g de s unendlichen Kettenbruche s

¥(x) = n(2x + 1 ) + n(2x + 3 ) + n(2x + 5 ) +

in geschlossener Form . Ibid. , vol . 33 , pp . 474-475 , 1859 . 56. Laurent , (a ) Not e su r les fractions continues. Nouvelle s Ann . d e

Math., ser . 2 , vol . 5 , pp . 540-552 , 1866 . This treats the continued fraction

x x x 1 + 1 + 1 + ' " '

E. Meyer , (b) Uebe r ein e Eigenschafb de s Kettenbruche s

x _ • • •. Archi v de r Math , un d Phys. , ser . 3 , vol . 5 , x"— x —•

p. 287 , 1903 . Meyer's results will be found on p . 548 of Laurent' s memoir

and differ s onl y i n that x has been replaced b y — 1/x 2. 57. Schlomilch . (a ) Uebe r de n Kettenbruc h fii r tan z. Zeitschrif t fu r

Math, un d Phys. , vol . 16 , pp . 259-260 , 1871 . Glaisher. (b) A continue d fractio n fo r ta n nx. Messenge r o f

Math., ser . 2 , vol . 3 , p . 137 , 1874 . (c) Not e o n continue d fraction s fo r ta n nx. Ibid. , ser . 2 , vol . 4 ,

pp. 65-58 , 1875 .


58. Schlomilch . Uebe r di e Kettenbruchentwickelun g fu r unvollstan -dige Gamma-function . Zeitschrif t fii r Math , un d Phys. , vol . 16, pp . 261-262 , 1871 .

This give s th e developmen t o f I t^" 1 e~fdt.

59. Schendel . Uebe r eine Kettenbruchentwickelung. Jour , fur Math., vol. 80 , pp . 95-96 , 1875 .

60. Lerch . Not e su r les expressious qui , dans diverses parties du plan, representent de s fonctions distinctes . Bull , des sciences Math, ser. 2 , vol . 10 , pp . 45-49 , 1886 .

61. Stieltjes . (a) Sur quelques integrates definies et leur developpement en fraction s continues . Quar . Jour , of pur e and applied Math. , vol. 24 , pp . 370-382 , 1890 .

(b) Not e su r quelques fractions continues. Ibid. , vol. 25 , pp. 198-200, 1891 .

62. Hermite . Su r le s polynome s de Legendre . Jour , fu r Math. , vol . 107, pp . 80-83 , 1891 .

This connect s DMJE**\x) wit h a continue d fraction .

IV. O N TH E CONNECTIO N O F CONTINUED FRACTION S WIT H DIFFEREN -

TIAL EQUATION S AN D INTEGRALS .

A. BiccaWs Differential Equation.

63. Euler . (a) D e fractionibus continuis observationes . Commentari i academise scientiaru m imperiali s Petropolitanse , vol . 11 , se e pp. 79-81 , 1739 .

(b) Analysi s facili s sequatione m Riccatiana m pe r fractionem con -tinuam resolvendi. M6moires de V Acad, imperiale des sciences de St . P6tersbourg , vol . 6 , pp . 12-29 , 1813 .

64. Lagrange . Su r Y usage de s fractions continue s dans l e calcnl inte -gral. Nouveau x M6m . d e l'Acad. roy . de s science s e t belles -lettres d e Berlin , 1776 , pp . 236-264, or Oeuvres, vol. 4, p. 301 ff.

One of th e fe w importan t earl y works . See 54 6 ; als o No. 66 a for work on diflferential equation s of the 1st order.

B. Miscellaneous Differential Equations of the Second Order.

In a numerous clas s of continue d fraction s the denominator s of the convergent ^ satisfy allie d (Heun , " gleichgruppige") diflfer -ential equations of the second order. Earl y instances are foun d in work s o f Gauss (No. 114) , Jacobi (No. 65) and Seine (No . 72). The theory , fro m tw o differen t aspects , i s furthest develope d i n 66a an d 76.

