COLLECTED MATHEMATICAL PAPERS
OF
HENRY J. S. SMITH
Bonbon
HENRY FROWDE
OXFORD UNIVERSITY PRESS WAREHOUSE
AMEN CORNER, E.C.
QUn> 2)orft
MACMILI.AN &: CO.. 66 FIFTH AVEXl'K
THE COLLECTED
MATHEMATICAL PAPERS
OF
HENRY JOHN STEPHEN SMITH
M.A., F.R.S.
LATE SAVILIAN PROFESSOR OF GEOMETRY IN THE UNIVERSITY OF OXFORD
EDITED BY
J. W. L. GLAISHER, Sc.D, F.R.S.
FELLOW OF TRINITY COLLEGE, CAMBRIDGE
WITH A MATHEMATICAL INTRODUCTION BY THE EDITOR, BIOGRAPHICAL SKETCHES
AND A PORTRAIT
IN TWO VOLUMES
VOLUME II
OXFORD
AT THE CLARENDON PRESS
1894
PRINTED AT THE CLARENDON PRESS
SI HORACE HART. PRINTER TO THE UNIVERSITY
cm
3
CONTENTS OF VOLUME II
PAGE
XXIII. Memoire sur Quelques Problemes Cubiques et Biquadratiques . . 1
Annali di Matematica, Ser. II. Vol. III. pp. 112-165, 218-242. M(3moire Couronne
par 1'Academie Royale des Sciences de Berlin, avec une moitiu du prix Steiner en
Tuillet 1868.
XXIV. Arithmetical Notes .67
Proceedings of the London Mathematical Society, Vol. IV. pp. 236-253. The three
papers which form these Notes were read on Jan. 9 and Feb. 13, 1878.
XXV. On the Integration of Discontinuous Functions 86
Proceedings of the London Mathematical Society, Vol. VI. pp. 140-158. Read
June 10, 1875.
XXVI. On the Higher Singularities of Plane Curves 101
Proceedings of the London Mathematical Society, Vol. VI. pp. 153-182. Read
June 10, 1875.
XXVII. Mathematical Notes 132
Proceedings of the London Mathematical Society, Vol. VII. pp. 287, 288. Read
Dec. 9, 1875. First printed in the Messenger of Mathematics, Vol. V. pp. 143, 144
(Jan. 1876).
XXVIII. Note on Continued Fractions 135
Messenger of Mathematics, Ser. II. Vol. VI. pp. 1-14 (May, 1876).
XXIX. Note on the Theory of the Pellian Equation, and of Binary Quadratic
Forms of a Positive Determinant 148
Proceedings of the London Mathematical Society, Vol. VII. pp. 199-208. Read
May 11, 1876.
VI CONTENTS
PAGE
XXX. On the Value of a Certain Arithmetical Determinant .... 161
Proceedings of the London Mathematical Society, Vol. VII. pp. 208-212. Read
May 11, 1876.
XXXI. On the Present State and Prospects of Some Branches of Pure Mathe-
matics ........... 166
Proceedings of the London Mathematical Society, Vol. VIII. pp. 6-29. Read
Nov. 9, 1876.
XXXII. On the Conditions of Perpendicularity in a Parallelepipedal System . 191
Proceedings of the London Mathematical Society, Vol. VIII. pp. 83-103. Read
Dec. 14, 1876.
XXXIII. On the Conditions of Perpendicularity in a Parallelepipedal System . 213
Philosophical Magazine, Ser. V. Vol. IV. pp. 18-25. Read before the Crystallological
Society, June 14, 1876.
XXXIV. Sur les Integrates Elliptiques Completes 221
Atti della R. Accademia dei Lincei. Transunti, Ser. III. Vol. I. pp. 42-44. Read
Jan. 7, 1877.
XXXV. Memoire sur les Equations Modulaires ...... 224
Atti della R. Accademia dei Lincei. Memorie della classe di Scienze fisiche, mate-
matiche e naturali. Ser. III. Vol. I. pp. 186-149. Read Feb. 4, 1877.
XXXVI. On the Singularities of the Modular Equations and Curves . . . 242
Proceedings of the London Mathematical Society, Vol. IX. pp. 242-272. Read
Feb. 14 and April 11, 1878.
XXXVII. Note on a Modular Equation of the Transformation of the Third Order 274
Proceedings of the London Mathematical Society, Vol. X. pp. 87-91. Read Feb. 18,
1879.
XXXVIII. Note on the Formula for the Multiplication of Four Theta Functions . 279
Proceedings of the London Mathematical Society, Vol. X. pp. 91-100. Read Feb. 13,
1879.
XXXIX. De Fractionibus Quibusdam Continuis ....... 287
Collectanea Mathematica (in memoriam Dominici Chelini), Milan, 1881, pp. 117-143.
The paper is^dated 1879.
XL. On some Discontinuous Series considered by Riemann .... 812
Messenger of Mathematics, Ser. II. Vol. XI. pp. 1-11 (May, 1881).
XLI. Notes on the Theory of Elliptic Transformation 321
Messenger of Mathematics, Ser. II. Vol. XII. pp. 49-99 (August-November, 1882).
OF VOLUME II Vli
PACE
XLII. Notes on the Theory of Elliptic Transformation 368
Messenger of Mathematics, Ser. II. Vol. XIII. pp. 1-54 (May-August, 1883).
XLIII. Memoir on the Theta and Omega Functions . . . . . . 415
XLIV. Memoire sur la Representation des Nombres par des Sommes de Cinq
Carres 623
From the Memoires presented par divers savants a I'AcadiSmie des Sciences de
1'Institut National de France, Vol. XXIX.
APPENDIX.
I. Address to the Mathematical and Physical Section of the British
Association at Bradford in 1873 681
II. Arithmetical Instruments ......... 691
III. Geometrical Instruments and Models 698
IV. Introduction to the Mathematical Papers of William Kingdon Clifford . 711
XXIII.
MEMOIRE SUR QUELQUES PROBLEMES CUBIQUES
ET BIQUADRATIQUES.
[Annali di Matematica, Ser. II. vol. iii. pp. 112-165, 218-242. M6moire Conronne par 1' Academic
Royale des Sciences de Berlin, avec une moiti6 du prix Steiner en Juillet 1868.]
UN admet qu'on peut r^soudre par de simples intersections de lignes droites
tout probleme de ge'ome'trie plane, qui n'a qu'une seule solution, et qui n'est pas
transcendental ; ou, pour nous exprimer avec plus de precision, tout probleme
dont la solution unique peut s'obtenir par des intersections de courbes geome-
triques d'un ordre quelconque fini. On admet de meme, que tout probleme quad-
ratique, qui n'est pas transcendental, peut se resoudre par des intersections de
lignes droites, et de sections coniques ; et, plus generalement, que tout probleme
qui n'a que n solutions, et qui n'est pas transcendental, peut se resoudre par des
intersections de droites, et de courbes de 1'ordre n. Et en effet, la verite de ces
theoremes parait decouler des premiers principes de 1'Algebre. Mais quand il
s'agit de probl ernes d'un degrd supe'rieur au second il se pre'sente un choix de
methodes, puisqu'on peut se servir, soit des intersections de droites par des
courbes dont 1'ordre n est egal au nombre de solutions du probleme, soit des
intersections de courbes dont 1'ordre est interme'diaire entre les deux nombres 1
et n 1. Ainsi, 1'on peut faire dependre la solution de tout probleme cubique ou
biquadratique, soit des intersections de courbes du troisieme ou du quatrieme
ordre par des droites, soit des intersections mutuelles de courbes du second ordre,
puisqu'on a la meme Evidence algebrique de la gene"ralit absolue des deux
VOL n. B
2 M^MOIRE SUB QUELQUES PROBLEMES
mdthodes. Or, c'est la derniere de ces deux m^thodes qui parait la plus simple,
et qui d e"te", & juste titre, pre'fdre'e par les gdometres. Ainsi, Ton a ramend la
recherche des points d'intersection d'une droite par une courbe du troisieme ou
du quatrieme ordre h, la recherche plus simple des points d'intersection de deux
coniques, tandis que personne, que nous sachions, n'a suivi la marche inverse, qui
k la ve"rit6 serait peu naturelle. II nous a done paru qu'il fallait avant tout
apporter quelque perfectionnement thdorique aux moyens dont on se sert pour
trouver les intersections de deux coniques donnees. Pour cela, nous avons cru
devoir imiter, autant que possible, les me"thodes dont on s'est servi si heureuse-
ment pour les problemes quadratiques. En effet, on sait que tout probleme, qui
n'admet que deux solutions, et qui n'est pas transcendental, peut etre re"solu par
les intersections de lignes droites, et d'une seule circonference de cercle, tracee
d'avance dans le plan. On doit ce beau r^sultat, base veritable de la partie
operative de la science, aux travaux des illustres fondateurs de la geome"trie
moderne ; il suffira de citer & cet e*gard le Traite des Proprietes Projectives de
M. Poncelet, et 1'ouvrage eldmentaire de Steiner, ayant pour titre : Die geome-
trischen Konstructionen ausgefilhrt mittelst der geraden Linie und eines fasten
Kreises. On suppose ordinairement que le centre du cercle tracd est donne, afin
d'avoir en meme temps la ligne droite a 1'infini, et les deux points imaginaires
conjugues oil cette droite est coupee par une circonference quelconque ; e"le"men ts
dont la connaissance est indispensable pour la solution d'un grand nombre de
problemes dont les donndes en dependent, soit explicitement, soit implicitement.
Nous ajouterons, qu'au lieu d'une circonference de cercle, on peut prendre pour
courbe auxiliaire une section conique quelconque completement de"crite, dont on
doit connaitre, non seulement le centre (ou deux diametres, si la courbe est para-
bolique) mais aussi le foyer.
En passant aux problemes d'un ordre plus e"leve, nous supposerons qu'une
seule section conique (qui d'ailleurs ne doit pas etre un cercle) soit completement
donne"e et ddcrite, et nous demontrerons qu'en se servant de cette courbe auxili-
aire on rdsout tous les problemes cubiques et biquadratiques avec la regie et le
compas seulement, en les ramenant, pour ainsi dire, dans les limites d^> la geome-
tric ele"mentaire. Nous ferons aussi remarquer que pour chaque probleme il
suffira de tracer une seule circonference de cercle, et qu'en supposant connus cinq
points seulement de la section conique, on peut remettre la description de cette
courbe jusqu'au moment oh Ton veut en determiner les points d'intersection par
le cercle qu'on doit tracer. II est bien entendu que, lorsqu'on veut ope"rer de
cette maniere, on doit supposer que la ligne droite & 1'infini, et les points cycliques
CTJBIQTTES ET BIQUADKATIQUES. 3
sur cette droite soient connus; c'est a dire que doivent etre donndes (1) un
parallelogramme quelconque, (2) deux angles droites, ou trois angles egaux
quelconques, ces angles dans les deux cas pouvant avoir ou le meme sommet, ou
des sommets differents, mais etant assujettis & ne pas former un parallelogramme.
Sans ces donnees, on ne pourrait determiner ni le centre du cercle, qu'on aura a
tracer, ni son rayon. La solution de chaque probleme cubique sera purement
lineaire, sauf le tracd du cercle ; mais pour les problemes biquadratiques il y aura
en general une construction quadratique, qu'on ne pourra eViter, mais qu'on
pourra effectuer, soit avant, soit apres le trace du cercle. Mais puisqu'au point
de vue pratique on abrege beaucoup d'operations lineaires en se servant d'une
section conique, nous n'insisterons point sur la linearite absolue des constructions,
et, le plus ordinairement, nous ferons usage en chaque probleme de la conique
auxiliaire des le commencement meme de la solution.
La methode que nous venons d'esquisser n'a rien de bien nouveau, puisqu'on
la trouve deja, comme resultat analytique, dans la Geometric de Descartes*.
Mais nous ne connaissons pas un seul probleme qui ait e"te" re"solu geometrique-
ment par cette me"thode. Les auteurs qui on traite de cette matiere, parmi
lesquels nous signalerons De la Hire, Maclaurin, et Joachimsthal f , nous parais-
sent ne pas avoir cherchd a de"montrer par des considerations de ge'ome'trie pure
la proposition si remarquable de Descartes, ni k s'en servir pour la solution ge"o-
metrique des problemes. Ainsi Maclaurin ne s'occupe-t-il guere que de la con-
struction gdomdtrique des racines des Equations algebriques ; De la Hire, qui
s'est propose* de trouver les normales a une section conique qui passent par un
point donne, n'a rattachd sa solution de ce probleme a aucune thdorie geome-
trique ; enfin, Joachimsthal, qui a donne une autre solution, d'une elegance admi-
rable, de ce meme probleme biquadratique, s'est encore servi de formules
analytiques, dont, comme il 1'avoue lui-m^me, sa construction geome"trique n'est
pas entierement affranchie. II restait done & trouver les liens qui rattachent la
proposition de Descartes aux theories modernes de la science, et a montrer le
parti qu'on peut en tirer pour les constructions actuelles de la ge"ometrie pure.
Nous nous empressons d'ajouter que dans ces developpements nous n'aurons qu'a
nous occuper de considerations tres ele"mentaires. En efFet, nous admettrons que
la solution de tout probleme cubique ou biquadratique peut se rdduire a la
determination des points d'intersection de deux coniques, qui ne sont pas de"crites,
mais dont chacune est determined par un nombre suffisant d'elements ; et nous
* Note I (p. 50). t Note II (p. 50).
B 2
4 MEMOIRE SUR QUELQUES PROBLEMES
ramfcnerons ce dernier probleme h, celui de trouver les points d'intersection d'une
circonference de cercle par la section conique de"crite d'avance, au moyen de
simples transformations homographiques ou correlatives, dont on pourra se servir
en plusieurs manieres difierentes.
Les problemes que nous aurons & rdsoudre, et surtout le probleme cubique
dont la solution est spe'cialement demandde par 1' Academic, impliquent ndces-
sairement des considerations relatives aux elements imaginaires. Nous jugeons
done k propos, avant de venir au sujet qui doit nous occuper principalement,
d'entrer dans quelques details sur cette matiere. Nous sommes loin de penser
que nous pourrions aj outer jl cet dgard quelque chose qui fut inconnue aux
geometres ; et nous n'avons aucune prevention de pouvoir edaircir la vraie nature
de ce phe'nomene singulier qu'on nomine I'imaginaire en gdomdtrie. Mais nous
avons remarque que les questions the"oriques qui concernent les imaginaires ont
ete traitees avec beaucoup plus de detail que les questions pratiques. On a bien
donne les moyens ndcessaires pour determiner les elements reels, qui dependent
pour ainsi dire imme'diatement d'elements imaginaires donnas. On sait, par
exemple, trouver les axes de symptose reels de deux couples de droites imagi-
naires conjugue'es, etc. ; mais il s'en faut beaucoup que de telles determinations
suffisent aux besoins actuels de la science. II parait resulter de considerations
algebriques tres elementaires que tout probleme qui peut se resoudre lorsq'on en
suppose les donne"es rdelles, devra egalement rester resoluble lorsqu'on substitue
deux elements imaginaires conjugues ^t deux elements quelconques reels qui
entrent symetriquement dans la question. Mais pour operer generalement cette
extension de la solution d'un probleme aux donnees reelles ^, la solution d'un
probleme aux donnees imaginaires, il faut souvent, du moins dans 1'etat actuel
de la science, isoler 1'un de 1'autre les deux elements d'un m6me couple d'ele-
ments imaginaires conjugues, afin de faire sur chacun d'eux une suite d'operations
plus ou moins longue, ayant pour resultat des couples d'elements imaginaires
conjugues, dont la combinaison donne enfin les elements reels qu'on cherche.
Cette isolation des elements imaginaires n'a rien d'absolu ; elle est relative & un
systeme de deux elements imaginaires conjugues arbitrairement ohoisis. En
effet, tant donnee une seule couple d'elements imaginaires conjugues, il parait
tout aussi impossible de distinguer entre ces elements que de distinguer entre
les racines de liquation
x 2 + 1 = 0.
Mais, de meme qu'en distinguant hypothetiquement entre les deux racines de
cette Equation (ce qu'on fait, par exemple, quand on les designe par +i et -t),
Art. 1.] CTJBIQUES ET BIQUADRATIQUES. 5
on parvient a distinguer en meme temps entre les deux racines imaginaires de
toute autre equation quadratique ; de meme, en cre'ant une fois pour toutes une
distinction fictive entre les deux elements d'une couple quelconque d'e'le'ments
imaginaires conjugues on arrive a etablir en m^me temps une distinction pareille
entre les deux elements de toute autre couple ; puis, en admettant cette isolation
relative, on peut ope"rer sur les elements imaginaires tout aussi bien que sur les
elements reels ; quoiqu'il faut avouer que, dans le cas des premiers, les operations
auxquelles on se trouve conduit sont presque toujours d'une longueur rebutante.
L'illustre et regrettable auteur de la Geametrie de Situation a savamment effectue
cette isolation hypothetique des elements imaginaires, en rattachant chaque ele-
ment d'une meme couple a 1'un des deux sens opposes qu'on peut observer en
toute formation ge'ome'trique qui contient des elements imaginaires. Mais cette
maniere de considerer la chose, quoique tres utile pour les developpements theo-
riques, nous semble se pre'ter aux constructions avec moins de simplicity que celle
que nous avons du prdferer pour notre but actuel.
D'apres ce qui vient d'etre dit, nous diviserons ce memoire en trois parties.
Dans la premiere nous traiterons des constructions dont les donne"es sont imagi-
naires ; dans la seconde nous ddmontrerons la the"oreme de Descartes ; dans la
troisieme nous etudierons divers problemes cubiques ou biquadratiques, et notam-
ment celui qui a ete signale par 1' Academic.
PREMIERE PARTIE.
1. Dans cette partie de notre travail nous ferons usage du mot grec dyade
pour exprimer 1'ensemble de deux elements conjugues imaginaires. Par chaque
dyade de points imaginaires il passe une seule droite re'elle ; elle sera pour nous
Yaxe de la dyade. Dans 1'espace, il existe, comme on sait, des dyades de droites
imaginaires qui ne se coupent pas, et qui ne passent par aucun point reel. Mais
les deux rayons d'une dyade de droites imaginaires, qui se trouvent dans un plan
reel, se coupent en un point qui est toujours reel, et que nous nommerons le
centre de la dyade de lignes droites. Pour plus de simplicity, nous conside-
rerons chaque dyade comme forme'e par les elements doubles d'une involution
reelle, dont les segments, ou les angles, empietent les uns sur les autres. Cette
definition d'une dyade aura toute la ge'ne'ralitd ne"cessaire, puisqu'on sait que
toute determination quadratique se re"duit, en derniere analyse, a la recherche
des elements doubles de deux systemes homographiques, elements qui sont en
me'me temps les elements doubles d'une involution, qu'on d^duit lineairement des
deux systemes homographiques, en prenant dans chacun des deux systemes,
6 MEMOIRE SUB QUELQUES PROBLEMES [Pt. I.
1'e'le'ment conjugu6 d'un mfime 6\6ment P, dont le conjugue harmonique par rap-
port a ces deux elements sera aussi le conjugue" dans 1'involution cherche"e. Nous
dirons qu'une dyade est donnce, quand on a deux couples d'eldments r6els de 1'in-
volution dont la dyade rdpresente les elements doubles.
Maintenant, si Ton prend une premiere dyade a r a 2 sur 1'axe A, et une
seconde dyade b t b 2 sur 1'axe B, different de A, qu'il coupe au point (A l} B^), les
deux couples de droites imaginaires a l b 1 , a, 2 b 2 , et a 1 b 2 , a z b 1} seront dvidemment
des dyades de droites. Soient P et Q les centres de ces dyades ; nous les appel-
lerons les centres d'homologie des dyades a, a 2 , ^ b 2 . On trouve line"airement la
droite PQ, en joignant les deux points A 2 , B 2 qui sont conjugues de (A 1} B^
dans les deux involutions qui determinent les dyades ia 2 , b 1 b 2 . Mais la con-
struction des points P, Q eux-memes est essentiellement quadratique. Soient
a i a 2> Pi ft* des couples de points appartenant aux deux involutions respectivement ;
les points P, Q seront les points doubles d'une involution, dont A 2 B 2 est une
premiere couple, et dont on trouve deux autres couples en prenant les intersec-
tions de PQ par les deux couples de droites o 1 |8 1 , a 2 ,8 2 et ajj8 2 , a 2 ^. Cependant,
on voit facilement que les segments de cette involution ne peuvent pas empieter
les uns sur les autres ; done les points PQ seront toujours re"els, comme on peut
voir a priori. L'un des points PQ etant donne, 1'autre s'en de"duit lineairement,
puisque PQ, A 2 B 2 sont quatre points harmoniques. Pour avoir une image nette
des deux dyades et de leurs deux centres d'homologie, on peut remarquer que
si ! a 2 est la dyade cy clique a Tinfini, Z>j b 2 est la dyade commune a un systeme
de cercles, ayant B pour axe radical, et PQ pour points limites. Nous ne nous
arreterons pas a la construction correlative des axes d'homologie, ou de symptose,
de deux dyades de droites imaginaires, qui n'ont pas le mdme centre.
En conside"rant deux dyades de points imaginaires, qui n'ont pas le meme
axe, comme homologiques par rapport a 1'un de leurs centres d'homologie, on
e"tablit une certaine correspondance entre les points des deux dyades, en telle
sorte que si 1'on ^change entr'eux les points imaginaires de 1'une, il faut en meme
temps ^changer entr'eux les points imaginaires de 1'autre. Puisqu'il y a deux
centres d'homologie, cette correspondance peut s'e"tablir de deux maiueres diffe-
rentes, qui se rapportent a ces deux centres respectivement. Ainsi nous dirons
que 1'homologie des deux dyades est donne"e par le centre P, ou par le centre Q,
selon que Ton a choisi le premier ou le second de ces deux centres pour etablir la
correspondance. Pour les dyades de droites imaginaires dont les centres sont
differents, on a une definition correlative.
Si Ton a deux dyades de mdme espbce, ayant le mdme centre, ou le meme
Art. 2.] CUBIQUES ET BIQUADRATIQUES. 7
axe, on doit etablir leur homologie d'une maniere indirecte, en prenant une troi-
sieme dyade, dont le centre ou 1'axe soit different, mais dont 1'homologie avec
chacune des deux premieres soit donnde.
L'homologie d'une dyade de points, et d'une dyade de droites, est donnee
imme'diatement, lorsque les droites de la seconde dyade passent par les points
correspondants de la premiere. En tout autre cas on etablit indirectement
1'homologie des deux dyades ; soit en determinant le centre d'homologie de la
premiere dyade, et de la dyade de points qui rdsulte de 1'intersection de la
seconde dyade par un axe quelconque reel, soit en determinant 1'axe d'homologie
de la seconde dyade et d'une dyade quelconque de droites imaginaires passant
par la premiere dyade *.
2. Quand on ne considere que deux dyades, independantes 1'une de 1'autre,
on peut choisir arbitrairement entre les deux manieres de determiner leur homo-
logie. Mais en considdrant un plus grand nombre de dyades, on voit qu'on peut
disposer comme on veut de 1'homologie de la seconde avec la premiere, de la troi-
sieme avec la premiere, et ainsi de suite, mais qu'alors 1'homologie de deux dyades
quelconques de cette serie sera completement ddterminee. Nous aurons done a
resoudre le probleme tres general, mais tres e"le"mentaire, que voici.
Etant donnee Fhomologie des dyades a l a 2 , \ b 2 , et aussi 1'homologie des
dyades a^ a 2 , Cj c 2 , trouver 1'homologie des dyades &i 6 2 , C T c 2 .
Ce probleme est toujours resoluble lineairement. II y a plusieurs cas a consi-
derer, mais pour abreger nous n'en considdrons que la moitie, d'oii Ton pourra
deduire la solution des autres en s'appuyant sur le principe de dualite.
I. Soient A, B, C les axes des trois dyades de points a l a z , \l>. 2 , CjC 2 ; nous
supposerons en premier lieu que ces droites soient toutes differentes entr'elles.
Soient Q et R les centres d'homologie de a^a 2> CjC 2 et de a l a 2 , ^b 2 respective-
ment. II s'agit de trouver P le centre d'homologie de b 1 b 2 , c t c 2 . On se rappelle
que si P* est le centre d'homologie opposd a P, la droite PP' peut se construire
lineairement f.
Premiere solution. Du point Q projetons les points de 1'axe A sur la droite
C, et du point R projetons les me"mes points sur la droite B. Soient a, a 2 deux
points conjugue"s de 1'involution determinant la dyade a 1 a 2 ; /3 X /3 2 et 71 7 2 les pro-
jections de ces points sur B et C respectivement. Les droites /8 1 7 1 , A>7 2 > e ^c.,
envelopperont une section conique, tangente aux deux droites B et C ; de plus,
les couples de tangentes, telles que ^71, J3 2 j 2 , formeront un systeme de tangentes
* Note III (p. 51). t Note IV (p. 54).
8 MEMOIRS SUE QUELQTJES PROBLEMES [Pt. I.
en involution de cette conique ; done les points d'intersection des deux tangentes
de chaque couple seront en ligne droite. Le pole de cette droite, par rapport a
la conique, est le point P cherchd. Pour le determiner, il suffira de trouver le
point de contact sur chacune des deux tangentes ft y lt /8 2 7 2 : la corde joignant
ces points ira couper PP au point P.
Si les trois axes A, B, C se coupent au meme point 0, les deux droites B, C,
qui sont deja homographiques par rapport aux points /3, /3 2 ..., y t 7 2 ..., deviendront
homologiques par rapport a ces memes points, puisque le point se correspondra
a lui-meme sur chacune des deux droites. Dans ce cas le centre d'homologie des
deux droites est pre'cise'ment le point cherche* P, qui se trouvera a 1'intersection
des droites PP ', QR. II y aurait aussi une simplification, si les trois droites QR,
B, C concouraient en un meme point.
Seconde solution. On determinera, comme nous avons fait dans la premiere
solution, deux divisions homographiques sur les droites B et C. L'axe * de ces
deux divisions coupera PP" an point P'. Ce point trouvd, la position du point
P s'en d6duira lindairement.
Cette solution se simplifie (1) si les divisions homographiques sur les deux
droites B et C deviennent homologiques, (2) si 1'axe de ces deux divisions se
confond avec PP'. En ce dernier cas, on dchange entr'eux, sur 1'un des deux
droites B et (7, les deux points conjuguds de chaque couple de Tin volution qui
determine la dyade que Ton considere sur cette droite. Alors les deux droites
deviennent homologiques, et leur centre d'homologie est le point P qu'on
cherche.
Si 1'axe A se confond avec 1'un des axes B et C, on pourrait encore trouver
1'homologie des dyades b l b 2 , c^c z , en faisant de Idgeres modifications dans les
solutions prdcedentes sans rien changer a leur principe. Mais on peut aussi
ope"rer de la maniere suivante. Soit Zj z 2 la dyade auxiliare qui etablit 1'homo-
logie de j 2 , &j 6 2 , dont on suppose les axes coincidents. L'homologie des dyades
a, a 2 , Zj z 2 sera donne"e, et aussi celle des dyades a x a 2 , c^ ; on en ddduira 1'homo-
logie des dyades z^, c l c 2 . Mais 1'homologie des dyades z\z t , &i& 2 est connue;
done on pourra trouver 1'homologie des dyades l> l 6 2 , c t c 2 . Cette soluuon suppose
seulement, que 1'axe de la dyade Zj z 2 ne se confonde pas avec C.
Si les trois axes A, B, C coincident, soit y l y 2 la dyade auxiliaire qui
etablit 1'homologie de a,a 2 , CjC 2 ; on trouvera par le cas precedent 1'homologie
' L'axe des deux divisions homographiques 0,0,..., /3j/3 2 ... est la droite lieu des points d'inter-
section des couples de droites telles que a,/3 2 , a a /3,.
Art. 3.] CUBIQTJES ET BIQUADRATIQUES. 9
de &! 6 2 , y l y 2 . Mais alors y 1 y. 2 sera une dyade auxiliaire qui etablira 1'homologie
de &j b. 2 , c\ c z-
II. Ce qui precede suffit pour tous les cas oh Ton ne considers que des
dyades de la meme espece : nous allons maintenant supposer que les trois dyades
a 1 a. 2 , &j&2, c 1 c. 2 soient d'especes differentes. Nous ne considererons que les deux
cas suivants, dans lesquels nous supposerons toujours que 1'homologie de a t a s ,
b l b 2 , et de a l a. 2 , c t c 2 , soit donnee, et que Ton cherche 1'homologie de b i b 2 , c l c 2 .
(i.) Soient a t a 2 , 6 X 6 2 des dyades de points, Cj c 2 une dyade de droites : soit
aussi y l y. 2 la dyade de points auxiliaires qui sert a etablir 1'homologie de a l a z ,
c l c i . Pour determiner 1'homologie de b^, c l c z , on n'aura qu'a trouver 1'homo-
logie de yiy. 2 , &i&2> ce qu'on pourra faire, puisqu'on connaitra 1'homologie de a x a a ,
&!& 2 > et aussi celle de a^a z , yiy. 2 .
(ii.) Soit ! 2 une dyade de droites, 6 X b. 2 , c t c 2 des dyades de points. De'-
signons par y l y. 2 , z l z 2 les dyades de points qui e'tablissent 1'homologie de a-^a^ avec
CjCjj et 6j& 3 respectivement. Dans la sdrie de dyades 6 X & 2 . ^z^, ^2/2, CjCa, on
connaitra 1'homologie de chacune avec celle qui la precede, done on pourra trouver
1'homologie de la premiere avec la derniere.
On peut dormer un dnonce" plus general du probleme de cet article.
'Soient !> b t b 2 , ..., xx 2 des dyades en nombre quelconque, de meme espece
ou d'especes differentes ; etant donnee 1'homologie de chacune d'elles avec celle qui
la suit immediatement, on peut trouver 1'homologie de la premiere avec la
derniere.'
3. Maintenant, prenons arbitrairement une dyade fixe, qui doit nous servir
comme terme de comparaison pour toutes les autres dyades que nous aurons a
consid^rer. Nous dirons qu'un e'le'ment imaginaire est donne, quand la dyade a
laquelle cet element appartient est donnee, 1'homologie de cette dyade avec la
dyade fixe etant aussi donnde. On voit que pour connaitre un element imagi-
naire, il faut connaitre (1) 1'axe, ou le centre, de la dyade a laquelle Telement
appartient, (2) deux couples d'e'le'ments reels de Finvolution qui determine cette
dyade, (3) 1'homologie de cette meme dyade avec la dyade fixe. II faut convenir
que de cette maniere on n'isole pas 1'un des deux elements imaginaires d'une
mfime dyade, et que, par consequent, 1'expression 'element imaginaire donne,'
n'est pas rigoureuse. Cependant on voit que si Ton ^change un element imagi-
naire donnd avec son conjugue", il faut en meme temps ^changer tout autre e'le'ment
donne avec son conjugue. Et c'est la tout ce qu'il faut pour qu'on puisse operer
avec des donndes imaginaires, de la meme maniere qu'avec des donndes reelles. En
effet, nous pourrons maintenant rdsoudre les deux problemes correlatifs que voici.
VOL. II. C
10 MEMOIRE SUR QUELQUES PROBLEMES [Pt. I.
Trouver le point $ inter section de deux droites donnees, dont I'une au moins
est supposee imaginaire.
Trouver la droite qui passe par deux points donnes, dont I'un au moins est
suppose imaginaire.
II suffira de considdrer le second de ces deux problemes. Solent done a 1} b lt
deux points imaginaires donnds, dont les axes ne se confondent pas. Puisque
les points a^, 6j sont donnds, les dyades a l a z , &i& 2 , auxquelles ils appartiennent,
sont aussi donndes, de me"me que 1'homologie de chacune d'elles avec la dyade
fixe. On peut done trouver lineairement 1'homologie de ces deux dyades. Soit
P leur centre d'homologie ; P sera le centre de la dyade k laquelle appartient la
droite cherchde (a 1} 6j). Les involutions qui correspondent aux dyades a l a 2 , l>J>z
ddterminent la m6me involution au point P; cette involution ddterminera la dyade
i, Pa. 2 b 2 . Enfin 1'homologie de cette dyade de droites avec chacune des dyades
!, a 2 6 2 est donnde immddiatement ; on peut done trouver son homologie avec
la dyade fixe ; ce qui acheve la determination de la droite (a 1 , 6j).
4. II rdsulte de la solution de ces deux problemes, que lorsqu'on peut
rdsoudre lindairement un probleme quelconque, dont les donnees ne contiennent
que des points et des droites rdelles, on pourra encore rdsoudre lineairement ce
meme probleme, apres qu'on aura substitud, en tout ou en partie, des droites et
des points imaginaires donnes, aux donnees reelles du probleme. On n'aura qu'^i
suivre de pas en pas la solution du probleme aux donnees reelles pour en conclure
la solution du probleme correspondant aux donndes imaginaires. Seulement,
puisque les de"terminations des droites joignant des points donne"s, et des points
d'intersection des droites donndes, qui sont des Postulata pour les cas rdels,
exigent des constructions ddtournees pour les cas imaginaires, on con9oit que la
mdthode, quoique parfaitement gendrale, doit conduire a des operations assez
longues.
Supposons qu'on ait re"solu lineairement un probleme dont les donnees soient
toutes rdelles ; si, parmi ces donndes, il y en a deux qui entrent symetriquement
dans la question, on pourra substituer une dyade donne'e & ces elements syme"-
triques, sans que le probleme cesse d'etre resoluble lineairement. En efiet, apres
cette substitution il n'y aura qu'une seule dyade inddpendante ; en la prenant
pour la dyade fixe, on trouvera lineairement 1'homologie de toute dyade derivee
qu'on aura & considdrer, et Ton n'aura qu'& opdrer sur des elements imaginaires
donnds. Mais il en serait autrement, si Ton voulait substituer & la fois plusieurs
dyades & la place d'un nombre dgal de couples de donnees reelles. Toutes ces
dyades seraient inddpendantes, et pour qu'on put appliquer la methode prdcd-
Art. 5.] CUBIQUES ET BIQUADRATIQUES. 11
dente, il faudrait qu'on en connut les centres, ou les axes d'homologie. Dans les
applications, on fera bien de determiner actuellement ces centres ou ces axes par
la construction que nous avons rappelee ci-dessus. Mais, puisque cette construc-
tion est quadratique, il importe de faire voir qu'on pourrait s'en passer & la
rigueur. Pour cela, il faut seulement qu'au lieu d'introduire toutes les dyades ii,
la fois, on les introduise 1'une apres 1'autre. Toutes les fois que Ton substituera
une seule dyade a deux donnees symetriques reelles, on aura un nouveau pro-
bleme, dont on pourra trouver la solution par ce qui precede. Mais on peut con-
siderer ce nouveau probleme comme n'impliquant que des donnees reelles, puisque
la dyade que nous y avons fait entrer, peut 6tre remplacee par deux couples de
1'involution qui la determine. On pourra done substituer une seconds dyade a
une seconde couple de donnees reelles, et ainsi de suite, sans qu'on ait besoin
d'aucune construction quadratique.
5. Nous allons maintenant indiquer quelques problemes dont on puisse trou-
ver les solutions par les principes precedents. Tout element, qui n'est pas dit
expressement etre reel, pourra etre imaginaire.
(i.) JZtant donnes trois points d' une' meme droite, ou trois droites passant par
un meme point, trouver le conjugue harmonique d'un de ces elements, par rapport
aux deux autres.
Soient 1; & u c x des points donnes d'une me'me droite ta l} et proposons nous
de trouver le point d l} conjugue" harmonique de c lt par rapport a a^. Soient
A, B, C les axes dyades a 1 a 2 , &i& 2 , c i^ 2 ; , /3, 7, les trois sommets du triangle
ABC. La droite A sera 1'axe d'homologie des dyades (aa lt aa 2 ) et a^ 2 ; pareille-
ment, B sera 1'axe d'homologie des dyades (fib,., /3& 2 ) et w^; on pourra done
determiner 1'axe d'homologie X des dyades (aa : , aa 2 ), (^b lt /86 2 ) ; soit x 1 x 2 la
dyade de points determined sur X par ces deux dyades, et D 1'axe d'homologie
des dyades (yx lf yx.^ et w 1 w 2 ; D sera 1'axe de la dyade cherchee d^dz, dont
1'homologie avec <a l u> 2 sera ^videmment connue.
(ii.) JZtant donnes, dans deux series homographiques, trois elements de Tune,
et les trois elements correspondants de 1'autre, trouver I' element de I'une des deux
series qui correspond a un element quelconque de 1'autre.
On resoudra line"airement ce probleme general, en imitant, de la maniere que
nous avons explique"e, les solutions qu'on a donnees du probleme dans le cas par-
ticulier ou les donne"es sont r^elles. Le principe de ces solutions (auxquelles on
peut donner des formes tres varides) consiste, comme on sait, a trouver une troi-
sieme serie qui soit homologique avec chacune des deux series donne"es.
On aura en meme temps la solution du probleme important,
C 2
12 MEMOIRE SUB QUELQUES PROBLEMES [it. I.
Btant donnes cinq Elements de m&me espece qui ddterminent une conique,
trouver autant d' elements de cette courbe qu'on voudra,
qui est au fond le meme que le pre"ce"dent.
II y a deux cas tres particuliers qui meritent quelque attention.
(1.) fitant donnees deux dyades de droites c^Og, b^ ayant le meme centre P,
et deux autres dyades a^ a 2 , fa fa, ayant le meme centre IT, et verifiant I' equation *
n. [ t a 2 , fa &]= p. [a, 2 , &A]
trouver la droite d'un des deux faisceaux qui correspond a une droite donnfa de
Vautre faisceau.
Pour que ce probleme n'admette qu'une seule solution il faut qu'on connaisse
1'axe d'homologie A des dyades a l a 2 , a x a 2 , et 1'axe d'homologie B des dyades 6 t 6 2 ,
fa fa. II est vrai que 1'un des deux axes dtant donne", on pourrait trouver 1'autre
line"airement, mais, pour abre"ger, nous supprimerons la demonstration de cette
assertion, et nous supposerons que les axes A et B soient tous les deux donnes f-
Soient p 1 p i , q q. 2 les dyades qu'on aura sur les droites A et B. En faisant passer
une conique par le points p l p 2 , qiq%, PII on ramenera le probleme & celui de
determiner autant de points qu'on voudra d'une conique dont on connait deux
dyades et deux points re"els. Mais, comme on sait, le theoreme de Carnot con-
duit a une solution line"aire de ce dernier probleme, mme en supposant qu'un
seul point reel de la conique soit donne.
(2.) Par un point P d'une conique dont on connait cinq points reels il passe
une droite imaginaire donnee ; trouver le second point ^intersection de cette droite
par la conique.
Soit (1) P un point reel. On sait trouver 1'axe de sympt6se reel de la
conique et. de la dyade h, laquelle appartient la droite imaginaire : le point d'in-
tersection de cet axe par la droite imaginaire donnee sera le point cherche'.
Soit (2) P un point imaginaire. L'axe de la dyade & laquelle ce point
appartient sera 1'un des axes de symptose de la conique et de la dyade h, laquelle
appartient la droite imaginaire donne"e. L'autre axe se trouvera line"airement et
fera connaitre la solution du probleme.
(iii.) Etant donn6 le trace complet d'une conique reelle, trouver les points d' in-
tersection d'une droite Wj par une conique C, reelle ou imaginaire, dont on donnait
cinq points.
Soit C reelle. On trouvera par une construction quadratique connue, pour
* Voir la note Art. 3, troisi&ine partie (p. 36). t Note V (p. 54).
Art. 5.] CUBIQUES ET BIQTJADRATIQUES. 13
laquelle on se servira de la conique tracee, les deux axes de sympt6se de C et de
la dyade de droites o^ w 2 ; les points d'intersection de ces deux axes par <o 1 seront
les points cherche's.
Soit C imaginaire. On ramenera la question a la recherche des points
doubles de deux divisions homographiques sur une m^me droite. On projetera
ces deux divisions sur la conique tracee ; et Ton cherchera 1'axe des deux divi-
sions projetees. Get axe sera en general imaginaire ; on determinera les points
oil il est coupe par la conique tracee ; ces points feront connaitre la solution du
probleme.
(iv.) Etant donnes cinq points imaginaires, trouver des coniques reelles
appartenant aufaisceau qui contient les coniques imaginaires determ inees par ces
cinq points, et par leurs cinq points conjugues.
Soit P un point reel, et proposons-nous de determiner la conique du faisceau
qui passe par le point P.
II importe d'observer que si 1'homologie des cinq dyades dtait inconnue
(hypothese que nous excluons, puisque nous supposons que les cinq points sont
donnes) il y aurait seize solutions correspondant a chaque point P, puisque les
cinq dyades determinent trente deux coniques dont chacune passe par 1'un de
deux points de chaque dyade. Cette remarque montre clairement 1'importance
qu'on doit attacher a 1'homologie des dyades.
Pour trouver la conique passant par P, on prendra un point quelconque rdel
q, et Ton en determinera la polaire relativement a 1'une des coniques imaginaires.
Cette de* termination sera toujours possible puisqu'elle n'exige que des operations
line"aires que nous avons de"ja enseigne* a faire. Soit Q le centre de la dyade a
laquelle appartient la polaire de q ; toute conique du faisceau divisera harmoni-
quement le segment re"el qQ. En prenant quatre points q lt q 2 , q 3 , q^, et leurs
quatre points reciproques Q lt Q z , Q 3 , Q t , on aura la solution du probleme, puis-
qu'on sait construire la conique qui passe par un point r^el donne, et coupe har-
moniquement quatre segments reels.
On resoudrait de la meme maniere le probleme suivant.
Etant donn&s deux systemes polaires reels, dont les coniques peuvent etre
imaginaires, trouver la conique qui passe par un point reel donne, et qui appar-
tient aufaisceau determine par les deux coniques.
Nous ferons remarquer a cette occasion qu'il y a deux especes diffe'rentes de
coniques imaginaires qu'il importe de distinguer entr'elles. Une conique imagi-
naire de la premiere espece n'a aucun point re"el ; elle est coupee par toute trans-
versale re"elle en deux points imaginaires conjugues ; elle a un systeme polaire
14 MEMOTEE SUB, QTTELQUES PROBlJlMES [Pt. I.
r^el. Une conique imaginaire de la seconds espece a une base forme'e, soit de
quatre points reels, soit de deux points re'els et d'une dyade, soit de deux dyades.