178 TH E BOSTO N COLLOQUIUM .

65. Jacobi . Untersuchun g iibe r di e Differentialgleichun g de r hyper -geometrischen Reihe. Nachlass . Jour , fur Math., vol. 56,1859 ; see in particular § 8, pp . 160-161 , o r Werke, vol . 6 , p . 184 .

66. Laguerre . (a ) Su r la reduction e n fraction s continue s d'un e frac -tion qui satisfai t k un e Equatio n differentiell e lineair e du pre -mier ordre dont les coefficients son t rationnels . Jour , de Math., ser. 4 , vol . 1 , pp . 135-165 , 1885 .

This i s a comprehensive memoi r whic h incorporates substan -tially al l th e followin g memoir s :

(b) Su r l a reductio n e n fraction s continue s d'un e class e asse z 6tendue d e fonctions . Comp . Rend. , vol . 8 7 (1878) , p . 923 , o r Oeuvres, vol . 1 , p . 322 .

(c) Sam e titl e a s (a) . Bull , d e l a Soc . Math , d e France , vol . 8 (1880), pp . 21-27 , o r Oeuvres, vol . 1 , p . 438 .

(d) Su r l a reductio n e n fraction continu e d'un e fraction qu i satis-fait k une Equation lin6aire du premier ordre a coefficients ration-nels. Comp . Rend. , vol . 9 8 (1884) , pp . 209-21 2 o r Oeuvres , vol. 1 , p . 445 .

67. Laguerre . (a ) Su r 1 ? approximation de s fonction s d'un e variabl e au moyen d e fraction s rationnelles . Bull , d e la Soc . Math , d e France, vol . 5 (1877), pp . 78-9 2 o r Oeuvres, vol . 1 , p . 277 .

(5) Su r le developpemen t e n fractio n continu e d e

arc t

e Ibid., vol . 5 (1877), pp . 95-9 9 o r Oeuvres, vol . 1 , p . 291 .

( x 4 - 1 \ w

Ibid., vol . 8 (1879), pp . 36-52 , o r Oeuvres, vol . 1 , p . 345 .

(d) Su r l a reductio n e n fraction s continue s d e e F<x>, F(x) desig -nant un polyn6me e n tier. Jour , d e Math. , ser . 3, vol. 6 (1880), pp. 99-110 , o r Oeuvres, vol . 1 , p . 325 .

(d') Same subject. Comp . Rend., vol. 87 (1878), p. 820, or Oeuvres, vol. 1 , p . 318 .

68. Humbert . Su r l a reductio n e n fractions continue s d'un e classe d e fonctions. Bull , d e l a Soc . Math , d e France , vol . 8 , pp . 182 -187, 1879-1880 .

69. Hermit e e t Fuchs . Su r u n developpemen t e n fractio n continue . Acta Math. , vol . 4 , pp . 89-92 , 1884 .

See also No. 20 , 3 4 ter, 71-76 .


C. Differential Equations of Order Higher than the Second.

70. Pincherle . Su r l a generatio n d e systeme s recurrent s a u moye n d'une equatio n lineair e differentielle . Act a Math., vol. 16 , pp . 341-363, 1892-3 .

See also No. 15 , 86 , 87 , 1246 .

D. The integral I J ±-!— • %/a % — 3

71. Heine , (a ) Uebe r Kettenbriiehe . Monatsbericht e de r k. preussi -schen Akad . der Wissenschafte n z u Berlin, 1866 , pp . 436-451 .

(a') Mittheilun g iiber Kettenbriiehe. Auszu g aus dem Monatsbe -richte, u . s . w . Jour , fur Math. , vol . 67 , pp . 315-326 , 1867 .

See also Nos . 12 , 26a , 28 , 45 , 102 , 113 , 118a .

E. Hyperelliptic and Similar Abelian Integrals.

72. Heine . Di e Lame'sche n Functione n verschiedene r Ordnungen . Jour, fu r Math. , vol . 60 , 1862 , pp . 252-303 ; i n particular pp. 256, 275 , 294-297 . O r see hi s Handbuch , vol . 1 (2 * Auf.), pp . 388-396 and 468.

73. Laguerre . Su r 1'approximation d'un e classe de transcendantes qu i comprennent comm e ca s particulie r le s integrate s byperellip -tiques. Comp . Rend. , vol . 84 , pp . 643-645 , 1877 .

(Not foun d i n vol . 1 . o f hi s Oeuvres. ) 74. Humbert . Su r V equation dift6rentiell e lineair e d u secon d ordre .

Jour, d e l'Ec . Pol y tech., vol . 29 , cahie r 48, pp . 207-220 , 1880 . 75. Heun . (a ) Di e Kugelfunctione n un d Lame'sche n Functione n al s

Determinanten. Dissertation , pp . 1-32 , Gottingen , 1881 . (b) Uebe r linear e Differentialgleichungen zweite r Ordnun g dere n

Losungen durc h de n Kettenbruchalgorithmu s verkniipf t sind . Habilitationsschrift. 1881 .

(c) Integratio n regulare r linearer Differentialgleichunge n zweite r Ordnung durch die Kettenbruchentwickelung von ganzen AbeF-schen Integrale n dritte r Gattung . Math . Ann. , vol . 30 , pp . 553-560, 1887 .

(d) Beitrag e zu r Theori e der Lam6'sche n Functionen . Math . Ann., vol . 33 , pp . 180-196 , 1889 .

The important group-propertie s o f th e continued fractio n ar e here brought ou t and are further develope d in No . 76 .

76. Va n Vleck. Zu r Kettenbruchentwickelun g hyperelliptische r un d ahnlicher Integrale . Dissertation , Gottinge n ; publishe d in the Amer. Jour , o f Math. , vol . 1 6 (1894), pp . 1-91 .


After developmen t first from an algebraic standpoint th e sub-ject is carried further by the method of conformal representation. The suggestio n o f thi s treatmen t i s given i n Klein's Differen -tialgleichungen, 1890-91 , vol . 1 , pp . 180-186 .

V. GENERALIZATIO N O F TH E ALGEBRAI C CONTINUE D FRACTION .

A. General Theory.

So far as I have been able to ascertain , th e first instance of the generalization i s containe d i n Hermite's memoir , No . 84 . Th e development o f a general theor y i s du e t o Fade and Pincherle. Nos. 77a , 776 , an d 79 a are especiall y recommended .

77. Pincherle . (a ) Saggi o d i un a generallizzazion e dell e frazion i con -tinue algebriche . Memoiri e dell a R . Accad . dell e Scienze dell ' Istituto d i Bologna , ser . 4 , vol . 10 , p . 513-538 , 1890 .

(a7) D i un'estension e dell ' algorithm o dell e frazion i continue . Rendiconti, R . Istitut o Lombard o d i Scienze e Lettere , ser . 2 , vol. 22 , pp . 555-558, 1889 .

{b) Sull a generalizzazione delle frazioni continu e algebrique. An -nali d i Mat. , ser . 2 , vol . 19 , pp . 75-95 , 1891 .

78. Hermite . Su r la generalisation de s fractions continue s algebriques . Annali d i Mat. , ser . 2 , vol . 21 , pp. 289-308 , 1893 .

79. Pade . (a) Su r l a generalisatio n de s fraction s continue s alge -briques. Jour , d e Math. , ser . 4 , vol . 10 , pp . 291-329 , 1894 .

(a7) Sam e subject . Comp . Rend. , vol . 118 , p . 848 , 1894 . 80. Bortolotti . U n contribut o alia teoria dell e forme linear i all e differ -

enze. Annal i d i Mat. , ser . 2 , vol . 23 , pp . 309-344 , 1895 . 81. Cordone . Sopr a un problem a fundamentale dell e teori a dell e fra -

zioni continu e algebrich e generalizzate . Rendicont i del Circolo di Palermo, vol . 12 , pp . 240-257 , 1898 .

Cordone seek s th e regula r algorithm s whic h ar e simila r t o those o f Pad e bu t occur i n connectio n wit h n series in descend -ing powers of x.