Elle coupe toute transversale rdelle (autre que celles qui joignent deux des points
de la base) en deux points qui ne sont ni tous les deux re'els, ni des points ima-
ginaires conjugue's. La polaire de tout p61e reel qui n'est pas un sommet du
triangle harmonique de la base, est imaginaire.
(v.) fitant donnees deux dyades a^a^, ft/Jjj, ayant le meme axe, et deux
autres dyades % a 2 , &! & 2 , ayant des axes A et B differents Tun de I'autre ; trouver
la conique 2 qui passe par les deux dyades a t a 2 , b^, et qui satisfait a I' equation
[!, a. 2 , &!, 6 2 ] = [(*!, a 2 , &, @ 2 ].
Pour que ce probleme soit line'aire il faut que 1'homologie des dyades c^ a 2 ,
a 1 a 2 , et aussi celle des dyades /?!&, &!& 2 , soient donndes. On prendra le centre
d'homologie de c^a.,, a 1 a 2 ; de ce centre on projetera ft/^ sur A ; et Ton de'ter-
minera le centre d'homologie de \b 2 , et de la projection de ft^ sur A. Ce
dernier centre sera un point de la conique cherche'e 2, qui sera des lors complete-
ment de'terminee.
Si Ton supposait que les dyades a^a z , jSj/Sg, n'ont pas le me'me axe, mais
qu'elles appartiennent a une conique reelle F, 1'expression [a^j, ^/3 2 ] etant
relative a cette courbe, on pourrait encore determiner la conique S par la con-
struction pre"ce"dente. On n'aurait qu'a projeter les dyades a^a z , ^^ z sur une
droite L, en prenant pour centre de projection un point quelconque de F.
On voit que les deux coniques F et S seront homographiques par rapport
aux points a^, ft^, et a l a 2 , &j6 2 . Soit un point donnd re"el de F; on trou-
vera de la maniere suivante le point correspondant x de la conique 2. Soient
a i2) Piflz, ' les projections de a^a^, faftz, sur la droite L: du centre d'homo-
logie des dyades 04 a 2 , a 1 a 2 on projetera /Qi/3 2 , ^' sur A : soient fi[, /8 2 , ^" les pro-
jections de ces points. On trouvera le centre d'homologie p des dyades ftlft'',,
&! 6 2 ; ce centre appartiendra a la conique 2, et le point x cherch^ sera le second
point d'intersection de la droite /o" par 2.
Si, au lieu d'un point re*el , une dyade i 2 > appartenant a F, etait donne'e,
on trouverait x l x 2 , la dyade correspondante de 2, par une construction toute
semblable. On aurait sur la droite A une dyade i' 2 , dont 1'homologie avec 3
serait connue ; puis on determinerait 1'axe de symptose de 2 et de la dyade des
droites pl p 2 ' ; ce qui suffirait pour faire connaitre la dyade x 1 x. 2 , et 1'homologie
de cette dyade avec la dyade donne'e j 2 .
On peut e*noncer le probleme que nous venons de resoudre de la maniere
suivante.
Art. 5.] CUBIQUES ET BIQUADRATIQUES. 15
jfitant donnees deux dyades d'une premiere figure correspondent a deux
dyades d'une seconde figure, transformer la premiere figure homographiquement
en la seconde.
(vi.) fitant donnees deux dyades a 1 a 2 , b^, ay ant des axes differ ents A et B,
et quatre points 1, 2, 3, 4 formant la base d'un faisceau de coniques reelles, trou-
ver la conique qui passe par les deux dyades a^a^, 6 X 6 2 et qui satisfait a I' equa-
tion
(1, 2, 3, 4) [!, 2 , &j, & 2 ] = 0i, 2 , &x, 6 2 ].
Soit F une conique quelconque passant par les deux dyades a l a 2 , fr^.
Qu'on prenne un point reel 7 de cette conique et que de ce point on projette sur
la conique les involutions qui sont determinees par le faisceau (1, 2, 3, 4) sur les
droites A, B. Soient p et q les poles des deux involutions qu'on aura maintenant
sur la conique ; les droites des deux faisceaux (p) et (q) correspondront anhar-
moniquement aux coniques du faisceau (1, 2, 3, 4) : done ces deux faisceaux de
droites seront homographiques ; soit C la conique, lieu des points d'intersection
des rayons correspondants. Qu'on determine les axes de symptose des dyades
(pa 1 , pci^), (qbi, qb 2 ) avec la conique C ; soient a^, &/3.J les points d'intersection
de ces axes par C ; les points p et q seront les centres d'homologie des dyades
a 1 a 2 , ajOa et b^, fiifi.} respectivement. Dans le faisceau (p) les droites imagi-
naires pa 1; pa% correspondront aux coniques (1, 2, 3, 4, a^, (1, 2, 3, 4, 2 ) ; de
meme dans le faisceau (q) les droites imaginaires qb lt qb. 2 correspondront aux
coniques (1, 2, 3, 4, fcj), (1, 2, 3, 4, 6 2 ). Done le rapport anharmonique des quatre
points a lt a 2 , /8 1} |8 2 appartenant a la conique C, sera dgal au rapport anhar-
monique
(1, 2, 3, 4) . [a l5 2 , & u 6J.
On ddterminera la conique qui passe par les deux dyades a lt a 2 , b 1} 6 2 , et qui
satisfait k liquation
[!, 2 , &!, 6J = [ai, a 2 , @ 1} j8j ;
cette conique sera celle qu'on cherche.
Cette solution est purement lindaire, puisqu'on ne trace pas les coniques dont
il y est question. On pourra 1'abreger en operant de la maniere suivante. Soit
u> le point d'intersection des droites A et B ; %, T\ les points ou ces memes droites
coupent pour la seconde fois la conique (1, 2, 3, 4, &>). Soit y un point quelcon-
que de la droite fy ; en prenant pour F la conique (a 1} a 2 , b lt & 2 , 7) les faisceaux
(p) et (q) deviendront homologiques, et les dyades a 1 a 2 , /3 1( 8 2 seront connues, des
que Ton aura determine' 1'axe d'homologie de ces deux faisceaux.
16 MEMOIRS SUE QUELQUES PROBLEMES [Pt. I.
Si Ton se permettait une construction quadratique, on pourrait encore rac-
courcir la solution. On trouverait un centre d'horaologie des deux involutions
de'termine'es par le faisceau (1, 2, 3, 4) sur les axes A et B ; en prenant ce centre
pour le point y, les deux points p et q viendraient se confondre en un seul point,
qui appartiendrait a la conique cherche'e. A la ve'rite', cette me"thode ne serait
pas applicable, si les centres d'homologie des deux involutions dtaient imagi-
naires.
Nous ajouterons quelques corollaires de ce probleme, qui nous seront utiles
plus tard.
(1.) Jlltant donnees quatre dyades a^a^, &!& 2 , CjC 2 , d^d^, et un point reel p,
trouver le point a> oppose d deux de ces dyades c^ c 2 , d^, par rapport a la courbe
cubique (a^, 2 , b lt 6 2 , c x , c 2 , d lf d z ,p).
On ddterminera la conique 2 qui passe par les points a 1} a a , 6 1( 6 2 , et qui
satisfait a 1'equation
(cj, c 2 , d lt d z ) . [!, a. 2> b lt & 2 ] = [i, 2 , \, 6J.
Ensuite on trouvera le point p' de cette conique pour lequel on a
(c lf c 2 , d lt d 2 ) [a n a 2 , b l9 & 2 , p] = [a lt a 2 , 6^ 6 2 , p'].
Pour cela, on remarquera que dans la solution gene"rale prdcedente (ou Ton
peut remplacer les points 1, 2, 3, 4 par les points c lt c 2 , d 1} d 2 ) on trouve imme-
diatement le point TT de la conique auxiliaire C, qui correspond a la conique
(cj, c 2 , d lt d 2 , p). Mais ce point trouve', on en de"duira le point p' par une con-
struction qui a 6t6 deja indiqude (v). Enfin, le point oil la droite pp' rencontre
pour la seconde fois la conique (a 1( a 2 , 6^ 6 2 , p') sera le point w cherchd.
Ainsi on pourra trouver lindairement autant de points qu'on voudra d'une
cubique dont on ne connaltra que quatre dyades de points imaginaires, et un seul
point rdel. Ce cas nous parait avoir 6t6 omis par les auteurs qui ont traitd des
constructions des courbes cubiques.
Si, au lieu d'un point rdel p, une dyade p l p 2 de points imaginaires e"tait
donne"e, on pourrait encore trouver la dyade ta l 2 , dont les dldments sont les points
opposes au systeme des quatre points c lt c 2 , d lt d%, par rapport aux ueux courbes
cubiques imaginaires
(!, a 2 , 6 lf & 2 , CL c 2 , d lt d z , PJ) et (a lt a 2 , b lt 6 2 , c^ c 2 , d lt d it p 2 ) *.
On commencera par trouver la dyade TJ^, qui appartient a la conique auxiliaire
* Note VI (p. 55).
Art. 1.] CTTBIQUES ET BIQUADBATIQTTES. 17
C, et dont les Elements correspondent aux coniques imaginaires (c l9 c 2) d it d z , p^
et (Cj, c 2 , d lt d. 2 , p 2 ). Le centre d'homologie de cette dyade et de p 1 p a sera un
point connu. Puis on determinera la dyade p\p' z , qui appartient a la conique 2,
et qui verifie 1'equation
(c lt c 2 , d 1} d 2 ).[a lt a 2 , b lt \, p lt p^\ = \a ly a 2 , b lf b 2 , p\, p.^].
Soit P le centre d'homologie des dyades p'ip' 2 , Pip 2 - Un des axes de symp-
tose de la dyade de droites (Ppi, PpJ), et de la conique 2, sera 1'axe de la dyade
p'ip' 2 ; 1'autre pourra se determiner line'airement ; les points d'intersection des
droites Pp lt Pp 2 par ce second axe de sympt6se seront les points a^wg cherchds.
(2.) Etant donnees quatre dyades de points imaginaires a a z , bi b 2> c : c z , d^,
trouver le neuvieme point 6, appartenant a toute courbe cubique qui passe par
les quatre dyades.
C'est ce qui se fera par une construction connue. Soient x, y deux points
reels; soient , *i les points opposes au systems c^, d t d z par rapport aux deux
courbes (a^ a 2 , b lt 6 2 , c lf c 2 , d lt d 2 , x) et (o%, a 2 , 6 1( & 2 , c 1( c 2 , d t , d % , y) respectivement,
^', 17' les points opposes au systeme 0^0,^, bj) 2 par rapport a ces monies courbes ;
les droites ', w se couperont au point 6.
SECONDE PAKTIE.
1. Soit 2 une section conique, qui ne soit ni dvanescente ni un cercle, et que
nous supposerons completement trac^e ; nous reviendrons plus tard sur le cas ou
Ton n'aurait qu'une partie de la courbe. Toutes les autres coniques dont nous
aurons k nous occuper ne seront point trace"es-(& moins qu'elles ne soient des
cercles, ou des couples de lignes droites) ; elles seront seulement de"finies par un
nombre suffisant de conditions. Voici le probleme que nous aurons k r^soudre.
' Etant donnees deux coniques S lt S. 2 , trouver leurs quatre points d'intersec-
tion, ou, ce qui revient au meme, leur triangle harmonique commun, avec la regie
et le compos seulement, mais en se servant de la conique tracee 2.'
Les coniques S 1} S. 2 peuvent 6tre imaginaires de premiere espece, ou imagi-
naires de seconde espece et conjugudes 1'une a 1'autre. Mais puisque, dans ces
deux cas, on peut construire autant de points qu'on voudra sur les coniques
reelles du faisceau (S l , S 2 ), nous pourrons supposer, sans perte de gdndralit^, que
S lt S 2 soient elles-m^mes rdelles.
II y a trois rdseaux de coniques qu'on est conduit naturellement &, considerer
relativement au faisceau (S lt S 2 ), ou plutot relativement k son triangle harmo-
nique afiy ; ce sont (1) le reseau des coniques circonscrites au triangle a/3y, (2) le
VOL. II. D
18 MEMOIRE SUR QUELQFES PROBLEMES [rt. n.
reseau des coniques inscrites a ce triangle, et (3) le reseau harmonique, c'est a
dire le reseau dea coniques dont ce meTne triangle est un triangle harmonique.
Une conique du premier reseau est de'termine'e par deux points ; une conique du
second re"seau par deux tangentes ; une conique du troisieme rdseau, soit par
deux points, soit par deux tangentes. Le second reseau contient quatre cercles,
dont la construction est dejk un probleme biquadratique. Mais chacun des autres
re"seaux ne contient qu'un seul cercle ; ce sont le cercle circonscrit, et le cercle
polaire, du triangle 0/87. On peut se servir de 1'un ou de 1'autre de ces deux
cercles pour resoudre notre probleme, et Ton est ainsi conduit a deux methodes
de solution diffe'rentes. D'apres la premiere methode on transforme homographi-
quement une conique quelconque du premier reseau dans la conique 2, ou bien on
transforme correlativement une conique du second re"seau en cette m6me conique.
D'apres la seconde methode, c'est sur une conique du troisieme re"seau que Ton
opere, et la transformation peut 6tre de 1'une ou de 1'autre espece. Dans tous
les cas on cherche le cercle unique du reseau transform^ : les intersections de ce
cercle avec la conique 2 font connaitre la solution du probleme.
2. Avant de donner les details ne"cessaires sur ces deux me"thodes, nous
allons resoudre quelques problemes preliminaires, qui se rapportent aux coniques
des trois re"seaux : surtout, nous ferons voir comment on peut construire, dans
tous les cas possibles, les cercles du premier et du troisieme re"seau.
Les coniques du re"seau circonscrit sont precisdment les coniques rdciproques
de droites, qui ont e'te considered par MM. Poncelet et Chasles. Ainsi chaque
conique K de ce rdseau est le lieu des points qui sont re"ciproques des points d'une
certaine droite k relativement au faisceau (Si, S 2 ) ; K est aussi le lieu des p61es
de k, relativement aux coniques du faisceau (Si, $ 2 ) ; enfin, 1'ensemble des lignes
K, k est la courbe Jacobienne du rdseau determine" par Si, S. z et la droite k, prise
deux fois. Pour construire la conique du rdseau circonscrit qui passe par deux
points pip a , on construit les points PiP 2 re"ciproques de Pip 2 ; la conique re"ci-
proque de la droite P l P z est la conique cherchde. De m^me, on pourra trouver
la conique du rdseau circonscrit qui soit tangente en un point donne" p k une
droite L. On prendra la conique reciproque de L ; sa tangente, au point re"ci-
proque de p, sera rdciproque de la conique cherchde. Pour les coniques du
second reseau on a des constructions correlatives ; mais la construction de la
conique or appartenant au reseau harmonique, et passant par deux points pip^,
est un peu moins facile. Soient Xi x z , y t y% deux couples de points qui divisent
harmoniquement le segment p r p 3 ; construisons les points r^ciproques X X 2 , YI Y 2 ,
et menons les droites X l X%, F a F a se coupant au point q. Soient encore abc les
Art. 2.] CTJBIQTJES ET BIQUADBATIQUES. 19
points ou la droite pp z rencontre les cotes du triangle 0/87 ; et ddsignons par
a'b'c' les points conjugues harmoniques de dbc par rapport a pip z . En consi-
derant Tin volution dont les points doubles sont p 1 p 2 , on voit que le quatrieme
point commun aux coniques (a, ft 7, x lt x 2 ), (a, ft 7, y lt y^, (c'est-a-dire le point Q
reciproque de q) est le point d'intersection des trois droites a a', fib', yc; mais ces
trois droites sont eVidemment les polaires de a, b, c, relativement k o- ; done le
point Q reciproque de q est le p61e de la droite p p 2 , relativement a cette coni-
que. Soit (a, ft 7, x, y) une conique du re"seau circonscrit qui passe par deux
points quelconques xy de la droite j>, p 2 ; cette conique est aussi une conique du
reseau circonscrit appartenant au faisceau (*Sj , o-) ; par consequent les points re*ci-
proques des points de (a, ft 7, x, y), considers relativement au faisceau (S lt a-),
seront en ligne droite. Pour avoir cette droite, on determinera les points XY,
re"ciproques de xy, qu'on sait trouver, puisqu'on connait les polaires de x, y relati-
vement aux coniques a- et S t . Soit 6 un point quelconque de (a, ft 7, x, y) ; les
polaires de 6, relativement a <r et S ly se croisent sur la droite XY; on connait
aussi le point ou la polaire de relativement k a- coupe la droite p l p 2 ; on pourra
done construire cette polaire, qui determinera completement la conique a-. On ne
doit pas prendre pour xy les points doubles de 1'involution determinde par (Si, d)
eur la droite p 1 p 2 ; ce qui ferait coincider la droite XY avec Pip 2 ; de mdme, on
ne doit pas prendre pour 6 le point re*ciproque du point d'intersection de XY,
PiP*-
De ce que nous avons dit on tire la construction suivante pour les deux
cercles. Soient x 1 x 2 , y-^y^ deux couples de points rectangulaires a l'infini ; X l X 2 ,
Y t Y 2 leurs points rtciproques relativement au faisceau (S lt S 2 ). Le centre Q du
cercle polaire est le point re*ciproque de 1'intersection q de X l X 2 , Y^Y^. Soient
r^ r z les deux autres points diagonaux du quadrilatere X l X 2 , Fj Y 2 . Le cercle
circonscrit est rdciproque de la droite r t r 2 ; de plus les points ^R^, rdciproques
de r^r z , sont les extr^mites oppos^es d'un meme diametre de ce cercle. En effet,
la droite ^ r 2 coupe la conique C, re"ciproque de la droite a l'infini, en deux points
imaginaires Q t Q 2 , qui sont re"ciproques des deux points wj > 2 de la dyade cyclique.
Car les points d'une conique re"ciproque d'une droite correspondent anharmoni-
quement aux points de cette droite ; de sorte qu'aux points doubles de 1'involu-
tion x 1 x 2 , y^y z sur la droite a l'infini correspondent les points doubles de 1'involu-
tion X l X 2 , FjF 2 sur la conique C. De plus, les deux points r t r 2 sont harmoni-
quement conj ague's aux points fij Q 2 ; done, sur le cercle re"ciproque de la droite
r 1 r 2 , les points ^ R 2 sont harmoniquement conjugu^s a wj a> 2 ; ce qui veut dire
que R l R 2 est un diamHre du cercle. On connait done le cercle circonscrit ;
D 2
20 MEMOIRE SUB QUELQUES PROBLEMES [pt. n.
quant au cercle polaire on achevera sa determination, soit en observant qu'il
coupe orthogonalement le cercle directeur du cercle circonscrit, soit en trouvant
par la me'thode que nous avons indiqude la polaire d'un point quelconque. Dana
les applications, on pourra toujours remplacer la conique non trace"e S, par la
conique tracee 2. On prendra pour x, y dans la construction pre'ce'dente les
points x 1 x t dont on s'est dej& servi, et dont on aura construit les polaires relati-
vement & 2. On abaissera de Q des perpendiculaires sur ces polaires ; la droite
qui joindra leurs pieds sera rdciproque de 1'hyperbole e*quilatere (a, ft <y, x lt x 2 )
par rapport au faisceau (2, a). Soit 6 un point quelconque de 1'hyperbole, ff le
point d'intersection de la droite re"ciproque de cette courbe par la polaire de
relativeraent a 2; la perpendiculaire abaissee de ff sur Q& sera la polaire de 6
relativement au cercle polaire. On arriverait aussi a des constructions assez
simples, en prenant pour xy, soit les deux points k 1'infini sur 2, soit les deux
points de la dyade cyclique. II est inutile d'aj outer qu'on ne peut determiner
lineairement aucun point du cercle polaire, ce cercle pouvant e"tre imaginaire.
On sait que le cercle circonscrit k un triangle hannonique de la conique S
coupe orthogonalement le cercle directeur de S; et que le cercle polaire d'un
triangle circonscrit k S coupe aussi orthogonalement ce me"me cercle directeur.
On pourrait done trouver le cercle circonscrit en construisant les cercles direc-
teurs de trois coniques quelconque du re"seau harmonique, et le cercle polaire en
construisant les cercles directeurs de trois coniques quelconques du re"seau in-
scrit. On aura aussi les rdsultats particuliers que voici, dont la plupart etaient
connus :
' Le lieu des centres des hyperboles dquilateres du rdseau circonscrit, est le
"cercle des neuf points " du triangle harmonique fondamental.
' Le h'eu des centres des hyperboles ^quilateres du re"seau harmonique est le
cercle circonscrit. Ces hyperboles se coupent aux quatre centres des cercles
inscrits au triangle harmonique.
' Le lieu des foyers des paraboles du re*seau inscrit est le cercle circonscrit.
' Les droites directrices des paraboles du rdseau inscrit se coupent au centre
du cercle polaire.
' Les droites directrices des paraboles du re*seau harmonique se coupent au
centre du cercle circonscrit.'
3. Ce qui precede suffit pour notre but actuel; cependant nous croyons
devoir aj outer quelques observations e'le'mentaires qui pourraient etre utiles en
d'autres occasions. On peut conside"rer les sommets du triangle harmonique
commun it deux coniques comme repre"sentant d'une certaine maniere les trois
Art. 3.] CUBIQUES ET BIQUADRATIQUES. 21
racines d'une equation cubique. Ainsi au probleme analytique 'Trouver les
fonctions symetriques des racines d'une Equation cubique ' correspondra le pro-
bleme geometrique ' Trouver les lieux et les enveloppes qui dependent syme'trique-
ment des trois elements du triangle harmonique commun a deux coniques non
tracees.' D'apres ce qui a e"te dit, on aura des moyens assez simples pour trouver
les points, les droites et les coniques qui dependent symetriquement des elements
du triangle, considere par rapport au systeme de deux points ou de deux droites
quelconques. Mais nous ne nous sommes pas occupd des lieux et des enveloppes
qui se rapportent au triangle conside"re relativement a un seul point ou a une
seule droite. Parmi tous ceux qu'on pourrait imaginer ce sont surtout les deri-
vees polaires du triangle (conside're' comme ligne soit du troisieme ordre, soit de
la troisieme classe) qui offrent quelque inte'ret. Pour abreger, nous ne parlerons
que du centre des distances moyennes du triangle, et de la droite a 1'infini con-
sideree relativement au triangle, puisqu'on tirera facilement de ce que nous aliens
dire tout ce qui est relatif a un point ou a une droite quelconque. En revenant
done sur la notation de 1'article 2, nous nous proposerons de trouver (1) le centre
des distances moyennes, (2) 1'ellipse minima circonscrite, (3) 1'ellipse maxima
inscrite.
Supposons que le point q soit harmoniquement conjugue a q par rapport aux
points X t X 2 ; si Q est le point rdciproque de q' relativement au faisceau (S lt S 2 ),
il est evident que QQ/ sera un diametre de 1'hyperbole e*quilatere (a, /3; y, x 1} x 2 ).
Soit le point milieu de R^R^, (7 le point milieu de OQ ; et 0' seront les
centres du cercle circonscrit, et du cercle des neuf points. Ce dernier cercle sera
completement determine", puisque le point milieu de QQ' sera un point de sa
circonference. Soit F le point harmoniquement conjugue a Q par rapport a 00':
F sera le centre des distances moyennes du triangle ; on sait, en effet, que le
centre des distances moyennes et le centre du cercle polaire sont les deux centres
de similitude du cercle circonscrit et du cercle des neuf points, et que les deux
centres de similitude divisent harmoniquement le segment 00'. On voit en
mfime temps, qu'en gene'ralisant convenablement cette construction, on pourra
trouver avec la regie seule le p61e d'une droite quelconque, et la polaire d'un
point quelconque par rapport au triangle a/3y. Les deux ellipses maxima et mini-
ma sont les coniques polaires du centre des distances moyennes et de la droite a
1'infini, par rapport au triangle. Mais elles sont aussi les coniques du premier et du
second reseau pour lesquelles la droite a 1'infini est la polaire du centre des distances
moyennes. II suffira done de re"soudre ce probleme ' Trouver la conique du premier
re"seau pour laquelle un point donnd F est le pole d'une droite donne"e A.'
22 MEMOIKE sun QUELQUES PBOBLEMES [pt. n.
Soit x l un point quelconque de A ; on trouvera la conique du troisieme
re'seau qui est tangente aux droitea a^F et A, et Ton menera du point F la
seconde tangente & cette conique, qui coupera A au point x 2 . Les deux points
Xi x 2 seront conjugue's par rapport & la conique qu'on cherche. Soient y l y z une
seconde paire de points conjugue's sur la mme droite ; prenons X 1 X 2 Y 1 Y i lea
points rdciproques de x l x 2 y l y 2 ; la diagonale du quadrilatere X 1 X 2 Y 1 Y 2 , qui ne
passe pas par le point d'intersection de X 1 X 2 , Y^Y 2 sera la droite re'ciproque de
la conique cherche'e. On peut aussi ope'rer de la maniere suivante. Soit x un
point quelconque de A ; toutes les coniques du re'seau circonscrit, qui coupent
harmoniquement le segment Fa, passent par le m&ne quatrieme point, dont nous
allons trouver le point re'ciproque. Soient F', X les points rdciproques de F, x ;
les coniques dont il s'agit auront pour droites r^ciproques des droites conjugue'es
h, la droite T'X par rapport k la conique re'ciproque de F x, puisque quatre points
harmoniques en ligne droite ont pour rdciproques quatre points harmoniques sur
la conique re'ciproque de la droite. Soient a 1 a 2 , b^ deux couples de points har-
moniquement conjugue's aux points F, x ; A^A Z , B^B Z les points re"ciproques cor-
respondants ; le point ou les cordes A 1 A 2 , B 1 B Z vont concourir sera le point
re'ciproque du quatrieme point d'intersection. En prenant un second point y sur
la droite A, on trouvera un autre point t\ tel que ; la droite fy sera re'ciproque
de la conique cherche'e.
On trouverait la conique du second re'seau, qui aurait un p61e et une polaire
donnds, par la construction correlative. Si c'dtait la conique du troisieme re'seau
qu'on voulait trouver, on prendrait la conique re'ciproque de la droite A, puis la
polaire de F' par rapport k cette conique, et en fin la conique re'ciproque de la
polaire. Cette derniere conique coupe la droite A aux monies points oil elle est
couple par la conique harmonique qu'on cherche. On pourra done trouver avec
la regie seule, autant de p61es et de polaires relativement a cette courbe que Ton
voudra ; puisque pour cela il n'est pas ndcessaire de connaitre les deux points oil
la courbe est coupe'e par la droite A, mais seulement 1'involution qui determine
ces deux points *.
Nous ajouterons encore un exemple de 1'application des principes pre'ce'dents.
Soit propose" de de'terminer le cercle qui passe par les pieds des perpendiculaires
abaisse'es d'un point donne" P sur les c6tes du triangle a/3y. On de"terminera la
conique du re'seau inscrit qui a pour foyer le point P. Le cercle decrit sur 1'axe
focal de cette conique comme diametre sera le cercle cherche'.
* Note VII (p. 55).
Art. 4.] CUBIQTTES ET BIQUADRATIQTJES. 23
Quoiqu'on puisse repre"senter les racines d'une Equation cubique par les trois
elements du triangle harmonique commun a deux sections coniques, il arrive
ordinairement que la solution d'un probleme cubique ne conduit pas a la recherche
d'un tel triangle, mais a la recherche de trois des points d'intersection de deux
sections coniques, dont un des points d'intersection est donne" d'avance ; ou a la
recherche des trois points communs a un re*seau de coniques circonscrites au
meme triangle. Or, le second de ces deux cas se re'duit au premier, puisqu'en
conside'rant les coniques d'un faisceau determine" par deux coniques du re"seau, on
voit qu'elles correspondent anharmoniquement aux points oil elles coupent une
troisieme conique du rdseau qui n'appartient pas au faisceau que Ton considere.
On pourra done determiner line"airement le quatrieme point commun k une
conique quelconque du faisceau et a la troisieme conique. De plus, e*tant donn^
un des points d'intersection P de deux coniques, le triangle a/3y est homologique
au triangle dbc, dont les sommets sont les trois autres points d'intersection ; il
est aussi inscrit k ce me'me triangle. Le centre d'homologie est le point P ; quant
k 1'axe d'homologie, c'est 1'axe de sympt6se d'une conique circonscrite a Pafiy et
d'une conique du re"seau harmonique qui est tangente a la premiere conique au
point P. L'homologie des deux triangles pourra done se determiner line'airement ;
et Ton aura ainsi une m^thode g^ndrale, dont on pourra se servir pour ddduire
les lieux et les enveloppes relatives a 1'un de ces triangles des lieux et des en-
veloppes relatives a 1'autre.
4. Premiere methode. On de"terminera cinq points a, b, c, d, e d'une conique
quelconque a- du re"seau circonscrit appartenant au systeme des deux coniques
(S l} S 2 ). On choisira k volontd trois points a'b'c' de la conique 2, qu'on fera
correspondre homographiquement a abc ; et Ton en ddterminera un quatrieme d'
de maniere que le rapport anharmonique des quatre points a He'd' de la conique
2 soit e"gal k celui des points abed de <r. Puis on transformera homographique-
ment les coniques (S 1} SJ) de maniere qu'aux points abed de la figure donne"e
correspondent les points ab'c'd' de la figure nouvelle. La conique a- se trans-
forme en 2, laquelle sera ainsi une conique du re"seau circonscrit appartenant aux
transform e"es de S t et S. 2 . Le cercle appartenant a ce re"seau coupera 2 en quatre
points, dont un qui est re"el sera connu d'avance, et pourra se construire line'aire-
ment, sans tracer ni le cercle, ni la conique 2. Les trois autres points (dont un
sera toujours re"el) seront les points de la nouvelle figure qui correspondent aux
sommets a/3-y du triangle harmonique commun & S 1 et S 2 . On pourra done
construire ce triangle ; ce qui permettra de trouver par une construction quadra-
tique les points d'intersection de S t et <S 2 .
24 MMOIRE SUB QUELQUES PROBLEMES [pt. n.
On voit que cette construction est purement line'aire, jusqu'au moment ou
Ton veut determiner les intersections du cercle et de 2. Mais on peut 1'abreger
de beaucoup, en se servant des le commencement du tracd de la conique 2. On
prendra deux points Pip 2 de cette conique, et Ton choisira pour er la conique
(a, ft, y, pi, p 2 ). Puis on conside'rera la droite p 1 p 2 comme axe d'homologie de a-
et 2 ; en se servant du tracd de 2 on pourra trouver les deux centres d'homologie
des deux courbes, appartenant a cet axe. On voit qu'on fera bien de prendre
pour Pip 2 deux points imaginaires conjugue's appartenant a 2 ; si Ton prenait
deux points re'els, il pourrait arriver que les points d'intersection de 2 et a- fussent
tous re'els, sans qu'il y eut aucune tangente commune; mais alors les deux
centres d'homologie appartenant a 1'axe Pip 2 pouraient devenir imaginaires. En
supposant done que Pip 2 soient imaginaires, on fera avec deux couples de Tin vo-
lution qui determine ces points les me'mes operations que nous avons faites avec
les deux couples de points rectangulaires x 1 x 2 , y 1 y 2 de 1'article 2 ; et, de me'me
que nous y avons trouvd les deux extr^mites d'un me'me diametre du cercle
circonscrit, on trouvera ici les deux extre'mite's re'elles aja 2 d'une corde de a-
passant par le p61e de la droite Pip 2 relativement a or. On joindra le point
d'intersection de a^, Pip 2 , au p61e de Pip 2 relativement a 2, par une droite qui
coupera 2 en deux points b^, qui seront toujours re'els, puisque le pole de p l p 2
est inte'rieur a 2 ; les intersections de a t b^ , a 2 b 2 , et de a, b 2 , a 2 & t seront les deux
centres d'homologie de a- et 2, appartenant a 1'axe d'homologie (pip 2 ). Avec
1'un de ces centres d'homologie, on de'terminera la droite X et la dyade 6 J 6 2 ,
homologues de la droite a 1'infini et de la dyade cyclique ; puis on trouvera le
segment re"el intercepte par la conique (a, ft, y, O lt 2 ) sur une droite passant par
le p61e de X relativement a cette conique : enfin on de'terminera le quatrieme
point d'intersection de (a, ft, y, O lt 6 2 ) avec (a, ft, y, p 1} p 2 ) ou or. En passant k la
figure homologique, la premiere de ces deux coniques deviendra un cercle, dont
on connaltra un diametre et une des intersections avec 2. On n'aura done qu'a
tracer le cercle pour avoir les trois points homologues des points cherches.
Si 2 est une ellipse on prendra pour p^p^ les deux points imaginaires ou
cette courbe est coupe'e par la droite a 1'infini. Dans ce cas, on determinera un
diametre du cercle circonscrit au triangle afty, et un diametre de a-, qui sera
1'ellipse circonscrite au meme triangle et homothe'tique avec 2. On trouvera les
deux centres de similitude des deux ellipses, en determinant les points extremes
d'un diametre de 2 parallele au diametre de a- ; ce qui exigerait une construction
quadratique, si 2 n'etait point trace'e. Alors, pour avoir la solution du probleme,
on n'aura qu'a tracer le cercle correspondant au cercle circonscrit a afty. Nous
Art. 5.] CUBIQUES ET BIQUADRATIQUES. 25
croyons que cette solution est peut-6tre la plus simple qu'on puisse trouver ;
cependant elle ne laisse pas d'etre fort longue. Pour 1'abrdger autant que
possible, on prendra les deux points & 1'infini sur les axes principaux de 2 pour
1'un des couples de points dont on se servira pour trouver un diametre, soit du
cercle circonscrit, soit de 1'ellipse a- ; le second couple ne pourra pas tre le m<lme
pour les deux courbes, mais on fera en sorte que les deux couples aient un point
commun. Ainsi il faudra chercher les points re"ciproques, par rapport au faisceau
donne, de dix points differents, afin d'avoir un diametre de chacune des deux
courbes, et leur quatrieme point d'intersection. Cette recherche sera un peu
penible, puisque la determination de chaque point re"ciproque exige la construc-
tion de la polaire d'un meme point par rapport a chacune des deux coniques non
tracers S t et S 2 .
On pourrait eViter toute construction quadratique, sans cesser de se servir
d'une conique homologue de 2, en prenant pour a- la conique du reseau circonscrit
qui est tangente a 2 en un point donne, puisqu'en ce cas, un des centres d'homo-
logie des deux coniques etant connu d'avance, on peut trouver 1'autre lineaire-
ment. C'est ce qu'on serait conduit naturellement k faire si la conique 2 etait
une parabole. On prendrait pour o- la parabole homothetique du reseau circon-
scrit ; on en trouverait trois points, dont un serait son quatrieme point d'inter-
section par le cercle circonscrit. En menant des tangentes & 2 paralleles aux
tangentes de a- en deux de ces trois points, et en joignant les points de contact
des tangentes homologues, on aurait le centre de similitude. En ce cas on
n'aurait a trouver que [neuf] points reciproques, mais, par compensation, la
determination du centre de similitude serait un peu plus compliquee.
5. Cette premiere mdthode va nous fournir une demonstration bien simple
de ce beau theoreme de Descartes, qu'il n'y a pas un arc de section conique si
petit qu'il ne suffise pas pour re"soudre ge"ometriquement tout probleme cubique
ou biquadratique. Prenons deux points a, 6 sur une conique quelconque <r du
reseau circonscrit ; faisons correspondre a ces deux points deux points a b' de la
partie tracee de 2. Soit a le sommet re"el, ou 1'un des sommets reels, du triangle
a/3-y. La conique a- est divise"e par la corde ab en deux parties ; en determinant,
avec une approximation me'me tres grossiere, la position du point a (ce qui se fera
dans tous les cas au moyen de quelques points qu'on prendra sur une conique
quelconque du reseau circonscrit, autre que <r) on distinguera Fare a a b, qui passe.
par a, de 1'autre partie de la courbe. Selon que Ton prend pour c un point de
1'arc aab, ou de 1'arc oppose, on prendra pour c un point de 1'arc trace ab', ou un
point de la conique 2 qui ne soit pas compris entre ces deux points. D'apres
VOL. II. E
26 MEMOIRE STIR QUELQUES PROBLEMES [Pt. II.
cela, les points de 1'arc trace" a b' correspondront sur la conique 2 aux points de
1'arc aab de la conique <r; done le point correspondant de a tombera entre les
points a, b' de 1'arc tracd, et y sera ddtermind par le cercle circonscrit, appar-
tenant k la figure transformed, qui viendra couper 1'arc a b' en ce point. Mais on
connait un second point d'intersection de ce cercle et de 2 ; c'est celui qu'on peut
construire lindairement ; done on pourra trouver par une construction line"aire la
sdcante commune, dont les intersections avec le cercle feront connaitre les deux
autres sommets du triangle a /3y.
6. Seconde methode, Lorsqu'on se sert du cercle polaire pour trouver les
intersections des coniques donne"es ^ et S 2 , on peut operer d'un grand nombre
de manieres diffe'rentes. Nous n'indiquerons ici que les constructions que nous
croyons les plus simples, mais nous n'osons pas affirmer qu'on ne pourrait en
trouver d'autres, dependant du meme principe ge"ne"ral, qui auraient quelque
avantage sur celles que nous aliens exposer.
(1.) Supposons que 0/87, le triangle harmonique commun de /% et S 2 , n'ait
qu'un sommet re"el. En ce cas, deux coniques quelconques rdelles du re'seau
harmonique se coupent en deux points re"els, et en deux points imaginaires
conjugues ; de plus, ce rdseau ne contient aucune conique imaginaire de la
premiere espece. D'apres cela, on peut opdrer avec une conique quelconque de.
ce reseau de la meme maniere qu'avec une conique circonscrite. Soit <r la co-
nique qu'on choisit ; on la transformera soit homographiquement, soit corrdlative-
ment, en 2, et apres avoir ddtermind le cercle appartenant au re'seau transforme,
on trouvera les deux intersections rdelles du cercle et de 2. On aura ainsi le
triangle harmonique commun & ces deux coniques, triangle qui est le correspon-
dant dans la nouvelle figure du triangle afiy dans 1'ancienne. On pourra aussi
employer une transformation homologique, au lieu de la transformation homo-
graphique gendrale ; seulement on remarquera que la ddtermination des centres
d'homologie de a- et 2 est loin d'etre aussi facile dans le cas actuel que dans le
cas d'une conique circonscrite. En eflfet, il convient de prendre pour a- une
conique du re'seau harmonique ayant avec 2 une secante ideale commune. Or,
on ne peut determiner lindairement aucun point d'une telle coniqve, quoiqu'on
sache trouver son systeme polaire. On aura done besoin d'une construction qua-
dratique pour trouver les points extremes d'une corde de a- passant par le pdle
de 1'axe d'homologie relativement a cette conique ; points dont on fait usage
pour trouver directement les deux centres d'homologie. II est vrai qu'on pourrait
operer avec une conique <r tangente a 2 en un point donne", mais la construction
d'une telle conique serait un peu penible.
Art. 7.] CUBIQTTES ET BIQUADRATIQTTES. 27
La methode que nous venons d'indiquer s'applique aussi au cas que nous
avons exclu, mais elle conduit quelquefois a un re"sultat inutile. En effet, quand
le triangle 0,87 est reel, les coniques du re"seau harmonique qu'on determine par
deux points imaginaires conjugue's peuvent 6tre imaginaires ; ainsi, le cercle du
reseau transforme peut etre imaginaire, ou, tout en restant re"el, il peut ne
recontrer 2 en aucun point reel. D'ailleurs, quand meme cette me'thode reus-
sirait, elle conduirait & des operations assez prolixes.
7. (2.) Supposons que les points d'intersection de S J} S 2 ne soient pas tous
imaginaires ; nous aliens voir qu'on pourra determiner ces points directement,
sans chercher pre"alablement le triangle harmonique.
Soit (1) 2 une ellipse, A^A^, B 1 B ses deux axes principaux. Qu'on mene
une transversale L qui coupe en deux points imaginaires une conique quelconque
rdelle S du faisceau (S t , S 2 ). Ce faisceau determine une involution sur la trans-
versale L, dont les points doubles MI M 2 sont necessairement reels. Des points MI ,
M 2 menons des tangentes k S; soient a l a 2 , b^ 2 les points de contact des tan-
gentes issues de MI et M 2 respectivement ; ces points, qui seront tous re'els, divise-
ront la conique S harmoniquement. Qu'on transforme la figure homographique-
ment de maniere qu'aux deux couples a^ a 2 , & x 6 2 correspondent les deux couples
A 1 A 2 , B 1 B 2 ; ce qui peut se faire de huit manieres difFerentes. La conique S
deviendra 2, puisqu'elle doit devenir une conique divisde harmoniquement par
les deux couples de points A 1 A Z , B 1 B 2 . De plus les quatre points d'intersection
de Si et S 2 se transformeront en quatre points de 1'ellipse 2, situ^s sur une meiwe
circonference de cercle. Car les points Mi/2> q u i son t des points conjuguds par
rapport & toutes les coniques du faisceau ((S'j, S 2 ), se transformeront en les points
situes k 1'infini sur les axes principaux de 2 ; done toutes les coniques du faisceau
transforme' auront leurs axes principaux paralleles & ceux de 2, et par consequent
se couperont sur une circonfdrence de cercle il. Pour avoir ce cercle, on cherchera
son centre par la construction ci-dessus ; puis on prendra un couple de points
re"ciproques PI P 2 par rapport au faisceau transform^ ; le cercle il coupera ortho-
gonalement le cercle dont Pj P 2 est un diametre. On peut aussi determiner il en
prenant trois couples de points re"ciproques par rapport au faisceau transform^ :
ce qui donnera trois cercles que il coupera orthogonalement.
Soit (2) 2 une hyperbole ay ant A t A 2 pour axe re"el, et passant par les
points 4>j 4> 2 k 1'infini. On coupera le faisceau (*% $ 2 ) par une transversale sur
laquclle on ait une involution aux points doubles rdels MI /" 2 - Soit S une conique
du faisceau qui coupe la transversale en des points re"els fa (f> 2 . L'un des points
sera exte"rieur il S; soit MI ce point, et de"signons par a t a 2 les points de
E 2
28 MEMOIRE SUR QUELQUES PROBLEMES [pt. n.
contact de tangentes h, S issues de MI ; la conique S sera divisde harmoniquement
par les couples de points fa fa, a r a 2 . En transformant la figure de maniere que
,, 02 deviennent A lt A 2 , et que fa, fa deviennent &1&2, S deviendra 2, et aux
points MI M 2 correspondront respectivement les points k 1'infini sur 1'axe imaginaire
et sur 1'axe reel de 2. Done les points d'intersection du faisceau transform^
appartiendront k une circonfe'rence de cercle, qu'on pourra determiner comme
precedemment.
Soit enfin (3) 2 une parabole, ayant A pour sommet et tangente au point
M k la ligne droite k 1'infini. On coupera toujours le faisceau (S l 2 ) P ar une
transversale sur laquelle on ait des points doubles reels MI M 2 . Soit S la conique
du faisceau qui touche la transversale au point MI ; de M 2 menons la seconde tan-
gente k S ; soit a son point de contact. Choisissons k volonte" deux points x et
X sur les coniques S et 2 respectivement ; soient y et Y deux autres points sur
ces me'mes coniques, tels que le rapport anharmonique des points MI , a, x, y de S
soit egal k celui des points M, A, X, Y de 2. En faisant corresponds homo-
graphiquement les points MA XY aux points naxy, on oura la meme construction
que dans les deux cas precedents.
Nous remarquerons qu'apres avoir mend la transversale on est plus restreint,
quant au choix de la conique qui doit etre transform^e en 2, quand cette courbe
est une parabole, que quand elle est une conique centrale. Mais, dans le premier
cas, la transformation de la conique en 2 peut etre ope're'e d'une infinite de
manieres differentes.
Cette rndthode offre de grands avantages. On n'opere que sur les coniques
du faisceau donne", sans avoir besoin d'aucune autre conique du re"seau harmo-
nique ; ce qui facilite beaucoup la construction. De plus, on arrive directement a
la solution du probleme biquadratique des intersections, et on ne la fait pas
dependre du probleme cubique du triangle harmonique. Quand 2 est une
parabole on peut prendre une droite tangente a 1'une des courbes S t et S. 2 , pour
la transversale qui coupe le faisceau ; en op^rant ainsi, on n'aura qu'une seule
construction quadratique it effectuer ; c'est celle qui est inevitable (k ce qu'il
parait) quand on veut trouver le rayon du cercle polaire au moyen duquel on
rdsout le probleme.
8. (3.) Quand aucun des points d'intersection de S t et S 2 n'est re"el, la deter-
mination du triangle harmonique precede necessairement la determination des
points d'intersection. En ce cas on peut faire usage de 1'un ou de 1'autre des
precedes suivants.
(i.) Prenons une droite tangente k ^ en un point quelconque x^ ; ce point
Art. 9.] CTTBIQTTES ET BIQUADRATIQTJES. 29
sera un des points doubles de 1'involution de'termine'e par le faisceau donne" sur la
droite ; soit x 2 le second point double, qui pourra e~tre trouve" line"airement ; en
designant par S 2 la conique du faisceau qui passe par x 2 , on aura deux coniques
du faisceau tangentes k la mme droite re"elle. Puisque ces coniques n'ont aucun
point reel commun, et qu'elles ont une tangente re"elle commune, elles en ont
quatre. Soient o-j et <r 2 les coniques re"ciproques polaires de $j et S 2 par rapport
a 2 ; o-j et o- 2 se couperont en quatre points re"els qu'on ddterminera par la
methode pre'ce'dente, dont 1'application sera tres facile, puisqu'un des quatre
points d'intersection e"tant connu d'avance, il suffira de trouver le centre du cercle
polaire pour determiner completement ce cercle. Les polaires des points d'inter-
section de o-j et o- 2 seront les tangentes communes de /S t et S 2 , et feront connaitre
le triangle harmonique du faisceau.
Au lieu de prendre les coniques re"ciproques polaires de S x et S 2 , il sera plus
facile, en beaucoup de cas, de faire usage de la transformation correlative suivante.
En supposant, pour abreger, que 2 soit une ellipse, soit L un point inte'rieur k S l}
soient n l , n z les rayons doubles reels du faisceau en involution determine" au point
L par les coniques S lf S 2 conside're'es comme formant un faisceau tangentiel.
Soient enfin a lt 6j et a 2 , 6 2 I GS deux couples de tangentes a S 1 aux points ou cette
conique coupe les droites M I; /" 2 - En faisant correspondre correlativement les
deux couples de sommets de 2 aux deux couples de droites a L b ly a 2 & 2 , on trans-
formera S t en 2, et les quatre tangentes communes de S 1 et S 2 deviendront
quatre points de 2, situe"s sur une mdme circonference de cercle, dont un sera
connu d'avance. On deduira la demonstration de celle de 1'article 7. Cette
methode pourra servir k trouver directement les tangentes communes de deux
coniques donne"es, en supposant toutefois qu'il y en a de re"elles.
(ii.) Deux coniques quelconques du reseau harmonique, qui se coupent en un
point reel, se coupent aussi, dans le cas qui nous occupe, en trois autres points
reels. On pourra done considerer, au lieu de S L et S 2 , deux des coniques du
re"seau harmonique qui passent par un mme point, et Ton appliquera la mdthode
de 1'article 7. Ici encore il y aura 1'avantage qu'on connaitra d'avance un des
points communs a 2 et au cercle polaire du reseau transform^.
9. (4.) Les cas du probleme general, ou il s'agit de trouver les intersections
d'une conique trace"e 2 avec une conique non traced S, me"rite quelque attention.
On pourra en ce cas se servir des principes de 1'article 7, et Ton transformera le
faisceau (S, 2) en un autre faisceau qui contienne a la fois la conique 2 et un
cercle, soit en transformant S en 2, soit en transformant 2 en elle-meme. Quand
le faisceau (S, 2) determine, sur la ligne droite a 1'infini, une involution ayant
30 MEMOIRE SUB QUELQUES PROBLEMES [pt. in.
des points doubles rdels (ce qui arrivera toujours si 2 est une ellipse ou une
parabole), on pourra ope'rer cette derniere transformation de maniere que la ligne
droite it 1'infini se corresponde & elle-meme. Pour cela, on n'aura qu'ii, prendre
la ligne droite & 1'infini pour la transversale L de 1'article 7, et la conique 2 elle
mfime pour la conique S. Lorsque 2 est une conique centrale, en transformera
les quatre points d'intersection de S et 2 en quatre points d'une me'me circon-
fdrence, en transformant 2 en elle-meme, de maniere que les points extremes d'un
certain couple de diametres conjugue's deviennent les sommets de la courbe.
Lorsque 2 est une parabole, on aura une transformation encore plus simple.
[Soit A le sommet de 2, a le, point de contact de la tangente k 2 mende paral-
lelement k la polaire par rapport a S du point & 1'infini sur 1'axe de 2 ; on n'aura
qu'a transformer 2 homologiquement en elle-meme, en prenant pour centre
d'homologie le point k 1'infini sur A a.]
TROISIEME PARTIE.
1. Probleme. Abaisser d'un point donne P des normales sur une conique
comptttement decrite.
Pour abre*ger le discours nous supposerons que la conique soit centrale. On
sait que ce probleme a 6t6 re"solu par Apollonius de Perge, qui a de'montre' que
les pieds des normales se trouvent sur une hyperbole dquilatere, passant par P
et par le centre de la conique, et ayant ses asymptotes paralleles aux axes
principaux de cette courbe. L'hyperbole est le lieu des points d'intersection des
diametres de la conique par des perpendiculaires abaissees de P sur les diametres
conjugue's.
En employant les diverses me'thodes pre'ce'dentes, on parviendrait & re-
soudre ce probleme d'un grand nombre de manieres differentes. Mais c'est
surtout la belle solution de Joachimsthal qui peut intdresser les gdometres ; c'est
pourquoi nous la reproduirons ici avec une demonstration fondle sur les principes
pre'ce'dents.
Soit 2 la conique trace'e ; F 1'hyperbole dquilatere d' Apollonius : S, C les
centres respectifs de ces deux courbes. Soit encore A l A 2 1'axe rdel, ou 1'un des
deux axes re"els, de 2 ; a le point k 1'infini sur cet axe, /8 le point k 1'infini sur
1'axe conjugud. Qu'on de*signe par y un point quelconque de F, et qu'on abaisse
du sommet A l une perpendiculaire sur la droite Py. Cette perpendiculaire, qui
sera en me'me temps parallele au diametre de 2 conjugud k Sy, coupera 2 en un
second point que nous de'signerons par <r. II est Evident qu'on pourra conside"rer
les deux courbes F et 2 comme homographiques par rapport aux deux sdries de
Art. 1.] CTIBIQUES ET BIQUADRATIQUES. 31
points 7 et a-. Qu'on transforms F en 2, et que, dans la seconde figure, 2' soit la
courbe correspondant k 2 dans la premiere. Nous aliens voir que les points
d'intersection de 2 et 2' appartiendront a une meme circonference. Aux points
a et /8 de F correspondront e"videmment les points A et A 2 de 2 ; d'ou Ton
conclut qu'au point C de la premiere figure correspond le point /3 de la seconde.
Soit L la droite de la premiere figure qui correspond & la droite & 1'infini de la
seconde : il faut que cette droite passe par C, puisque le point correspondant de
C est & 1'infini. En se rappelant la propriete caracteristique des points de F, on
verra que, si 2 est une ellipse, la droite L est 1'axe d'homologie de la dyade
asymptotique de 2 et de la dyade des perpendiculaires abaissdes de P sur la
dyade asymptotique ; et que les points imaginaires, oh L est coupe's par 1'une ou
1'autre de ces dyades, appartiennent & F. Soit D le point a 1'infini de la droite L ;
les deux points CD seront eVidemment des points conjugues par rapport k la
conique F ; done, au point D de la premiere figure il correspondra le point a de la
seconde. Mais les points CD sont aussi des points conjugues par rapport a 2,
puisqu'ils sont hannoniquement conjugue's aux points oil la droite L rencontre la
dyade asymptotique de 2 ; d'ofr il s'ensuit que L est parallele & la polaire de C
par rapport a 2. Par consequent, les points a, /3 seront des points reciproques
par rapport aux deux courbes 2 et 2' ; done il passe une circonference de cercle
par les points d'intersection de 2 et 2'. La demonstration serait tout aussi
simple si Ton supposait que 2 fut une hyperbole. Soit O le cercle appartenant
au faisceau (2, 2^ ; pour le determiner completement, il faudra trouver trois
couples de points re"ciproques par rapport & ce faisceau. Pour cela, soit s le point
de 2 qui correspond au point S de F, et qui se trouve sur la perpendiculaire
abaisse"e de A l sur PS; les droites sA lt sA 2 de la seconde figure correspondront
aux droites Sa, Sft de la premiere, c'est-a-dire, aux deux axes de 2. Qu'on
prenne les points reciproques de et /3 par rapport au systeme des deux courbes
2 et F ; les points de la seconde figure qui correspondront & ces deux points
reciproques seront eVidemment les points d'intersection de la tangente A^fi par
sA 2 , et de la tangente A 2 f3 par sA l ; points qu'on ddsignera par a t et a 2 . Done
les deux points A l a l , et les deux points A 2 a 2 seront rdciproques par rapport au
systeme (2, 2'). Le cercle coupera orthogonalement les cercles (A 1 a 1 } et
(A 2 a 2 ) ddcrits sur A^a^ et A 2 a 2 comme diametres ; mais il est eVident que ces
cercles coupent orthogonalement le cercle (A 1 A 2 } ; de plus la droite qui joint leurs
centres est precise"ment la tangente a 2 au point s. On aura done le beau
resultat de'montre' par Joachimsthal que les cercles (A l A^ et (Q) ont pour axe
radical la tangente a 2 au point s.
32 M^MOIKE SUR QUELQUES PROBLEMES [Pt. III.
Pour achever la determination du cercle Q, il reste k trouver un troisieme
couple de points rdciproques par rapport a 2 et 2'. C'est ce qu'on peut faire
d'une infinite" de manieres differentes. Soit, par exemple, F le pied de la per-
pendiculaire abaisse'e du point P sur la polaire de ce point relativement a 2 ; P,
F seront des points re"ciproques par rapport au systeme (2, T), puisque PF est la
tangente a F au point P. On trouvera le point p de 2 qui correspond au point
P de r en menant par le point A une parallele &, la polaire de P relativement a
2. Pour trouver le point /de la seconde figure qui correspond au point F de la
premiere, on menera dans la conique 2 la corde A 1 f parallele k la polaire de F;
1'intersection de la droite sf par la tangente a 2 au point p sera le point f
cherche'. Le cercle ft coupera orthogonalement le cercle (pf), et se trouvera par
cela completement determine'.
On pourrait aussi se servir de la me'thode suivante. Qu'on fasse varier le
point P sur la droite fixe SP, et qu'on prenne sur cette droite SP' = - SP. Les
points s et p resteront fixes ; les cercles ii passeront toujours par les points d'in-
tersection de la tangente au point s et du cercle (A 1 AJ). Aux deux points P et
P' correspondra le meme cercle ; de plus les cercles O varieront anharmoniquement
avec les segments de Tin volution PI PI, P 2 Pz, etc. On en tirera la construction
suivante. Soit TT le centre du cercle Q qui passe par le point p ; et qu'on designe
par p 1'intersection de SP par la polaire de P. Le centre du cercle O cherche
sera le point d'intersection de la ligne des centres par une droite mene'e de P
parallelement & ptr *.
2. On sait qu'un grand nombre de problemes cubiques et biquadratiques
conduisent k I'e'tude des correspondences ddtermine'es par les Equations
x 1 (A 1 y* + 2B 1 y 1 y 2 +C 1 y% + x. 2 (A 2 y* + 2B 2 y 1 y 2 +C 2 yl) = 0, . . . (1)
x\ (A, y\ + 2JB lVl y 2 + C, yl) + 2x,x 2 (A 2 y\ + 2B 2 y, y 2 + C 2 y\)
iy2 +C 3 y) = 0, ... (2)
l) = 0, ... (3)
/-v nt
dans lesquelles on peut supposer que les quotients , - - sont des rapports an-
X 2 2/2
harmoniques qui de'terminent la position des points variables x et y, soit sur une
droite, soit sur une conique. Une thdorie complete de ces correspondances
* Note VILE (p. 55).
Art. 2.] CUBIQUES ET BIQUADEATIQUES. 33
depasserait de beaucoup les limites de ce mdmoire. II nous suffira de placer ici
quelques observations qui sont d'une grande importance dans cette theorie.
(1.) Soient a, b, c trois points d'une conique determines par 1'equation
cubique
3 Bx\x z + 3 Cx^ x\ + Dx\ = 0.
Soient aussi ABC le triangle circonscrit, L 1'axe d'homologie des deux triangles
ABC, dbc ; a, b', c les points d'intersection des droites Aa, Bb, Cc par la co-
nique. Alors les trois points a'b'c' sont les points determines par le co variant
cubique de F; et les deux points d'intersection de L par la conique sont de'ter-
mine's par le covariant quadratique.
Supposons qu'on donne sur une droite trois points P, Q, R, et trois autres
points p, q, r harmoniquement derives des premiers par rapport k un systeme F
de trois points inconnus. Proposons-nous de determiner (1) le point s harmoni-
quement derivd d'un point donne S par rapport au systeme F, (2) les deux points
covariants de ce systeme, (3) le systeme de deux points S l S 2 harmoniquement
derives d'un point donne" s, (4) les trois points inconnus eux-memes. De ces
problemes le premier n'est que lineaire, le second et le troisieme sont quadra-
tiques, le quatrieme est cubique. On projetera les points PQR, pqr sur une
conique, en prenant pour centre de projection un point de la conique ; nous
designerons les points projetes par les mmes lettres. Soient P'QfR les p61es
des droites Pp, Qq, Rr relativement a cette conique. Le point X, pole de la
droite L, satisfait & 1'equation
X.[P', P, q, Q, R, K\ = \P, p, Q, q, R, r];
done ce point pourra etre determine lineairement, puisqu'on pourra trouver deux
sections coniques dont il sera le quatrieme point d'intersection, les trois autres
etant connus. Le point s sera determine lineairement par 1'equation
X.[P, Q,R, S] = [p,q,r, s];
de m6me, en supposant que s soit donne, la droite \S, determine par cette
equation anharmonique coupera la conique aux deux points $!$ 2 . Enfin, en
prenant un point quelconque w de la conique, et en faisant correspondre anhar-
moniquement les faisceaux X . [P, Q, R, ...] et a>. [p, q, r, ...], on aura une section
conique S qui coupera 2 au point connu w, et aux trois points inconnus. On
pourra se passer, comme on voit, du trace de la conique 2, si Ton ne veut deter-
miner que les points s et X, et la droite L.
Dans la solution du probleme cubique on remarquera que le point o> peut
VOL. H. F
34: MEMOIRE SUB QUELQUES PKOBLEMES [pt. m.
e*tre pris &, volontd BUT la conique 2 ; on pourra mdme determiner d'avance la
position de ce point de sorte qu'on puisse faire passer une circonfe"rence de cercle
par les quatre points , a, /3, 7. Pour cela, on observera que les points w cor-
respondent anharmoniquement aux coniques du faisceau (a, /3, y, X). On prendra
trois positions du point w, et Ton de'terminera les points rectangulaires & 1'infini
appartenant aux axes principaux des coniques correspondantes (a, /8, y, X, <o) ; ou,
plutot, les trois points p harmoniquement conjuguds k un point fixe par rapport &
ces trois systemes de points rectangulaires. II est Evident que les points p cor-
respondront anharmoniquement aux points ; done, en de"signant par a- le conju-
gue" harmonique du point fixe par rapport aux points a 1'infini appartenant aux
axes principaux de 2, le point sera de'termine' lindairement par 1'^quation
[fi, P*> Pa, -] = [i, W 2 3> <>]
(2.) Dans 1'dquation (2), qu'on conside"rera relativement i\ une conique 2, on
fera varier le point y, ou ce qui revient au meme, le rapport y^'.y z - Les cercles
qui joignent les deux points de'termine's par les valeurs correspondantes de a^ : x s
envelopperont une section conique X. Pareillement, on aura une section conique
Y, enveloppe des droites joignant les points y correspondant &, une meme position
de x. Soit 6 un point quelconque de 2 ; soient n\ 12 I GS deux points qui cor-
respondent si 0, conside're' comme appartenant k la s^rie des x ; ^^ 2 les deux points
correspondant au me'ine point 6, conside're' comme appartenant & la se"rie des y.
Qu'on prenne la corde ^^2, conjugue'e harmonique de <r6 par rapport aux deux
droites ^ i7 2 > 1 ^2 se coupant au point a-. Cette corde sera la polaire de par
rapport k une troisieme section conique, que nous ddsignerons par 0. Pour les
deux coniques X et Y, les droites enveloppantes, ou, si Ton veut, les points de
contact sur ces droites, correspondront anharmoniquement aux points de la co-
nique 2.
II y a trois problemes biquadratiques qui se prdsentent naturellement, quand
on considere la correspondance doublement quadratique (2).
Trouver les quatre points (x) pour lesquels les deux points (y) correspondents
deviennent coincidents.
Trouver les quatre points dont chacun represente deux points (y), qui sont
devenus coincidents.
Trouver les points oit le point (x) coincide avec I'un des points (y) cor-
respondents.
Qu'on prenne sur la conique 2 les quatre points de contact des tangentes
communes a 2 et X ; chacun de ces points reprdsentera deux points y devenus
Art. 2.] CTJBIQUES ET BIQUADRATIQTJES. 35
coincidents. On en de"duira les quatres points x, auxquels correspondent ces
quatre points doubles, en se servant de la relation anharmonique que nous avons
indique'e. Enfin le point x coincide avec 1'un des points y correspondants aux
quatres points de rencontre de 2 et 0.
La the'orie de liquation (2) se simplifie, si elle est syme'trique relativement
aux deux series de points x et y. En ce cas, les deux coniques X et Y coincident
1'une avec 1'autre, et avec la conique, polaire re"ciproque de 2 par rapport k 0.
Les points d'intersection des deux coniques X et 2 sont pre'cise'ment les points x
pour lesquels les points y correspondants deviennent coincidents ; et les points de
contact sur 2 des tangentes communes de 2 et X sont ces doubles points. En
supposant toujours que liquation (2) soit syme'trique, prenons les points y l y 2
correspondant & un point quelconque x : soit x' 1'un de ces deux points ; 1'un des
deux points correspondant k x' sera x, 1'autre sera un nouveau point x". Deter-
minons successivement de la meme maniere les points x'", x"", ... ; il peut arriver,
comme on sait, qu'apres un nombre fini d'operations on retombe a la fin sur le
point de depart x. Ces cas particuliers ont etd beaucoup e'tudie's par les geo-
metres ; mais c'est surtout le cas ou Ton aurait x'" = x, qui est important (ainsi que
nous allons voir) pour la the'orie des problemes du troisieme et du quatrieme ordre.
Les problemes line'aires et quadratiques qui se rattachent k I'e'quation (2)
peuvent se re'soudre en beaucoup de cas par les me'thodes connues. Par exemple,
Ton voit qu'dtant donne's huit points x, et un point y correspondant k chacun de
ces points, les deux points y correspondant k un point x quelconque doivent
s'obtenir par une construction quadratique. Et, en efiet, on peut ope*rer cette
construction, en se servant des proprie'te's des courbes du quatrieme ordre, ayant
deux points doubles.
(3.) Conside'rons la correspondance de"finie par I'e'quation (3) ; et supposons
que cette Equation soit relative a une conique 2. A chaque point x correspon-
dront trois points de la sdrie y : et Ton aura ainsi une sdrie de triangles inscrits a
2. Mais cette se'rie de triangles sera en me'me temps circonscrite k une seconde
conique ; elle sera de plus une se'rie de triangles harmoniques par rapport a une
troisieme conique. Ainsi, cette se'rie de triangles pourra e~tre de"finie par une
correspondance quadratique double, qui sera syme'trique, et dans laquelle le troi-
sieme e"le"ment derivd coincidera avec 1'e'le'ment d'ou Ton est parti. II s'ensuit
que la plupart des questions relatives k 1'involution cubique definie par 1'dquation
(3) pourront 6tre re"duites aux recherches analogues relatives k cette espece par-
ticuliere de correspondances quadratiques doubles *.
* Note IX (p. 59).
F 2
36 MEMOIRE sun QUELQUES PROBLEMES [pt. in.
Ct (* l~)(*
3. Nous ddsignerons par [a, b, c, d] le quotient J'-T~J> qui est un des rapports
anharmoniques des quatre points a, b, c, d en ligne droite ; de mdme, nous repre*-
n r n , .sin aPc sin bPc ,
senterons par P. [a, b, c, d] le rapport anharmomque - *-, : - , , des quatre
droites Pa, Pb, PC, Pd se coupant au meme point P *. Cela pose", nous aurons
les deux lemmes suivants.
Lemme I. Soient
Pp, Pp lt Pp 2 , Qq, Qq 1} Qq 2 , Rr, Rr u Rr 2 ,
neuf droites donne"es ; le lieu d'un point x, qui satisfait & 1'equation
P. [x, p, p l} p 2 ] x Q . [x, q, q 1} q a ] x R. [x, r, r lt r 3 ] = a, ... (1)
a e"tant une constante, est une courbe cubique passant par les neuf points d'inter-
section des droites Ppi, Qq : , Rr avec les droites Pp 2 , Qq 2 , Rr 2 .
Demonstration. (1). Le lieu du point x qui satisfait h, 1'dquation
Q.[x, q, q lt q z ]xR.[x, r, r lf r 2 ] = n, ...... (2)
M dtant une constante, est une conique (/u) passant par les quatre points d'inter-
section des droites Qq l} Rr t avec les droites Qq 2 , Rr z . Soit Rp le rayon du
faisceau (R) qui satisfait h, 1'dquation
R-[p,r, r lt rJ = M ............ (3)
En divisant membre k membre 1'equation (2) par 1'equation (3), on aura
Q . [x, q, q^q^xR. [x, p, r l5 rj = 1,
ou, ce qui revient au mdme,
R . [x, q, q lt q 2 ] = R. [x, p, r 2 , r,],
Equation qui ddmontre ce qui a dte" avancd.
* En general, si a, b, c, d sont quatre elements quelconques, dont on peut definir le rapport
anharmonique (par exemple, quatre points d'un meme conique, ou quatre courbes d'un mfeme faisceau)
nous exprimerons ce rapport pas la fonnule [a, b, c, Z] ; et, toutes les fois qu'il sera necessaire do
distinguer entr'eux les divers rapports anharmoniques du mme systeme de quatre cllments, la formule
[a, b, e, rfj d6signera pour nous le rapport anharmonique analogue au rapport : j-j de quatre points
(( < I (>' I
en ligne droite. Nous nous servirons, avec quelques giSometres, des parentheses pous exprimer des
courbes passant par des points donnes ; ainsi (a, b) sera la droite qui joint les points a, b ; (a, b, c, d, e)
sera la conique des cinq points a, b, c, d, e. Si P est la base d'un faisceau de courbes d'ordre quelcon-
que, (P, a) sera la courbe de ce faisceau qui passe par le point a; et P. [a, b, c, rf] sera le rapport
auharmouique des quatre courbes (P, a), (P, b), (P, c), (P, d).
Art. 3.] CUBIQTJES ET BIQUADRATIQUES. 37
(2.) En donnant k /u des valeurs successives differentes, les coniques (/*) cor-
respondantes, lieux des points x, seront toutes circonscrites au meme quadrilatere.
Or, ces coniques correspondront anharmoniquement aux valeurs de M. En effet,
supposons que Qx soit une direction fixe ; on sait que les coniques correspondront
anharmoniquement aux points ou elles coupent cette droite. En de"signanfe par p,
r t , r 2 les points d'intersection de Qx par Rp, Rr lt Rr 2 , la position du point x,
appartenant a la conique (/), sera de'termine'e par 1'equation
[x, p, r 2 , r^ = Q.[x, q, q lt q 2 ~],
dont le second membre est une constante C. En multipliant les deux membres
de cette Equation par les deux membres de liquation
l* = [p,r, r 1} r 2 ],
on aura
[X, T, T l , T 2 J ,-, j
d'ou il s'ensuit que le rapport anharmonique de quatre points x est egal au
rapport anharmonique des valeurs correspondantes de /u.
(3.) La position de la droite Px, qui satisfait k 1'equation
7i r -\ a
P.[x,p, Pl , P*\ = ->
variera anharmoniquement avec les valeurs de p.. Done le lieu des points d'inter-
section d'une conique (n) par la droite correspondant a la meme valeur de n sera
une courbe cubique ; mais il est Evident que ce sont pre"cise"ment ces points
d'intersection qui satisfont a 1'dquation (1). On voit d'ailleurs que la courbe
cubique passera par cinq des neuf points d'intersection des droites donndes ; elle
passera aussi par les quatre autres, puisque dans la demonstration on peut
echanger entre eux les faisceaux (P), (Q), (R).
Lemme II. Soient S l , S 2 , S 3 trois courbes du mme ordre ; A l} A 2 , A 3 ,
B!, B 2 , B z les courbes des faisceaux (S 2 , S 3 ), (S 3> SJ, (S L , S 2 ) qui passent par les
points a et /3 respectivement ; on aura 1'equation
[S 2 , S 3 ,A 1} B,] x [S,, S 1} A 2 , B 2 ] x [8,, S 2 , A 3 , B 3 ] = + 1. (3)
Demonstration. Puisqu'il y a toujours une courbe du faisceau (S. 2 , 83) qui
appartient en meTne temps au faisceau ddtermind par deux courbes quelconques
des faisceaux (>S' 3 , SJ, (S lt S 2 ), il y a une courbe commune aux faisceaux (S 2 , S 3 ),
(A 2 , A 3 ). Mais cette courbe commune ne peut etre autre que A l} puisque A l est
une courbe du faisceau (,,, S 3 ), et qu'elle passe par a, un des points d'intersection
38 MEMOIRE SUR QUELQUES PROBLEMES [Pt. III.
de A s , A a . Done les trois courbes A lt A 2 , A s appartiennent au meme faisceau ;
le me 1 me raisonnement e'applique aux trois courbes B l} J5 2 , J5 3 . Solent a et ft les
points ou vont concourir les droites polaires d'un meme point par rapport a
A lt A t , A 3 , et a B lt B 2 , B 3 respectivement ; soient aussi <r l , a- 2 , a- 3 les points
d'intersection de (a, /3) par les polaires de relativement a S 1} S 2 , S 3 ; on aura
evidemment
[S 2> S 3 , A l} J = [<r 8 , <r 3 , a, 0]
[$!, $2, A 3 , B 3 ] = [(7^ (T 2 , 0, ft],
valeurs qui satisfont identiquement a 1'dquation (3).
Avant de terminer ces pre"limmaires nous rappellerons qu'e'tant donne trois
points sur chacune de deux cubiques, et en outre six des neuf points d'intersec-
tion des deux courbes, on trouve aise'ment les trois autres points d'intersection
par une construction cubique qu'on doit a M. Chasles. Soient 1, 2, ..., 6 les six
points d'intersection donnas, 7, 8, 9 les trois points cherchds ; on determine
line"airement deux coniques telles que (5, 6, 7, 8, 9), (4, 6, 7, 8, 9), dont les quatre
points d'intersection sont le point connu 6, et les trois points cherches. Pareille-
ment, e"tant donne" trois des points d'intersection d'une conique et d'une cubique,
qu'on suppose de'termine'es toutes les deux par un nombre suffisant de points, on
trouvera les trois autres points d'intersection par une construction cubique facile.
Soient a, b, c les points donnes, d, S, des points donne's de la cubique et de la
conique respectivehient, w le point oppose" au systeme a, 6, c, d, relativement a la
cubique. En de"signant par x un point quelconque de la cubique, les coniques
(a, b, c, d, x) et les droites (w, x) se correspondront anharmoniquement. Soit le
quatrieme point d'intersection de la conique donne'e par (a, b, c, d, x) ; les deux
faisceaux (3, ), (, x) seront homographiques, et la conique, lieu des points
d'intersections des rayons correspondants, coupera la conique donne'e au point
connu S et, en outre, aux trois points cherches.
4. PROBLEMS. Jfitant donne treize des points d'intersection de deux courbes
du quatri&me ordre, trouver les trois autres.
Nous supposerons que les treize points soient tels qu'on peut faire passer
actuellement par ces points une vraie courbe du quatrieme ordre. Nous exclu-
rons done absolument les cas ou Ton aurait, soit cinq points en ligne droite, soit
neuf points sur une me'me conique, soit treize points sur une meme cubique.
Mais, afin de simplifier la discussion ge'ne'rale, nous en exclurons aussi, pour le
moment, les cas ou Ton aurait
Art. 4.] CUBIQUES ET BIQUADRATIQUES. 39
(1) Neuf points formant la base d'un faisceau de cubiques.
(2) Huit points sur une meme conique.
(3) Onze points sur une meme cubique.
(4) Quatre points en ligne droite.
Pour tous ces cas la solution ge"nerale se simplifie plus ou moins ; nous les consi-
dererons se"parement plus tard. Enfin, nous supposerons que les treize points
soient tous reels, et tous diffe'rents ; nous reviendrons ci-apres sur les cas oil Ton
aurait des points imaginaires.
Soient 1, 2, 3, ..., 13 les treize points donne*s, 14, 15, 16 les points qu'il
s'agit de trouver.
Prenons six points quelconques des treize points, par exemple les points
8, 9, 10, 11, 12, 13 ; nous allons montrer comment on peut determiner la cubique
qui passe par ces six points, et par les trois points inconnus.
Pour cela, nous prenons un quelconque des six points que nous avons choisis,
par exemple le point 8 ; nous le joignons aux sept points 1, ..., 7, et nous consi-
ddrons le systeme des huit points 1, ..., 8 comme formant la base P 8 d'un faisceau
de courbes cubiques. De"terminons le point p s , de sorte que les cinq droites
(P*, 9), (PS, 10), (Ps, 11), (p, 12), (^ 8 , 13)
correspondent anharmoniquement aux courbes cubiques
(P 8) 9), (P 8 , 10), (P 8 , 11), (P 8 , 12), (P 8 ,13).
La determination du point p g se fera lindairement par une construction sur
laquelle nous reviendrons plus tard ; nous dirons que ce point est biquadratique-
ment oppose aux points 1, ..., 8 de la base P B . La courbe du quatrieme ordre,
qu'on peut faire passer par les treize points et par le point p s , aura pour faisceaux
generateurs le faisceau de droites ( p^), et le faisceau de courbes cubiques (P 8 ) ; de
plus, les droites
(p s , 14), (p s , 15), (p s , 16)
du premier faisceau correspondront aux courbes cubiques
(P 8 , 14), (P 8 , 15), (P 8 , 16)
du second faisceau, puisque 14, 15, 16 sont des points de la courbe du quatrieme
ordre. Substituons successivement au point 8 deux autres points du systeme de
six points, par exemple les points 9 et 10 ; soient P 9 , P 10 les bases cubiques qu'on
aura ainsi, p 3 , p lf) les points biquadratiquement opposes & ces bases ; nous allons
40 MMOIRE SUB QUELQTTES PROBLEMES [Pt. in.
voir que les trols points opposes p a , p 9 , p 10 , et les trois points d'intersection des
trois couples de droites
#,(10), Pio (9); ^o(8),^(lO); j> 8 (9),.P,(8),
sont des points de la cubique cherche'e, qui sera des lors completement de'termine'e
puisqu'on en connaitra douze points. Conside'rons la courbe cubique, lieu des
points x qui satisfont k 1'^quation
p s . [x, 11, 9, 10] xp g . [x, 11, 10, 8] x_p 10 . [x, 11, 8, 9] = + 1.
D'apres le lemme I, cette courbe passe par les points p s , p%, p w , 8, 9, 10, et par
les trois points d'intersection des trois couples de droites
(_p 9 , 10), (_p 10 , 9) ; (p w , 8), (^ 8 , 10) ; (p s , 9), (j> 9 , 8) ;
elle passe en outre par le point 11, puisque chacun des trois rapports anharmo-
niques devient e"gal jl Funit^ positive si Ton fait coincider x avec ce point. Mais
les points 12, 13, 14, 15, 16 appartiennent aussi k la meme courbe. En effet,
soit un quelconque de ces points ; d'apres la relation anharmonique qui subsiste
entre les faisceaux (p 8 ), (P 8 ) ; (p 9 ), (P 9 ) ; (p M ), (P 10 ), on aura
p s .&n, 9, io]=p g .[en, 9, 10],
p 9 .K, 11, 10, 8] = P 9 .[en, 10, 8],
p 10 .[,ll, 8,'9] = P M .[eil, 8, 9].
Mais le produit des seconds membres de ces Equations est 1'unitd positive ; comme
il r^sulte du lemme II, en y e*crivant
^ = (P 9 ,10) = (P 10) 9),
i8 2 =(P M , 8) = (P 8 ,10),
S 3 = (P 8 , 9) = (P 9 , 8).
II resulte de ce qui precede que la cubique, qui passe par les neuf points
8, ..., 16, passe aussi par les neuf points biquadratiquement opposes aux systemes
de huit points, qu'on obtient en joignant successivement aux sept points 1, ..., 7
chacun des huit points 8, ..., 16; et par les trente-six points d'intersection des
couples de droites (p a , /3), (p^, a), en de'signant par a, ft deux nombres indgaux
de la sdrie 9, ..., 16. De ces cinquante-quatre points, on en connaitra ving-sept,
qui ne dependent pas des points inconnus 14, 15, 16.
Pour avoir une autre cubique passant par ces trois points, nous remarquons
que, par hypothese, la cubique (7j des neuf points 8, ..., 16, ne peut pas passer
Art. 5.] CUBIQUES ET BIQTJADBATIQUES. 41
par tous les sept points 1, ..., 7. Soit done 7 un de ces points qui n'appartient
pas a Cj ; on echangera entr'eux dans la construction pr^cddente le point 7 et un
point quelconque 8 des six points 8, ..., 13 ; et Ton determinera ainsi la cubique
C 2 des neuf points 7, 9, ..., 16. Les cinq points 9, ..., 13 seront des points com-
muns aux deux courbes C x et (7 2 ; le point biquadratiquement oppose aux points
1, ..., 8 sera un sixieme point commun ; enfin, les points qu'il s'agit de trouver
seront precise'ment les trois autres points communs. Puisque par hypothese les
huit points 1, ..., 8 n'appartiennent pas tous a une me" me conique, on pourra dire
autant des huit points 9, ..., 16. Soient 11, 12, 13 trois des points 9, ..., 13 qui
n'appartiennent pas a une mdme conique avec les points inconnus. Qu'on deter-
mine les deux coniques (12, 13, ..., 16) et (11, 13, ..., 16), se coupant au point
connu 13 ; les trois autres points d'intersection de ces courbes seront finalement
les points chercb.es.
On remarquera que les deux courbes du quatrieme ordre dont nous nous
sommes servis dans la demonstration pre'ce'dente, ne sont pas deux courbes quel-
conques du faisceau determine par les treize points donne"s. Chacune des deux
courbes est assujettie a passer par le neuvieme point appartenant a la base cu-
bique forme'e par un systeme de huit points choisis parmi les treize points donnds *.
5. La solution pre'ce'dente depend essentiellement de la determination des
points biquadratiquement opposes aux divers systemes de huit points qu'on peut
former avec les treize points donnas. Soit P = [l, 2, 3, ..., 8] 1'un quelconque de
ces systemes ; pour avoir le point oppose biquadratiquement a P, il nous faudra
avant tout un systeme de cinq points, ou de cinq droites, qui correspondent
anharmoniquement aux cubiques (P, 9), (P, 10), (P, 11), (P, 12), (P, 13). A. cet
effet, on pourrait se servir, comme on sait, soit des tangentes a ces courbes en un
point quelconque de la base P, soit de leurs points d'intersection par une des
droites qui joignent deux des points P, soit enfin de cinq droites, polaires d'un
mgrne point par rapport aux cinq courbes. Mais nous preferons la methods
suivante, qui conduit a des operations moins penibles. On choisira parmi les
points P un systeme Q de quatre points quelconques 1, 2, 3, 4, et 1'on deter-
minera pour chacune des cinq cubiques le point oppose au systeme Q. Pour cela
on determinera la conique F qui passe par les quatre points 5, 6, 7, 8, et qui
admet le rapport anharmonique Q . [5, 6, 7, 8] ; puis, on prendra sur cette conique
des points 9', 10', 11', 12', 13', tels qu'on ait
[5, 6, 7, 8, 9', 10', 11', 12', 13>g.[5, 6, 7, 8, 9, 10, 11, 12, 13];
* Note X (p. 63).
VOL. II. G
42 MEMOIRE SUR QUELQUES PROBLEMES [pt. in.
les intersections de F par les droites
(9, 9-), (10, 10'), (11, 11'), (12, 12'), (13, 13-)
seront les points opposes & Q, appartenant respectivement aux cubiques
(P, 9), (P, 10), (P, 11), (P, 12), (P, 13).
Nous les de"signerons par 9 , N>, w u , i 2 , i3> et nous nous en servirons pour un
systeme de points correspondants anharmoniquement aux cinq cubiques. Ensuite,
on de"terminera la conique 2 qui passe par quatre points quelconques 9, 10, 11, 12
des cinq points 9, 10, 11, 12, 13, et qui admet le rapport anharmonique
[w 9 , o) 10 , w u , <> 12 ]. Soit 13" le point de cette conique qui satisfait k liquation
[9, 10, 11, 12, 13"] = [o> 8 , o> 10 , o> u , w 12 , ft> 13 ];
le point d'intersection de 2 par la droite (13, 13") est le point oppos biquadra-
tiquement au systeme P. On voit que la construction revient au fond k la con-
struction si connue du point oppose* k un systeme de quatre points appartenant a
une courbe cubique.
La construction ne reussit pas si trois des points Q sont en ligne droite, mais
elle ne devient que plus facile si trois des points 5, 6, 7, 8, ou bien trois des
points 9, 10, 11, 12, 13 sont en ligne droite, puisqu'alors 1'une ou 1'autre des
coniques F, 2 est remplacee par un systeme de deux droites. On peut done
toujours faire en sorte que le systeme Q ne contienne pas trois points en ligne
droite. Cependant, s'il y avait quatre points en ligne droite parmi les points P,
on ferait bien de les prendre pour le systeme Q, puisque en ce cas la de"termina-
tion du point biquadratiquement oppos au systeme P se re"duirait tout simple-
ment k la determination du point oppose au systeme des quatre points 5, 6, 7, 8,
relativement k la courbe cubique (5, 6, ..., 13). Encore, s'il y avait quatre des
cinq points 9, 10, 11, 12, 13 en ligne droite, la construction precedente ne serait
plus applicable. En eflfet, dans ce cas il n'y a aucun point oppose" biquadratique-
ment au systeme P, k moins que la condition
P. [9, 10, 11, 12] = [9, 10, 11, 12]
ne soit satisfaite par les quatre points en ligne droite. En supposant que cette
condition eut lieu, on ddterminerait sur la droite (9, ..., 12) le point 13' qui
satisfait & 1'dquation
P. [9, 10, 11, 12, 13] = [9, 10, 11, 12, 13-];
et on trouverait que tout point de la droite (13, 13 7 ) aurait la propri^td carac-
tdristique d'un point opposd biquadratiquement au systbme P.
Art. 6.] CUBIQTJES ET BIQUADBATIQTTES. 43
La construction cesse encore d'etre applicable si les points P et- trois des
points 9, ..., 13 appartiennent & une meme cubique. Elle deviendrait inde"ter-
minee si 1'un des cinq points 9, ..., 13 etait le neuvieme point appartenant & la
base cubique P. Ainsi en supposant que 13 appartint & cette base, tout point de
la conique 2 serait oppose biquadratiquement au systeme P. C'est ce qui arri-
verait aussi si Ton avait la relation
P.[9, ..., 13] = [9, .... 13],
ce dernier symbole se rapportant k la conique qu'on peut mener par les cinq
points.
6. Nous allons maintenant revenir sur les cas particuliers que nous avons
exclus de la discussion generate (Art. 4). Dans tous ces cas, comme on a pu voir
par ce qui pre'ce'de, on pourra simplifier la determination des points opposes
biquadratiquement k certains systemes de huit points, si toutefois ces points ne
cessent pas d'exister.