B. Convergence of the Generalized Algorithm.

82. Pincherle . Contribut o alia generalizzazione dell e frazioni continue . Memoirie della R. Accad. dell e Scienz e dell' Istituto di Bologna , ser. 5 , vol . 4 , pp . 297-320 , 1894 .

83. W . Fran z Meyer , (a ) Uebe r kettenbruchahnliche n Algorithmen . Verhand. de s ersten internationale n Mathematiker-Kongresse s in Zurich , pp . 168-181 , 189 8 ; see i n particular § 7.

DIVEEGENT SERIE S AN D CONTINUE D FRACTIONS . 18 1

(a') Zu r Theori e de r kettenbruehahnliche n Algorithmen . Schrif -ten de r phys-okonomische n Gesellschaf t z u Konigsberg , vol . 38, pp . 57-66 , 1897 .

C. Special Cases of the Algorithm. 84. Hermite . Su r la fonction exponentielle . Comp . Rend., vol . 77 , pp.

18-24, 74-79 , 226-233 , 285-293 , 1873 . This i s the famou s wor k provin g th e transcendenc e o f e.

85. Hermite . (a ) Su r V expression U s in # + Fco s x + W. Extrai t d'une lettr e a Monsieu r Pau l Gordan . Jour , fu r Math. , vol . 76, pp . 303-311 , 1873 .

(b) Su r quelque s approximation s algebriques . Ibid. , vol . 76 , pp . 342-344, 1873 .

(c) Su r quelques equations differentielle s lin6aires . Extrai t d'un e lettre a M. L . Fuch s d e Gottingue . Ibid. , vol . 79 , pp. 324-338, 1875.

86. Laguerre . Su r la fonction exponentielle . Bull , d e la Soc. Math, d e France, vol . 8 (1880) , pp . 11-18 , o r Oeuvres , vol . 1 , p . 336 .

87. Humbert , (a ) Su r un e generalisatio n d e l a theori e de s fraction s continues algebriques . Bull , d e l a Soc . Math , d e France , vol . 8, pp . 191-19 6 ; vol . 9 , pp . 24-30 , 1879-1881 .

(b) Su r l a fonctio n (x — 1)\ Ibid. , vol . 9 , pp . 56-58 , 1880-81 . 88. Pincherle . Sull a rappresentazion e approssimat a d i un a funzion e

mediante irrazional i quadratici . Rendiconti , R . Istitut o Lom -bardo d i Scienz e e Lettere , ser . 2 , vol . 23 , pp . 373-376 , 1890 .

89. Pincherle . (a) Un a nuov a estension e dell e funzion i sferiche . Memoirie dell a R . Accad . dell e Scienz e dell'Istitut o di Bologna , ser. 5 , vol . 1 , pp . 337-370 , 1890 .

(a') Sull a generalizzazion e dell e funzion i sferiche . Bologn a Ren -diconti, 1891-92 , pp . 31-34 .

(b) U n sistem a d'integral i ellittic i coosiderat i com e funzion i dell'invariante assoluto . Att i dell a R . Accad . de i Lincei , ser . 4, vol . 7 l} pp . 74-80 , 1891 .

90. Bortolotti . (a ) Sui sistem i ricorrent i de l 3 ° ordin e e d i n particolar e sui sistemi periodici . Rendicont i de l Circolo di Palermo, vol. 5 , pp. 129-151 , 1891.

(b) Sull a generalizzazion e dell e frazion i continu e algebrich e peri -odiche. Ibid. , vol . 6 , pp . 1-13 , 1892 .

VI. Series of Polynomials (Naherungsnenner). The serie s

-^— = £ (2 n + l)Q^(v)P^(u)


was first given by Heine in Crelle' s Jour. , vol . 4 2 (1851) , p . 72 . See als o hi s Handbuch , vol . 1 , pp . 78-79 , 197-200 . Amon g the numerous works relating t o expansion s i n terms o f Kugel-functionen erster und zweiter Gattung ma y be mentioned :

91. Bauer . Vo n de n Coefficiente n de r Reihe n vo n Kugelfunctione n einer Variablen . Jour , fu r Math. , vol . 56 , pp . 101-121 , 1859 .

92. C . G. Neumann. Uebe r die Entwickelun g eine r Functio n mit imag-inarem Argumente nach den Kugelfunctionen erste r und zweiter Gattung, Halle , 1862 .

93. Thome . Uebe r di e Reihe n welch e nac h Kugelfunctione n fort -schreiten. Jour , fu r Math. , vol . 66 , pp . 337-343 , 1866 .

94. Laurent . Memoir e sur les fonctions de Legendre. Jour , de Math., ser. 3 , vol . 1 , pp . 373-398 , 1875 .

See th e comment s b y Hein e i n vol . 2 , pp . 155-157 , als o b y Darboux and Lauren t in the sam e vol. , pp . 240 , 420 .

Numerous memoir s relate t o serie s i n terms o f the polynom -ials arisin g fro m th e expansio n o f ( 1 — 2ax + a 2)". I t suffice s here t o refe r t o th e Encyklopadi e de r Math . Wissenschaften , I A 10 , §31.

95. Frobenius . Uebe r di e Entwicklun g analytische r Functione n i n Reihen, di e nac h gegebene n Functione n fortschreiten . Jour , fur Math. , vol . 73 , pp. 1-30 , 1871 .

An interesting memoir . 96. Darboux . Su r V approximation des fonctions d e tres-grand s nom -

bres e t sur une classe 6tendu e de developpements e n serie , Part 2. Jour , d e Math. , ser . 3, vol . 4 , pp . 377-416 , 1878 .

97. Gegenbaue r Uebe r Kettenbruche. Wiene r Berichte, vol. 80, Abth. 2, pp . 763-775 , 1880 .

98. Poincare . (a ) Su r le s equation s lin£aire s au x differentielle s ordi -naires e t au x differences finies. Amer . Jour , o f Math. , vol . 7 , pp. 243-257 , 1885 .

This gives an importan t criterion fo r the convergence of serie s of polynomials. Se e lectur e 4 .

(a') Su r le s series des polynomes. Comp . Rend. , vol . 56 , p . 637 , 1883.

99. O n the series %A n(x — ax)(z — a2) • • • (x — an). A serie s of thi s for m i s employe d i n Newton' s interpolatio n

formula, Philosophis e naturali s principia , boo k 3 , lemm a V . See the Encyklopadi e de r Math. Wissenschafoen, I D 3 , § 3. A similar us e i s made b y


Cauchy. (a ) Su r le s fonction s interpolates . Comp . Rend. , vol . 11, pp . 775-789,1841 .

See next No. 95 . Peano. (b) Sull e funzion i interpolari . Att i dell a R . Accad . dell e

Scienze d i Torino, vol . 18 , pp . 573-580 , 1883 . Bendixson. (c ) Su r une extension k l'infin i d e l a formul e d J inter-

polation de Gauss . Act a Math. , vol . 9 , pp . 1-34 , 1886 . (c') Su r l a formul e d'interpolatio n d e Lagrange . Comp . Rend. ,

vol. 10 1 (1885), pp. 1050-105 3 and 1129-1131 . Pincherle. (d) Sull'interpolazione. Memoiri e della R. Accad . dell e

Scienze d i Bologna , ser . 5 , vol . 3 , pp. 293-318 . (See a "not e historique " b y Enestrom , Comp . Rend. , vol .

103, p . 523 , 1886) . See also No. 103 .

100. Pincherle , Su r l e developpemen t d'un e fonctio n analytiqu e e n s6rie de polynomes. Comp . Rend., vol . 107 , p . 986 , 1888 .

101. Pincherle . R6sum 6 d e quelques r6sultat s relatif s k l a th6orie de s syst&mes recurrent s d e fonctions . Mathematica l Papers , Chi -cago Congress , 1893 , pp . 278-287 .