(1.) Supposons que les neuf points 1, 2, ..., 9 forment la base P d'un faisceau
de courbes cubiques. La conique 2, qui passe par les points 10, ..., 13, et admet
le rapport anharmonique P. [10, 11, 12, 13] passera aussi par les points 14, 15, 16.
Car on pourra faire passer par les treize points et par un point quelconque a- de
cette conique, une courbe S du quatrieme ordre qui aura pour faisceaux ge"ne"ra-
teurs le faisceau de cubiques (P), et le faisceau de droites (o-). Les courbes 2 et
S se couperont en huit points, dont a-, 10, 11, 12, 13 seront cinq; nous allons
voir que les trois autres seront prdcisement les points inconnus 14, 15, 16. -Soit,
en effet, x un des trois points d'intersection de 2 et 8, autres que a-, 10, 11, 12,
13 ; en prenant x pour centre du faisceau ge"ne"rateur, on aura une courbe X du
quatrieme ordre, qui passera par x et par les treize points, mais qui ne pourra pas
co'incider avec S, puisque, en designant par a un point quelconque qui n'appar-
tient pas & la conique 2, les deux rapports anharmoniques
x. [10, 11, 12, a], <r. [10, 11, 12, a]
ne sauraient 3tre egaux. Done le point x est bien un des trois points 14, 15, 16,
puisq'il appartient en meme temps aux deux courbes S et X. On arriverait au
m6me resultat en s'appuyant sur la proposition gendrale de 1'article 4, d'ou Ton
conclurait qu'un point quelconque de 2 appartient k la cubique de'termine'e par
les sept points 10, ..., 16 et deux quelconques des points P ; c'est-a-dire que cette
cubique est composde de 2 et de la droite qui joint les deux points. Pour com-
pldter la solution du probleme, on de"terminera par la me"thode ge'ne'rale une
cubique qui passera par les trois points inconnus, par trois des points 10, ..., 13,
G 2
44 MEMOIRE SUB QUELQUES PROBLEMES [Pt. IH.
et par trois des points P. Trois des points d'intersection de cette cubique par 2
seront connus ; done on trouvera les trois autres par la construction cubique que
nous avons dejk indiqude.
Nous ferons remarquer que si les points 10, ..., 13 appartiennent, deux a
deux, k deux courbes cubiques du faisceau (P), la solution du probleme sera
lindaire. En eflet, soient (P, 10, 11), (P, 12, 13) les deux cubiques ; on pourra
conside'rer la droite (12, 13) et la cubique (P, 10, 11), prises ensemble, comme
une courbe du quatrieme ordre passant par les treize points ; pareillement la
droite (10, 11) et la cubique (P, 12, 13) composeront une autre courbe du qua-
trieme ordre passant par les me'mes points ; done le point d'intersection de
(10, 11) et (12, 13) sera un des points cherchds; et 1'un des deux autres sera le
troisieme point d'intersection de la cubique (P, 10, 11) par (10, 11), point qu'on
sait determiner line"airement.
La construction ne sera que quadratique, si trois des points qui n'appar-
tiennent pas a P sont en ligne droite. Soient 10, 11, 12 trois points d'une
mme droite X ; soit aussi a le point de X qui satisfait k 1'dquation
[10, 11, 12, ] = P.[10, 11, 12, 13];
la conique 2 sera remplace'e par 1'ensemble des deux droites X, et (13, a). Qu'on
determine par la mdthode gdndrale ci-dessus une cubique qui passe par les trois
cherche"s, par trois des points P, par deux des points 10, 11, 12, et enfin par 13 ;
un des points cherche"s sera le troisieme point d'intersection de la cubique par X ;
celui-ci se trouvera line'airement ; les deux autres seront les deux points d'inter-
section, autres que 13, de la droite (13, a) par la cubique ; on les aura par une
construction quadratique.
Enfin, si tous les quatre points 10, 11, 12, 13 appartiennent k la meme
droite, il n'y aura aucune vraie courbe du quatrieme ordre qui pourra passer par
les treize points donne"s, k moins que la condition
P. [10, 11, 12, 13] = [10, 11, 12, 13]
ne soit ve'rifie'e ; done tous les fois que cette relation ne subsistera pas, ce cas
sera un de ceux que nous devons rejeter. De plus, quand meme la condition se
trouverait re'alise'e, on pourra faire abstraction du point 13, puisque toute courbe
du quatrieme ordre qui passe par les points 1, ..., 12, passera ne"cessairement par
ce point. II faudra done qu'un quatorzieme point soit donn pour que les seize
points soient completement de'termine's ; mais ce point e"tant donne", les deux
autres se trouveront par la construction quadratique pre'ce'dente.
Nous avons dejk remarqu^ qu'en de"signant par P le systeme 1, ..., 8, tout
Art. 6.] CUBIQUES ET BIQUADRATIQTTES. 45
point appartenant & la conique (9, ..., 13) est oppose" biquadratiquement k P, si
Ton a la relation
P. [9, 10, 11, 12, 13] = [9, 10, 11, 12, 13].
Ce cas se reduit immddiatement & celui que nous venons de traiter. En effet, le
neuvieme point appartenant &, la base cubique P, est eVidemment un des trois
points cherche's, puisqu'il est un point commun k deux courbes du quatrieme
ordre, passant toutes les deux par les treize points, et ayant le me'me faisceau
gene"rateur de courbes cubiques (P), mais ayant des points diffe'rents biquadra-
tiquement opposes k ces faisceaux. Done on trouvera lindairement un des trois
points 14, 15, 16 ; on obtiendra les autres par une construction quadratique,
puisqu'on connaitra quatre des points d'intersection de la conique (9, ..., 13) par
une cubique qu'on fera passer par les deux points inconnus, par quatre des points
9, ..., 13, et par trois des points de la base cubique.
(2.) Supposons que les huit points 1, ..., 8 appartiennent tous k une meme
conique (P). La couique <? = (9, 10, 11, 12, 13) sera une premiere conique passant
par les points inconnus 14, 15, 16. Soit C une cubique de'termine'e par les trois
points inconnus, par trois des points 9, ..., 12, et par trois des points P. On
pourra determiner cette cubique par la methode ge"ne"rale, et puisqu'on connaitra
trois des points d'intersection de C par <r, on pourra trouver une seconde conique
a-', qui coupera a- en quatre points, dont un sera connu d'avance, tandis que les
autres seront les points cherche's.
(3.) Supposons que onze des treize points appartiennent k une me'me cubique
(P); soient 1, 2, ..., 11 ces onze points, que nous ddsignerons par P. On sait
que toute courbe du quatrieme ordre, qui passe par onze points d'une cubique,
rencontre la cubique en un douzieme point fixe. Ce point fixe sera un des trois
points cherche's ; nous le ddsignerons par 14 ; il pourra se determiner lindaire-
ment ; de plus, cette determination de"pendra uniquement des onze points P, et
nullement des points 12, 13. On pourra done substituer k ces deux points deux
autres points quelconques choisis k volonte", pourvu qu'ils ne soient pas situs's sur
la cubique (P). Nous prendrons actuellement, au lieu de 12 et 13, deux points
12' et 13' qui forment avec sept des onze points les neuf points basiques Q d'un
faisceau de courbes cubiques, auquel la cubique (P) n'appartient pas. Soient
1, 2, ..., 7 les sept points ; on prendra pour 12' un point quelconque qui n'appar-
tient pas k (P) ; 13' sera le neuvieme point appartenant & la base cubique 1, ..., 7,
12', mais il ne sera pas ne*cessaire de le trouver, quisqu'on n'en fait aucun usage
dans la construction. On determinera la conique qui passe par les quatre points
8, 9, 10, 11, et qui admet le rapport anharmonique Q.[8, 9, 10, 11]: cette co-
46 MEMOIRE SUB QUELQUES PROBLEMES [rt. m.
nique passera par le point 14 cherchd, puisqu'elle doit passer (Art. 6, 1) par lea
trois points qui competent la base biquadratique 1, ..., 11, 12', 13'. En dchan-
geant entre eux un des quatre points et un des sept points, on aura une seconde
conique, qui coupera la premiere en trois points connus d'avance ; le quatrieme
point d'intersection sera le point 14. Pour trouver les points 15 et 16, on con-
siderera un systeme de huit points 0, compost des deux points 12 et 13, et de
six points quelconques des points P. Soient 1, ..., 6 ces six points, et de"signons
par w le point biquadratiquement oppose" a Q. La cubique (P) et la droite
(12, 13) composent une courbe du quatrieme ordre passant par les treize points
donnes. Done les points 15, 16 sont les deux points d'intersection (autrea que
12, 13) de la droite (12, 13) par la courbe du quatrieme ordre qui a pour faisceaux
ge'ne'rateurs le faisceau de cubiques (O) et le faisceau de droites (). Done enfin
les points 15, 16 sont les points doubles des deux divisions homographiques
determinees par les deux faisceaux sur la droite (12, 13).
Si douze des points donnas se trouvaient sur une mdme cubique, il faudrait
qu'un quelconque de ces douze points fut determine) par les onze autres de la
maniere que nous avons indique'e ; autrement on ne pourrait faire passer aucune
vraie courbe du quatrieme ordre par ces points. Mais en ce cas il faudrait aussi
qu'un quatorzieme point fut donne", afin de determiner completement le systeme
des seize points ; alors la construction des points 15 et 16 serait la m6me que ci-
dessus.
(4.) Lorsque quatre des treize points sont en ligne droite, on aura tout
d'abord une cubique passant par les trois points inconnus, puisque ces points
appartiendront dvidemment a la cubique des neuf autres points. Mais on peut
aussi ope"rer de la maniere suivante, sans faire usage de cette cubique. Soient
10, 11, 12, 13 les quatre points en ligne droite; les points p w , p llt p l2 , p l3 pour-
ront se determiner par la mdthode generate de 1'article 5. Ces points, ainsi que
les points d'intersection des droites (PI O , 11), (pu., 10), etc., appartiendront a la
conique (8, 9, 14, 15, 16), qui se trouvera ainsi completement de'termine'e. De
meme on pourra determiner la conique (7, 9, 14, 15, 16), dont les intersections
avec la premiere conique feront connaitre la solution du probleme.
7. II y a encore quelques cas particuliers du probleme qui ne sont pas
depourvus d'inte'ret, mais dont la discussion, d'ailleurs tres facile, depasserait les
limites que nous nous sommes prescrites. Mais nous ne saurions nous dispenser
de placer ici les observations suivantes qui serviront & e"claircir la solution g6n6-
rale.
(1.) La determination de la cubique (8, ..., 16) exige la connaissance de deux
Art. 7.] CUBIQUES ET BIQUADKATIQUES. 47
seulement des points p a , p 9 , ..., p 13 . En effet, quand on aura trouvd les deux
points p g et p 9 on aura neuf points de la cubique, puisque le point d'intersection
des droites (p g , 9), (p 9 , 8) appartient aussi a cette courbe. De plus, tant que le
probleme reste cubique, il ne peut arriver que les neuf points forment la base
d'un faisceau de courbes cubiques. Pour que le point d'intersection de (p s , 9),
(p g , 8) fut le neuvieme point appartenant & la base cubique 8, ..., 13, p s , p 9> il
faudrait que les six points p g , 9, ..., 13 appartinssent a une mme conique. Or
la conique (p s , 10, ..., 13) n'est autre que la conique qui satisfait a la relation
P 8 .[10, 11, 12, 13] = [10, 11, 12, 13];
en outre, x etant le point de cette conique qui verifie 1'equation
P 8 .[9, 10, 11, 12, 13] = [>, 10, 11, 12, 13],
p % sera le second point d'intersection de la droite (x, 9) avec la conique. Done,
si le point 9 appartient lui-mdme a la conique, p s viendra se confondre avec 9,
mais la position limite de la droite qui joindra ces deux points coincidents sera
toujours la droite (x, 9) ; d'ou il s'ensuit que les six points p g , 9, ..., 13 (dont les
deux premiers sont coincidents) ne peuvent pas etre census appartenir a une
m^me conique, a moins que le point x ne coincide avec 9. Mais si cela arrivait,
1'dquation
P 8 .[9, 10, 11, 12, 13] = [9, 10, 11, 12, 13]
serait satisfaite ; c'est-k-dire, le neuvieme point appartenant & la base P 8 serait
un des trois points cherche's, et la determination des deux autres ne serait que
quadratique.
On conclura aussi, de ce qui vient d'etre dit, que si p g , 1'un des deux points
opposes qu'on aura a determiner, venait k coincider avec 1'un des points 9, ..., 10,
la cubique (8, ..., 16) n'en serait pas moins completement determine'e, puisqu'on
aurait remplace' deux points par une tangente et son point de contact.
Nous ajouterons que si les deux points p g et p 9 , tout en restant determines
1'un et 1'autre, venaient a se confondre en un seul point p, ce point serait un des
trois points cherche's ; puisqu'en considdrant successivement les deux bases P 8 et
P 9 (qui auraient le me'me point oppose p, mais qui ne pourraient pas 6tre iden-
tiques, parceque nous avons suppose que la position du point oppose p n'est pas
indeterminee), on aurait deux courbes du quatrieme ordre, passant par les treize
points, et se coupant en outre au point p.
Done on conclura gdneralement qu'afin d'avoir les deux cubiques, dont on a
besoin pour determiner les trois points cherches, il suffira de trouver trois points
48 MEMOIRE SUE QUELQUES PBOBLEMES [Pt. III.
opposes biquadratiquement a trois systemes de huit points, choisis convenable-
ment parmi les treize points donnas.
(2.) Puisque les deux courbes cubiques (8, ..., 16) et (7, 9, ..., 16) se coupent
au point p s , il est Evident que p s est le neuvieme point qui appartient st la base
cubique 9, ..., 16. Nous aurons done le the*oreme que voici :
' Si Ton partage les seize points d'une base biquadratique en deux systemes
de huit points, le point biquadratiquement oppose" a Tun de ces systemes appar-
tient en mme temps a la base cubique de'termine'e par 1'autre systeme. La
courbe du faisceau qui passe par le neuvieme point appartenant a 1'un des deux
systemes, passe aussi par le neuvieme point appartenant a 1'autre systeme.'
8. Nous allons maintenant supposer que quelques uns des treize points de-
viennent imaginaires. II suffira de considdrer les deux cas (1) ou Ton n'aurait
que trois points rdels, (2) oil Ton n'aurait qu'un seul point re"el. Pour abre"ger,
nous supposerons que la position des treize points soit tout-a-fait ge'ne'rale, et
nous ne nous occuperons pas encore des circonstances spe"ciales que nous avons
conside're'es dans 1'article 6.
(1.) Supposons que 1, 2 ; 3, 4 ; 5, 6 ; 10, 11 ; 12, 13 soient des dyades de
points imaginaires, mais 7, 8, 9 soient des points re"els. La determination des
points a, b, c biquadratiquement opposes aux bases cubiques (1, ..., 6, 8, 9),
(1, ..., 6, 9, 7), (1, ..., 6, 7, 8), que nous de"signerons par A, B, C, se fera a peu
pres comme si les treize points e"taient tous re"els. En effet, d'apres ce que nous
avons dit dans la premiere partie de ce me'moire, on saura determiner (1) la co-
nique F de 1'article 5 ; (2) le point reel ,, et les dyades a> 10 a>n> W i2 ft) i3 appartenant a
cette conique et correspondant anharmoniquement a la cubique re"elle (C, 9) et
aux cubiques imaginaires conjugue'es (C, 10), (C, 11) et (C, 12), (C, 13) ; (3) la
conique qui passe par les points 10, ..., 13, et qui satisfait k liquation
[10, 11, 12, 13] = [co 10 , n , 12 , Wl3 ]=(7.[10, 11, 12, 13];
(4) le point 9' de cette conique qui vdrifie la relation
[9', 10, 11, 12, 13] = [o, 9 , o, 10 , co u , co 12 , o> 13 ]= (7. [9, 10, 11, 12, 13] ;
(5) enfin, le point c cherchd, oil la droite (9, 9') rencontre pour la seconde fois la
conique 2. Les points a, b, c une fois trouve"s, les cubiques (8, 9, 10, ..,16),
(9, 7, 10, ..., 16), (7, 8, 10, ..., 16), dont deux suffisent pour notre but, seront
completement de'termine'es. En effet, on connaitra neuf points, dont trois sont
re"els, de chacune de ces courbes ; et nous avons vu que ces neuf points ne peuvent
pas appartenir a une meme base cubique.
(2.) On e*tendra la solution au cas oh Ton n'aurait qu'un seul point reel 7, au
Art. 8.] CTJBIQTJES ET BIQUADRATIQTJES. 49
moyen des principes ge"neraux que nous avons e'tablis dans la premiere partie.
La cubique (8, ..., 16) sera reelle ; la determination de cette courbe se rdduira a
celle des points b, c qui seront des points imaginaires conjugue's. Pour les trou-
ver il faudra substituer dans la solution precedente la dyade 8, 9 aux deux points
reels que nous avons designes par les me'mes nombres. Quoiqu'en cette solution
il soit question de points imaginaires, elle ne consiste actuellement que d'une
certaine suite d'operations lindaires, portant sur des points et des droites rdelles.
Done, en substituant aux points re"els 8, 9 la dyade de points imaginaires, on
parviendra h, determiner lineairement les deux points imaginaires c, b biquadra-
tiquement opposes aux systemes 1, ..., 7, 8 et 1, ..., 7, 9 ; il est d'ailleurs Evident
que ces deux points appartiendront a la merne dyade, dont on connaitra 1'homo-
logie avec la dyade 8, 9. Le centre d'homologie des deux dyades appartiendra
lui-meme a la cubique cherchee ; il sera le seul point reel qu'on en connaitra.
Pour eviter la consideration de courbes cubiques imaginaires, on pourra substituer
successivement a la dyade 8, 9 les dyades 1, 2 et 3, 4 ; on aura ainsi trois courbes
cubiques reelles se coupant en quatre points connus. On determinera, par la
me'thode de M. Chasles, les trois coniques dont chacune passe par les cinq points
d'intersection inconnus de deux de ces trois courbes. Ces coniques se couperont
aux trois points cherche's, qu'on pourra des lors determiner par une construction
cubique.
VOL. II.
APPENDICE.
NOTE I (p. 3).
Descartes, Geometric, livre troisieme ((Euvres de Descartes, dd. Cousin, vol v. p. 409).
' Or, quand on est assure* que le probleme propose' est solide, soit que 1' Equation par
laquelle on le cherche monte au carre* de carre', soit qu'elle ne monte que jusques au cube,
on peut toujours en trouver la racine par 1'une des trois sections coniques, laquelle que ce
soit, ou meme par quelque partie de 1'une d'elles, tant petite qu'elle puisse etre, en ne se
servant au reste que de lignes droites et de cercles. Mais je me contenterai de donner une
regie ge'ne'rale pour les trouver toutes par le moyen d'une parabole, a cause qu'elle est en
quelque facon la plus simple.'
NOTE II (p. 3).
De la Hire, La construction des equations analytiques, opuscule qui fait partie (pp. 297-452) des
Nouveaux elements des sections coniques, Paris, 1679. Le probleme des normales d'une
section conique sert a 1'auteur pour exemple de la me'thode alg^brique ge'ne'rale.
Maclaurin, Treatise of Algebra, London, 1748, p. 352.
Joacbimsthal, Journal de Crelle, vol. xxvi, p. 172, vol. xlviii, p. 377. Voici la critique que
fait Joacbimsthal de la solution qui lui est propre, et de celle qu'il attribue a De la
Hire.
' On sait que chaque probleme qui depend d'une Equation du quatrieme iegre', se re'sout
par la regie et le compas, en supposant une seule section conique completement de'crite.
C'est sur ce principe que repose la solution du probleme en question, due a De la Hire.
Tandis que le gdometre grec qui a traite" le premier cette question, Apollonius de Perge, se
sert, outre la conique donne'e, d'une hyperbole equilatere, De la Hire ne fait usage que d'une
circonfe'rence de cercle, dont les intersections avec la courbe ont, a un facteur pres, les
mSmes abscisses que les pieds des normales. Mais cette me'thode offre plusieurs inconvd-
nients qu'il est bon de signaler. En premier lieu, telle que De la Hire 1'a represented, sa
Note HI.] APPENDICE. 51
methode n'est qu'un r^sultat de calcul, et ne se rattache & aucune proposition de geometric ;
ensuite le choix de 1'inconnue comporte n^cessairement une ambiguite, et, en dernier lieu,
les formules qui determinent la position et le rayon de la circonfe'rence sont trop compli-
quees pour se preter ais^ment aux constructions graphiques. On adressera peut-etre, et a
juste titre, ce dernier reproche egalement a la nouvelle solution qu'on va lire ; et ndanmoins
je n'hesite pas a la publier ; car, quoiqu'elle ne soit pas encore une solution definitive, les
propositions si simples sur lesquelles elle repose, pourront servir de point de depart pour
arriver a une solution purement g^ome'trique du probleme dont il s'agit.'
La solution, dont parle ici Joachimsthal, est due non pas a De la Hire, mais a M. E.
Catalan. II est vrai que M. Catalan ne parle que d'une reproduction avec quelques simpli-
fications de la solution ancienne (Nouvelles Annales de Maihematiqnes, per MM. Terquem et
Gerono, vol. vii, p. 332) ; mais il est pervenu a une solution qui est entierement diffe'rente
de celle de De la Hire, et qui nous a paru plus elegante, et surtout plus naturelle. Nous
n'avons pas cherche Interpretation g^om^trique des formules analytiques un peu compli-
que'es dont s'est servi De la Hire ; mais nous sommes parvenus a traiter par les me'thodes de
la geometric pure la solution de M. Catalan, aussi bien que celle de Joachimsthal. (Voir la
note VHI.)
NOTE III (p. 7).
Soient p^p v ^fo deux couples de 1'involution qui determine une dyade donnde Aj A 2 ;
nous dirons que cette dyade est represented par \_p\pv gifo]- Soit encore [a^a^i ^1^2] une
representation d'une seconde dyade HiH 2 ; et supposons que liquation anharmonique
[ft. Pz> 2i> &] = [], <*2> yi, 9y\
soit satisfaite. Cette Equation entraine ne'cessairement 1'une ou 1'autre des deux Equations
suivantes
[ft>.?>2> ?i. ?2> x i. A 2]=fo> 2 y\> y*> MI. F Z ].
[ft. PV ?i> ? 2 . A 2> A i]=[ aT i. x v y\> y*i MI. MB] ;
mais il importe d'observer que ces Equations ne sauraient avoir lieu toutes les deux
ensemble. Selon que la premiere ou la seconde Equation est satisfaite, nous dirons que les
elements imaginaires AjAj, y^y^, ou bien les elements imaginaires AgAj, f/ 1 /u 2 . sont repre"-
sent^s homographiquement paries systemes [Pip 2 , q\l2\> [ x i iK ^ V\y^- D'apres la definition
que nous avons donnde de 1'homologie des dyades, il est Evident que lorsqu'on connait 1'ho-
mologie de deux dyades, on connait en meme temps deux representations homographiques
des elements homologues de ces dyades. Et, rdciproquement, il rdsulte de la solution du
probleme fondamental de 1'article II, que Ton peut trouver lineairement le centre ou 1'axe
d'homologie de deux dyades de meme espece, dont les axes ou les centres ne sont pas coin-
cidents, lorsqu'on connait deux representations homographiques des elements imaginaires
qu'on veut regarder comme homologues dans ces deux dyades. De plus, on verra ci-apres
qu'en designant par a l a 2 , b-J)^, c^c^, trois dyades quelconques, on peut determiner lineaire-
ment deux representations homographiques de b^b^, c^c^, lorsqu'on connait deux representa-
tions homographiques de a l a z , d l 6 2 , et deux representations homographiques de a^, c l c 2 .
D'apres cela, on peut se servir des representations homographiques pour definir 1'homologie
des dyades, en disant que 1'homologie de deux dyades est donnee, lorsqu'on connait une
H 2
52 APPENDICE. [.Vote III.
representation homographique de ces dyades. Cette maniere de definir 1'homologie des
dyades a le double avantage de ne pas exiger 1'emploi de dyades auxiliaires, ni dans le cas
de deux dyades dont les axes ou les centres sont coincidents, ni dans le cas de deux dyades
d'espece diffe'rente, et de se preter facilement a la theorie des dyades de droites qui ne se
rencontrent pas dans 1'espace.
Ce que nous venons de dire se rattache imme'diatement a la belle theorie des imagi-
naires, qui a forme 1'objet principal des savantes recherches de 1'auteur de la Geometrie de
Situation *. Nous croyons faire plaisir aux lecteurs de 1'ouvrage de cet excellent ge'ometre,
en ajoutant les observations suivantes, qui feront voir combien nous nous sommes peu
de la route qu'il a trace'e.
(1.) Etant donne deux representations homographiques
des elements imaginaires AjA 2 , /Xjj^i et une autre representation quelconque [r l r 2 , #!#.,] de
la premiere de ces deux dyades ; on pourra determiner lineairement les elements !2> i\v z ,
qui satisfont a 1' Equation anharmonique
et on aura ainsi une nouvelle representation [u l v 2 , v l v 2 ] des elements ftj/s^. qui sera homo-
graphique a la representation [^ r. 2 , s l # 2 ] des elements AjAg. Soient done a^a. 2 , b l b 2 ,c l c. 2
trois dyades quelconques donnees ; [A] , [j?] deux representations homographiques des eie-
ments a^a^, \l^ ; [^j, [Cj] deux representations homographiques des elements a 1 a 2 > c i c 2'
On determinera une representation [<?] des elements c x c 2 qui soit homographique a la repre-
sentation [A] des elements z a 2 ; et on aura ainsi deux representations homographiques [.B],
[C], des elements ^b^ c l c 2 .
(2.) fitant donne une representation [pip. 2 , q\qz\ d'une dyade \^\ 2 , et un troisieme
couple quelconque ^ r 2 de 1'involution qui determine cette dyade, il existe toujours un qua-
trieme couple de cette meme involution, qui satisfait a liquation anharmonique
\.PV Pv ?!' &> X l> X 2] = [^ r 2> *1 *2' A l. X 2]'
On trouve lineairement ce quatrieme couple en prenant pour x^g^ les elements conjugues a
P 2 p l dans 1'involution determinee par q^r^ q-2, r \- On peut done trouver une infinite de
representations des elements imaginaires A a A 2 , homographiques a une representation donnee.
(3.) II convient d'attribuer un signe algebrique determine a chaque representation d'une
dyade ; ainsi nous dirons que la representation [Pip 2 , q\ ? 2 ] ^ e ^ a dyade A t A 2 est positive ou
negative, selon que le sens du mouvement continu indique par Piq^p 2 est positif ou negatif.
II s'ensuit de la que deux representations homographiques d'une meme dyade ont le meme
signe, ou des signes contraires, selon que les elements imaginaires se correspondent directe-
ment ou reciproquement dans les deux representations. En effet, en designant par
[.PiPf ?i?a]) [ r i r 2> *i*z] deux representations homographiques de la dyade AjA 2 , on a, dans
le premier de ces deux cas, liquation anharmonique
[/>i> Pv ?i> ?2> A u *2\=[ r \y r 2> *i *> A i> A a] 5
d'ou il s'ensuit que le'sens indique par p l qij'z est le meme que le sens indique par r l s 1 r 2 ,
* Staudt, Beitrage zur Geometric der Lage (Niirnbfrg, 1856).
> T ote III.] APPENDICE. 53
puisque deux elements correspondants, qui parcourent deux divisions homographiques, dont
les Elements doubles sont imaginaires, se mouvent toujours dans le meme sens. Done, en ce
premier cas, les deux representations sont de meme signe ; tandis que dans le second cas, on
aura 1' equation
[Pl> P2' 2l> &> A l> A 2] = l/l> f 2> *1 *2> A 2> A l]>
ou, ce qui est la meme chose, 1' Equation
Cette equation implique que le sens indiqud par p l q^ p 2 coincide avec le sens indique" par
* 1 r 1 # 2 ; d'ou Ton conclura que les deux sens indiquds par p l q l p z et r-^^r^ sont opposes, c'est
a dire, que les deux representations [pip. 2 , 3i? 2 ], \f-L r z> *i*zl son * ^ e s ig nes contraires.
(4.) Etant donne deux systemes ge'ome'triques, dont chacun consiste d'une s^rie d'eie-
ments simplement infinie, et dont 1'un depend lindairement de 1'autre, a chacun des deux
sens qu'on peut attribuer a la succession des elements dans le premier, il correspondra un
sens determine dans 1'autre. C'est ainsi qu'en rattachant chacun des deux elements d'une
meme dyade a 1'un des deux sens opposes que Ton peut concevoir dans le systeme geome-
trique auquel cette dyade appartient, de maniere a etablir une distinction fictive entre les
deux elements de la dyade, on etablit en meme temps une distinction correspondante entre
les deux elements d'une dyade quelconque lineairement derivee de la premiere ; puisque les
elements homologues des deux dyades peuvent etre censes appartenir aux sens correspon-
dants dans les deux systemes dont elles font partie.
Comme exemple tres particulier de cette remarque generale, considerons deux dyades
de points AjA 2 , ju a ^ 2 ayant le meme axe LM. Soient [p l p 2 , ?i? 2 ], \_ x \ x v ^i.^J deux repre-
sentations homographiques des elements imaginaires AjA 2 , juj^. Puisque ces deux repre-
sentations peuvent etre de meme signe ou de signes contraires, et puisque la correspondance
des elements imaginaires est differente dans les deux cas, on voit tres clairement dans ce cas
particulier, qu'en considerant les deux sens opposes dans lesquels un point peut decrire une
droite, on arrive a distinguer a priori, non pas entre les deux elements d'une meme dyade
(ce qui serait absurde), mais entre les deux manieres d'etablir 1'homologie de deux dyades.
D'ailleurs on peut observer que, dans ce meme cas particulier, il y a une autre distinction
tres marquee entre les deux manieres d'etablir 1'homologie des dyades donnees. En eiFet, si
les deux representations homographiques donnees des elements AjAg, /^M 2 son t de meme signe,
les points doubles de Tin volution (A^i, A, 2 /x 2 ) seront imaginaires, et les points doubles de
Tin volution (A^, A^) seront reels, tandis que, dans le cas contraire, les points doubles de
la premiere involution seront reels, et ceux de la seconde seront imaginaires. Pour demon-
trer cette assertion, prenons un point quelconque sur la circonference d'une conique, et de ce
point, comme centre de projection, projetons les points de la droite LM sur la courbe. Pour
plus de simplicite, nous designerons les projections de ces points par les memes lettres que
les points eux-memes. Les dyades Aj A 2 , ^ ^ 2 considers sur la conique, donnent lieu a deux
centres d'homologie, dont 1'un est interieur, et 1'autre exterieur a la courbe. Soit i2 le centre
interieur, et menons les cordes x^L^, y 2 !l 2 , y^ni, y^^\ [&&> fi^] sera une represen-
tation de la dyade A^, homographique a la representation [pip 2 , ft^] de cette meme
dyade, et aussi a la representation \x^x z , y^y^\ de la dyade /x^; de plus, le sens de ^m( a
est le meme que le sens de x 1 y 1 a; 2 , puisque i2 est un point interieur. Done la representation
54 APPKSTHCE. [Note IV.
[fi &> fi ''2] comporte le meme signe que la representation [ft p z , q^ j 2 ] , ou le signe contraire,
selon que les deux representations [a?i# 2 , y\yi\> [PiPi> ?i?z] son t de meme signe, ou de
signes contrairea. Par consequent [fjf 2 > lifj e * [-"i"^ y\y?\ son ' deux representations
homographiques des elements Aj A 2 , /"i /^ dans le premier cas, et des elements A 2 Aj , ^ jx, dans
le second. Mais cela revient a dire que li est le point d'intersection des cordes imaginaires
A 1 // 1 , A 2 /x 2 dans le premier cas, et des cordes Aj^, A 2 f/j dans le second. Et de la, en reve-
nant des points de la conique a ceux de la droite, on conclut immediatement la verite de la
proposition qu'il fallait demontrer.
NOTE IV (p. 7).
Les deux solutions que nous avons donnees de ce probleme ne different pas essentielle-
ment, puisque la droite, lieu des points d'intersection des couples de tangentes que Ton
considere dans la premiere solution, est en meme temps 1'axe des deux divisions homogra-
phiques que Ton obtient en echangeant entre eux sur 1'une des droites B, C, les deux points
de chaque couple de 1'involution qui determine la dyade sur cette droite. La solution
suivante, peu differente d'ailleurs des autres, nous parait aussi simple qu'on peut le desirer.
En se servant des centres d'homologie donnes on obtient deux representations homo-
graphiques [PiPz, /S'j/S'j], [y^, /i/ 2 ] des elements imaginaires 6 l b 2 , c 1 c z ; puis on deter-
mine les axes des quatre systemes homographiques que voici,
Les deux premieres de ces droites se croisent au point P ; les deux dernieres au point f.
Si le premier ou le second systeme devient homologique, le centre d'homologie est le point
P'; pareillement, si le troisieme ou le quatrieme systeme devient homologique, le centre
d'homologie est le point P.
On peut encore remarquer que si les dyades a^a^, b^b^ c l c z appartiennent a une meme
section conique, les six systemes de trois points, P'QR, PQ'R', PQ'Ji, P'QB', PQE', P'Q'R,
seront chacun en ligne droite.
NOTE V (p. 12).
Supposons seulement que 1'axe d'homologie A de a x 2 , a t a 2 soit donne. Faisons passer
une conique reelle 2 par Pn, et par la dyade de points determine par a 1 a 2 , ou OjOj, sur
1'axe A. Soient k^k^ K^K^ les dyades de points determines sur 2 par les dyades donnees
^i ^z> Pi J2- L'axe A passera par 1'intersection des droites imaginaires conjuguees k^ 2 , k 2 j ;
ce qui suffit pour faire voir qu'on peut distinguer lineairement entre les deux centres d'ho-
mologie de /fcj 2 , icj (c 2 , et pourtant entre les deux axes d'homologie de b l b 2 , /3j (8 2 . La con-
struction est entierement lineaire, puisque pour determiner les dyades k 1 k. i , l * 2 il n'est pas
necessaire de tracer la conique 2.
Note Till.] APPENDICE. 55
NOTE VI (p. 16).
La determination de la dyade Wjirg ne prdsente aucune difficult^ theorique. Soit
[a^a^, yy^ une representation donnde de la dyade Pip 2 ; <r une conique qui passe par cette
dyade. D'un point quelconque r^el de <r projetons sur cette conique 1'involution de'termine'e
par les coniquea du faisceau [c lt c 2 , d lt d 2 ~\ sur 1'axe de p^p^. Soit -rr le pole de 1'involution
qu'on aura ainsi sur la conique <r. Aux quatre rayons is. [x 1 , x 2 , y^ y.^\ il correspondra
anharmoniquement quatre coniques du faisceau. Soient 1( 2 > r] 1 , ?j 2 les points de la conique
auxiliaire C, qui correspondent anharmoniquement a ces quatre coniques ; [fj f 2 , ^ j 2 ] sera
une representation de la dyade Wj ir 2 , et cette representation sera homographique a la repre-
sentation donned [a^a^i V\V^ de ^ a dyade
NOTE VH (p. 22).
Toute conique du reseau circonscrit qui passe par F rencontre la droite A en deux
points hannoniquement conjugues par rapport a la conique cherchee. De meme, les deux
tangentes menees du point F a une conique du reseau inscrit, qui est tangente a la droite A,
sont deux droites conjuguees par rapport a cette meme conique du reseau harmonique. On
peut done trouver tres simplement 1'involution que determine cette conique, soit au point F,
soit sur la droite A ; et, d'apres ce que nous avons dit dans 1'article precedent, cela suffit
pour determiner le systeme polaire de la courbe.
Les expressions ' ellipse minima circonscrite,' ' ellipse maxima inscrite,' dont nous nous
sommes servis dans cet article, sont relatives au cas d'un triangle harmonique reel. Lorsque
ce triangle est imaginaire, les coniques polaires du centre des distances moyennes, et de la
droite a 1'infini, sont toutes les deux des hyperboles.
NOTE VIII (p. 32).
Soient X, Ties projections orthogonales de P sur les axes AA 2 , B^B^ respectivement ;
ST la tangente a F au point S. Menons la corde **' perpendiculaire a A 1 A 2 . D'apres la
definition de F, ST est le diametre de 2 conjugue aux cordes perpendiculaires a SP. Done
ST est parallele a A 2 s ; de plus, puisque 1'hyperbole F est equilatere, les droites bissectrices
de Tangle CST sont paralleles aux asymptotes de la courbe ; done SC est parallele a A 2 /.
On voit en meme temps que SC est le diametre de 2 conjugue aux cordes perpendiculaires a
XY, puisque XY et SP font des angles egaux avec les axes de 2. Mais la droite XY passe
elle-meme par C, puisqu'elle est une des diagonales du quadrilatere SPafi, inscrit a F. Done
C est le point d'intersection de XY avec le diametre de 2 conjugue aux cordes perpendicu-
laires a XY, ou, si Ton veut, avec le diametre parallele a A 2 s' ; et la polaire de C par rapport
a 2 est la perpendiculaire abaissee du pole de XY sur cette droite elle-meme. Ces determi-
nations nous seront utiles plus tard.
Puisque nous tenons a faire voir que la solution de Joachimsthal ne conduit pas a des
operations impraticables a cause de leur longueur, nous indiquerons ici la maniere de les
effectuer, qui nous parait la plus simple. On supposera connus les axes et 1'un des foyers H
de la conique centrale 2 ; on prendra pour A^ A z 1'axe focal. En se servant d'un equerre, on
56 APPENDICE. [Note VHI.
menera la corde A 1 * perpendiculaire a PS, la corde A 2 p perpendiculaire a A l #, la corde ##'
perpendiculaire a A 1 A 2 ; enfin, la normale et la tangente a 2 au point * (on salt quo cela
peut se faire tres simplement avec I'dquerre). Soit ju le point d'intersection de 1'axe focal
avec la normale ; a-, tj les projections orthogonales de S, II sur la tangente, Y la projection
orthogonale de P sur 1'axe conjugud, Y l le point de ce meme axe pour lequel Tangle YHY 1
cat droit ; enfin, A^p^ dtant mende perpendiculaire a PY l , soit y le point d'intersection des
cordes p 1 p. 2 > *-^> e * " 1* projection orthogonale de y sur **'. Le centre to du cercle de
Joachimsthal sera le point d'intersection des droites Op, Sv ; le rayon de ce cercle sera o>jj.
On de'crira le cercle, et on joindra les points d'intersection des deux courbes au point A l par
des cordes A 1 x: les normales issues du point P seront perpendiculaires a ces cordes, et leurs
pieds se trouveront sur les diainfetres de 2 paralleles aux cordes A 2 x. D'apres ce que nous
avons dit, on verifiera facilement les details de cette construction, dans laquelle on n'aura
besoin du compas qu'au moment ou Ton veut tracer le cercle ii. En efiet, puisque D se
transforme en a, et C en ft, les axes principaux de la conique transformed 2' sont les droites
qui correspondent au diametre de 2 qui passe par C, et a la polaire de C. L'un de ces deux
axes est *#'; c'est celui qui correspond a SC; 1'autre est yd, puisque y correspond au point
YI, qui se trouve sur la polaire de C. Done 6 est le centre de 2'; et, puisque la perpendicu-
laire abaisse'e de 6 sur sa polaire par rapport a 2 doit passer par to, et aussi par p., la con-
struction du point to se trouve justifie'e.
On remarquera qu'en ge'ne'ral, e'tant donne' un point quelconque Z, pour en trouver le
point correspondant dans la figure transformed, on menera les cordes A l p l , A 2 s 1 , dont la
premiere est perpendiculaire a PZ, et la seconde est parallele a SZ; les cordes *# 1; pp t se
croiseront au point z. Et, en eflet, c'est ainsi que nous avons determine' les points / et y,
correspondant aux points F et Y lt dans les constructions pre'ce'dentes.
La solution analytique du probleme qui nous occupe, donne'e il y a presque deux cents
ans par De la Hire, a dtd 1'objet de recherches int^ressantes par M. E. Catalan (Nouvelles
Annulet de Matkematiques par MM. Terquem et Gerono, Vol. vii. p. 332 et 396, annde 1848).
Nous aliens voir que la solution de M. Catalan, aussi bien que celle de Joachimsthal, se
de'duit naturellement de la me'thode dont nous nous sommes servis.
Soient j, n 2 , n 3 , 4 les pieds des normales abaissdes de P sur 2; a ces quatre points
M. Catalan substitue quatre autres n'i,n' 2 ,n' 3 ,n' 4> qui sont les points d'intersection de 2 par
une circonfeYence de cercle, et dont les abscisses, mesur^es du centre de 2 sur 1'un des deux
axes, sont proportionnelles aux abscisses de n lt n 2 ,n 3 ,n t *. Soient toujours A 1 A Z , S^S^ les
axes de 2 ; a, /3 les points a 1'infini sur ces axes respectivement ; en supposant que A l A. 2
soit 1'axe des abscisses, on aura I'e'quation anharmonique
ft.[a,S,n 1 ,n 2 ,n 3 ,n t ]=ft.[a > S,)i\,n' 2 ,n,' a ,n 4 ] (A)
Qu'on transforme la figure de maniere qu'aux points n de la figure donne'e correspondent
les points n' de la nouvelle figure. Puisque les points /3, n lt n 2 , n s , n f appartiennent a la
* Dans la solution plus compliqu^e de De la Hire ce sont les ordonne"es des points n qu'on fait proportionnelles
aux abscisses des points . De plus, au lieu du centre de la conique, on prend pour origine un point tel que les
sommes des ordonne'es des points ', et des abscisses des points n, s'cVanouissent se'pare'ment. On voit qu'il en doit
nSsulter une construction ge'ome'trique entierement diffe'rente de celle que nous allons diJduire de 1'analyse de M. Catalan.
D'ailleurs, nous avons reconnu que la solution de De la Hire ne s'applique pas au cas on il y aurait quatre normales
rtellos : niais, uialgru cet inconvenient, cette solution nous parait un'ritor une otmlc plus approfondie.