102. Blumenthal . Uebe r di e Entwickelun g eine r willkiirliche n Funk -tion nac h den Nenner n de s Kettenbruches fu r

'_* z — £ *

Dissertation, Gottingen , 1898 . The mos t advance d developmen t o f thi s subjec t i s found i n

the wor k o f Blumenthal and Pincherle. 103. Laurent . Su r le s serie s d e polynomes . Jour , d e Math. , ser . 5 ,

vol. 8 , pp . 309-328 , 1902 . 104. Stekloff . Su r l e developpemen t d'un e fonctio n don6 e e n s&ie s

procMant suivant le s polynomes d e Tchebichef f et , e n particul -ier, suivan t le s polynome s d e Jacobi . Jour , fu r Math. , vol . 125, pp . 207-236 , 1903 .

See also Nos. 20 , 70 , 71 . 104 bis. Rouche . Memoire su r l e developpemen t de s fonctions e n serie s

ordonnees suivan t le s denominateurs de s reduite s d'un e frac -tion continue . Jour , d e l'E c Polytech. , cahie r 37 , pp. 1-34 .

This memoi r has a close connection with th e work o f Tcheby -chef.

VII. On the Roots of the Numerators and Denominators of the Convergents. 105. Sylvester , (a) O n a remarkable modificatio n o f Sturm' s theorem .

Phil. Mag. , ser . 4 , vol . 5 , pp . 446-456 , 1853 .


(b) Not e o n a remarkable modification o f Sturm's theore m and o n a new rule fo r finding superior and inferior limit s to the root s of an equation . Ibid. , vol . 6 , pp . 14-20 , 1853 .

(c) O n a ne w rul e fo r finding superio r an d inferio r limit s t o th e real root s of any algebraic equation . Ibid. , vol . 6 , pp. 138-140 , 1853.

(d) Note o n the ne w rul e o f limits . Ibid. , vol . 6 , pp . 210-213 , 1853.

(e) O n a theory o f the syzygetic relation s o f tw o rationa l integra l functions, comprisin g a n applicatio n t o th e theor y o f Sturm' s functions, an d that o f th e greates t algebrai c commo n measure . Phil. Trans. , 185 3 ; se e in particular p. 49 6 ff.

( / ) Th6orem e sur les limites des racines r6elles des equations alg6-briques. Nouvelle s Ann . de Math. , ser. 1 , vol. 12 , pp. 286-287 , 1853.

(g) Pou r trouver une limit e superieur e et une limite inferieure des racines r6elles d'une equation quelconque. Ibid. , ser. 1 , vol. 12 , pp. 329-336 , 1853 .

106. Laguerre . Su r quelques propriet y de s equation s algebrique s qu i ont toutes le s racines reelles. Nouvelle s Ann. d e Math., ser. 2 , vol. 1 9 (1880), pp . 224-239 , o r Oeuvres, vol . 1 , pp . 113-118 .

Laguerre considers her e th e root s of th e numerator s and de -nominators of the approximants forf(x) an d 1 If(x) when f(x) i s a polynomial wit h rea l roots .

107. Gegenbauer . (a ) Uebe r algebraische Gleichunge n welch e nu r reel e Wurzeln besitzen . Wiene r Berichte , vol . 8 4 (1882) , Abt . 2 , see in particular pp . 1106-1107 .

(6) Uebe r algebraisch e Gleichunge n welch e ein e bestimmt e An -zahl complexe r Wulzel n besitzen . Ibid , vol . 87 , pp . 264-270 , 1883.

108. Markoff . Su r le s racine s d e certaine s Equations . Math . Ann. , vol. 27 , pp . 143-150 , 1886 .

108 Ws. Hurwitz . Uebe r di e NuUstelle n de r Bessel'sche n Function . Math. Ann. , vol . 33 , pp . 246-266 , 1889 .

Although th e functions considere d i n thi s memoi r are of a special character , th e memoir is mentioned her e o n account o f the methods employed .

109. Porter . O n the root s of functions connecte d b y a linea r recurren t relation o f th e secon d order . Annal s o f Math. , ser . 2 , vol . 3 , pp. 55-70 , 1902 .