Note VIII.] APPENDICE. 57
conique F, on conclut de I'dquation (A), que /3 appartiendra a r', la transformed de F. Or,
r' ne peut etre qu'une parabole, dont 1'axe est parallele a B 1 B 2 . Car le faisceau (r", 2),
dont les points n' forment la base, contient par hypotbese un cercle ; done a/3 sont les points
doubles de 1'involution que ce faisceau determine sur la ligne droite a 1'infini, et la conique
du faisceau, qui passe par /3, y touche cette droite. Cela pose", il s'ensuit de 1'dquation (A)
que le point /3 de F' correspond au point a de T ; done a se transforme en /3, et 1'asymptote
Co. devient la droite a 1'infini a/3. Soient , b les projections de C sur les axes A l A.,, B^ B 2 ;
a se transformant en /3, b se transforme en a, puisque a b, /3 a sont des points re'ciproques par
rapport aux faisceaux (2, F), (2, F"), respectivement. De plus, d'apres 1'^quation (A), le
point 5 doit se transformer en un point 5' situe* sur {3S; done A 1 A 2 , ou 8 a, se transforme en
S'fi, ou _3j B 2 . Soit 2j la conique du faisceau (2, F) qui se transforme en 2 ; il faut que pour
cette conique les droites aS, aC soient des droites conjugue'es. Cette condition determine I I
sans ambiguit^, puisque, des deux coniques du faisceau (2, F) qui y satisfont, 1'une est F,
qui ne se transforme pas en 2. En de*signant par X l Y^ les points d'intersection des axes de
2 avec la polaire de C, soient <?, x les points place's syme'triquement a C, X lt par rapport a
1'axe S ly 2 , y le point place* syme'triquement a }\ par rapport a 1'axe A l A 2 . La conique du
faisceau (2, F) pour laquelle C est le pole de xy ne peut etre autre que 2 t . Car la polaire de
c par rapport a F est 1'axe A 1 A 2 , puisque Sc est tangent a T au point S; et la polaire de c
par rapport a 2 doit passer par x, a cause de la situation symdtrique des points CX l , ex ;
done c, x sont des points re'ciproques par rapport au faisceau (2, F), et la conique de ce fai-
sceau par rapport a laquelle C est pole de xy, aura Cc, ou Ca, pour polaire de x ; c'est a dire
que aS, aC seront des droites conjugue'es par rapport a cette conique. II re'sulte de la que
a l>x sera un triangle harmonique par rapport a 2 1( et que pour satisfaire a toutes les con-
ditions du probleme, il suffira de transformer 2j en 2, de maniere que le triangle alx de-
vienne le triangle /3a& Soit (1) 2 une ellipse ; les involutions determines par le faisceau
(2, T) sur les axes de 2 ont eVidemment des points doubles imaginaires ; done toute conique
du faisceau rencontre ces droites en deux points re"els. Soient Aj A 2 les points d'intersection
de 2j avec A 1 J 2 ; x sera le point milieu du segment A-, A 2 ; mais S, qui est le centre de
1'involution, se trouvera aussi sur ce mme segment ; done les droites 6S, lx seront situ^es
dans le meme angle forme' par les droites 6\ l , >A 2 , tangentes 2 t en Aj, A 2 . Mais 1'axe bS
rencontre 2 t en deux points re"els; done aussi Ix rencontre 2j en deux points re*els v l v 2 .
D'ailleurs, les points AjA 2 , v r v 2 sont quatre points harmoniques de 2 : ; done en transformant
homographiquement v lt v 2 en A l , A 2 , et A 1; A 2 en H lt B 2 (ce qui peut se faire par une quel-
conque de quatre transformations diff^rentes), on transformera 2, en 2, abx en fia.S, F en une
parabole ayant son axe parallele a l 2 , et, enfin, les quatre points n en quatre points n'
situ^s sur la circonfe'rence d'un cercle et satisfaisant a Fe'quation (A). Quelque soit 1'axe
qu'on a choisi pour A 1 A 2> ce sera toujours la meme conique 2 t qui se transformera en 2 ;
puisque la definition que nous avons trouve" pour 2j est syme"trique par rapport aux deux
axes. Et Ton peut ajouter que dans les quatre transformations, relatives a un meme axe, ce
sera toujours la mSine conique qui se transformera en un cercle, et que les quatre cercles
resultants, ainsi que les quatre paraboles F', seront place's syme'triquement par rapport aux
axes principaux de 2. Si (2) 2 est une hyperbole, on voit d'abord qu'il faudra prendre
pour A l A. 2 1'axe qui ne rencontre pas la courbe. Car si les sommets A { A 2 dtaient rdels, les
points /iA 2 seraient rdels, comme dans le cas de 1'ellipse, et il faudrait transformer deux
points reels AjA 2 en deux points imaginaires B 1 S 2 . On doit done supposerque les sommets
VOL. II. I
58 APPENDICE. [.Vote VIII.
A l A* sont imaginaires ; en ce cas, on aura a transformer lea points A t \ 2 en deux points re'els
-B, B^ mais, pour que les points A t A 2 restent eux-memes re'els, il faudra que Sjf soit plus
grand que SA*. Qu'on mene des perpendiculaires aux asymptotes do 2 par les points ou
ces droites rencontrent la tangente a 1'un des sommets B l , _B 2 ; et qu'ensuite par les points
d' intersection de ces perpendiculaires avec 1'axe conjugue' on mene des paralleles a 1'axe
focal. La condition ci-dessus revient a dire que le point P doit etre compris entre les deux
paralleles qu'on a trace'es. D'ailleurs, cette limitation de la me'thode de M. Catalan rdsulte
clairement des fonnules analytiques dont il 1'a fait de'pendre. Lorsque la condition de pos-
sibilite' est satisfaite, les angles A 1 ^A 2 ,/Ste empietent 1'un sur 1'autre ; mais 1'axe IS ren-
contre la conique 2, en deux points re'els ; done Ix ne la rencontre pas, et I a la coupe en
deux points re'els MI/V En faisant correspondre A t A 2 a B 1 B 2 , et ^^ 2 aux deux points a
1'infini sur les asymptotes de 2, on aura quatre transformations diffe'rentes, dont chacune
pourra servir pour la solution du probleme.
Lorsqu'on veut faire usage de cette me'thode, on commencera par la determination de
C, et de la polaire de ce point par rapport a 2 ; on aura ainsi les points a, d, c, x. Pour
avoir les points Aj A 2 , on projettera du point B l sur 1'axe A l A 2 les deux extre'mite's du
diametre de 2 conjugue' a la corde l x. De meme, si 2 est une ellipse, on trouvera 4> l <J> 2 ,
les deux points d'intersection de B 1 B 2 avec 2 t , en projetant du point A l sur B l B i les ex-
tre'mite's du diametre parallele a A^ b. En faisant correspondre A x a B l} A 2 a B z , on prendra
sur S l B 2 les points a', S', qui satisfont a liquation anharmonique
[a, A 1; A 2 , a, S] = [ft, S lt -8 2 , ', S'],
et Ton menera par S' une parallele a A l A 2 . Cette parallele correspond a B l B 2 ; elle ren-
contre 2 en deux points re'els <f>' l <(> f 2 , correspondant a <j>i<}> 2 . On fixera a volont^ la cor-
respondance de <p\ <f>' 2 , <t>i<t> 2 , et on prendra le point /3' qui satisfait a I'dquation
[S, <t> lt <f> 2 , fi] = [S', 4>\, <f>' 2> ft],
et qui, par consequent, correspond a ft. Le point C', correspondant a C, est eVidemment le
point a 1'infini sur la droite a'/3'; done D', qui correspond a D, et qui est le point rdciproque
de C' par rapport au faisceau (2, T"), se trouve a 1'intersection de /3'/3 avec le diametre de 2
conjugue" a a' '/?. Du point If abaissons sur a'j3' une perpendiculaire ; soit to le point d'in-
tersection de cette perpendiculaire avec une droite So>, qui fait avec 1'un des axes princi-
paux le memo angle que la perpendiculaire, mais de 1'autre cote de cet axe. Le cercle, dont
oj est le centre, et qui coupe orthogonalernent le cercle dont off? est le diametre, appartient
au faisceau transform^. C'est ce qu'on veYifie en observant que ex, aft, CD sont des couples
de points r^ciproques par rapport au faisceau (2, T), et qu'en ddsignant par c le point a
1'infini harmoniquement conjugue' a C' par rapport a a/3, les points correspondants c'S, aft',
C'D', sont dgalement des couples de points rdciproques par rapport au faisceau (2, r").
Pour avoir le rapport des abscisses des points n' aux abscisses des points correspondants
, projetons P sur les deux axes. En d&ignant, comme nous avons fait plus haut, ces pro-
jections par X, Y, prenons Y' le point correspondant a Y; le rapport cherche' sera celui de
S'Y' a SX.
Avant de terminer cette longue note, nous indiquerons une troisieme mdthode, qui ne
s'applique pas a 1'hyperbole, mais qui conduit a une solution assez simple pour le cas de
1'ellipse. Soient a t a 2> 6 t 6 2 les diametres conjugu^s ^gaux de 1'ellipse 2. Qu'on transforme
Note IX.] APPENDICE. 59
la courbe en elle-meme, de maniere que, la ligne a 1'infini restant la meme, les axes A l A 2 ,
B 1 B 2 deviennent les diametres e'gaux a^a^, b 1 b 2 . C'est ce qu'on peut faire par quatre
transformations diffe'rentes, en supposant, pour abreger, qu'on e'change entre eux les points
a 1'infini de la courbe. Mais quelle soit la transformation qu'on choisit, le faisceau, qui cor-
respond au faisceau (2, T), contiendra un cercle, puisque les points doubles de 1'involution
que ce faisceau de'terminera sur la ligne droite a 1'infini, seront les points rectangulaires a/3.
Soient done a l a 2 , b^ b 2 les points qui correspondent a A t A 2 , JB 1 B 2 respectivement ; et soit X
un point donne" de la courbe. On trouvera le point correspondant # en menant la corde Xx
parallele a A l a l ; on aura ainsi le diametre Sx correspondant au diametre SX; de plus, la
droite qui joint deux points correspondants de ces diametres sera parallele a Xx ; done on
pourra trouver tres facilement, dans 1'une des deux figures, le point correspondant a un
point donne" de 1'autre. Soient toujours a, b les projections orthogonales de C sur les axes
A l A 2 , B : B. 2 respectivement, a, V les points correspondants dans la nouvelle figure. Qu'on
abaisse de a', b' des perpendiculaires sur b l b 2 , a x a 2 respectivement ; le point de concours de
ces droites sera le centre o> du cercle li qu'on cherche. Avec le rayon Sa lt de'crivons un
cercle concentrique a 2 ; soit d l d 2 le diametre de ce cercle qui fait un angle droit avec S<a ;
les points d lt d. 2 appartiendront a la circonfe'rence de ii. On tire cette derniere conclusion
d'un the'oreme qui n'est qu'un cas particulier d'un autre plus ge'ne'ral, mais qui vaut la peine
d'etre enonce' :
'Toute hyperbole, ayant ses asymptotes paralleles a Sa lt Sb lt et passant par S, ddter-
mine par ses intersections avec 2 une circonfe'renee de cercle, qui coupe orthogonalement le
cercle imaginaire, dont S eat le centre, et Sa\ le carre" du rayon.'
On peut encore remarquer que la somme des angles excentriques de deux points cor-
respondants x et X est e'gale a un multiple impair de -a. Cela ve'rifie que les points cor-
respondants aux pieds des normales appartiennent a une meme circonfe'rence.
NOTE IX (p. 35).
La the'orie des correspondances (1), (2), (3), a 6t6 e'tudie'e par M. Battaglini, dans un
excellent Mdmoire (' Suite forme binarie dei primi quattro gradi, appartenenti ad una forma lernaria
quadratica,' Giornale di Matematiche, vol. v. p. 39), qui, malheureusement, nous dtait encore
inconnu, lorsque nous ecrivions 1'article pr^c^dent. Cependant, nous placerons ici quelques
remarques additionnelles, qui ne sont pas sans inte'ret pour les constructions grapbiques.
La the'orie g^om^trique de la correspondance (2) peut etre pr^sentfe de la maniere
suivante. Soient (A) et (B} deux systemes correlatifs dans le plan d'une conique donne'e 2 ;
et soient X et Y les coniques qui correspondent dans les systemes A et B a la conique 2,
conside're'e comme appartenant aux systemes et A respectivement. Soit la conique des
poles des deux systemes correlatifs, c'est a dire, la conique lieu des points qui se trouvent
sur leurs droites correlatives ; de meme soit ()' la conique des polaires, ou la conique enve-
loppe des droites qui passent par leurs points correlatifs. En conside'rant un point quel-
conque y de 2 comme appartenant au second systeme, la droite correlative rencontrera 2
en deux points x, qui seront lie's au point y, par une Equation de la forme (2). D'apres cela,
on aura les thdoremes suivants qui donnent immddiatement la solution des problemes bi-
quadratiques dependant de la correspondance (2).
I 2
60 APPENDICE. [Note IX.
(1!) Les points d'intersection de X et 2 sont les quatre points x pour lesquels il y a
coincidence des points y correspondants : et les quatre points de contact avec 2 des tan-
gentes communes a Y et 2 sont les points y, qui sont devenus coincidents. La correlation
des deux systemes (A) et (B) fera connaitre 1'un des deux systemes de quatre points lors-
qu'on aura trouve" 1'autre. Pareillement, les points d'intersection de Y et 2, et les points de
contact avec 2 des tangentes communes a X et 2, sont respectivement les points y dont les
correspondants coincident, et les points coincidents eux-memes.
(2.) Les quatre points d'intersection de et 2 sont des points de coincidence d'un
point x avec 1'un des points correspondants y, et les tangentes mene'es a 0' de 1'un quelcon-
que de ces quatre points (tangentes dont 1'une est aussi tangente a X, et 1'autre a Y),
rencontrent la conique 2 en deux points, qui sont les points correspondants a 0, autres que
6 lui-meme.
On remarquera qu'une seule construction biquadratique suffit pour trouver, soit les
points x 1 x 2 , soit les points y t y 2 , qui deviennent coincidents, soit enfin les points qui cor-
respondent a ces points coincidents dans chacun des deux systemes ; puisque, ayant trouve"
les points d'intersection de deux coniques, on n'a besoin que d'une construction quadratique
pour trouver leurs tangentes communes. Mais, pour trouver les points x, qui coincident
avec un de leurs points correspondants y, il faut une construction biquadratique nouvelle.
Le probleme line'aire ' Etant dorms' huit points x, et un point y correspondant a chacun
de ces points, trouver la droite y l y 2 correspondante a un point quelconque x ' peut s'enoncer
plus ge'ne'ralement de la maniere suivante, 'Etant donne* huit points dans 1'une de deux
figures correlatives, et huit autres points situs's respectivement sur les droites correlatives
des premiers points, determiner la correlation des deux figures.' Or, c'est de ce probleme
que depend (ainsi que 1'ont fait voir MM. Seydewitz et Schroter) la construction de la
surface du second ordre qui passe par neuf points donne*s. On en trouvera la solution com-
plete dans le M^moire de M. Schroter (Journal de Crelle-Borchardt, vol. Ixii. p. 215).
Si liquation (2) est syme'trique, les coniques X et Y coincident, de meme que les
coniques et ', et les deux systemes (A) et (B) deviennent polaires re"ciproques par
rapport a et 0'. En ce cas particulier la solution du probleme line'aire est tout-a-fait
tSldmentaire.
En passant maintenant a la correspondance cubique, ddfinie par I'e'quation (3), suppo-
sons que yy'y" soient les trois points correspondants a un meme point x, YY' Y" le triangle
des tangentes a 2 en ces trois points. L'axe d'homologie des triangles yy'y", YY'Y"
enveloppera une section conique C l ; pareillement, le lieu du centre d'homologie de ces deux
triangles sera une seconde conique Gj, polaire re'ciproque de C l par rapport a 2. Soit, de
plus, a la conique inscrite aux triangles yy'y", ii la conique par rapport a laquelle ces mSmes
triangles sont harmoniques, de sorte que ii est une des coniques r^ciproquantes de 2 et a.
Les trois coniques li, C 2 , 2 ont les memes points d'intersection ; ces points sont en meme
temps les points de contact avec 2 des tangentes communes aux trois courbes <r, C l , 2 ; de
plus, la tangente a ii en un quelconque de ces points rencontre 2, pour la seconde fois, en
un des points d'intersection de 2 avec a, et y est tangente a cette derniere conique. Les
points de coincidence de deux des points y, qui correspondent a un meme point x, sont
e'videmment les points de contact avec 2 des tangentes communes a 2 et a- ; d'ou Ton voit
que, pour trouver ces points, il suffit de connaitre 1'une quelconque des quatre coniques
auxiliaires C lt C z , <r, Q.. Les axes d'homologie des triangles yy'y" et YY'Y", considers
Note IX.] APPENDICE. 61
comme des tangentes a C 1 , et les centres d'homologie de ces m&nes triangles, considers
comme des points de C 2 , correspondent anharmoniquement soit aux triangles yy'y", soit aux
points x ; cette observation servira pour determiner les points x qui correspondent aux
quatre triangles eVanescents que nous venons de trouver. La determination des points x,
qui coincident avec un des points correspondants y, se fait un peu diffeYemment. Solent P,
Q deux points quelconques de 2, P' un point qui n'appartient pas a cette conique. Les
coniques du faisceau (P, P', y, /, y") correspondront anharmoniquement aux points x ; par
consequent, le lieu geome"trique des intersections des droites Qx et des coniques correspon-
dantes (P, P", y, y', y") sera une courbe cubique, qui passera par les points P et Q, et qui, en
outre, rencontrera 2 en quatre points, qui sont ceux qu'on cherche. On les determinera en
se servant de la construction biquadratique indiqude par M. Chasles.
Comme verification des resultats precedents, nous ajouterons quelques unes des princi-
pales formules analytiques qui se rattachent a la theorie des correspondances (2) et (3).
Soient p, q, r les coordonnees homogenes d'un point quelconque du plan que Ton considers ;
on prendra I'equation^r q z = pour 1'equation de la conique 2, et on representera par 0j 2 ,
6 l 2 , 2 2 les coordonnees d'un point quelconque de cette conique. Les deux systemes cor-
reiatifs, dont depend la correspondance (2), seront definis par 1'equation
2 ) = ;
et en mettant dans cette equation les coordonnees du point 6, soit pour p 1 q l r l , soit pour
P 2 q 2 r 2 , on aura 1'equation, soit de la droite r^Jfe) s it de la droite a f 2 - Done 1'equation de
la conique A", enveloppe de x 2 , sera
et, pareillement, 1'equation de Y, enveloppe de T?!^, sera
(A l p + ZSrf + Ci r) (A 3 p + 2S 3 q+ C 3 r) = (A
Enfin, on aura pour la conique des poles 1'equation
et pour la conique des polaires 1'equation
0'=4A0-<I> 2 =0,
en designant par A le determinant
*
et par 4> la fonction lineaire
Passons a la correspondance (3). Soit toujours 2 = prq 2 = 1'equation de la conique
2, et designons par P, Q, R, S, T, U les determinants du systeme
62 APPENDICE. [Note IX.
pris dans leur ordrc naturel, et par
flj , I/! , C^,
a !2 "J2> C 12>
2 > *2 C 2
les neuf quantity's
Solent, de plus,
\2 x X
A l> A l A 2'
i, Mi Ma, M|,
les coordonn&s de deux des trois points y qui correspondent & un meme point x. En elimi-
nant , et x 2 , et divisant par 3(A 1 ft-j A 2 Mi)> on aura I'e'quation suivante :
qui est celle d'une correspondance quadi-atique double. Cette correspondance est ^videm-
ment syme'trique ; elle est aussi triangulaire, puisque, la fonction
E=PU+KS-QT
dtant identiquement zeYo, la conique enveloppe de la droite qui joint les deux points A cor-
respondants a un point p donnd, c'est a dire la conique
ne differe pas de la conique enveloppe de la droite qui joint le point p a 1'un ou 1'autre des
deux points correspondants A ; en efiet, on trouve pour 1'dquation de cette derniere conique
Cela posd, la conique <r est la conique inscrite a tous les triangles du systeme ; et
=
est liquation de la conique par rapport a laquelle ces memes triangles sont des triangles
harmoniques. Liquation
exprime la correspondance doublement quadratique, mais non symdtrique, qui a lieu entre
un point donne" x et les deux points covariants du triangle correspondant. Done 1' Equation
de la conique C lt enveloppe de la corde qui joint ces deux points, sera
c i = 4(a lj p + ^?H-c 1 r)(a 2 ^+ 2 ? + e 2 r)-(a 12j p + S 12 ^ + <? 12 r) 2 = 0;
et on aura 1'expression suivante pour liquation de la conique C 2 , polaire rdciproque de
par rapport a 2,
Note X.]
APPENDICE.
63
r, j 2 , a z
12 ,
P,
P,
P, c\, %
= 0.
Chaque terme de cette derniere Equation est divisible par le determinant
D
i >
en supprimant ce facteur constant, on a 1'expression plus simple :
C. 2 = 30 + (R-3S) 2 = 0.
Les fonctions D et .R 35 sont tres connues; 1'eVanescence de la premiere implique que les
axes d'homologie des triangles yy'y", YY'Y" passent tous par un mme point, et que le
systeme contient un triangle dont les trois sommets se confondent en un seul ; 1'eVanescence
de la seconde exprime que deux triangles quelconques du systeme sont harmoniquement
conjugue's 1'un a 1'autre, et que, par consequent, le systeme donne" coincide avec le systeme
harinonique correspondant. (Voir le Me"moire de M. Battaglini, pp. 44-49.)
NOTE X (p. 41).
Si les sept points 1, ..., 7, appartiennent tous a une meme conique, la determination de
la cubique (9, ..., 16) ne reussit pas. En effet, dans ce cas, le point biquadratiquement
opposd au systeme (1, ..., 7, a) est le point a lui-meme ; ou, plus exactement, il n'y a aucun
point biquadratiquement oppose a ce systeme, puisqu'il n'y a aucune courbe biquadratique
passant par les treize points, et ayant (1,..., 7, a) pour base de courbes cubiques generatrices.
II faudra done, dans cette hypothese particuliere, eviter de faire usage de la cubique
(9, ..., 16), ce qui sera toujours possible.
Si les dix points 1, ..., 10 appartiennent a une meme cubique, la determination de la
cubique (9, ..., 16) devient illusoire, et doit etre remplacee par une autre, puisque, dans ce
cas, les trois systemes de points (/> 8) 9, 10), (p g , 10, 8), (JB IO , 8, 9), sont respectivement en
ligne droite ; d'ou il resulte que 1'equation anharmonique
ft . [x, 11, 9, 10] x p 9 . [x, 11, 10, 8] x JB IO . [x, 11, 8, 9] = + 1
devient identique quel que soit le point x, et ne peut servir a definir aucun lieu geometrique.
La meme chose arriverait si les droites p^p w , PwfT> PsPa passaient par les points 8, 9, 10
respectivement. Pour eviter cet inconvenient, on prendra arbitrairement les points 8, 9, et
on determinera les points p s , jo 9 avant de choisir le point 10. La droite jp s , jo 9 peut bien
passer par un des points 10, ..., 13, mais elle ne peut pas passer par deux de ces points a, /3,
puisqu'il s'ensuivrait de cette supposition que les neuf points 1, ..., 7, a, ft ferment la base
d'un faisceau de courbes cubiques, ou bien que les onze points 1, ..., 9, a, @ appartiennent a
une meme cubique. II y aura done au moins trois des points 10, ..., 13, qui ne se trouvent
pas sur la droite p s p t ; de plus, de ces trois points il y aura au moins deux qui ne peuvent
pas appartenir a la cubique (1, ..., 9) ; on prendra a volonte 1'un ou 1'autre pour le point 10.
II correspond une analyse tres simple a la demonstration geometrique du theoreme
64 APPENDICE. [Note X.
de cet article. Solent (a, 6), (a, b, c, d, e), ..., les fonctions algeljriques qui, e'gale'es a ze*ro,
donnent les equations de la droite (a, b), de la conique (a, b, c, d,e),..., et ainsi de suite.
En d&ignant par X une constante inde'termine'e, les courbes cubiques du faisceau
(1,...,8,9) = \(1,...,8, 10);
correspondent anharmoniquement aux droites
(ft, 9) X (ft, 10) j
et puisque les fonctions (P 8 , 9), (/>, 9), ..., qui entrent dans ces Equations, peuvent etre
multiplies par des constantes quelconques, on peut faire en sorte quo la droite (p 9 , 11) cor-
responde a la cubique (P s , 11). Cela pose",
(1, ...,7,8, 9) (ft, 10) =(!,.. .,7, 8, 10) (ft, 9) ...... (a)
sera I'e'quation d'une courbe biquadratique qui passe par le point jo g et par les treize points
donnas. Pareillement, les courbes biquadratiques
(l,...,7,9,10)(ft, 8) = (!,.. .,7, 9, 8) ( A , 10) ....... (4)
(1,...,7, 10,8) ( Ao , 9) = (!,... ,7, 10,9) ( Ao , 8) ...... (c)
passeront par les memes treize points, et par les points /, p w> respectivement. En multi-
pliant ces trois Equations, membre a membre, et divisant par le produit
on aura 1'e^uation cubique
(A, 10) (ft, 8) (ft,, 9) = (ft, 9) ( ft , 10) (ft,, 8) ....... (d)
II rdsulte du choix que nous avons fait du point 10, que les deux membres de cette Equation
ne peuvent pas etre identiques, a -moms que le triangle Pg,p 9 ,p 10 ne coincide avec le triangle
8, 9, 10. Mais, en supposant toujours que les sept points 1, ..., 7 n'appartiennent pas tous
a une m6me conique, on remarquera que, si cette coincidence a lieu, les courbes biquadra-
tiques (a), (b), (c) doivent avoir des points doubles aux points 8, 9, 10 respectivement ; c'est a
dire, que les trois points chercbe's coincident avec ces memes points, et que toutes les courbes
biquadratiques qui passent par les treize points ont en ces trois points des tangentes com-
munes, dont on trouve facilement la direction, en se servant des courbes (a), (b), (c). En
revenant done au cas ge'ne'ral, 1'^quation (d) sera 1'^quation d'une courbe cubique, qui passe
eVidemment par les six points 8, 9, 10, p s , p g , p lo et par les trois points d'intersection de
(p s , 10), (j 10 , 8), (ft, 9); (p 10 , 9), (p 9 , 10). Mais cette courbe passe aussi par les points
11, ..., 16. Soit un de ces points; la chose est eVidente, si n'appartient a aucune des
courbes cubiques
(1,...,7, 8,9), (1,..., 7,9, 10), (1,..., 7, 10,8);
ensuite, si appartient a une seulement de ces courbes, par exemple a la premiere, sera le
point d'intersection de (p 9 , 8) et (p B , 9), et ne cessera point d'appartenir a (d) ; enfin, si
appartient a la fois a deux des memes courbes cubiques (ce qui par hypotbese De peut pas
arriver, a moins que ne soit un des trois points inconnus), sera le neuvieme point appar-
tenant a une base cubique donne'e, et pourra etre de'termine' line"aireinent, sans qu'il soit
ne"cessaire de chercher la cubique (d).
Note X.] APPENDICE. 65
II est assez remarquable quo la solution que nous avons donnde du probleme de cet
article, s'applique aussi a cet autre probleme plus general :
' Etant donne 4 3 des 4 points d'intersection d'une courbe d'ordre n avec une courbe
biquadratique, trouver les trois autres points.' On suppose n > 4.
Prenons 4 8 des 4w 3 points donne"s ; ajoutons-y
points choisis arbitrairement, et considerons 1' ensemble des r^ 1 points comme
determinant la base P 4n _ 8 d'un faisceau de courbes d'ordre 1. En considerant cette base
par rapport aux cinq points 4 7, 4 6, 4 5, 4 4, 4 3, on aura un point oppose
#4,1-8 > qui sera le centre d'un faisceau de droites correspondant anharmoniquement aux
courbes du faisceau P 4B _ 9 . Le lieu des intersections des lignes correspondantes des deux
faisceaux sera une courbe de 1'ordre w, qui passera par les 4 3 points donnes, et, par
consequent, par les trois points cherche's. En permutant cycliquement les points 4 8,
4 7, 4 6 (sans changer autrement la base des courbes de 1'ordre n 1), on aura deux
autres courbes de 1'ordre n passant, comme la premiere, par les 4 points. De la en suivant,
soit notre demonstration geometrique, soit 1'analyse precedente, on conclura que la courbe
cubique des neuf points 4w 8, ..., 4 passe aussi par les points opposes p 4n - e , /> 4B _ 7 , etc ...,
et par les points d'intersection tels que celui des droites (/> 4B - 8 , 4 7) et (p tn -^, 4>n 8).
On voit done que tout se reduit a la determination des points opposes, determination qui
sera facile, lorsqu'on connaitra les rapports anharmoniques des faisceaux
P 4B _ 8 .[4-7, 4 6, 4 5, 4-4, 4w 3].
Or, il n'est pas douteux, qu'etant donne des points en nombre suffisant pour determiner une
courbe geometrique d'ordre w, on ne puisse trouver lineairement, soit la tangente a cette
courbe en un point donne, soit la droite polaire d'un point donne par rapport a la courbe ;
mais, puisqu'il parait qu'on n'a pas encore cherche la solution de ce probleme general de
geometric lineaire, nous ferons voir qu'on peut s'en passer ici, en se servant d'une methode
particuliere qui se presente naturellement. Considerons les points 1, ..., 8, comme determi-
nant la base A d'un faisceau de courbes cubiques. Soit (a) la conique qui satisfait a liqua-
tion
(9, 10, 11, 12) = A.[9, 10, 11, 12].
En prenant successivement differents points a de cette conique pour des points opposes a la
base A, on aura un faisceau de courbes biquadratiques, correspondant anharmoniquement
aux points a. Soit B la base du faisceau biquadratique ; il est evident, qu'etant donne un
point quelconque x, on pourra trouver lineairement le point a correspondant a la courbe
(B, x). Determinons la conique (b) qui satisfait a liquation
(13, 14, 15, 16) = B. [13, 14, 15, 16].
Prenons un point quelconque b de (b) ; nous le considerons comme un point oppose a la
base biquadratique B ; et nous aurons de la sorte un faisceau de courbes du cinquieme ordre
qui correspondront anharmoniquement aux points b. Soit C la base de ce nouveau faisceau ;
pour trouver lineairement le point b qui correspond a une courbe quelconque (C, x) du
faisceau, on determine sur la conique (b) le point so' qui satisfait a liquation anharmonique
[13, 14, 15, 16, of} = B. [13, 14, 15, 16, of} ;
VOL. II. K
66 APPENDICE. [Note X.
le point b est le second point d'intersection de la conique par la droite xx'. Apres avoir
determine une troisieme conique (c), qui satisfait & liquation
(17, 18, 19, 20) = C.[17, 18, 19, 20],
on sera conduit a considdrer une base D de courbes du sixieme ordre, et, en continuant de
la sorte, on arrivera enfin k une base M de courbes de 1'ordre n 1 ; cette base comprendra
tous les points 1, 2, 3 4 8, puisque chaque base de la se'rie ascendante comprend
evidemment tous les points de la base pre'ce'dente ; de plus, les courbes du faisceau M corre-
spondront anharmoniquement aux points d'une certaine conique (I) passant par les points
4 11, 4>n 10, 4 9, 4 8; de sorte qu'e'tant donne un point quelconque x, on pourra
trouver line'airement le point I de cette conique qui correspond a la courbe (M, x). Enfin,
on determinera la conique (m) qui satisfait k 1'^quation
(4-7, 4-6, 4 5, 4-4) = Jf.(4w 7, 4-6, 4-5, 4-4),
et le point z' 4n _ 3 de cette derniere conique qui correspond & la courbe (M, 4 3) ; le second
point d'intersection de la conique (m) avec la droite (4 3, m' tn _ a ) sera le point /> 4n - 8 , qu'il
s'agissait de trouver.
On peut dire que cette construction est composee de transformations homographiques
successives. On commence par determiner la conique T qui satisfait a I'e'quation
[5,6, 7, 8] = (1,2, 3, 4). [5, 6, 7, 8].
Soit v'\y = 9, ..., 4 3] le point de cette conique qui corresponde anharmoniquement a la
conique (1, 2, 3, 4, v) : le second point d'intersection de la droite (v, v') avec F est le point co,
de T qui correspond anharmoniquement & la cubique (A, v). On transforme homographique-
ment la figure de maniere que les points o> 9 , o> 10 , a) 11( o> 12 deviennent les points 9, 10, 11, 12 ;
la transformee de F sera (a); soit a' v \y = 13, ..., 4>n 3] le point de cette derniere conique
qui correspond au point <a v de la conique F ; le second point d'intersection de la droite
(v, a\) avec (a) sera le point <z,,, qui correspond anharmoniquement a la courbe biquadratique
(B, v). On transforme encore la figure de maniere que les points a 13 , a u , a ls , a 16 deviennent
les points 13, 14, 15, 16 ; on determine la conique (4) correspondant &, (a), et le point
V v \v = 17, ..., 4n 3], correspondant a ; on mene la droite (v, b',), qui determine le point
b,, correspondant anharmoniquement & la courbe (C, v) ; et en continuant cette se'rie uni-
forine d" operations lindaires on parvient enfin a determiner le point % n _ 3 de la conique (m),
qui n'est autre que le point oppose jo 4n _ 8 .
XXIV.
ARITHMETICAL NOTES.
[Proceedings of the London Mathematical Society, vol. iv. pp. 236-253. The three papers which
form these Notes were read on January 9 and February 13, 1873.]
I. On the Arithmetical Invariants of a Rectangular Matrix, of which the
Constituents are Integral Numbers.
1. .LET || - 1| represent a rectangular matrix of the type n x (n + m), and let
V, be the greatest common divisor of the ==== r x ^ :
minor determinants of order s which appertain to it ; so that, in particular, V,, is
the greatest common divisor of the -^ ^ - determinants of the matrix ; the
Iln. llm
numbers V n , V B _ 1} ..., Vj are the arithmetical invariants of the matrix. We
suppose that the matrix is asyzygetic ; i.e. that the -== == - determinants are
not all equal to zero. We then have the theorem (Memoir, p. 388 sqq. *) :
Theorem (a). ' The quotient - is the greatest common divisor of the
_l
quotients obtained by dividing each minor determinant of order s by the greatest
common divisor of its own first minors.'
Let p be a prime dividing V,-, and let I i be the exponent of the highest
power of p which divides V< ; any minor of order i which is divisible by p 1 **"
* The references in this and the two following notes are to a Memoir ' On Systems of Linear
Indeterminate Equations and Congruences' (Phil. Trans, vol. cli. pp. 293-326.) [This Memoir is
No. XII of vol. i, pp. 367-406. The references in the text are to the pages of the Memoir as printed
in vol. i.]
K 2
68 ARITHMETICAL NOTES. [l.
may be said to be divisible in excess by p". Using this abbreviated mode of ex-
pression, we may enunciate the two following corollaries :
Corollary (b). ' Any minor determinant, which is not divisible in excess by
p, contains at least one first minor which is not divisible in excess by p.'
Corollary (c). ' Any minor determinant, which is not divisible in excess by
p", contains at least one first minor which is not divisible in excess by p"'
Or, which is the same thing,
' If all the first minors of a given minor are divisible in excess by p", the
given minor is itself divisible in excess by p".'
Of these corollaries, the first is a particular case of the second ; and the
second is only a re-statement, in other words, of the theorem (a).
2. It is the object of this note to establish a theorem, which may be re-
garded as reciprocal to the theorem (a).
Theorem (A). 'The fraction -~ is the greatest common divisor of the
fractions obtained by dividing each minor of order s 1 by the greatest common
divisor of its first majors ; or, which is the same thing, the integral number _ -
is the least common denominator of these fractions.'
Any square matrix which contains a given square matrix is here called, for
brevity, a major of that matrix ; if the given square matrix be of order s l, its
first majors are the square matrices of order s which contain it.
The theorem (A) admits of two corollaries, corresponding to the corollaries
(b) and (c) :
Corollary (B). ' Any minor determinant, which is not divisible in excess by
p, is contained in at least one first major which is not divisible in excess by p.'
Corollary (C). ' Any minor determinant, which is not divisible in excess by
p", is contained in at least one first major which is not divisible in excess by p"'
Or, which is the same thing,
' If all the first majors of a given minor are divisible in excess by p a , the
given number is itself divisible in excess by p".
3. To prove the theorem (A), we consider, in the first place, a square matrix
|| a - 1| of the type n x n, so that V n = 2 + a u a^ . . . a nn ; and we represent the minor
determinant -~ by A^, so that the reciprocal matrix is ||-4</ll- The determi-
** n-l
nant of this matrix is V , and the greatest common divisor of its minor deter-
n
minants of order i is V x V n _j.
Art. 4.] ARITHMETICAL INVARIANTS OP A MATRIX. 69
Tin
represent any one of the M 2 minors of order s - 1 which appertain to the matrix
||O0|| ; let k^ v \y = 1, 2, 3, ..., N 2 ] represent any one of the first majors of k^ ; let
d^ be the greatest common divisor of these N 2 numbers ; lastly, let K^ and K^
represent the minors which, in the reciprocal matrix \\A {j -,\\, are reciprocal to the
minors k^ and k^ in the matrix || - 1| ; so that, for example, if
If / j n n n J\ ^* t A A A
IL ~~~ *^"* <i \Jv-t 1 22 ** 1 * _ 1 1 ** It ~~~ ^^ "l -il a a ^J- BJ.1 J_1 ^L
Thus the N 2 first minors of K^ are precisely the determinants K^ v ; we also have
i
the equations ^T M = V x^, K^ v = V x^_ y ; so that, applying the theorem
(a) to the matrix |[^ tf ||, we find that the greatest common divisor of the M*
integral numbers,
8 * 1 f,
tY7 v Z* "1 f V7 v/VT V7 v -
V ^\ A/yJ ~ I ^"u I v rt J '
n n **/
is the quotient
T7
[V xV._,]-r[ V xV.] = V n x^
n
that is to say, the greatest common divisor of the M 2 fractions -* is the fraction
-=p- in accordance with the theorem (A). The corollaries (B) and (C) may be
immediately verified by observing that, if either of them were supposed untrue
k
for any given minor k^, the corresponding fraction -~ would acquire a denomina-
/*
tor which could not be a sub-multiple of
4. To extend the demonstration to the case of an oblong matrix of the type
n x (m + n), we retain the preceding notations, so far as they are applicable ; and
we put M l = ==-. -T f=T7 r% > N 1 = n + m s+\; so that the values of
Tl(s-l).Tl(n + m-s + l)'
n and v are now [1, 2, ..., My. M^\ and [1, 2, ..., NxN^ respectively. Let Q be
a common multiple of the My.M l numbers d^, and let us complete the given
rectangular matrix into a square one by adding rows of constituents each of
which is divisible by Q. In the resulting square matrix, any given minor k^
(appertaining to the given rectangular matrix) will have N\ first majors ; but it
70 AEITHMETICAL NOTES. [l.
is evident that the greatest common divisor of these Nl first majors will be the
same as that of the NX. N! first majors which appertain to the given rectangular
k
matrix. Hence the MxM l fractions -f, obtained by dividing each minor of
a,,
order s 1 in the given matrix by the greatest common divisor of its first majors,
occurs among the M\ fractions similarly derived from the completed square
matrix ; also the numbers V, and V,_! are evidently the same for both matrices.
Applying, therefore, the theorem (A) to the square matrix, we see that - ' is a
k
common divisor of the M x M r fractions -f ', that it is the greatest common
a,,
divisor of these fractions, may be proved by considering a minor determinant of
order s in the given matrix, which is not divisible in excess by p ; this minor de-
terminant contains [by the corollary (&)] a first minor which is itself not divisible
in excess by p ; if k^ be the first minor, the denominator of the corresponding
k
fraction -/ necessarily contains the prime p raised to the power I e I e _ 1 .
dp
It will be observed that, whether we consider a rectangular or a square
matrix, the corollary (C) is an immediate consequence from the theorem (A) ; but
the theorem (A) does not follow conversely from the corollary (C). For the
absence of factors prime to =1-^- from the least common denominator of the frac-
tions -r is asserted in the theorem, but is not asserted in the corollary.
dp
5. Every common divisor of the first minors of a given minor is evidently a
common divisor of the first majors of that minor; and, if the matrix be square, a
consideration of the reciprocal matrix shows that, conversely, every common
divisor, prime to the determinant of the matrix, of the first majors of any given
minor, divides the first minors of that minor. We thus obtain the self-reciprocal
theorem :
Theorem (dD). ' In any square matrix, the greatest common divisor of the
first minors of any given minor is identical with the greatest common divisor of
its first majors, so far as factors prime to the determinant of the matrix are con-
cerned.'
This theorem is not universally true in the case of oblong matrices, at least
if we suppose that, in its enunciation, the words ' determinant of the matrix ' are
replaced by the words 'greatest common divisor of the determinants of the
matrix.' If, for example, ||a|| is a square matrix of order n 1, and ||6|| is a
Art. 5.] SYSTEMS OP LINEAR CONGRUENCES. 71
matrix of the type n x (n + m), of which the determinants are relatively prime,
the symbol
may serve to represent a matrix of the type nx(2n + m 1),
in which the first n 1 constituents of the uppermost row are zeros. In this
matrix the greatest common divisor of the first majors of |a| is evidently |aj
itself; and this greatest common divisor is prime to the greatest common divisor
of the determinants of the matrix, because these determinants are relatively
prime ; but unless ||a|] is an unit matrix, its determinant cannot be equal to the
greatest common divisor of its own first minors.
6. It may be added, that the properties to which this note refers admit of
being stated in a generalized form. Thus, we may replace the enunciations of
the theorems (a), (A), and (dD) by the following :
Theorem (a). ' The quotient is the greatest common divisor of the
V-
quotients obtained by dividing each minor determinant of order s by the greatest
common divisor of its own first minors of order s i.'
Theorem (A'). ' The fraction -- 1 is the greatest common divisor of the
fractions obtained by dividing each minor of order s i by the greatest common
divisor of its own majors of order s.'
Theorem (dD'). ' In any square matrix, the greatest common divisor of the
minors of order s i appertaining to a given minor of order s is identical with the
greatest common divisor of the majors of order * + i appertaining to the given
minor, so far as factors prime to the determinant of the matrix are concerned.'
A very slight modification of the proof of the theorem (a), (Memoir, pp. 397-
399,) supplies a proof of the theorem (a), and from it the theorems (A') and (dD')
may be inferred by means of the methods employed in this note to establish the
theorems (A) and (dD).