See also Nos. 20 , 26a , 31 , 32a, 45 , 56 , 71 , 74 , 76 , 87a , 118a .


VIII. Approximation to a Function at More Than One Point. Connection of Continued Fractions with the Theory of Interpolation* Under No . 9 9 hav e bee n alread y classifie d variou s work s

which relat e t o simultaneou s approximatio n a t severa l points . In addition , th e followin g memoir s ma y als o b e consulted :

110. Cauchy . Su r l a formul e d e Lagrang e relati v a interpolation . Analyse Alg. , p . 528 , o r Oeuvres , ser . 2 , vol. 3 , pp . 429-433 .

111. Jacobi . Uebe r di e Darstellun g ein e Reih e gegebne r Werth e durc h eine gebrochene rational e Function . Jour , fu r Math. , vol . 30 , pp. 127-156 , 1846 , o r Werke , vol . 3 , p . 479 .

112. Pade . Su r l'extensio n de s propriete s de s r6duite s d'un e fonctio n aux fraction s d'interpolatio n d e Cauchy . Comp . Rend. , vol . 130, p . 697 , 1900 .

See also Nos. 95 , 99 .

For genera l work s upo n interpolatio n whic h brin g ou t th e relation o f th e subjec t t o continue d fractions , se e Heine' s Handbuch de r Kugelfunctionen , vol . 2 , an d MarkofF s Differ -enzenrechnung (deutsch e Uebersetzung) , chap . 1 , 6 , 7 ; als o th e following memoir :

113. Posse . Su r quelque s application s de s fraction s continue s alg6 -briques. Pp . 1-175 , 1886 .

114. Gauss . Methodu s nov a integraliu m valore s pe r approximatione m inveniendi. Werke , vol . 3 , pp . 165-196 , 1816 .

115. Christoffel . Uebe r di e Gaussisch e Quadratu r un d ein e Verallge -meinerung derselben. Jour , fu r Math. , vol. 55, pp. 61-82,1858 .

116. Mehler . Bemerkunge n zur Theorie der mechanischen Quadraturen . Ibid., vol . 63 , pp. 152-157 , 1864.

117. Posse . Su r le s quadratures . Nouvelle s Ann . d e Math. , ser . 2 , vol. 14 , pp . 49-62 , 1875 .

118. Stieltjes . (a ) Quelque s recherche s su r l a theori e de s quadrature s dites m^caniques . Ann . d e l 'Ec . Nor. , ser . 3 , vol. 1 , pp . 409 -426, 1884 .

We find her e th e origi n o f his notabl e 189 4 memoir, No . 26a . (a') Su r 1'evaluatio n approch^ e de s integrales . Comp . Rend. ,

vol. 97 , pp . 74 0 und 798 , 1883.

(b) Not e su r Y integrat e I f(x)G(x)dx. •/a

Nouv. Ann . d e Math. , ser . 3 , vol . 7 , pp . 161-171 , 1888 . 119. Markoff . Su r l a m^thod e d e Gaus s pou r l e calcu l approch e de s in -

tegrates. Math . Ann. , vol . 25 , pp. 427-432 , 1885 .


120. Pincherle . S u alcun e form e approssimat e pe r la rappresentazione di funzioni. Memoiri e della K. Accad. delle Scienz e dell'Istituto di Bologna , ser . 4 , vol . 10 , pp . 77-88 , 1889 .

121. Tchebychef . A brie f sketch o f the memoirs below will b e found o n pp. 17-2 0 o f Vassilief's memoi r o n "P . L . Tchebyche f e t so n oeuvre scientifique. "

(a) Sur les fractions continues. Jour , d e Math., ser. 2 , vol . 3 , pp . 289-323, 1858 , o r Oeuvres, vol . 1 , p . 203-230 .

(b) Su r une formuled'analyse . Bull . Phys . Math , de l'Acad. de s sciences d e St . P6tersbourg , vol. 13 , pp. 210-211,1854 , or Oeuv-res, vol . 1 , pp . 701-702 .

(c) Su r un e nouvell e serie . Ibid. , vol . 17 , pp . 257-261 , 1858 , o r Oeuvres, vol . 1 , pp . 381-384 .