II. On Systems of Linear Congruences.
1. Let A iil x l + A it tXt+...+A i>n x n = A i n+1 (modM), . . . . (1)
i = l,2, 3, .... n,
represent a system of n linear congruences ; let
Dn, A.-1, , A,
72 ARITHMETICAL NOTES. [ll.
respectively denote the arithmetical invariants of the augmented and unaug-
mented matrices of the system ; i.e. of the matrices
Mi, i = l, 2, ..., n,
U y-i, a, ...,
let c?, and <J 8 be the greatest common divisors of Jf with -~- , and of M with
V
- - respectively ; lastly, let d = d 1 xd 2 x ... x.d n , S = S 1 xS 2 x ... x S n . We then
V-i
have the two theorems (Memoir, p. 399) :
' The necessary and sufficient condition for the resolubility of the system (1)
isd = S.'
' When this condition is satisfied, the number of its incongruous solutions is
d=s:
There are similar theorems (Memoir, p. 402) relating to defective and re-
dundant systems of congruences. Some observations which may serve in certain
cases to facilitate the applications of the theory are contained in the present
note.
2. Consider separately the powers of the different prime numbers dividing
the modulus M . Let p be one of these primes ; and let /u, a s , a, be the exponents
of the highest powers of p which divide M, D t , V g respectively. Then, because
-JT+TT-, ~~^~-, V.-rA> and =-- -=- j^- , are all integral (Memoir,
*' -^-l V 8 V 8-l V -l LJ t-\
pp. 396-398), we have the inequalities
^---^ a l-o; ...... (5)
........... (6)
*-! .......... (7)
Let <t a a a _ 1 be the first term in the series (4) which is less than /u ; so that, if
n <*_!< jw, we have a- n, and in every other case
<r + l-a(r^M >,-<,_! ......... (8)
We may then replace the two theorems of the Memoir by the two fol-
lowing :
Art. 3.] SYSTEMS OF LINEAR, CONGRUENCES. 73
' The necessary and sufficient condition for the resolubility of the congru-
ences (1), considered with regard to the modulus yp, is
, = ,.' ........... (9)
'When this condition is satisfied, the number of incongruous solutions is
^a,, + (n-<r)/^>
For the condition of resolubility is (Memoir, p. 400) that the greatest com-
mon divisor of the numbers
p } p'n-sn, pon-t+tp, ..., p"?, ....... (10)
should be the same as the greatest common divisor of the numbers
The greatest common divisor of the numbers (10) is p", + (-)r for it follows
from the inequalities (8) and (4) that, in the series (10), the exponent a ff + (n o-)//.
is less than any exponent which follows it, and not greater than any which pre-
cedes it. Again, if a a = a a , p** +<-*) P i s also the greatest common divisor of the
numbers (11) ; for it is equal to one of them, and it certainly divides all of them,
because it divides all the numbers (10), each of which, by virtue of the inequali-
ties (6), divides one of the numbers (11). But if a a >a a , the greatest common
divisor of the numbers (11) is a power of p having an exponent higher than
a a + (n ar)fji. For the inequality a a >a a , combined with the inequality (7),
implies the successive inequalities <r + i >a <r+i> a a+2 >a a + 2, > a >a ; so that
every exponent in the series (11), which precedes a a +(n a)n, surpasses the cor-
responding exponent in (10), and therefore certainly surpasses a a + (n a-) p..
And again, the exponents following a a + (n <r) /u. in the series (11) also surpass
a a + (n <r)fj., for these exponents are not less than the corresponding exponents
in (10), each of which is greater than a a + (n <r) /u. Thus the condition for the
equality of the two greatest common divisors, i.e. for the resolubility of the con-
gruences (1), is a a = a a . And, by transforming the proposed system (Memoir,
p. 401) into an equivalent system of the type
(12)
it is immediately ascertained that, when this condition is satisfied, the number
of incongruous solutions is equal to p<, + (-)>*, i.e. to the common value of the
two greatest common divisors.
3. The criterion of resolubility just established differs in form only from that
VOL. II. L
74 AKITHMETICAL NOTES. [ll.
given in the Memoir ; and the coincidence of the two may be shown by observing
that the equation a a = a a , combined with the inequalities (7), implies the equa-
tions a a _ l = a a _ l , a a _ a = a a _2, ..., a^-a^. As here stated, the criterion of resolu-
bility, and the expression for the number of solutions, depend only on the
numbers a-, V,,, D a , and do not involve explicitly the consideration of the two
complete series of in variants. When n surpasses <* -i> the condition of
resolubility is a n = a n , and the number of solutions is />". In this case we only
need to determine the invariants V and D n , or rather the exponents of the
highest powers of p dividing those numbers. But when M^ -!, it is
necessary to calculate successively the invariants D n , D n-1 , D n _ 2 , ... in order to
ascertain the index o- for which the inequalities (8) are satisfied. It will be ob-
served, however, that since every minor of order s, which is not divisible in
excess by p, contains a minor of order s 1 which is itself not divisible in excess
by p, it is always possible to restrict the examination of the minors of order s 1
(which is required in order to obtain the value of D s -\) to those which are con-
tained in a single minor of order s.
We may also notice that if, in the series of differences
a t a, is the first which is equal to zero, p a >+i- a > is the highest power of p for
which the proposed congruences are resoluble.
4. In the case of a redundant or defective system, as well as in the case of a
system such as (1), which is neither defective nor redundant, it will be found
that the equation of condition d = S may be replaced by the equation a a = a a .
And in all these cases alike, whenever the system admits of any solution at all,
the number of incongruous solutions is p a * +(n ~ a)l *, n denoting the number of the
indeterminates.
5. In the case of a redundant system, the condition d = S, or a a = a a , although
necessary is not sufficient (Memoir, p. 404). The complete condition of resolu-
bility in this case is that the greatest common divisor of the numbers
should be the same as the greatest common divisor of the numbers
And the identity of these two greatest common divisors implies not only (as in
Art. 2) the equation a a = a a , but also the inequality
a n ; .......... (15)
Art. 5.]
SYSTEMS OF LINEAK CONGRUENCES,
75
since, if this condition be not satisfied, a n+l will be less than any of the expo-
nents following it in the series (13), and therefore less than any exponent in
the series (14).
The two conditions expressed by the equation a a = a a , and the inequality
(15), are sufficient as well as necessary. But it is remarkable that the exponent
M not only satisfies the inequality (15), but also the inequality
, (16)
which is, in general, closer than (15), because a n ^a n . To prove this, consider, in
the given redundant system, any set of n + 1 congruences such that the determi-
nant of their augmented matrix is not divisible in excess by p ; and multiply
these n + 1 congruences, taken in order, by the determinants of their unaug-
mented matrix. The sum of the products is the determinant of the augmented
matrix ; we infer that this determinant is divisible by p a + **, i.e. that a n + 1 ^ a n + /*
Since (as has been said) the two conditions of resolubility, taken together,
are sufficient, they must involve the inequality (16). To deduce it from them,
we may employ the following lemma :
: Let
i = l, 2, ..., n,
j = l, 2, .... n + m,
represent any square or oblong matrix of integral numbers, and let \\q\\ be a
partial matrix consisting of any s of the n horizontal rows of \\Q\\ ; also let Q n ,
Qn-n J ?> <?_!> he the arithmetical invariants of \\Q\\ and 1 1 q \[ respectively;
then, if r is any number not surpassing s, ^~ n is divisible by -*-'
To establish this lemma, let us suppose that \\q\\ consists of the last s rows
of \\Q\\, and let us replace \\Q\\ by a compound matrix of the form \\Q'\\ x \{Q"\\, in
which | \Q" 1 1 is a prime matrix of the same type as \\Q\\, and \\Q'\\ is a square
matrix of the form
Ji I- I- If
"I I "'1, 2> "'I.SJ ) "%)
Or) Z* If
y '"2 > ^2, 3 > > / * / 2, 1
0, 0, Jl 3 , ..., 3,,
0, 0, 0,
Then the greatest common divisor of any horizontal row of minors in ||Q|| is the
same as the greatest common divisor of the corresponding row of minors in \\Q'\\
L 2
76 ABITHMETICAL NOTES. [ll.
(Memoir, p. 389) ; it is sufficient, therefore, to verify the lemma in the case of
this last matrix. In it we have
and if we multiply h^ x h 2 x . . . x h n _, = by any minor of order s r contained
in the last s rows of H^H, we obtain a product which is either a minor of \\ty\\ of
the order n - r, or else is equal to zero ; this product is therefore in every case
/~\ /*"! ^v
divisible by Q n . r ; i.e. x q,_ r is divisible by Q n _ r , or -^- is divisible by --
To apply this lemma to the case which we are considering here, we change
n into n + 1, and we put s = n, r = n a-. Attending only to the powers of p, we
thus find
But a a -a a , and a a + j - a a ^ M ; therefore +1 - a n ^ M.
6. When the determinants of the augmented or unaugmented matrix of a
proposed system of congruences vanish, we may add multiples of the modulus to
the constituents of these matrices, so as to obtain determinants which do not
vanish. We proceed, however, to show that this preliminary operation is un-
necessary, and that the preceding theory is immediately applicable, notwith-
standing the evanescence of the determinants.
If, in any matrix of which the arithmetical invariants are V n , V n _!, ..., V x
the minor determinants of order s 4- 1 are all equal to zero, we must attribute the
same value to the invariant V 8+1 ; and vice versd, if V, +1 = 0, the minor determi-
nants of order s + 1 must themselves be all equal to zero. Let V, +1 = 0, V 8 > 0, so
that V, +r = (r=l, 2, .... n-s), V 8 _ r >0 (r = 0, 1, 2, ..., s-1). The quotients
^~, ^^-, --, ^r are determinate positive integers, and the quotient -~ is
",_! V,_ 2 V V
zero; but the quotients =~, ..., ^=~ , assume the indeterminate form -. To
Vf,.,.! V n _!
these quotients we attribute the value zero : this may be regarded as an arbi-
trary definition of the value of these indeterminate numbers ; it is, however,
suggested by the observation that if the constituents of the matrix, instead of
being integral numbers, were rational and integral functions of an indeterminate
quantity, every factor of V,- would also be a factor of V,- + 1 , and would be con-
tained in V <+1 oftener than in V< ; i.e. -~^ would vanish with V,-.
Art. 7.]
SYSTEMS OF LINEAR CONGRUENCES.
77
7. Let us first suppose that the proposed system of congruences,
A i , l z 1 + A i>2 x 2 +...+A i , n+m x n+m = A n+m+l (modp"), . . . (17)
i = l, 2, 3, ..., n, m = 0,
is not redundant ; and let the invariants V n , V n-1 , ... of its unaugmented matrix
vanish as far as V g+1 inclusively, but let V g not vanish. The integers
<*, a,
n> a n 1 > a s+l
<* a.
-1. " 8 + l- a 8
a n~ a n-l> '> a s + 2~ a i + l
may then be regarded as greater than any assigned number ; and if, in the
inequalities (8), which determine the index <r, we adopt this interpretation of
these symbols, the criterion of resolubility, and the expression for the number of
solutions, will continue applicable to the system (17) without any modification
whatever. For the criterion of resolubility this is evident, because the demon-
stration of Art. 2 subsists unchanged. To prove the same thing for the formula
which expresses the number of solutions, we employ the following lemma :
"}' *' "" n '
.7 = 1, 2, ..., n+m,
'The matrix [^H
(in which V 8 + 1 vanishes, but V 8 does not vanish,) satisfies an equation of the form
III xfl4,|jx dri-
ft* o
0,
(A)
in which ||/3|| is an unit matrix of the type n x n, ||y|) an unit matrix of the type
(n + ra) x (n + m),
0,
a matrix of the type n x (n + m), which has all its con-
stituents equal to zero, except those composing the partial matrix |j^||; this
partial matrix being of the type s x 5, and having for its arithmetical invariants
V,, V,.!, ..., V .'
Consider the system
0, ..... (18)
i=l, 2, 3, .... n,
which is equivalent to s independent equations; and take for ||7|| any unit
78 ARITHMETICAL NOTES. [n.
matrix of the type (n + m) x (n + m), in which the last n + m s columns are the
n + m s rows of a fundamental set of solutions of the system (18). The com-
pound matrix ||<4||x||y|| will be of the form ||<ty, 0|{, where ||c -|| is a partial
matrix of the type n x s, and ||0|| is a partial matrix, composed entirely of zeros,
of the type n x (n + m s) ; so that jjc^, 0|| is of the same type as ||.4||. Again,
consider the system
1 J 1 2, J 2 n, j n > \ /
j = l, 2, 3, ...,*,
of which the equations are independent ; and take for ||/3|| any unit matrix of the
type n x n, in which the n s lowest rows are the n s rows of a fundamental set
of solutions of the system (19). The compound matrix ||/3|| x ||<4|| x \\y\\ is then of
the form j* w>
If therefore, in the system (17), we transform the indeterminates by the
substitution
(a it 11 i
\yi> yzi > yn+m/f
and at the same time replace the congruences themselves by others linearly
derived from them by multiplication with the constituents of the rows of [|j8||, we
obtain an equivalent system of the form
..... (20)
j = l, 2, ..., s,
in which the indeterminates y e+ i, ..., ?/+ do not appear explicitly. By the
theorem of Art. 2, the number of solutions of (20) is p^ + ^-^f- and to each of
these solutions there correspond p (n +m ->>i* solutions of (17), because each of the
indeterminates y t + i, ..., y n + m ma y have any one of $? values. Thus the number
of solutions of (17) i & p, + (+ *-<?.
8. Next, let the proposed system of congruences be redundant ; and, as
before, let V g + 1 vanish, but not V 8 . In this case also the demonstration relating
to the criteria of resolubility subsists unchanged ; to determine the number of
solutions, we select from the proposed system a partial system of s equations such
that the determinants of its unaugmented matrix are not divisible in excess by
p. Every solution of this partial system satisfies the remaining congruences of
the proposed system ; i.e. the number of solutions of the proposed system is
pa,* (-<r) Mj if n b e t n? number of indeterminates.
9. It is worth while to observe that, in the equation (A), we may attribute
Art. 10.]
SYSTEMS OF LINEAR CONGRUENCES.
79
to the partial matrix ||<fo|| the simple form
||v|| are reducing units of ||<fo||, so that
_
8 _x' V 3 _ 2 ' -' V
For if \\u\\ and
v, v,.,
WV._,' ' V
we may replace \\u\\ and ||v||, which are both of the type s x s, by unit matrices of
the types n x n and (n + m) x (n + m) respectively ; and may then compound these
substituted units with the unit matrices ||/3|| and ||7||. The unit matrices thus
substituted for ||tt|| and \\v\\ contain |[tt|| and ||w|| as partial matrices; their forms
are sufficiently indicated by the symbols
u,
0,
J
where / and J are
matrices of the types n x (n s) and (n + m s) x (n + m) respectively, and are
subject to no restriction other than that implied by the equations
u,
0,
= 1,
J
= 1.
Another transformation of a matrix, in which the first n s invariants vanish,
is sometimes useful ; viz., we may write
V 8 V,., V
MI-NX
0,0, ..., o,
.-,'?._,' ' Vo
where \\oo\\ is an unit matrix of the type n~x.n, and \\Q\\ is a matrix of the same
type as \\A j| , of which the s lowest rows form a prime matrix, while the others
are arbitrary. The proof of this transformation presents no difficulty, and may
be omitted here.
10. When the absolute terms in a linear system of congruences are all con-
gruous to zero, the criteria of resolubility are always satisfied, as they ought to
be, because the system is satisfied by attributing to the indeterminates values
congruous to zero, and therefore certainly admits of one solution at least. In this
case, however, it is usually of importance to determine, not the whole number of
solutions, but the number of solutions prime to the modulus (i.e. the number of
solutions in which one at least of the indeterminates is prime to the modulus).
Reducing the proposed congruences to an equivalent system of the form (12),
which can always be done by the transformations indicated in the Memoir (p.
401), and in Art. 7 of the present note, we find that if N is the whole number of
solutions, N-rp n ~" is the number of solutions in which every indeterminate has
a value divisible by p, so that N (l ^17) is the number of solutions prime to
80 ARITHMETICAL NOTES. [in.
the modulus. And, in general, if k { is the number of terms in the series (5)
which satisfy the inequality a t+1 a,^n i, so that, in particular, k = n er, the
number of solutions in which every indeterminate has a value divisible by p { will
be found to be N &+**+ + *<-'.
11. The transformation of a given square matrix indicated by the equation
-x
^t-,|*=i,|s=! f ...,^,^ X ||/3|| . . . (21)
v n-l v n-2 v n-S v l v c
forms the basis of many of the demonstrations in this and the preceding note.
It has been already pointed out (Memoir, p. 392) that this equation always
admits of an infinite number of solutions : this observation may be verified as
follows. If \\Ufj\\ is any unit matrix of which the constituents are defined by the
equations
u i,j = \j> j&>
the matrix ||vyl|, of which the constituents are defined by the equations
will also be an unit matrix ; and, if |[ o||, ||/S|j be two given unit matrices satisfy-
ing the equation (27), that equation will also be satisfied by the unit matrices
comprehended in the formulae
IMIxIMI, H|-'
and by them only
III. On an Arithmetical Demonstration of a Theorem in the
Integral Calculus.
l - Let a<i2/i + a,-22/2+.- ,' n = <)
i=l,2,3,...,n )
represent a system of linear equations, in which the numbers a^ and b t are inte-
gral, the determinant 7 = 1%! being different from zero and positive. If, con-
sidering the numbers 6,- with regard to the modulus V, we attribute to them in
Art. 2.] THEOREM IN THE INTEGRAL CALCULUS. 81
succession the V" systems of values of which they are susceptible with regard to
that modulus, we have the arithmetical theorem (Memoir, p. 405) :
(i.) 'The system (A) is resoluble in integral numbers y for precisely V 71 " 1 of
the V" systems 6.'
2. Let 2 represent a complex, or multiplicity, of order n, of which the inde-
terminates y lt y 2 , ..., y n are independent. Any system of values assigned to the
indeterminates is called a point of 2 ; and if the values are integral, the point is
an integer point. Let
[x fl , x i2 , x i3 , ..., o;,- n j
= 1, 2, 3, ..., n,
represent n given integer points of S ; we shall suppose these n points to be
asyzygetic, so that the determinant V = | x^ \ is different from zero, and may be
assumed to be positive. We then obtain the theorem :
(ii.) 'The number of integer points [X 1} X 2 , ..., X n ] which satisfy the in-
equalities
LI dx fl " 2 dxa '" dx in =
t = l, 2,3, ...,,
is precisely V.'
For, by the theorem (i.), the equations
(B)
t = l, 2, 3, ..., n
of which the determinant is V"" 1 , and in which each of the numbers B { is sup-
posed to receive in succession every value from 1 to V"" 1 inclusive, are resoluble
in integral numbers X for V ( "~ 1)2 of these V"^- 1 ' systems of values. But the
solution of the equation (B) is given by the system *
. . . + B n x nj ,
If therefore the equations (B) are resoluble in integral numbers for any
particular system of values of the numbers B { , considered with regard to the
modulus V, they are resoluble for the V n(n ~ 2) systems of values congruous to that
particular system for the modulus V, but incongruous to one another for the
modulus V" 1 . Hence, if k be the number of systems of values, taken with
regard to the modulus V, for which the equations (B) are resoluble in integral
VOL. II. M
82 ARITHMETICAL NOTES. [ill.
numbers, we have kx v n(n ~ 2) = V (n ~ 1)! , or & = V ; a result which establishes the
theorem (ii.).
3. Another proof of the same theorem may be obtained as follows ; The
number of systems of values for which the equations (B) are resoluble in integral
numbers is equal to the number of solutions of the congruences
B l x^ + B 2 x 2j +...+B n x nj = 0, (mod V),
j=l, 2,3, ...,,
in which B lt B 2 , ..., B n are the indeterminates. And the number of solutions of
this system of congruences is V. (Memoir, p. 399, or Arts. 1-3 of the preceding
note.)
4. Let 0,-j, y, .... y.-J
i = l, 2, 3, ..., n,
represent any n asyzygetic points of 2 ; and let X,- be a positive common measure
of the n quantities y lj} y 2j , ..., y nj , so that we have n z equations of the form
yfj^XfjX Aj, in which the numbers x (j are integral, and |a; -| = V is different from
zero and positive.
j"
Let D = | y^\ = V x II A,- ; the number of points of 2, which are of the type
7=1
[XjXj, X 2 \ 2 , ..., X n \ n ], X lf X 2 , ... denoting integral numbers, and which satisfy
the inequalities
is, by the theorem (ii.), equal to Z)-^ITX = V; that is to say, the product IT A,
taken as many times as there are points (X^, X 2 \ 2 , ..., X n \ H ) satisfying the
conditions (C), is equal to D. Let \, \, ..., A n be infinitesimal, and let U repre-
sent the integral f...fdY l dY z ...dY n , extended to all values of Y lt Y 2 , ..., Y n
which satisfy the inequalities
we have, by the definition of a definite integral, U = D.
5. This well-known result (which may be described as a generalization of
the expression for the volume of a parallelepiped in space referred to three
rectangular axes) may of course be otherwise obtained by employing the formula
Art. 6.] THEOEEM IN THE INTEGRAL CALCULUS. 83
for the transformation of a multiple integral ; viz., if we put
dD dD dD
u i -M ~3 r 2 "7 ' + + %n -} -i
%a <*y %,-
the limits of u { are and D, and the functional determinant is D n ~ l ; hence
fD fD pD
/ .../ / du 1 du 2 ...du n = D n - 1 xU, or Z>= Z7.
/O JQ ''O
But as we have obtained the value of the definite integral U by a direct
process, we may employ the result to demonstrate the formula for the transform-
ation of a multiple integral ; and this it may be worth while to do, because some
of the demonstrations which have been given of that formula are not free from
obscurity.
6. For convenience of expression, we may regard the indeterminates X 1} X 2 ,
X 3 , ..., X n as rectangular coordinates of a homogeneous and isotropic space of n
dimensions. On this supposition we may define the volume unit of 2 to be that
portion of 2 which is comprised between the limits
oislfci,
i=l, 2,3, ..., n;
so that dX l dX 2 . . . dX n represents an element of 2 having the form (in n dimen-
sions) of a rectangular parallelepiped. In any multiple integral f FcZ2, of order
n, we must regard the integration as extending over a certain ' integral space,' or
portion of 2 ; this integral space is to be resolved into infinitesimal elements,
which may be of any form whatever, but which must exhaust the whole of it ;
each element is to be multiplied by the function F of its coordinates ; and the
definite integral is the sum obtained by adding the results. When the indeter-
minates are transformed by the equations
x 3>
we have to consider a new space a-, in which the new indeterminates x lt x 2 ,...,x H
may be regarded as rectangular coordinates ; to each element of the first space
corresponds an element of the second space, and vice versd ; the equations (D),
whether by their own nature or by restrictions imposed on their interpretation,
being supposed to establish a ' one-to-one ' correspondence between the two integral
spaces. The problem of the transformation of a multiple integral then reduces
M 2
ARITHMETICAL NOTES.
[III.
itself to the determination of the ratio of corresponding elementary spaces d 2
and d <r. By the nature of infinitesimal magnitudes, this ratio is independent of
the particular form assigned to either element (the form of one of course deter-
mines the form of the other). Let us take for the element d a- the parallelepiped
(in n dimensions) of which one vertex is at the point \x lt x 2 , ..., ], and of which
the adjacent n vertices are
the quantities S f Xj being infinitesimal and asyzygetic ; so that, by the theorem of
Art. 4, the volume of the parallelepiped (which is an infinitesimal of order n) is
\S iXj \. If
/dX;
the element d"Z, corresponding to the element dv^\^ t Xj\, is evidently |<J,-.X}|.
7 -T7-
j - ; i.e. the ratio of corresponding spaces is the functional
3
determinant ; a conclusion which establishes the theorem of transformation.
7. If we decompose the spaces 2 and a- into elementary parallelepipeds of
the type dX l dX 2 ...dX a , dx t dx 2 ... dx n , we may write the equation
dX {
fUdS=fUx
in the form
X da-
f...ffUdX l dX 2 ...dX n =f..,ffUx
x dx l dx 2 ... dx n .
In this equation dX l dX 2 . . . dX n and dx^ dx 2 . . . dx n are not corresponding ele-
ments ; but the substitution of elements which do not correspond for elements
which do correspond is admissible, because the value of a definite integral is not
altered by substituting one mode of decomposition into elements for another. It
may be added that the symbol fUd^Z, which involves no hypothesis as to the
form of the element d 2, supplies the most general and abstract expression of the
definite integral; while the symbol f...ffU dX^dX 2 ... dX n , which indicates a
particular mode of decomposition into elements, suggests at the same time a par-
ticular method of obtaining its value by successive integration.
8. It will be seen that, when a definite order is established among the inde-
terminates (X) and (x), the ratio of corresponding elements in the two spaces is
Art. 8.]
THEOREM IN THE INTEGEAL CALCULUS.
85
determined in sign as well as in magnitude ; but that if two indeterminates in
either set be interchanged, the sign of this ratio is also changed. The Jacobian
. dXi
loci
= and
= oo (which for our present purpose we take together)
may be regarded, in general, as dividing the space 2 into two regions A and B,
is positive, and in the second negative. Similarly, a-
L
= 00, =0, into two regions a
in the first of which
is divided by the corresponding loci
X:
dX t
and b respectively corresponding to A and B, in such a manner that correspond-
ing elements have the same sign in the regions A, a, and opposite signs in the
regions B, b. But in using the formula for the transformation of a multiple
integral, it is in general convenient to give to the functional determinant its
absolute value, thus considering the elements of both spaces as positive through-
out. And when the space over which the integration extends is traversed or
bounded by the Jacobian loci, it is always necessary to examine the circum-
stances which present themselves at these loci.
XXV.
ON THE INTEGKATION OF DISCONTINUOUS FUNCTIONS.
[Proceedings of the London Mathematical Society, vol. vi. pp. 140-153. Read June 10, 1875.]
1. XtlEMANN, in his Memoir 'Ueber die Darstellbarkeit einer Function
durch eine Trigonometrische Reihe ' (Abhandlungen der k. Gesellschaft der
Wissenschaften zu Gottingen, vol. xiii., p. 87), has given an important theorem
which serves to determine whether a function f(x) which is discontinuous, but
not infinite, between the finite limits a and b, does or does not admit of integra-
tion between those limits, the variable x, as well as the limits a and 6, being
supposed real. Some further discussion of this theorem would seem to be
desirable, partly because, in one particular at least, Riemann's demonstration is
wanting in formal accuracy, and partly because the theorem itself appears to
have been misunderstood, and to have been made the basis of erroneous in-
ferences.
2. Let d be any given positive quantity, and let the interval b a be
divided into any segments whatever, S 1 = x 1 a, S 2 = x 2 -x l , ..., S n = b-x n _ 1 ,
subject only to the condition that none of these segments surpasses d. We may
term d the norm of the division ; it is evident that there is an infinite number
of different divisions having a given norm ; and that a division appertaining to
any given norm, appertains also to every greater norm. Let e lt e 2 , ..., e B be
positive proper fractions; if, when the norm d is diminished indefinitely, the
sum
S = o\ f(a + l ty + Jg/fo + e 2 S 2 ) + . . . + S n f (_! + e n S n )
converges to a definite limit, whatever be the mode of division, and whatever be
fb
the fractions e 2 , e 2 , ..., e n , that limit is represented by the symbol / f(x)dx, and
WT
Art. 3.] ON THE INTEGRATION OP DISCONTINUOUS FUNCTIONS. 87
the function/ (x) is said to admit of integration between the limits a and b. We
shall call the values of f(x) corresponding to the points of any segment the
ordinates of that segment ; by the ordinate difference of a segment we shall
understand the difference between the greatest and least ordinates of the
segment. For any given division S 1} S a , ..,, S n , the greatest value of S is
obtained by taking the maximum ordinate of each segment, and the least value
of S by taking the minimum ordinate of each segment ; if Z>,- is the ordinate
difference of the segment S { , the difference 6 between these two values of S is
But, for a given norm d, the greatest value of S, and the least value of S, will in
general result, not from one and the same division, but from two different
divisions, each of them having the given norm. Hence the difference between
the greatest and the least values that S can acquire for a given norm, is, in
general, greater than the greatest of the differences 9. To satisfy ourselves, in
any given case, that S converges to a definite limit, when d is diminished without
limit, we must be sure that 6 diminishes without limit ; and it is not enough to
show (as the form of Riemann's proof would seem to imply) that 6 diminishes
without limit, even if this should be shown for every division having the norm d.
3. Let A (d) be the greatest value of S appertaining to a given norm d, and
let B (d) be the least value of S appertaining to the same norm. If d 1 and d 2 are
any two norms, of which d l is greater than d 2 , it is evident that A (c^) ^ A (d 2 ),
B (d^ ^ B (d 2 ), because every division appertaining to the norm d 2 also appertains
to the norm d 1 . And it may be proved (although, for brevity, we omit the
demonstration here) that, given any norm d l} we can always assign a norm d s , less
than d-i, which shall satisfy the inequalities A (d^ > A (d 2 ), B (d^ < B (<4) ; except
only when the function is such that the maximum (or minimum) ordinate is the
same, throughout the whole interval, for all segments however small. In this
excepted case, which is one by no means inconceivable, the value of A (d), [or of
B (cZ),] is independent of d, and is simply h (b a), where h is the maximum (or
minimum) ordinate common to all segments of the interval b a. In all other
cases, it is possible to assign a series of norms, decreasing without limit, and such
that the corresponding maximum values of S form a decreasing series, while the
corresponding minimum values of S form an increasing series.
Besides the maximum and minimum values of S corresponding to a given
norm, we have also to consider the maximum and minimum values of S cor-
responding to a given division. Let P (d) be the maximum value of S appertain-
ing to a given division of norm d, and let Q(d') be the minimum value of S
88 ON THE INTEGRATION OP DISCONTINUOUS FUNCTIONS. [Art. 4.
appertaining to a different division, having the same norm or a different norm.
It is important to observe that we shall always have P(d)> Q(d'}, the sign of
equality being inadmissible, except when the function is such as to be represented
geometrically by a single segment, or a system of segments, parallel to the axis
of x. Leaving out of consideration the excepted case, we may enunciate the
theorem ' The least value of S that can be obtained by taking, in any division
whatever, the greatest ordinate of each segment, is greater than the greatest
value that can be obtained by taking, in any division whatever, the least ordinate
of each segment.' To prove this theorem, let the two divisions, which give the
values P(d) and Q(d f ], be simultaneously applied to the interval b a. To
obtain P(d), each segment in the resulting division will have to be multiplied by
its greatest ordinate, or by a still greater ordinate in some adjacent segment ;
whereas to obtain Q (d') each segment will have to be multiplied by its least
ordinate, or by a still less ordinate. It follows that we have, in general,
P(d) > Q(d'). If, however, we regard the interval b a as composed of segments
l lt 4, ..., each of which has for its extremities points which are also extremities
of segments in each of the two given divisions, we shall find that the inequality
P (d) > Q (cT) must be replaced by the equality P(d) = Q (d'), if it should so
happen that the maximum ordinate of each segment I is the same as its minimum
ordinate ; i.e., if the function f(x) is represented geometrically by a series of
segments parallel to the axis of x, and respectively equal to the segments
lit 1 2 , ....
4. Again, let B'(d) be the least value of S corresponding to the division
which gives A (d) ; and let A'(d) be the greatest value of S corresponding to the
division which gives B (d) ; it is evident from what has been said that we shall
have the inequalities
A(d)>A'(d)>B'(d)>B(d).
Now
A (d) - B (d) = [A (d) - B' (d)] + [A' (d) - B (d)] - [A' (d) - & (d)]
and A' (d) ^B'(d);
therefore A(d)-B(d)< [A (d) - B' (d)] + [A r (d) - B (d)}.
Hence, to prove the evanescence of A (d) B (d) or 6, it suffices to prove the
evanescence of A (d) B'(d), and of A'(d) B(d), which are, in fact the two
values of 9 corresponding to the two divisions which give the absolutely greatest
and least values of S for the norm d.
5. The theorem of Eiemann may be enunciated as follows :
' Let a- be any given quantity, however small ; if, in every division of norm d,
Art. 7.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 89
the sum of the segments, of which the ordinate differences surpass <r, diminishes
without limit, as d diminishes without limit, the function admits of integration ;
and, vice versd, if the function admits of integration, the sum of these segments
diminishes without limit with d.'
The following (with a slight modification suggested by the preceding con-
siderations) is Rlemann's demonstration of the first part of the theorem :
Let s t be the sum of the segments which, in the division corresponding to
A (d) and B'(d), have ordinate differences surpassing a- ; and let be the greatest
ordinate difference in any division appertaining to the norm d ; Q is necessarily
finite, because all the ordinates are finite. The contribution of the segments s x
to the difference A (d) B' (d) cannot surpass s 1 x Q, and the contribution of the
remaining segments cannot surpass a- x (b a s^ ; i.e.,
A(d)-B'(d)^s 1 xQ + <r(b-a- s,).
Similarly, if s. 2 is the sum of the segments which, in the division corresponding to
A' (d) and B(d), have ordinate differences surpassing cr,
A'(d) - B(d) < s 2 x + a- (b - a - s 2 ).
Adding these two inequalities, we find
A d-J8d^s + s Q-<
But a- may be taken as small as we please, and, by hypothesis, however small a-
may be, d can always be taken so small as to render s 1 and s 2 as small as we
please ; i.e., the difference A (d) B (d) = 6 diminishes without limit with d, and
f(x) admits of integration between the limits a and b.
6. Biemann's demonstration of the second part of the theorem requires no
modification. For, if S converges to a definite limit, must be comminuent with
d, and, d fortiori, each of the quantities must be comminuent with d. But,
evidently, in any given division in which s is the sum of the segments having
ordinate differences which surpass a-, <rs<0. Hence, however small the given
a
quantity a- may be, we can always, by taking d small enough, make - less than
any assigned quantity ; i.e., if S converges to a definite limit, s must dimmish
without limit at the same time with d.
7. It will be observed that, in order to establish the convergence of S to a
definite limit, it is sufficient to know that the sum of the segments, having
ordinate differences surpassing cr, is comminuent with d in each of two specified
divisions [viz., in the division which gives A (d) the maximum value of S, and in
VOL. II. N
90 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 8.
that which gives B (d) the minimum value of S]. Hence, if these two sums are
commitment with d, the corresponding sum in any other division of norm d is
also comminuent with d.
S. Let us suppose that the function f(x) has any number of discontinuities
between a and b ; and let there be ^ (?) points at which there are discontinuities
surpassing a-. (We say that a discontinuity surpassing a- exists at a given point,
when any segment, however small, being taken which includes that point, the
ordinate difference of the segment surpasses <r.) If \(/ (<r) has a finite and assign-
able value for every value of <r, however small, the condition of integrability is
certainly satisfied, even if ^ (T) increase without limit, when a- diminishes with-
out limit. For, in any division of norm d, the sum of the segments having
ordinate differences which surpass a-, cannot surpass 2dx^(cr); and, however
small a- may be, d can be taken so small that 2dx~^(<r) shall be less than
any quantity that can be assigned. As an example, we may take the function
considered by Riemann, viz.,
,. . (ar) (2x) (3 a;)
/(*) = V + V + 9 + ~"
where, by (x) we are to understand the (positive or negative) excess of x above
the whole number nearest to x ; or, if x lies half-way between two whole num-
bers, the arithmetical mean between the two differences ^ and ^, i.e., zero. In
(YV\
this function, if x = - , where m and 2n are relatively prime, we have
2n
Thus the number of discontinuities in any given interval is infinitely great. But
the number of discontinuities which in any given interval surpass a given
quantity <r, is always finite. For example, the number of discontinuities between
and 1 which surpass a-, is equal to the number of irreducible proper fractions,
o
having even denominators 2n, which verify the inequality - ~ 2 >(T ' or> ^ 'P ( m )
o )!/
be the number of numbers not surpassing m and prime to m, and if h be the
greatest integer not surpassing - = , the number of discontinuities in ques-
tion is
Art. 11.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 91
which is evidently finite for any given value of a-, although it increases without
limit when a- diminishes without limit.
9. Next, let us suppose that f(x) in the interval b a has an infinite num-
ber of discontinuities surpassing a given quantity a-. The points at which these
discontinuities occur may either ' completely fill ' one or more finite portions of the
interval b a, or there may be no finite portion of that interval which is ' com-
pletely filled ' by them. A system of points is said to ' fill completely ' a given
interval when, any segment of the interval being taken, however small, one point
at least of the system lies on that segment. Thus the rational points on any
line, i.e., the points of which the abscissae are rational, completely fill any seg-
ment whatever upon the line. We may observe that the assertion, that any
given segment of an interval contains at least one point of a given system, is
equivalent to the assertion that any given segment contains an infinite number
[i.e., a number greater than any that can be assigned] of the points of the system.
For we may divide the given segment into as many parts as we please, and each
of them must contain at least one point of the system.
10. When the points at which there occiir discontinuities surpassing <r com-
pletely fill any finite portion of the interval b a, the function f(x) is certainly
incapable of integration. For, if I be the total length of the segments which are
completely filled, we have evidently 9 > <rl for any division of any norm d ; i.e., it
is impossible that should diminish without limit with d.
But points may exist in an infinite number within a finite interval, without
completely filling any portion of that interval. Whenever this happens, it must
be possible in any given segment of the interval, however small, to take a finite
part such that it shall contain no point of the system ; otherwise, the segment in
question would be completely filled. We give a few examples of such systems of
points, the limits of the interval being in each case and 1. We shall say, for
brevity, that points are in close order on any segment when they completely fill
it, and in loose order when they do not completely fill it, or any part of it how-
ever small.
11. (i.) Let the system of points be defined by the equation x--, a being
any positive integer. It will be seen, (1) that these points are infinite in
number ; (2) that they are indefinitely condensed in the vicinity of the origin ;
(3) that they are in loose order over the whole interval, no segment, even in
the immediate vicinity of the origin, being completely filled. For if d lie any
given quantity, however small, we can always find a finite integral number such
N 2
92 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 12.
all
that -<d, and then the finite spaces ( , ) , ( , ) , &c. . . .
m \m + l m' \m + 2 m + 1/
lie on the segment (0, d), and are all free from points of the system, if we leave
their initial and terminal points out of account.
12. (ii.) Let the system of points be defined by the equation x = - - H ,
where a t and a 2 are any positive integers. Here, it is evident that the points are
indefinitely condensed in the vicinity of each of the points of the system (i).
But it can also be shown that they are in loose order over the whole interval
from to 1. Let x = L 1} x = L 2 , (L 1 <L 2 ,) be two consecutive points of the
system (i) ; let M be any positive quantity whatever, and consider the segment
(- * , Z 2 j. If x = 1 lies on this segment, we must have - - ^ L lt
- ^ L lt because no point of the system (i) lies on the interval (L 1} L.^ ; and also
Cva
1 1 ^ jj , . m, . r ,.
- + - - ; whence a, < T f-, a,< -_ ^. Ihese inequalities show
a, a. 2 ~ (JL + 1 ~ L. 2 LI ~ L 2 - L v
that, if, from the beginning of any free segment in the system (i), we cut off as
small a part as we please (which we may do by taking /x great enough), the
remaining portion of that segment will contain only a finite number of points
belonging to the system (ii). And this suffices to prove that the points of the
system are in loose order ; for if d be any segment, however small, situated any-
where in the interval from to 1, we can certainly find on this segment a part
free from points of the system (i), and, by what has just been proved, parts of
that part will be free from points of the system (ii).
13. (Hi.) Let a system of points P e+l be defined by the equation
11 1
x = + +... + ,
where a lt a 2 , ..., a, +l are positive integers. Assuming (what has just been proved
for s = 2) that the system P, is in loose order over the whole interval from to 1,
we shall prove the same thing for the system P g+1 . Let x = L lt x = L 2 be any
two consecutive points of the system P t ; and consider as before the interval
( - 2 , Z 2 ). If the point P s+ i, or x= ----- 1 ---- h + , lies on this inter-
\ fi + i <*! a z a +\
val, we must have, besides the inequality
L...
a, a
Art. 14.] ON THE INTEGEATION OF DISCONTINUOUS FUNCTIONS. 93
the s + 1 inequalities included in the formula
11 1 1
-++...+ <A+ .
1 2 S + 1~ ;
because no point of the system P a can be between L v and L, 2 . These inequalities
give
whence we may infer, precisely as in the case in which s = 2, that the points
P s + i are in loose order over the whole of interval from to 1.
14. Lety(x) be a function, which coincides with a given continuous function
(f> (x) for all values of x between and 1, except at the points P g + 1 ; and let the
difference between f(x) and <p (x) at those points not exceed the finite quantity a-.
It may be shown thatf(x) is integrable between the limits and 1, and that
r l r l
I f(x) dx = (f) (x) dx.
For, take any small interval from to S ; the points P which lie outside it,
between 8 to 1, are finite in number and at finite distances from one another.
Let there be Aj of them ; from each of them measure a space ^ to the right ;
the number of points P 2 , lying outside of the measured spaces S + \S l} is neces-
sarily finite ; and these points are at finite distances from one another. Let their
number be A 2 , and measure a distance S 3 to the right of each of them. Proceed-
ing in this way, we shall obtain measured spaces amounting in all to
Let e be any given quantity however small ; and in the preceding construction
let
x< f_ i ^ s
< o /- o\ > 1 < o / o\ A ' 2
we shall thus have H <^e. Let d be the least of the spaces S, S lt S 2 , ..., S e + 1 ',
it may be shown that, in any division of norm d, the sum of the segments con-
taining points P t+l cannot exceed 3H. For all the points P s + l lie on the
measured spaces ; and supposing (which is the most unfavourable case) that one
of those spaces begins and ends with a point P i+ i, we can at most triple it, by
imagining a segment equal to d placed on each side of it. Thus, in every division
of norm d, the sum of the segments containing the points of discontinuity is less
94 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 15.
r 1
than e ; whence we infer, by Biemann's theorem, that / f(x) dx has the same
r 1 '
value as / d> (x) dx.
JQ
15. (iv.) Let m be any given integral number greater than 2. Divide the
interval from to 1 into m equal parts ; and exempt the last segment from any
subsequent division. Divide each of the remaining m 1 segments into m equal
parts ; and exempt the last segment of each from any subsequent division. If
this operation be continued ad infinitum, wfe shall obtain an infinite number of
points of division P upon the line from to 1. These points are in loose order :
for if d be any segment however small, situated anywhere in the interval from
to 1, we may take an index k which satisfies the inequality - - < \d ; and then
determine a segment of the type fj 5-) lying entirely on the segment d.
But this segment is either itself an exempted segment or its m th part is so. It
will be seen that, after k operations, the sum of the exempted segments amounts
/ 1\*
to 1 ( 1 -- ) ; so that, as k increases without limit, the points of division P
\ m/
occur upon segments which occupy only an infinitesimal portion of the interval
from to 1. And it may be inferred that a function, having any finite discon-
tinuities at the points P, would be integrable. For, if d be any given small
quantity, let the index k be determined by the inequalities T > d > r ; the
m" mr +
number N of excepted segments which surpass ^ is
and the sum of the remaining segments is
It is evident that in any division of norm d, the sum of the segments containing
points P cannot exceed
1 \ k ~ l
- ) + 2Nd.
/ \ -
But, as d decreases, and k increases, without limit, (l J and 2Nd, which
is less than 5-, both decrease without limit ; i.e., in any division of norm d, the
Art. 17.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 95
sum of the segments containing points of discontinuity diminishes without limit
with d ; and the function is integrable.
16. (v.) Let us now, as in the last example, divide the interval from to 1
into m equal parts, exempting the last segment from any further division ; let us
divide each of the remaining m 1 segments by m 2 , exempting the last segment
of each segment ; let us again divide each of the remaining (m 1) (m 2 1) seg-
ments by m 3 , exempting the last segment of each segment ; and so on continually.
After k 1 operations we shall have
N= 1 + (m - 1) + (m - 1) (m 2 - 1) + ... + (m- 1) (m 2 - l)...(m fc - 2 - 1)
exempted segments, of which the sum will be
This sum, when k is increased without limit, approximates to the finite limit
\ E( ): where E ( ) is the Eulerian product 1 ( 1 -- r ) . and is cer-
\m/ V7H,/ J - LI V m,'
tainly different from zero. The points of division Q exist in loose order over the
whole interval. For, if d be any small segment of that interval, and if
m lk(k-i)<2 d ' a segment of the type ( m ,^_ 1) , m %-i)) can b found lying
entirely on the segment d, and this segment is either itself exempted, or its
/ 1 \ th
( j) part is exempted. But a function having finite discontinuities at the
points Q would be incapable of integration. For, if d be any norm, and
S < , fc(t _ 1) < d, in the division
(which is a division of norm d], the sum of the segments containing points of
discontinuity is
which approximates to the finite limit E ( ) when d is diminished, and k is
increased without limit.
17. The result obtained in the last example deserves attention, because it is
opposed to a theory of discontinuous functions, which has received the sanction
of an eminent geometer, Dr. Hermann Hankel, whose recent death at an early
96 ON THE INTEGRATION OP DISCONTINUOUS FUNCTIONS. [Art. 17.
age is a great loss to mathematical science. In an interesting memoir (' Unter-
suchungen ueber die unendlich oft oscillirenden und unstetigen Functionen,'
Tubingen, 1870), Dr. Hankel has laid down the distinction, here adopted from
him, between a system of points which completely fill a segment, and a system
of points which do not completely fill any segment, but lie in loose order. [The
term employed by Dr. Hankel is ' zerstreut ' ; the use of the equivalent English
words ' dispersed ' or ' scattered ' has been avoided in the presenb note, because
they might seem to exclude the sort of condensation in the vicinity of a finite or
infinite number of points, which, as we have seen in the examples (i.), (ii.), (iii.),
may present itself in the case of systems of points in loose order.] Dr. Hankel
then asserts (see p. 26) that, when a system of points is in loose order on a line,
the line may be so divided as to make the sum of the segments containing the
points less than any assignable line. The proof of this assertion is, in effect, as
follows : Divide the line into segments, of which each contains a point of the
system, and imagine each segment to be diminished to its n^ part, yet so as still
to have upon it the point of the system which it contained before. The sum of
the segments can thus be made less than the n ib part of the whole line ; i.e., less
than any line that can be assigned, because we may suppose n as great as we
please. It must be conceded that this demonstration is rigorous, if the number
of points in the system is finite ; but the construction indicated ceases to convey
any clear image to the mind, as soon as the number of points becomes infinite.
If we are allowed to divide the line from to 1, in example (iii.), in such a man-
ler as to include every point (-P + 1 ) in a segment of its own, these segments, in
nu - vicinity of the points P t , will have to be less than any line that can be
^d ; and, if such a mode of division is admissible, it is difficult to see why
i not also be considered admissible so to divide the line as to include
and the sional point in a segment of its own : in which case Dr. Hankel's
would extend to systems of points in close order, as well as to
ose order. But whether we do or do not admit the truth of
proposition, the use which he makes of it (p. 31) to establish
j, i , ,-i r \1 /
of Biemann's criterion to a certain class of functions would
points P cannot e. eoug TQ p roye that Ri emann > 8 con dition of integrability
T en discontinuous function, we have to show that, given
however small, the sum of the segments, which, in-
But, as d decreases, and norm d, contain the points of discontinuity, is evan-
. , 2N , , , 's evident that this cannot be shown, if we confine
is less than r , both deer
W lodes of division in which some of the segments
Art. 19.] ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. 97
are from the very beginning assumed to be less than any quantity that can be
assigned.
While, therefore, we may safely admit the theorem that no function can be
integrable which has discontinuities, surpassing a given quantity er, at an infinite
number of points forming a system in close order ; the converse assertion that,
when the system of points of discontinuity is in loose order, the function is inte-
grable, would seem to be established by no satisfactory demonstration, and to be
negatived by the result obtained in example (v.).
18. Another proposition, contained in the same memoir (p. 28), appears open
to a similar objection. It may be admitted that a function f(x) having discon-
tinuities, which surpass a given quantity a- however small, only at points which
form a system in loose order, is necessarily continuous over finite portions of any
interval however small. But it would seem to be untrue that such a function is
necessarily continuous in the vicinity of any one of its points of discontinuity.
If, for example, /(--H -- ) ^-, and /(a;) = 0, for every other value of x, it is
^ OEj 0&2 '
evident that, however small the given quantity e may be, the difference
oscillates an infinite number of times between the values and 1, as 8 decreases
from e to ; i.e., the function f(x) is discontinuous in the vicinity of the point
to the right.
a,
19. We add a few remarks which may serve still further to illustrate the
meaning and use of Riemann's theorem.
(i.) The problem, ' Given a system of points upon an interval (a, V), to find,
among all divisions of norm d, that in which the segments containing the points
have the maximum sum,' is perfectly determinate. We may say that a point of
the system is isolated, when it is separated from the next preceding and next
following point by a distance > 2 d. Similarly a group of points may be said
to be isolated, when the distance between any two consecutive points of the
group is less than 2d, but the distance between the extreme points of the
group, and those which immediately precede and follow it, is greater than 2d.
It is evident that, for any given value of d, the given system of points
resolves itself into a finite number of isolated groups. The first and last
point of each group determine a segment ; on either side of each of these
segments, and on either side of each isolated point, we may place a segment
VOL. n. o
98 ON THE INTEGRATION OF DISCONTINUOUS FUNCTIONS. [Art. 20.
equal to d. The sum of the segments thus obtained is the maximum sum
required.
It will be observed that in this solution each point of the system is regarded
as double ; i.e., as capable of affecting two segments at once, one on each side of
it. If the discontinuity of a function at any point can be removed by changing
the value of the function at that point only, for example, if f(x 0) = a,
f(x) = a + <r,f(x + Q) = a, the point must be regarded as single (its contribution
to the difference of Art. 2 would be only a- x d). But if the values of the
function preceding and following the point of discontinuity are different (i.e., if
f(x 0) = a, f(x + 0) = a + <r), the point of discontinuity produces a double effect,
its contribution to the difference being 2 <r x d. Similarly, in the case of
functions which, like cos C-j in the vicinity of the origin, admit of an infinite
number of maxima and minima within a finite interval, the contribution to of
each point at which there is a maximum, or minimum, is two-fold. For the
practical application of Riemann's criterion, the distinction between points pro-
ducing a one-fold effect and those producing a two-fold effect is immaterial.
20. (ii.) When a function, which is discontinuous but never infinite, does not
r*
admit of integration between the limits a and 6, the symbol / f(x) dx becomes
indeterminate. But the maximum and minimum values attributable to that
symbol are perfectly determinate ; and if it should become advisable to attribute
a definite value to the symbol, we might select for that purpose the arithmetical
mean between these two extreme values. If, for continually decreasing values
of d, we calculate the corresponding maximum values of the sum S of Art. 2,
these values will, as shall now be shown, converge to a determinate limit A. And
similarly the successive minimum values of S will converge to a determinate
limit B, different from A in the case under consideration. The difference A B
is, of course, the limit of the successive differences 0.
From the two sets of inequalities
A (dj > A (d 2 ) > A (d.,) > ...,
B(d l )<B(d 2 )<B(d 3 )<...,
combined with the inequality A (d n ) > B (d n ), which holds for any value of n
however great, we infer that each of the two series,
A (d,) - A (d,), A (c/ 2 ) - A (d 3 ), A (d s ) - A (d,), . . .,
B(d 2 )-B(d 1 },
Art. 21.] ON THE INTEGRATION OP DISCONTINUOUS FUNCTIONS. 99
consists of positive terms, and that, however many terms of either series we add
together, we can never surpass A (dj) A (d x ) in the first, and B (d x ) B (dj in
the second ; i.e., in neither of them can we ever surpass A (d^ B(dj). But if a
series of positive terms be such that the sum of any number of its terms, how-
ever great, can never surpass a given finite quantity, the sum of the first n terms
of the series converges to a finite and determinate limit, when n is increased
without limit (see Riemann, Vorlesungen, pp. 39, 40). The sums A (d^ A (d n ),
B (d n ) B (d^, therefore converge to finite and determinate limits ; or, which is
the same thing, the two series of terms
A(d 2 ), A(d 3 ), ...,
B(d 2 ), B(d 3 ),...,
converge to finite and determinate limits.
If, for example, the function f(x) have the value <r, at every point of the
system considered in Art. 16, example (v.), and the value o- 2 < <TI at every other
point ; we shall find
B = (T 2 , A = <T 2 + (ff l CT 2 ) X -
21. (iii.) Riemann's criterion of integrability is applicable to the case of any
multiple integral extended over a finite space. For example, in the case of a
triple integral, we must imagine the whole space of the integration divided into
small spaces such that any one of them could be comprehended within a sphere
of a diameter d ; and any such division into spaces is a division of norm d. The
criterion of integrability, then, is that, in any division whatever of norm d, the
sum of the spaces in which the ordinate-difFerences surpass a given quantity a-,
must diminish without limit with d. The ordinate-difference of any space is, of
course, the difference between the greatest and least values of the function
within the space.
Considering, for simplicity, the case of two dimensions only, we observe that
the space of integration may not only contain points of discontinuity finite or
infinite in number, but may be intersected by curves of discontinuity. The
function may have values differing by a finite quantity on either side of such a
curve ; or its values at points along the curve may be discontinuous, or both of
these kinds of discontinuity may be combined at the same curve. If L (<r), the
total length of the curves at which the discontinuities surpass <r, be finite, the
function can be integrated over the given space ; since, if we draw curves parallel
to the curves of discontinuity and at a distance d from them on either side, the
o 2
100 ON THE INTEGRATION OP DISCONTINUOUS FUNCTIONS. [Art. 21.
area of the channel-like spaces thus obtained will be 2dL(<r), and will surpass
the greatest sum of spaces, including the curves in any division of norm d. But
the function may be integrable even if the total length of the curves of discon-
tinuity is infinite ; because an infinite number of contiguous curves may be
enclosed in one and the same channel. And, provided that the curves can all be
included in channels of which the length is L, and of which the breadth S is com-
minuent with d, the condition that L x S should be comminuent with d, will
suffice to ensure the integrability of the function.
XXVI.
ON THE HIGHER SINGULARITIES OF PLANE CURVES.
[Proceedings of the London Mathematical Society, vol. vi. pp. 153-182. Read June 10, 1875.]
J.HE ordinary singularities of a plane curve are its double points and double
tangents, its stationary points and stationary tangents ; or, as they have been
also called, its nodes and links, its cusps and inflexions. The fundamental
theorem, that any of the so-called higher singularities of a plane curve may be
regarded as equivalent to a certain number of ordinary singularities of each of
these four kinds, has been enunciated by Professor Cayley, who has also given a
method for determining in every case the four indices , T, K, i, proper to any given
singularity.
Several enquiries, which appear to possess some interest, are suggested by
this theorem. Among them we may mention the two following
(1) It is important to prove that the indices of singularity, as defined by
Professor Cayley, satisfy the equations of Plucker; and that the 'genus' or
' deficiency ' of the plane curve is correctly given by these indices.
(2). It is also of interest to examine whether any given singularity can be
actually formed by the coalescence of the ordinary singularities to which it is
regarded as equivalent ; in other words, whether a singularity of which the
indices are S, T, K, t, and which is therefore to be regarded as equivalent to <5
double points, T double tangents, * cusps, and / inflexions, possesses a penultimate
form, in which all these singularities exist, distinct from one another, but in-
finitely close together.
The present paper relates chiefly to the first of these enquiries ; the second
is reserved for a future communication.
1. Consider a plane curve C of order m and class n, defined by an equation
102 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 2.
F(j),q)=0 between the parameters of two pencils, of which the corresponding
rays intersect on C, and which are represented by equations of the form
p (QP) + (QE) = 0, q (PQ) + (PR) = ; P, Q, R denoting the three vertices of a
triangle, (PQ) = 0, (QR) = 0, (PR) = the equations of its sides. It is convenient
to suppose that Q and P, the centres of the two pencils, have no speciality of
position with regard to C ; or, more precisely, that neither Q nor P lies on the
curve, nor on any singular line appertaining to the curve. Under the general
name of singular lines we include (1) lines joining two singular points, (2) singular
tangents, (3) tangents at singular points, (4) tangents passing through a singular
point ; we shall also suppose that PQ is not a tangent to C, and does not pass
through any singular point. Thus to every finite value of p there will correspond
m finite values of q, and vice versd ; and, in particular, to any singular point on
the curve there will correspond a finite pair of values of p and q. To an infinite
value of q there will correspond m infinite values of p, and vice versd ; these
answer to the m intersections of PQ with the curve, no two of which, by
hypothesis, are coincident. We may, if we please, project the line PQ to an
infinite distance, and regard p and q as Cartesian coordinates ; we prefer, how-
ever, for our present purpose, to consider them as parametric ratios ; i.e., as
purely numerical quantities (real or complex).
2. Let f(q) be the discriminant of F(p,q) = 0, considered as an equation
of the order m in p ; we may suppose the coefficient of p m , which is certainly
different from zero, to be unity. The first polar of P with regard to C is
J -HI
-j = 0, and _/((?) is the resultant of the elimination of p from F and -= , so that
the roots off(q) = are the parameters of the lines drawn from P to the points
of intersection of C with the first polar of P. Attending to the suppositions
which have been made as to the situation of P and Q relatively to the curve C,
we infer (a) that f(q) has no infinite roots, and is therefore of the full order
m(m l) in q; (/3) that f(q) has n, and only n, non-multiple roots q' ; (y) that
for each of these n roots q' the equation F( p, (f) = acquires two equal roots p',
its remaining roots being all different from p', and from one another ; () that q'
is not a multiple root of the equation F(p', q) = 0. The n sets ( p', q') give the n
points of contact of tangents from P ; the remaining factor of f(q), viz.,
fi (?) /(<!) ~=~ n (q q), consists exclusively of multiple factors, and appertains to
the singular points of the curve. The index of its order, i.e., m(ml) n, we
may term the total discriminantal index of the singular points of the curve.
Let q be a root of f (q) = of multiplicity v ; the equation F (p, q ) = has but
Art. 3.] ON THE HIGHEK SINGULARITIES OF PLANE CURVES. 103
one multiple root ; let this be p , and let its multiplicity be ft. ; then (p , q ) is a
singular point O on the curve, of which the order (i.e., the least number of points
in which it is cut by any straight line passing through it) is n, and of which v
may be termed the discriminantal index. It is evident that the number of
singular points is equal to the number of unequal roots of f^ (q) = 0, and that the
total discriminantal index is equal to the sum of the discriminantal indices of the
separate singular points. We shall presently (Art. 8) see that the discriminantal
index of a singular point can in general be further subdivided into parts, apper-
taining respectively to the different branches of the curve which pass through
the point, taken singly, and in pairs.
3. It is a well-known theorem of Cauchy, that so long as the analytical
modulus of q q is less than the least of the modules of any of the quantities
<7i <?< where q is any root off(q) = other than q , the m roots of the equation
F (p, q) = are developable in convergent series of the form
p-p = A+A (q-q )+A 1 (q-qo) a i + A 2 (q-q ) a *+..., ... (A)
the exponents a lt a 2 , ... being rational and positive numbers, which satisfy the
inequalities I<a 1 <a 2 < Of the equations (A), m fj. give the values of p
corresponding to the m fj. points not in the vicinity of 0, in which C is cut by
the line (). The series in the right-hand members of these m M equations we
shall designate by A lt A 2 , ..., ^4 m _ M : we observe that in them the quantities A
are all different from one another and from zero ; because (q ), not being a singular
line, intersects C in m p. points, which are different from one another and from
; also, in these equations, the exponents a lt a 2 , a 3 , ... are integral. In the
remaining /* equations, which give the developments appertaining to the branches
of C that pass through 0, the quantities A are all equal to zero : these equations
divide themselves into groups of conjugate equations, the equations of any one
group being of the type
where the numerators /3 are positive, integral, and increasing ; A is less than /3 l5
and is the least common denominator of the fractional exponents ; Q> is any root
of W A = 1 : so that, if we use one and the same value of the radical in all the A
equations of the group, they will differ from one another only by containing
different values of <a ; each of the n equations defines a branch of the curve pass-
ing through 0. If A = 1, the branch is linear or of order 1 ; if A > 1, the A con-
jugate equations are regarded by Professor Cayley as defining A partial branches
104 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 4.
forming a single superlinear branch of order A ; in every case the sum of the
orders of the branches is equal to the order of the point, i.e., 2A = / u. The
coefficients B are all different from zero, and the indices ~ are all greater than
unity, because neither (p ) nor (q ) is one of the tangents at 0; but these
coefficients and indices are not necessarily different in two developments belong-
ing to two different linear or superlinear branches indeed any two such de-
velopments may coincide for any finite number of terms ; and to ascertain the
true nature of any singular point it is indispensable to continue the developments
until they all become different from one another. The series in the right-hand
members of the n equations we denote by B 1} B 2 , B 3 , ..., B^.
4. The series A and B of the preceding article are absolutely convergent
within the assigned limits ; i.e., any one of these series would continue to be
convergent within these limits if its terms were replaced by their analytical
modules. For the multiplication of two absolutely convergent series we have the
theorem :
' If the product of two given absolutely convergent series, proceeding by
ascending powers of a variable, be arranged in a series proceeding in the same
manner, this series is absolutely convergent for all values of the variable for
which the given series are absolutely convergent, and its sum is equal to the pro-
duct of the sums of the given series.' (Cauchy, ' Analyse Alge"brique,' cap. vi.)
Multiplying together the m series p p A, and p p B, we obtain, by
virtue of this theorem, the equation
F( P , q) = n (p - Po - Z) x n (p - Po - B).
This equation is an identity ; i.e., if the multiplication be actually effected on the
right-hand side, all powers of q q a above the m th will disappear, and the terms
that remain will be precisely the terms of F(p +pp , q + q q<>) or F(p, q).
But an arithmetical equality between the two sides of the equation subsists only
so long as the analytical modulus of q q does not surpass the limit assigned in
Cauchy's theorem (Art. 3). Subject to the same limitation, f(q) is equal to the
product of the squares of the differences of every two of the series A and B.
5. The number of the intersections at any point of two branches of the
same curve, or of different curves, which pass through the point, and which are
there represented by equations of the form
Art. 6.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 105
is defined by Professor Cayley to be the number which expresses the order of
evanescence of p m p\ i.e., the integral or fractional exponent X for which
d) _ (2>
... X has a finite limit, when q q is diminished without limit. We mav
(j-ft)*
justify this definition by proving that, whenever two curves C^ and (? 2 have a
multiple intersection at any point, its multiplicity is correctly obtained by adding
together the numbers (as thus defined) of the intersections of each branch of C t
by each branch of (7 2 . If we suppose (as we may do) that the points P and Q
have no speciality of position with regard to the curves C^ and C. z considered as
one curve, the resultant &(q) of the equations C l (p,q) = and C 2 (p, q) = Q is
of the order m 1 x m 2 ; and if MI branches of C^ and yu 2 branches of C. 2 pass through
O, we shall have, for C lt 7% MI equations A 1 , and MI equations B ; and similarly,
for C 2 , m 2 M 2 equations A (2 \ and M 2 equations B (2 \ Denoting by II (.B (1) - B ( ' z) )
the product of the MI x /u 2 differences obtained by subtracting them in succession,
each series B (2} from each series B (l \ and by X the number of intersections, as
above defined, of any one branch of Cj by any one branch of (7 2 , we see that the
limit of n (5 (1) - 5< 2) ) + (q- q )** is finite. But * (q) = U (B (l) - B (2} ), the sign of
multiplication now extending to all the m t x ra 2 differences obtained by consider-
ing the m^ series A (1) and B (l \ and the m 2 series A 2 and B 2 ; and of these m l x m 2
differences, none, except the MI x n 2 differences already considered, are evanescent
with q q (for the hypothesis that P and Q have no speciality of position with
regard to the system of the two curves Cj and C 2 implies that none of the con-
stants A w can be equal to any of the constants A (2} ). Hence 2 X is the multi-
plicity of the factor q q in 3> (q) i.e., since (p , g ) is the only intersection of
Cj and C 2 which lies on (qj), 2 X is the multiplicity of that intersection.
If we regard the equation F(p, q) = as determining a correspondence of
points on a line, the coincidences of corresponding points (except indeed the
coincidences p = q = oo) answer in number and multiplicity to the intersections of
C by the straight line p = q. We are thus led to a theorem given by M. Zeuthen
('Bulletin des Sciences Mathe"matiques,' Vol. V., p. 186).
6. As it is only the hypothesis that the points P and Q have no speciality
of position with regard to C which gives us a right to assert that every one of
the developments B contains a term linear in q q , and no term in which the
exponent of q q is less than unity, it is worth while to see how far the results
of the preceding article can be depended on when this hypothesis is dispensed
with. It will be found that if (p ) is one of the tangents at O, i.e., if one, or
more, of the coefficients B is zero, the discriminantal index of the point is still
VOL. II. P
106 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 7.
equal to the order of evanescence of IT (B i J5,) 2 . But this conclusion would no
longer hold, if (q ) were one of the tangents at 0. In this case the developments
appertaining to the branches to which (qj) is a tangent would contain powers of
q q inferior to unity; and the order of evanescence of II (^ Bj) 2 would
exceed the discriminantal index of by T, if n T is the number of tangents
other than (5,,) which can be drawn to the curve from P. But the multiplicity
of the intersection of two different curves, at a point which is singular for one or
both of them is correctly obtained by the process of Art. 5, even when the
developments contain positive powers of (q q ) inferior to unity. Thus, in the
i
curve p p = (q <? ) a > a being an integer, the order of evanescence of II [B t B$
is a 1, whereas the point is not a singular point at all, and has consequently a
discriminantal index equal to zero : its tangent (c^) however is a singular tan-
gent, and counts as a 1 tangents drawn from P. On the other hand, if we
i i
consider the two curves (p -p ) = (q- q) a , p-po-(q <?o) b > i n which a and 6 are
both integers and b < a, the order of evanescence of II (B { Bjf is 6 ; and this is
the multiplicity of the intersection at (p , q ).
7. We can now prove that the discriminantal index of the singular point O
is equal to twice the number of the intersections of C by itself at that point ;
and, again, that this discriminantal index is equal to the number of the inter-
sections at the same point of C by its first polar with regard to any point not
having a special position. For (1), considering the M. equations B, we see that
twice the number of intersections of C by itself at the point (_p , (?o)> is the order
of evanescence of II (B t Bj) 2 , the sign of multiplication extending to all the
l/tx(yu 1) differences; or, observing that f(q)-r H (B { BJ) 2 is a product of
\m(m 1) ^fj.(n 1) squared differences, none of which vanish with q q ,
twice the number of intersections of C by itself at the point (p , q ) is equal to
the order of evanescence of f(q) with q q^, i-e., to the discriminantal index v.
And (2), since the polar of P is -5 = 0, and since the resultant of ^=0 and
dF P
-j = is f(q), we infer (Art. 5) that v is the number of intersections at (p Q , <? )
of C by its polar with regard to P.
8. Considering a superlinear branch, of which the component branches are
defined by the A equations p
p-p, = B, ) (q-q Q ) + B 1 ^(q-q^+..., ...... (A)
let Aj be the greatest common divisor of A and & ; and, if y 1 is the first of the
Art. 8.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 107
numbers ft which is not divisible by A lt let A 2 be the greatest common divisor of
Aj and 7l ; if, again, 72 is the first of the numbers ft which is not divisible by A a ,
let A 3 be the greatest common divisor of A 2 and 72 , and so on continually. Since
the numbers ft have no common divisor with A, we shall at last arrive in the
series A 1; A 2 , A 3 , ... at a term equal to unity, when the series will terminate : and
twice the number of the intersections of the superlinear branch by itself will be
expressed by the formula
in which 7 is written for &. For if a> denote any given root of the equation
<oA = 1, of its remaining roots x there are A f 1 which verify the equations
xi = u>">, xTi = ei)Ti, ..., ofH-i = to"fi-i ; because A f is the greatest common divisor of A,
% 7i> > 7<-i : similarly there are A f+1 1 roots other than 1 which verify the
same equations, and in addition the equation a;?* = oX Thus, of the A (A 1)
differences obtained by subtracting each of the series (A) in turn from every other,
*Vj
there are A (A,.- A i + 1 ) which are of the order ^ ; i.e., 2N='2y i (A i A,. +1 ). The
value of N depends, therefore, not on every exponent in the series (A), but only
on certain critical exponents j- , in the denominators of which, when reduced to
their lowest terms, a new factor appears for the first time. The number 2JV,
which is the ' discriminantal index ' of the superlinear branch is, not itself neces-
sarily even, but the difference 2^ (A 1) is always even, since we have
and in this expression, if A is uneven, so also are A 15 A 2 , ... ; if A is even, let A,-
be the first of the numbers A 1; A 2 , ... which is uneven ; then 7f-1 is uneven, and
so are all the subsequent numbers A,-^, A i+2 , ... . In either case, therefore, every
term in the expression of 2^ (A 1) is even.
Again, if two superlinear branches of the orders A and A' have the same
tangent, let (q-qd) h be the lowest power of q-q<, which has not the same
coefficient B in the two sets of series (A) and (A') : it may, of course, in one of
these sets have a zero coefficient. Then the terms of lower exponent are common
to the two sets ; and if the exponents be reduced to their least common denomi-
nator, these initial terms will be of the form
a, a a a.
(q - 5.) + B, 6". (q - q,} d + &*& fe ~ V.tf + + B i fr (q -
P 2
108 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 9.
where d is a common divisor of A and A', is any root of 0^ = 1, and -4 is the
a
exponent next inferior to h. The number of intersections of the two superlinear
branches is then
the numbers a-, cr l , a-%, ..., d 1} d%, ... (of which in particular <r = a 1 ) being deter-
mined from the series of exponents -j, in the same way that the numbers
8
y, y 1} ..., A 1( A 2 , ... were determined from the series of exponents . For if 6
represent a given root of the equation 6 d = 1, the -j- roots of the equation CO A = 1,
A d
which satisfy the equation w d = 6, will give the same initial terms ; and we may
thus divide the equations (A) into -j- groups, each containing d equations ; the
equations of the same group differing from one another by containing different
values of 0, but the different groups not differing from one another, so far as the
A f
initial terms are concerned. Similarly we may divide the equations (A 7 ) into -,-
groups. Considering only one group of each set, we find (by the same reasoning
as before) for the order of the product of the d x d differences obtained from
them, the expression
the additional term hd appearing because we have now to take into account the
d differences in which all the initial terms vanish : the result, multiplied by
-T x -T , gives the value of N'.
Lastly, when two superlinear branches have not the same tangent, the num-
ber of their intersections is evidently N" = AA'. By means of these formulae the
discriminantal indices of the branches at any singular point, taken by themselves
or in pairs, may always be obtained as soon as the developments appertaining to
the branches have been found. The sum of these separate discriminantal indices
is of course the discriminantal index of the point, or i/ = 22JV+227V' + 22/V".
9. Every singular point of a plane curve is regarded by Professor Cayley as
being equivalent in a certain manner to S common nodes and K common cusps ;
and, correlatively, every singular tangent as equivalent to T double tangents and
i inflexional tangents. For any superlinear branch of order A passing through a
Art. 10.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 109
singular point, the cuspidal index K is by definition A - 1 ; thus, for a linear
branch K = 0. The cuspidal index of a singular point is the sum of the cuspidal
indices of the several superlinear branches passing through it ; so that, for any
singular point, K = 2 (A- 1) = M -X, if /* is the order (Art. 2) of the point, and X
the number of distinct linear, or superlinear, branches passing through it. The
nodal index S for a singular point, and for its branches, taken singly or in pairs,
is defined, not directly, but by equating 2S + 3 K to the discriminantal index;
thus, for any superlinear branch of order A, we have
which is always even (Art. 8), and positive, except when 7 = 8, A = 2, in which
case S = 0, and the superlinear branch is a common cusp.
For r and t we have correlative definitions.
10. Adopting these definitions, we have now to prove that the numbers Z<5,
2/c, ZT, Zt (the summations extending to all the singularities of the curve) satisfy
the equations of Pliicker, and further that the deficiency of the curve is correctly
given by the formula
H=(m- 1) (m - 2) - ZJ- Z*.
It is sufficient to establish the four equations,
(i.) = m(m-l)-2Z<$-3ZK,
(ii.) m = w (n-l)-2Zr-32i,
(iii.) r=(m
(iv.) jy=-J(w
because the three equations (i.), (ii.), and (iii.) = (iv.) are equivalent to the six
equations of Pliicker. But the equation (i.) has been already proved ; for we
have found (Art. 2) that n = m(m 1) 2/; and by definition Zi/ = Z (2 S + 3 K).
The equation (ii.) is the correlative of (i.) and needs no separate proof. In the
equations (iii.) and (iv.) it is important to take a definition of H which does not
involve any special supposition as to the nature of the singularities appertaining
to the curve. The simplest, though not the most direct, course is to adopt the
method of Riemann, and to define 2H+1 as the index of multiplicity of con-
nexion of the m-leaved spirally connected surface [Q], which is such that if the
complex values of q be represented upon it in the usual manner, p may be
regarded as a one-valued function of q. In any such surface the index of multi-
plicity of connexion 2H+1, the number of leaves m, and the number of spires
(spiral points, windungs-punkte) N&re connected by the equation N= 2H+ 2m 2.
This equation Riemann himself demonstrates by comparing the values of certain
110 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 11.
contour-integrals (' Theorie der Abelschen Functionen,' Art. 7). But he observes
that it is entirely independent of considerations of magnitude, and that it belongs
properly to the geometry of situation. The demonstration of it from this point
of view, which has been given by M. Neumann (' Vorlesungen,' p. 309, $ 99), is also
independent of any supposition as to the special nature of the singularities of the
curve C ; and is therefore available for our present purpose. But we may observe
that the algebraical demonstration of the same equation, which is given by MM.
Clebsch and Gordan (in their 'Theorie der Abelschen Functionen,' p. 54, 16),
would here be inadmissible, because in that demonstration it is expressly sup-
posed that the singular points of C are only common nodes and cusps. (See the
note at p. 11, loc. sit.)
It is not difficult to find the number of spires N on the surface [(?]. There
is a one-fold spire for every tangent from P to C; for, if (p , q ) be the point of
contact of any such tangent, we have for values of q in the vicinity of q two
conjugate developments of the type
in which B l is different from zero ; all the other developments (Art. 3) being of
the type (A), because the point P has no speciality of position. Again, there is
a (A l)-fold spire for any singular branch which is superlinear and of order A ;
this is apparent from the form of the A developments appertaining to the branch
(see Riemann, loc. cit., Art. 6 ; M. Puiseux, ' Liouville,' 1st series, Vol. XV. pp.
384-404).
We have therefore ^V=n + 2 (A 1) =71 + 2*, and Eiemann's equation be-
comes n + Sr = 2//+2(m-l);
or, since n + 2/t = m (m 1) 2 2^ 2 2*,
H= (m- 1) (m-2) - 2J- Sc,
which is the equation (iii.) Again, it is an immediate consequence of Riemann's
definition of the number H (see his ' Abelsche Functionen,' Art. 11) that this
number remains unchanged by any unicursal transformation of the equation
F (p, q) = 0. But (as has been already observed by MM. Clebsch and Gordan)
any tangential equation of the curve C may be regarded as an unicursal trans-
formation of the equation F (p, q) = 0, because the points and tangents of a curve
correspond to one another one to one. The equation (iii.), therefore, involves the
equation (iv.) ; a result which, as we have seen, implies that the six equations of
Pliicker are satisfied by the numbers 2^, 2/c, 2r, 2<.
11. The indices T and i appertaining to any superlinear branch at a singular
Art. 11.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. Ill
point, and the number of tangents common to two osculating superlinear branches,
may be ascertained directly from the point-equations B, without actually forming
the corresponding line-equations. To prove this, we shall establish a relation
which subsists between certain terms in the two sets of equations.
If Q and R are given constants, p = Qq + R is the equation of a straight line
in the system of parametric point-coordinates which we have been employing.
In passing to line-coordinates, we may take Q and R as the coordinates of this
straight line ; and we may regard Q and R as the parameters of two ranges of
points, lying on the lines PQ, PR, respectively, and represented by equations of
the form
the line p = Q^q + R^ or (Q , R^) being the line joining the points determined in
the two ranges by the values Q , R of the parameters. If to the hypothesis of
Art. 1 we add the supposition that PR is not a tangent to C, and does not pass
through any singular point of C, the line-equation of (7, which we may represent
by < (Q, R) = 0, will have the same sort of freedom from speciality which has
been already attributed to the point-equation F(p,q) = Q- The parameters of
the tangent to C at the point (p, q) are
Let (p, q) be a point lying on the branch B, of which the point-equation is
p -po = B (q - g ) + B, uPi (q - g ) A + . . . ;
and suppose (p, q) different from (p , g- ), but sufficiently near to it (Art. 3) to
ensure the convergence of the m series A and B. Writing
where M is a product of factors, none of which can vanish at the point (p, q),
because no singular point other than (p 9 , q<,) exists within the range of values
attributed to q, we find ^5 >g
\d'' "* 3\d'
Putting Bo=Q , PQ q B^^R^,
so that ((?, -R ) is the tangent at (p 0> q t ) to B, we obtain the equations
112 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 12.
which determine the parameters Q, R of the tangent at any point of B. If we
further write
these equations become
F*i-* = *,-*[! 4-^-^ + ^3-0, + ...], ..... ( a)
+ ............. 08)
where we have written <r i for ^~tt , and p i for (\ M ^. It will be observed
Pi -D 1 A /
that w has disappeared from these equations, which therefore appertain equally
to all the A branches composing the superlinear branch B. To obtain the tan-
gential equation of that branch, i.e., the expansion of R R in a series proceed-
ing by powers of Q Q , we have three operations to perform. First, we have to
raise each side of the equation (a) to the power -5 - - ; we thus obtain an ex-
pansion of the form
eY=Z(l + At > a + B%> +...}, ....... (Y)
6 denoting any root of the equation 0^i~ A = 1. Secondly, we have to revert the
series (Y), so as to obtain the series
Z=eY{i+A'(eY) a! +B'(eYy+...} ....... ()
Lastly, we have to substitute, in the equation (/3), for its value given by the
series () ; the final result being of the form
'>*+...; ....... (Z)
C /A \0i-A
or, if n = (on) >
A, A,
fr(Q-QjA-*+.... (H)
12. Certain of the terms of H, and indeed precisely those critical terms
upon which the determination of r and * depends, can be assigned a priori by
the help of the following considerations.
(i.) If a, b, c, ...,l, ... are positive and integral numbers, arranged in order
of magnitude, of which I is such that it cannot be formed by addition of any
multiples of the numbers which precede it, the coefficient of x l in the expansion
of [\J<- (x)Y, where er is any real exponent, and
^ (x) = 1 + Ax? + Bo? + Cx? + . . . + Lx l + . . . ,
is <rL; and, in particular, if all the numbers preceding I are multiples of any
Art. 12.] ON THE HIGHER SINGULARITIES OP PLANE CURVES. 113
number a, of which I is not itself a multiple, a supposition which implies that I
cannot be formed by addition of multiples of a, b, c, ..., I is the least exponent in
the development of [\^ (#)]", which is not divisible by a.
(ii.) If the series y = x^ (x) be reverted so as to obtain the equation
the exponents a 1} b lt c 1} ... are all formed by addition of multiples of a, b, c, ____
For, if this is not so, let /^ be the least exponent in ^ (y), which cannot be
formed by adding multiples of a, b, c, ... ; on substituting x^(x) for y in
y^i (y)' a substitution which ought to have x for its result, we find that the
coefficient of cc*i + 1 is H^ ; i.e. H^ = 0, or the exponent ^ does not occur in ^ (y).
Again, if the exponent I in \f<- (a;) cannot be formed by adding multiples of the
exponents which precede it, the coefficient L of y 1 in ^ (y) is L ; for, on
making the same substitution as before, the coefficient of x l + l is found to be
L l + L; i.e. L^ = L. And, in particular, if I is the lowest exponent in ^ (x)
which is not divisible by a, I is also the lowest exponent in ^ (y) which is not
divisible by a.
Let Pf = y, be one of the critical exponents 7, y l} ... considered in Art. 8 ;
then all the differences /3 2 fa, |8 3 /3 1} ... up to /:?,_! ft, are divisible by A 8-1 ;
but fa fa is not divisible by A,^. Therefore, by (i.), the coefficient of ft-ft. in
6Y <r-
the development of -=- is ' ; by (ii.) the coefficient of (0F)ft-ft. in the ex-
c, Pi~ A
pansion of ^y is -r , and P i P l is the least exponent in that expansion
which is not divisible by A.-.j ; finally, on substituting in the equation (/3), we
see that the term H(QYY i in the development of Z can arise only from the
terms p l fi and p t ^ in (/3) ; its coefficient H is therefore p l _' + /,- = B t ;
ft T. ' l ~
and the coefficient of 0^ (Q - Q )^-^, or e' 1 ' (Q - ^ ) 7 ~ A , m tne expansion ofR-R
is fj. e iB f . Nor can any exponent preceding --2= have a numerator which is not
divisible by A,_!.
Observing that the greatest common divisor of ft A, and fa , is the same as
that of A and fa, we infer from this result that the numbers Aj, A 2 , A s , ... ; 7, 7,,
7 2 , are the same for the series H as for the series B ; and since the numbers
7, 7j, 7 2 , . . . have no common divisor with A, neither have they with ft A = 7 A ;
i.e., 7- A is the least common denominator of the exponents of H. Hence we
have, writing 7 A = A,,
VOL. II. Q
114 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 13.
or, subtracting the discriminantal index 2 S +
an equation which establishes a relation between the four indices of the super-
linear branch.
13. If we consider any term whatever in the series B, for example the term
a
(i) = B i aPi(q q ) > ; we shall in general find a corresponding term (/) in the series
Q
//, containing QQt raised to the power -~ t : (/) may be considered as the
sum of two parts, / x and / 2 , of which the first, I lt arises from the term (i) itself,
the other, I 2 , from the terms preceding (i) ; (I) being in no way affected by the
terms following (i). If /3,- is one of the critical exponents, we have just seen that
/j = 0, /! = /u e < B f (Q - <?o) 01 ~ A . If /8,- is not one of the critical exponents, the first
of these equations ceases to subsist, but the second remains true, and its proof
requires only a slight modification of the reasoning in Art. 12. Now let two
series B, appertaining to two different superlinear branches, which have a com-
mon tangent, coincide as far as the term (i), but exclusively of it ; the two cor-
responding series //will coincide as far as the term (/), but exclusively of it ;
we suppose i > 0. That all terms preceding (/) will coincide in the two develop-
ments H is evident, for these terms arise solely from the terms preceding (i),
which are identical in the two developments B. And the terms (/) themselves
ft
are different : for the difference of the two terms (i) is (B i B'^) &A (q g ) A , where
one of the two B it B' t may be zero, but the difference B { B" { is by hypothesis
not zero ; and the difference of the two terms (/) is
* x(Q- Q,) _ jr _ /;,
for these two terms have the same part / 2 .
Let D be the number of points, T the number of tangents common to the
two branches B at the point (p , q ) ; T is given by the formula
- Ii1t ....
which is derived from the expression for N' in Art. 8, by writing 7 A for A,
Art. 14.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 115
/
-f = , = - ,
A A d
/
y' A' for A', and <r d for d. Observing that -f = , = - , whence
/-A' _ A' y-A _ A A
'
(r ~ =
, ,. ,, ' t
= y =i +K +2,
-=- , = ,
a K+ I K + 1
we find r-.D
+ !) = A, A,'- AA'.
We have supposed in the demonstration that i > 1, or that the two developments
ofp PQ B^(q g ) coincide for at least one term. But, for the validity of the
formulae, it is only necessary that the first exponent should be the same in the
two developments ; and indeed the last two expressions for TD hold universally
for any two superlinear branches having a common tangent.
14. The species of a superlinear singularity may be regarded as defined by
the series of numbers A and A,; \, Aj, ..., y lt y 2 , ..., so that two superlinear
singularities, for which these indices have the same values, may be considered as
belonging to the same species. A rougher classification, however, which is some-
tunes useful, may be obtained in the following way. Leaving out of sight the
case in which two superlinear singularities present themselves as conjugate
imaginaries, and attending only to the case of a real superlinearity, we may dis-
tinguish four varieties differing from one another in the appearance which they
present to the eye. (See a Memoir by M. Stolz, ' Mathematische Annalen,' Vol.
VIII., p. 440.)
(i.) A uneven, A, uneven ; no apparent cusp or inflexion.
(ii.) A even, A, uneven ; an apparent cusp, no apparent inflexion.