(d) Su r V interpolation pa r la m6thode des moindres carr6s . Mem . de PAcad . de s science s d e St . Petersbourg , ser . 7 , vol . 1 , pp . 1-24, 1859 , o r Oeuvres, vol . 1 , pp . 473-498 .

(e) Su r le d&veloppement des fonctions k ui\e seule variable. Bull , de PAcad . imp . de s sciences de St . Peiersbourg , ser . 7 , vol . 1 , pp. J 94-199, 1860 , o r Oeuvres , vol . 1 , pp . 501-508 .

IX. MISCELLANEOUS .

122. Tchebychef . (a ) Su r les fraction s continue s algebriques. Jour , d e Math., ser . 2 , vol. 10 , pp. 353-358,1865 , o r Oeuvres, vol. 1 , pp. 611-614.

(b) Su r le d^veloppemen t de s fonction s e n series a l'aide de s frac -tions continues , 1866 . Oeuvres , vol . 1 , pp . 617-636 .

(c) Su r le s expression s approchees , lineare s pa r rappor t a deu x polynomes. Bull , de s sciences Math , e t Astron. , ser . 2 , vol . 1 , pp. 289 , 382 ; 1877 .

Hermite. (d) Su r une extensio n donne e a l a th6ori e des fraction s continues pa r M . Tchebychef . Jour , fu r Math. , vol . 88 , pp . 12-13, 1880 .

123. Tchebychef . (a ) Su r le s valeur s limite s des integrales. Jour , d e Math., ser . 2 , vol . 19 , pp . 157-160 , 1874 .

(&) Sur la representation de s valeurs limites des integrale s par des residus integraux (1885) . Acta . Math . vol . 9 , pp . 35-56 , 1887 .

Markoff. (c ) Demonstratio n d e certaines in6galit6s de M . Tcheby -chef. Math . Ann. , vol . 24 , pp . 172-178 , 1884 .

(d) Nouvelle s application s de s fractions continues . Math . Ann. , vol. 47 , pp . 579-597 , 1896 .

124. Laguerre . (a ) Su r l e developpemen t d e (x — z)m suivan t le s puis -sances d e (z 2 — 1). Comp . Kend. , vol . 8 6 (1878) , p . 956 , o r Oeuvres, vol . 1 , p . 315 .


(b) Su r l e d6veloppemen t d'un e fonctio n suivan t le s puissance s d'une polynome . Jour , fur Math. , vol. 88 (1880); i n particular , p. 37 , o r Oeuvres, vol . 1 , p . 298 .

(c) Sam e subject . Comp . Rend. , vol . 86 , (1878 ) p. 383 , or Oeuv-res, vol . 1 , p . 295 .

(d) Su r quelques th6or6mes d e M . Hermite . Extrai t d'un e lettr e addressee h M. Borchardt . Jour , fu r Math. , vol. 8 9 (1880), pp. 340-342, o r Oeuvres, vol . 1 , p . 360 .

125. Sylvester . Preuv e qu e TT n e peu t pa s 6tr e racin e d'un e Equatio n alg6brique k coefficient s entiers . Comp . Rend. , vol . I l l , pp . 866-871, 1890 .

A fundamenta l erro r i n th e proo f ha s bee n pointe d ou t b y Markoff. Se e p. 38 6 of vol. 3 0 of the Fortschritt e de r Math .

126. Gegenbauer . Uebe r di e Naherungsnenne r regulare r Kettenbriiche . Monatshefte fa r Math, un d Phys. , vol . 6 , pp . 209-219 , 1895 .

127. Bortolotti . Sull a rappresentazione approssimat a d i funzion i alge -briche per mezz o d i funzion i razionale . Att i dell a R . Accad . dei Lincei , ser . 5 , vol . 1 1? pp. 57-64 , 1899 .

ADDENDUM T O I A .

128. Euler . D e fractionibus continui s dissertatio . Comment . Petrop. , vol. 9 , p . 12 9 ff., 1737 .

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