(iii.) A uneven, A, even ; an apparent inflexion, no apparent cusp.
(iv.) A even, A, even ; an apparent cusp, and an apparent inflexion.
The form of (ii.) is that of the common or keratoid cusp ; (iv.) has the form of the
cusp of the second species, or rhamphoid cusp. There is an apparent inflexion at
the rhamphoid cusp, because, if a person describing the curve continuously passes
through the cusp, the concavity of the curve is to his right after he has passed
through the cusp, if it was to his left hand before, and vice versd. We may
further observe that, in case (iv.), A and A,, being both even, have a common
Q 2
116 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 15.
measure ; thus A 2 > 1, and the superlinearity is composite. The cases (ii.) and
(iii.) are correlative ; the cases (i.) and (iv.) are their own correlatives.
15. The curvature of a curve at two points infinitely near to a given super-
linear point, and at equal distances from it on either side, is always the same ;
and is infinite, finite, or zero, according as A> A,, A = A,, or A< A,. Thus, in
each of the cases (i.) and (iv.), there are three sub- varieties of form ; and two in
each of the cases (iii.) and (ii.). The following are the simplest examples of each
of these suh- varieties : for the sake of completeness, the cases in which either of
the two numbers A or A, is unity, are included.
(i.) A and A, uneven,
A > A, : y = x$; y-x^.
A = A, : y = x*
A < A, : y = x* ; y = x%.
(ii.) A even, A, uneven,
A > A, : y = x% ; y = x$.
A < A, : y = xi
(iii.) A uneven, A, even,
A > A, : y = x$.
A < A, ; y = x 3 ; y = x*.
(iv.) A and A, even,
A > A, : y =
A = A, : 2/ =
A < A, : y = x a + x%.
It should be noticed that in the equation y = x% + x^, the only independent
radical is x*, and that x^ is to be interpreted as (x*) 2 . Thus, supposing x posi-
tive, and understanding by ^/cc 3 and ,^/x 7 the real and positive values of the
radicals, we have for the four partial branches the equations
y= Jx*+*/x\ y= *Jx*-yx\
y=-Jx*-i*fx\ y= -^/x 3 + i^/x\
of which the first two appertain to a real rhamphoid cusp. If we were to change
the sign of ^/x 3 , we should pass from the equation
U=(y 2 + x 3 ) 2 - x 3 (x 2 + 2y)* = 0,
which is the rationalised equivalent of y = x% + x$, to the equation
V= (y 2 + x 3 ) 2 - x 3 (x 2 + 2 y) 2 = 0,
Art. 17.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 117
which is the rationalised equivalent of y = x% + x%. It is, of course, quite pos-
sible that two developments, such as y= + x% + x$+ ..., may both belong to the
same curve (as indeed they do both belong to the curve UV+ as 15 <f) (x, y) = 0), but
such a curve would have two distinct superlinear branches touching one another
at the point x = 0, y = 0.
16. Let O be any point whatever on a curve line ; let the arc OP = a-, P being
a point on the curve infinitely near to ; let M be the orthogonal projection of
P on the tangent at ; and let the tangents at and P intersect at T, making
the infinitesimal angle w. Then it will be found that
A, _ logo, _ OT _ LTPO _ logMP
A ~ log <r ~ TP ~ L TOP " logOM ~
The fraction ^ which admits of these various geometrical interpretations may
perhaps be called the logarithmic curvature of the curve at the point 0. At any
ordinary point it is unity ; and in a geometrical curve it is always rational, but
in a transcendental curve it may have any value rational or irrational.
Since A or K + 1 is the number of points in which the superlinear branch is
cut by any line passing through 0, other than its tangent at the point 0, we
infer that, correlatively, i + 1 or A, is the number of tangents drawn to the super-
linear branch from any point on the tangent at 0, other than itself. Thus, if
d be the discriminantal index of 0, or the number of points in which the curve
is cut at by the polar of any arbitrary point, d + A, is the number of points in
which the curve is cut at by the polar of any point on the tangent at 0, other
than itself; there is, of course, a correlative definition of c? + A. Lastly, since
A + A, is the number of points common at to the tangent and the curve, it is
also, correlatively, the number of tangents drawn from O to touch the curve at
that point. Thus the polar of the point intersects the curve at in d + A + A,
points, and the tangent at counts as d + A + A, tangents common to the curve,
and to the tangential polar of OT with regard to the curve. For the numbers
Aj, A 2 , ..., 71, 7 E , ... no simple geometrical definition has as yet presented itself.
17. The proof of Pliicker's formulas, which is indicated in Art. 10, may
appear very indirect. Some further observations on these formulae, and on the
various modes of demonstrating them, may not be out of place.
(1.) If we write D = 2 (2J + 3*), J- 2 (2r + 8t), I- Si, jBT- Ss, | (T- ,S) - 0,
Pliicker's formulae become
n = m(m l) D,
m = n (n -1)-T;
118 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17.
Q = m (m - 2) - D,
-Q = n (n -2)-T;
giving T- D = n 2 - m a , Q = 3 (n - m),
Q* - 2fl 2 (T+ D) - 4 Q (T- D) + (T- D) 2 = 0.
It is thus apparent that Pliicker's equations do not contain either K or / separ-
ately, but only the difference / K.
(2.) The discriminantal index d = 2S + 3ic of any given point is defined geo-
metrically as the number of intersections of the polar of an arbitrary point with
the curve at the given point. But the definition which we have given in Art. 9
of the cuspidal index K is an analytical one, and does not readily admit of inter-
pretation in coordinate geometry. The Hessian does not serve to define either
i or K, for in all the cases that have as yet been rigorously investigated, it lias
been found that the number of intersections of the Hessian with the curve at a
point of discriminantal index d is 3 d + I K, so that, even if the number of these
intersections at any singular point should be determined by a general method,
we should only obtain a definition of the difference i K. Again, if several super-
linear branches have a common tangent OT at the point 0, it will be seen that
the geometrical definitions of Art. 16 only give the numbers 2(< + 1) and 2(/c + 1) ;
viz., if d is the total discriminantal index of all the branches intersecting at O,
the first polar of any point on OT (other than 0) intersects the curve at O in
d + 2 (j + 1) points ; the polar of intersects the curve at O in d + 2 (t + K + 2)
points; and there are correlative definitions of the numbers d + 2 ( + !), and
d + ~2(i + K + 2). By combining these definitions, we obtain a geometrical defini-
tion of the difference 2 (i K), the summation extending to all the branches which
touch one another at 0. But here it is to be observed (1) that to deduce the
values of 2t and 2/c from those of 2 (i + 1) and 2 (K + 1), we should require to
determine the number X of distinct superlinear branches which touch T at O ;
and (2) that, even if 2t and 2* were known, it would still remain to determine
the decomposition of these sums, and to assign the partial indices appertaining
to each of the X branches ; whereas no determination of the number > , or of the
indices t and K of each separate superlinear branch, has as yet been obtained by
considering the intersections of the given curve with any concomitant or system
of concomitants.
(3.) The difficulty, which thus presents itself in obtaining a definition of the
indices t and K, ceases to exist when we leave the domain of coordinate geometry,
and consider either the analytical expansions, or the geometrical representations
Art. 17.] ON THE HIGHER SINGULARITIES OP PLANE CURVES. 119
(depending on principles foreign to coordinate geometry) which correspond to
those expansions. If several superlinear branches touch one another at a given
point, the analytical expansions separate them, and assign the cuspidal and in-
flexional indices proper to each of them. If we apply to the equation F(p, q) =
the geometrical methods of double algebra, the cuspidal indices appear in the
cycles of values of p, which present themselves at the points answering to the
discriminantal values of q. (See the memoir of M. Puiseux, ' Liouville/ Vol. XV.,
p. 384.) If, instead of the simple plane of double algebra, we use the multiple
plane of Biemann, the cuspidal indices are represented by the spires which con-
nect the leaves of the multiple plane. But it is important to remember that, in
employing the methods of double algebra, and d fortiori in employing the
surfaces of Biemann, we are entirely abandoning the methods of coordinate and
projective geometry. The present question is perhaps not directly affected by
the fundamental distinction between the ' infinite ' of double algebra, which is a
point, and the infinite of projective geometry, which is a straight line. But the
duality, characteristic of projective geometry, is lost in double algebra ; so that,
when the complex values of p and q which satisfy the equation F(p, q) = are
regarded as developed on a plane, or on one of Biemann's surfaces, we do indeed
obtain a direct representation of the cuspidal index K, but no corresponding
representation (unless we first transform the equation into its reciprocal) of the
correlative index t. Indeed, it may be asserted that, whereas the character of
any given superlinearity mainly depends on a series of indices A = /c + l, A, = t + 1,
A,, AS, ..., 71, 7 2 , ..., the modes of geometrical representation, to which we are
here referring, offer a sensible image of the first of these indices only. If we
employ a simple plane, any one of the A values of p, which come to coincide with
one another at the discriminantal point, must describe A elementary contours
around that point before it acquires again its original value. If for simplicity we
suppose that A 2 = 1, the A values of p, which form the cycle, will divide them-
selves into Aj sub-cycles, each containing values ; and any value, belonging to
one of these sub-cycles, will acquire approximately its original value, after de-
scribing elementary contours around the discriminantal point, the order of the
error being -^ if the order of the infinitesimal radius be taken as unity. And
upon this approximate return to the original value depends the only indication
which the method affords of the existence of sub-cycles, and of the values of the
120 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17.
numbers A, and 7,. If we employ the multiple plane of Riemann, we may per-
haps represent the relations of the A expansions to one another by taking a A r
leaved plane, repeated times, and having a spire of order A 1, so arranged
A
that after revolutions we return to the same A r leaved plane upon which we
AI
were when we set out, but not to the same leaf of that plane. And we can give
to this image a certain amount of clearness by supposing that the A t leaves of
any A^leaved plane are infinitely nearer to one another than are any two of the
repetitions of the Aj-leaved plane.
"i
(4.) The demonstrations of Pllicker's formulae, which are usually given,
apply only to the case in which the singularities are simple ; the cases of multiple
points, or multiple tangents, or of branches having contact with one another of
any order, being made to depend, by the method of limits, on the simple cases of
double points, or double tangents (see Dr. Salmon's 'Higher Plane Curves,' p. 53).
But these demonstrations do not admit of immediate extension to the case of the
higher singularities properly so called, because it has not as yet been established,
in any general manner, that a higher singularity may be regarded as the limit of
an equivalent number of lower singularities situated infinitely near to one another.
It would seem that Plticker himself was well aware of the incompleteness (in this
respect) of the demonstration of his equations ; for he supplements that demon-
stration by separately considering the case of a common cusp of the second
species. Assuming the equation n = m(m 1) D, and its reciprocal, (about the
rigorous proof of which there is no doubt,) we have only to establish one other
equation of the system. Two different methods are given by Pllicker (' Theorie
der Algebraischen Curven,' Part ii., Arts. 77-81) : (i.) He establishes directly the
theorem that, at a cusp of the second species, the curve
d*F /dF^ d-F dF F d*F
dp 2 \dq ' dp dq dp dq dq* \dp
(which may be used for our present purpose instead of the Hessian) intersects
the given curve in 3d + 1 K = 15 points. We have already stated that, in all the
cases which have been examined hitherto, the number of intersections of the
Hessian with the curve at any point has been found to be 3 d + 1 K ; but no
general demonstration of this theorem has as yet been given. The only method
at present known for determining the number of intersections of two curves at a
point which is singular on each of them, consists in obtaining the developments
Art. 17.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 121
of the various branches of the two curves at the point, and in comparing these
developments with one another. The discussion in Art. 18 of the development
of the polar curve in the vicinity of a superlinear branch, may serve to show that
the corresponding enquiry in the case of the Hessian is one of considerable
intricacy, (ii.) The other method employed by Pliicker depends on a determi-
nation of the number of double tangents lost by a curve of the fourth order in
consequence of the presence of a cusp of the second species. In the absence of
any demonstration that a higher singularity can be regarded as the limit of
simple singularities existing infinitely near to one another, it is difficult to see
how this mode of proof can be rendered universally applicable.
(5.) We have seen (Art. 10) that the theorem of the invariance of the num-
ber \ (m 1) (m 2) Z<5 2/c in any unicursal transformation of the curve
suffices to establish the equation
and thus to complete the proof of the formulae of Pliicker. Among the demon-
strations of this theorem which have been given in recent times that of MM.
Bertini and Zeuthen (' Giornale di Mathematica,' Vol. VII., p. 105 ; ' Mathematische
Annalen,' Vol. III., p. 150; Dr. Salmon's 'Higher Plane Curves,' p. 314) is
remarkable for its simplicity ; and appears, as we shall now attempt to show, to
admit of extension to the case in which the curves have any singularities what-
ever. We begin by assuming that when a curve is subjected to an unicursal, or
one-to-one transformation, the continuity of its branches is invariably preserved,
even when the position of these branches with regard to one another has under-
gone great distortion. For example, if a curve have two branches intersecting at
the point O, these two branches will certainly be represented by two correspond-
ing branches in the transformed curve ; but these two branches may have no
point of intersection, and the point may be represented by two different points
one on each of the two branches. Again, two branches which osculate one
another with any degree of approximation may be transformed into branches
having no contact and no point in common. But a superlinear branch behaves
as one branch, and always is transformed into one branch and one only. Consider,
for example, a real branch which is superlinear at 0, and suppose for simplicity
that no other branch passes through O ; whatever be the nature of the super-
linearity, we have one continuous branch passing through O, and if a point
describe this branch, the track of the image point in the transformed figure can-
not be anything but one continuous branch.
Let Ci, C 2 be two curves of the orders r m l ,m z , and of the classes n 1( n 2 lying
VOL. II. R
122 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 17.
in the same plane and corresponding to one another unicursally ; and let P l , P a
be points upon them corresponding unicursally. Taking two arbitrary points
S lt S 2 , we consider, with M. Zeuthen, the locus F of the intersection of the rays
S 1 P l , S Z P 2 ; an d we propose to determine the number of tangents that can be
drawn to F from each of the two points S T and S 2 . We may suppose that <Si/S a
cuts each of the two curves in points which do not have singular points of the
other curve for their corresponding points ; then it is evident that F will have m^
ordinary branches passing through S lt and m, l ordinary branches passing through
S 2 . We may further suppose that the n^ tangents drawn from S 1 to C, are none
of them singular tangents, and that to the points of contact of these % tangents
there answer on C 2 points having no singularity : each of these tangents will
then be a tangent of F, but not a singular tangent of that curve. Beside the
2m 2 + 7i 1 tangents, which we have now drawn from S t to F, there may be others,
coinciding in direction with the rays running from $ x to the singular points of C l .
Let Xi be a superlinear point on C lt having the cuspidal index ^ ; and to X l let
X. 2 answer on C 2 , the cuspidal index of X z being K Z , where *r 2 >0. We may sup-
pose at first that only one branch passes through X l and only one through X z .
The ray S l X l meets C l at X 1 in precisely K I + 1 coincident points, because S t X 1
is not a tangent at X 1 ; similarly S 2 X 2 is not a tangent at X 2 , but meets C. 2 in
precisely K 2 + 1 points at X. 2 , since we may attribute to S 2 the requisite generality
of position with regard to C 2 . Thus, if Q is the intersection of S l X lt S.,X 2 , the
locus F is intersected at Q /cj + 1 times by S l X l , and ir 2 + l times by S. 2 X 2 . The
points of the curve F answer, one to one, to the points of Q or C 2 ; thus at Q there
is but one branch answering to the one branch at X l} or to the one branch at X 2 .
If K 1 = K. 2 , the cuspidal index of this branch is K l = K 2 , while its inflexional index
remains unknown. If ^ > K 2 , its cuspidal index is /c 2 , its inflexional index is
/T! K 2 1 ; similarly, if /c 2 > K I} these indices are ^ and K 2 K l 1; i.e., in the first
case, SL X l counts ^ /c 2 times as a tangent to F at Q, and S 2 X 2 is not a tangent
at all ; in the second case, S 2 X 2 counts K Z jc t times as a tangent, and S l X l is not
a tangent at all. When c a = * 2 > neither /Sj X-^ nor S 2 X 2 are tangents. The pre-
ceding reasoning will not be affected, if we now introduce the supposition that
several linear or superlinear branches intersect or osculate at X lt and that
branches corresponding to some or all of them pass through X 2 . Several
branches will now pass through Q, but each of them may be considered separ-
ately, and the number of times that it is touched by S r Q or S 2 Q may be ascer-
tained as above. Equating the results appertaining to the points S l and S. 2> we
now obtain
Art. 17.] ON THE HIGHEE SINGULAEITIES OF PLANE CURVES. 123
2 m, 2 + n-t + 2' ( KI -K 2 ) =
where 2' extends only to those differences which are positive. Written in the
f rm %! + 2^ - 2 m t = n 2 + 2- 2 - 2 m 2 ,
this equation coincides with the formula
i (Wi - 1) K - 2) - 2 ft + Kl ) = 1 (m, - 1) (m 2 - 2) - 2 ( 2 + KZ ),
which it was required to prove.
The assumption, which we have explicitly made, that a linear or superlinear
branch is always transformed by a one-to-one transformation into one branch,
and one only, is indispensable in the preceding proof; as upon it depends the
determination of the number of times that F is touched by S l X l or S 2 X 2 . In
the case of a real branch transformed by a real transformation, the assumption
may be regarded as evident ; in the general case, we should have to consider^
instead of two plane curves, the two corresponding surfaces of Rlemann. For
our immediate purpose, however, we do not need to establish the assumption as
universally true in all cases ; because the only one-to-one transformation (beside
that of Ci or C 2 into F) which is here employed is the transformation by polar
reciprocation ; and the investigation of Art. 1 1 affords a direct proof that in this
transformation any one linear or superlinear branch is always transformed into
one branch (linear or superlinear).
(6.) Abandoning for a time the hypotheses of Art. 1, let us suppose that P
is a singular point on the curve C, Q retaining its generality of position. And
first let P be a point through which only one superlinear branch passes, having
the indices K = A 1, i = A, 1 ; let us also suppose that no singular tangent of C
(other than the tangent at P) passes through P. The order of p in the equation
F (p, q) = is now m A, instead of m ; and the number of tangents that can be
drawn from P to the curve C (other than the coincident tangents at P itself) is
n A A, (see Art. 16), instead of n. To all the singular points of C, other than
P, there will appertain developments of precisely the same form as in the case in
which P has no speciality of position. Let q^ be the value of q corresponding to
the tangent at P ; the parameters of the point P are p = oo, q = q a . We cannot
therefore, in examining the superlinear branch at P, develope p in a series pro-
ceeding by powers of q q<>', but we may so develope - , or any linear function
of p, such as - ~ , which assumes a finite value p = r , when p= oo. The ex-
a + bj> b
ponents in any such development will have A, instead of A, for their least com-
B 2
124 ON THE HIGHER SINGULARITIES OP PLANE CURVES. [Art. 17.
mon denominator, because the tangent to C at P meets the curve (Art. 16) in
c ~\~ dfy
A + A, points, so that, if # = <?<)> A/ of the m A values of - r- become equal to
tt ~r up
p 9 . Setting out from the given equation F(p, q) = 0, let us form the develop-
ments appertaining to all the singular discriminantal values of q ; and in each
group of conjugate developments let us consider the greatest common divisor Q
of its exponents. The sum 2 (Q 1) will be equal to 2* + A, A,, instead of 2/c ;
and the three numbers, by which we have now replaced m, n, and 2/e, will satisfy
the equation
(w - A - A,) + ( Z* + A, - A) - 2 (m - A) = n + 2/c - 2 m.
The cases in which (a) more than one branch passes through P, (/3) one or more
singular tangents pass through P, (7) Q as well as P has some speciality of
position with regard to C, may all be treated by the same method. In any of
these cases, let E (p) be the highest exponent of p in the equation F ( p, q) = ;
and let w (p) = 2 (6 1), the sign of summation now extending to all the discri-
minantal values of q, so that 2 (6 1) contains an unit for every ordinary tangent
that can be drawn from P to touch the curve elsewhere. If any of the discri-
minantal values of q, or any of the corresponding equal values of p, are infinite,
we are to employ linear functions of p and q themselves, in forming the develop-
ments from which we are to infer the numbers 6. We shall thus obtain the
equation - = - = ic-2m, ..... (E)
from which, as Clebsch has shown, the general theorem of the invariance of the
deficiency may be immediately deduced. (See a Memoir by M. Nother, ' Mathe-
matische Annalen,' Vol. VIII., p. 497.)
In the memoir to which we have just referred, M. Nother offers a demon-
stration of the equation (B). But this demonstration is perhaps not wholly free
from obscurity. (See the words, p. 499, loc. cit., ' Dieses findet . . . ergiebt,' with
the accompanying reference to the Gottingen ' Nachrichten.') A similar remark
applies to a second demonstration, in the same memoir, of the invariance of the
deficiency. [See p. 501, ' Man hat aber dann . . . das Glied 2^ fa 7).']
M. Nother has returned to the same subject, in a recent memoir of great
interest (' Mathematische Annalen,' Vol. IX., p. 166), in which he considers the
resolution of a higher singularity by successive applications of a simple quadratic
transformation, and infers (though by a method which can hardly be accepted as
rigorous) that any higher singularity may be regarded as the limit of a certain
number of lower singularities situated infinitely near to one another. We may
Art. 18.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 125
observe (a) that the use of a quadratic transformation for the resolution of com-
plicated singularities is due to Cramer (' Analyse des Lignes Courbes ') ; (/3) that to
establish the complete system of the formulae of Pliicker, M. N other selects the
same three equations, which we have been led to employ in the present paper
[viz., the equations (i.), (ii.), and (iii.) = (iv.), of Art. 10].
18. The expansions of Arts. 3 and 4 enable us to examine the relation of a
curve at a singular point to its polar curves. Putting for brevity p p^ = n,
From the expression of F l (n, ) as a product of m factor-series, we infer that if,
on writing K^ % for n in F 1 (n, )> we obtain a result of which the order of evan-
escence with is higher than n, i\ = K l + . . . is the beginning of one at least of
the expansions B. Again, let us substitute for / in F l (q, ) an expression of the
form
in which 1 < a 2 < a 3 < . . . < a v . If the order of evanescence of F 1 (K, ) with ex-
periences an abrupt diminution when either a v or K v (the exponent and coefficient
of the last term of K) is affected by any small variation, the terms K are the
initial terms of one at least of the expansions B. This observation (which admits
of some useful applications) enables us to deduce the developments appertaining
dF
to the polar curve -j- , in the vicinity of the point (p 9 , q^, from the develop-
ments appertaining to C.
Let k of the developments B coincide with one another and with K, as far
as the term K v "' inclusively, so that for any one of these k developments we
, /<> ........ (K)
the terms X,. ' not being all identical.
Put V=i-K, U
then F l (n,$ = Mx<t>(V), and
M being a product of m k factors, viz., of the m n factors n A, and of those
m k factors r\ B which do not coincide with >? K as far as the term K v " in-
clusively. Suppose, at first, that 1 1} 1 2 , ..., l k are all unequal, and arranged in
order of magnitude ; it is easily ascertained that the first terms, in the expansions
126 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 18.
of the roots of (f)'(V) = are
dF
Substitute for 17 in - an expression of the form tj - K+ II " A , where h > , and
if is independent of . If H^ is not the same as any one of the quantities F,,
F 2 , ..., F fc _!, the order of evanescence of <(F) - T surpasses that of Md>'(V) ;
r71f
for the order of evanescence of Jf cannot surpass that of j by a number
greater than a v , whereas the order of < (F), on the supposition that none of the
equations H h = F f is satisfied, surpasses the order of <'(F), at least by one of
the numbers 1 1} 1 2 , ..., l k . If we now suppose H and h to vary continuously, the
order of evanescence of (^)'(F) is abruptly increased when H h comes to coincide
with any one of the roots F , F 15 ..., V k _ l ; and, since the order of evanescence of
dF
M remains unchanged, that of -^ is also increased abruptly. Hence k 1 of the
U>]
rl J?
developments appertaining to -v are of the type
Again, suppose that s of the indices I are equal ; let, for example, the s lowest
indices be equal ; then s roots of the equation <$>'(V) = are of the form H f ^ l + ...,
where I = ^ = Z 2 = . . . = l t ; and if
the s coefficients H f are the roots of the equation
If the s equal indices ^ . . . 1 2 are followed by another set of s' indices equal to one
another and to I', I' being > I, put
then the equation <'(F) = has /roots of the form 11'^''+..., the coefficients
II- being the roots of the equation
and so on continually. Lastly, considering any group of equal indices I, for
Art. 18.] ON THE HIGHEE SINGULARITIES OF PLANE CURVES. 127
example the group Z 8 + 1 , l s+2 , ..., ?, + ,., let a- of the corresponding coefficients X be
supposed equal (in which case <r of the developments K coincide with one another
for one term at least after K v ) ; the corresponding equation (&') will have or 1
roots (and no more) equal to one another and to the equal coefficients X ; so that
<r 1 of the developments appertaining to the polar will coincide, as far as the
term next after K v a ", with the <r developments appertaining to C. To carry on
these <r 1 developments until their complete separation from one another, we
must repeat the preceding process as often as may be necessary, using in the first
instance K+ X J instead of K, and confining our attention to the <r developments,
appertaining to C, in which K+ X' are the initial terms.
As the roots of the equations \f<-($) = 0, ^i(Q) = 0, ... are all different from
zero, so also are the roots of the equations (6), (6'}, ..., except when the highest
index I is one of a group of equal indices. In this case, if \J/- (0) = II (9 X), the
sign of multiplication extending only to those coefficients X t - which occur in terms
having the greatest exponent I, the last of the equations (6) is of the form
v// (0) = 0, and r of its roots may be equal to zero. When this happens, in the r
polar developments corresponding to the zero roots, the terms K are not followed
by a term of the form H l , but by a term of higher exponent. To determine this
term in each of the r developments, we must use, in forming ^ (6), not simply the
quantities X,-, but as many terms of the series X,. + X; '<+... as may be necessary.
The zero roots of \f/ (0) = are then replaced by roots of the form H a , a being
positive, and the initial terms of the r polar developments are given by the
formula K+H? + a .
We shall employ the preceding method to examine the nature of the polar
branches in the vicinity of a superlinear branch. We suppose the superlinear
branch to be of the type
[A, A 15 A 2 , ..., A 8 , A g + 1 = l; y lf y t , ..., ?.] ;
and we consider only the case in which this superlinear branch (A) is not touched
by any other branch. The polar has A 1 branches (A') touching the superlinear
branch. Their developments coincide with one another, and with those of (A),
- A
as far as the term [X A ] exclusively. But at this term 1 of them cease to
7
osculate any branch of (A) ; they do not contain the term [a; A ], which is replaced
in each of them by a term of higher exponent, yet so that the aggregate of the
A 7 A
1 exponents cannot exceed 1. The remaining (Aj 1) branches
128 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 19.
divide themselves into groups of Aj 1 each. The ^ 1 branches of each
group are identical with one another, and with Ag of the branches (A), as far as
1? A
the term [# A ] exclusively. At this term ^ 1 branches out of each group
cease to osculate any branch of (A), and the remaining - (A 2 1) divide them-
A ^
selves, in the same way as before, into - groups of A 2 1 each ; the branches of
each group being identical with one another, and with A 2 of the branches of (A),
72
as far as the term (o; A ) exclusively. In this way we obtain the following theorem
in which i is to have every value from to s, both inclusively.
' The polar curve of an arbitrary point has branches which form
A t ~r I | .
- 1 superlinear branches of the type
These superlinear branches coincide with one another, and with the branches of
n
(A) as far as the term [X A ] exclusively ; instead of the term [# A ] each of them
contains a term of higher exponent ; the 1 superlinear branches may, but
do not necessarily, group themselves into higher superlinear branches.'
19. The development appertaining to a superlinear branch can always be
obtained from the equation of the curve by successive applications of the
' analytical triangle.' The process has been described by M. Puiseux in his im-
portant memoir ' Recherches sur les fonctions algebriques.' (' Liouville,' Vol. XV.,
p. 384; see also a paper by M. de la Gournerie, ibid., 2nd series, Vol. XIV.,
p. 425, Vol. XV., p. 1.) We propose to conclude the present paper by showing
how the numbers 7, ^ , . . . , A, A t , . . . present themselves in the course of the
operation. Putting, as in Art. 18, r\ for p p , % for q q<,, we first of all write
the equation F^, ) = i n the form u lt + u lt + l + ..., where u^ is a homogeneous
function of and n of the order p., which is that of the singular point. If
(17 _C ) is a multiple factor of u^ the line n B^ is touched by branches (linear
or superlinear) of which the aggregate order is a. Put 17 - B^ = v ; the resulting
equation between v and will give precisely a values of v in which the order of v
surpasses that of . Form, by the analytical triangle, the equations (of the
Art. 19.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 129
aggregate order a in v) which give the initial terms of the expansions of these a
values. These equations are of the type
where X and v are relatively prime, X > v, and '2a l v = a; they are always obtained
linearly, except when there are s of them in which the numbers a 1} X, v are all
the same ; in which case the analytical triangle determines an equation, of order
s, having constant coefficients, of which the roots are the s quantities K. There
are four cases to be considered : (i.) 0^ = 1, v = 1 ; (ii.) a^ = 1, v > 1 ; (iii.) a l >1,
v = 1; (iv.) ! > 1, v > 1. (i.) To the equation v K^ x = 0, X > 1, answers a linear
branch which, considered by itself, has no point-singularity (if X is > 2, it is an
inflexion), (ii.) To the equation v v K^ = Q answers a superlinear branch of
which the character is defined by the equations A = v, A x = 1, y = X ; its develop-
i
ment proceeds by integral powers of ", and the successive terms are obtained
linearly by the analytical triangle, (iii.) To the equation (v K*) a i = answer
Oj branches, which may be all linear, but which also may group themselves in
whole or in part into superlinear branches ; if A, A', A", . . . are the orders of these
separate linear or superlinear branches, we have 2A = a^ (iv.) To the equation
(if K%*)"i = answer a x v branches, which may belong to a^ distinct superlinear
branches of the type (A = v, A x = l, -y = X); these superlinear branches may how-
ever themselves be grouped, wholly or in part, into branches of higher super-
linearity ; if A, A', A", . . . are the orders of the distinct superlinear branches,
these numbers are all divisible by v, and Z = j ; we have also for every one of
A 7
them = v, -J-- = X, A! having to be determined subsequently for each of them
Aj A,
separately. With the cases (i.) and (ii.) we have nothing further to do ; the case
(iii.) may be regarded as included under (iv.) ; we therefore continue the process
i \ i
in this last case only. Put v K" > v = v l , " representing any one determinate
value of the radical ; and form by the analytical triangle equations of the type
(tf-JEU'^-O,
of which the aggregate order in v t is a ly and which give the initial terms of
x
those ! values of v r , of which the order surpasses that of " ; we have of course
- > - , or ' > X ; X! and v l are relatively prime, but we observe that \ is not
v v
l
VOL. II. S
130 ON THE HIGHER SINGULARITIES OF PLANE CURVES. [Art. 19.
necessarily prime to v. We consider the same four cases as before, (i.) To the
*i
equation Vj KI" =0, or more properly to the v equations comprehended in it,
answers a superlinear branch of the type (A = v, AJ = 1, 7 = v). (ii.) To the v equa-
^>
tions v? KI " = there also answers a single superlinear branch for which
A = w lt A! = v lt A 2 = 1 ; 7 = Xv, , y l = \ ; i.e., a superlinear branch of the type
( = v, ^="1, A 2 = l; 7 = XA 15 -y^X^)-
In this case, as well as in (i.), the discussion of the superlinearity is complete.
\
(iii.) To the v equations (v l K^ ") a 2 = there may answer a 2 superlinear branches
of the type ( - =v, A t = l; 7 = XA 1 ); or these may group themselves in any
1 A
manner into higher superlinear branches for each of which = v, y = \\; the
AI
numbers \ (which have to be determined for each branch separately), satisfying
the condition 2A 1 = a 2 . (iv.) To the equations (v[ l K^Y* answer a certain
number of superlinear branches, for each of which
A A,
while A 2 and the subsequent numbers of the series have still to be determined,
and may be different for each of them ; we have however the equation
The process, which we need not follow further, may be considered to germinate
for any particular development, when that development is separated from every
other, and can be continued linearly. This will happen when, in the series
aa t a 2 ..., we arrive at a term equal to unity. And we shall eventually arrive
at such a term ; for, though the second of two consecutive indices a may be as
great as the first (the equation (v K V Y 1 may, for example, at the next step
in the process, lead to only one equation ; and this may be of the type
so that we should have a 2 = c^), yet it is impossible for two branches to osculate
one another indefinitely, because the discriminantal index is necessarily finite.
Art. 19.] ON THE HIGHER SINGULARITIES OF PLANE CURVES. 131
If a, be the first of the indices a which is equal to unity, we have
and the development appertaining to the superlinear branch is of the type
8 2
XXVII.
MATHEMATICAL NOTES.
[Proceedings of the London Mathematical Society, vol. vii. pp. 237-238. Eead December 9, 1875.
First printed in the Messenger of Mathematics, vol. v. pp. 143-144 (January 1876)].
(i.) tjN a Problem of Eisenstein's. If p is an uneven prime, the function
j) -|
4 -- = Z can always be expressed in the form Y 2 ( )i (p ~ l) pX 2 , where X
X ~~ A
and Y are rational and integral functions of x having integral coefficients. This
is a theorem of Gauss. Eisenstein's problem (' Crelle's Journal,' vol. xxvii., p. 83)
is 'To determine the cases in which the equation Z= Y 2 ( )i (l> ~ 1 ' > pX 2 admits
of a multiplicity of solutions, and to ascertain the law connecting the various
solutions, when there is more than one.' The solution of this problem is as
follows : If \T, Z7] is any solution whatever in integral numbers of the equation
T' 2 ( )^ (p ~ 1) pU 2 = 4i, and [X, F] is any one given solution of Gauss' equation,
then all the solutions of Gauss' equation are comprised in the formula
Thus, if p = 4w + 3, the equation admits of but one solution (the four solutions
[ + X, Y~\ being regarded as but one) except in the case p = 3, when it admits
of three; if p = 4n + l, the equation admits of an infinite number of solutions.
That the functions [1 (TX+pUY), $(UX+TY)] are all of them solutions
of Gauss's equation, is evident ; the proof that this formula comprises all the
solutions of the equation is less elementary, because it depends on the irreduci-
bility of the function Z. There exists a general theory of the representation of
i-ational and integral functions of x by quadratic forms ; such representation
being, of course, only possible when the given function of x is capable of resolu-
tion into two factors by the adjunction of a quadratic surd.
MATHEMATICAL NOTES. 133
(ii.) On the Joint Invariants of Two Conies or Two Quadrics. Let P and Q
be two conies, and let 123 be any triangle self-conjugate with regard to P. Let
also P 1} P 2 , P 3 be the rectangles of the points 1, 2, 3 with regard to the conic P,
these rectangles being taken upon transversals measured in any fixed direction ;
and let $j , Q 2 , Q 3 have similar meanings with regard to the conic Q, the direction
of the transversals being also fixed. Then the expression -^ + -j/ + -- has the
*! *i * 8
same value for all self-conjugate triangles of P, and is, in fact, that invariant
of P, Q which is linear with regard to Q and quadratic with regard to P, and
the evanescence of which expresses that Q harmonically circumscribes P. The
corresponding theorem in the geometry of the straight line is ' If Q t Q 2 , P! P 2 are
two pairs of fixed points on a line, and if A l A 2 is any pair of harmonic conjugates
of PjPg, the value of the expression / p 1 / p 2 + ~ " p" ^w i g independent
A-l JL i . AI -i 2 -"-Z-LI -"-2 *%
of the particular pair A 1 A 2 considered.' From this theorem the result given
above for two conies follows immediately ; from it the corresponding property for
two quadrics may be inferred, viz. ^ + ^ + ^ + ^ = constant ; and so on for
* I " 2 "* "l
quadratic functions containing any number of indeterminates.
(iii.) On the Equation P x D = constant, of the Geodesic Lines of an Ellipsoid.
From this equation (in which P is the perpendicular from the centre upon the
tangent plane at any point of the geodesic, and D is the semi-diameter parallel
to the tangent line of the geodesic), it is convenient to be able to infer directly
the principal properties of the geodesic line, without having first to transform
the equation into M. Liouville's form /x 2 cos 2 i + v 2 sin 2 i a?. In Dr. Salmon's
' Geometry of Three Dimensions,' the theorem of the constancy of the sum or
difference of the geodesic radii vectores, drawn from any point of a line of
curvature to two umbilics, is thus demonstrated. And it is worth while to add
(though it is very improbable that the point has not been noticed before), that
a proof of the theorem, that two geodesic tangents of a line of curvature, which
intersect at right angles, intersect on a sphero-conic, may similarly be obtained
without transforming the equation. Let Q be the point where the two geodesic
tangents intersect at right angles, O the centre of the ellipsoid ; let c = OQ, and
let a, b be the semi-axes of the central section parallel to the tangent plane at
Q. The two geodesies make angles of 45 with the lines of curvature at Q ;
2a' 2 b 2
hence, for either of these geodesic lines, D 2 = j- . Let Q' be a second point
134 MATHEMATICAL NOTES.
where two geodesic tangents to the same line of curvature intersect at right
2 P 2 a 2 b 2 2 P' 2 a' 2 b' 2
angles ; then - rr- = ^ j-^- , because P x D has the same value for all
(t ~f~ o Ct ~r I)
geodesic lines touching the same line of curvature. But P 2 a 2 b 2 = P 2 a' 2 b' 2
because parallelepipeds circumscribing an ellipsoid with their faces parallel to
conjugate diametral planes are equal. Hence a 2 + & 2 = a' 2 + b' 2 . But also
therefore c = c and Q and Q' lie on the same sphero-conic.
XXVIII.
'NOTE ON CONTINUED FRACTIONS.*
[Messenger of Mathematics, Ser. n. vol. vi. pp. 1-14 (May 1876)].
1. .LET = /*, H --- , p and q being two numbers relatively prime.
q
of which p is the greater. Writing, for convenience, P = - and Q = - , we divide
a line 01 of unit length (measured from left to right) into p equal parts at the
points IP, 2P, 3P, ..., (p l)P; and also into q equal parts at the points 1$,
2 Q, 3Q, ..., (q l)Q. We do not reckon either as a point P or as a point Q,
but we reckon 1 both as a point P and as a point Q, so that we have in all p
points P, and q points Q, of which none are coincident, excepting the two ex-
treme points, which coincide at 1.
2. It is the purpose of this note to show that the arrangement of the points
P and Q upon the line 01, or, which is the same thing, the arrangement in order
3C 77
of magnitude of the proper fractions - and - , may be inferred from the develop-
f) . f)
ment of in a continued fraction ; and that, vice versd, the development of
may be inferred from an inspection of the arrangement of the points. An
example will serve to explain the nature of the relation which we have to
establish.
3. Let p = 39, q = 17, so that we have the development ff = 2 + - - ;
O "T ^ "l ~
the arrangement of the points P and Q is indicated in the following scheme, in
' The substance of this note was commnnicated to the Mathematical Section of the British
Association, at the Bristol meeting in 1875.
136 NOTE ON CONTINUED FRACTIONS. [Art. 3.
which transverse lines are placed at the close of each of the sequences* to be
presently defined.
P, 2P, Q | 3P, 4P, 2 Q | 5P, 6P,SQ\ 7P, || 8P, 9P, 4<? |,
10P, IIP, 5|12P, 13P, 6|14P||15P, 16P, 7g|||,
17P, 18P, 8<)|19P, 20P, 9Q,|21P, 22P, 10#|23P||
24P, 25P, 11|26P, 27P, 12#|28P, 29P, 13Q|30P,|||
31P, 32P, 14#||| 33P, 34P, 15#|35P, 36P, 16Q[
37P, 38P, 17|39P||||.
In this scheme, because ^ = 2, we have two points P before we come to a
point Q ; the sequence PPQ, which consists of MI points P followed by a point
Q, we term a sequence of order 1 ; this sequence is repeated three times, because
M 2 = 3, and is then followed by a single point P (which is a sequence of order
zero) ; a sequence, such as PPQ | PPQ \ PPQ \ P, consisting of /u 2 sequences of
order 1, followed by a sequence of order zero, we term a sequence of order 2 ; it
contains MI M 2 + 1 points P, /u 2 points Q ; this sequence of order 3 is, in the scheme
before us, repeated twice, because /u 3 = 2, and is then followed by a sequence of
order 1 ; the sequence thus obtained, consisting of fi 3 sequences of order 2, fol-
lowed by a sequence of order 1, we term a sequence of order 3. This sequence,
containing MI i" 2 f^ + /*i + Ms = 16 points P, and n 2 im 3 +l = 7 points Q, is in the
* These sequences have been already noticed by M. Christofiel, in an interesting paper entitled
1 Observatio Arithmetica,' (Annali di Mathematica, 2nd series, vol. vi., p. 148), with which I unfortun-
ately did not become acquainted until my own investigation was completed. M. Christoffel considers
the least positive remainders of the series of numbers q, 2q, 3q, ... for the modulus p, and designates
any remainder by the symbol c or d, according as it is less or greater than the remainder immediately
following. It is easily seen that the sequences of the symbols c and d coincide with the sequences of
the points P and Q. For if the remainder of sq is greater than the remainder of (s+ l)q, we shall
have, for some integral value of h, the inequalities
(h l)p<sq<hp< (+l)5'< (h+l)p,
s h s+1
whence - < - < -
p q p
or the point hQ lies between the points sP and (s+1) P. And, again, if the remainder of sq is less
than the remainder of (s+ l)q, we have the inequalities
(h-l)p <sq< hp, (h- l)p < (s+ l)q <hp,
which give immediately
h-\ s h s+1
-< -> - > -- .
q p q p
proving that no point Q can lie between the points sP and (+ 1) P.
Art. 5.] NOTE ON CONTINUED FRACTIONS. 137
Bcheme repeated twice, because // 4 = 2, and is followed by a sequence of order 2.
We thus obtain a sequence of order 4, consisting of M4 sequences of order 3,
followed by a sequence of order 2, and containing
Ml fJ-2 f*3 M 4 + Ml M 2 + /*l ^4 + ^3^4+1 = 39 pOUltS P,
and /* 2 MS n t + M 2 + M 4 = 1 7 points <?.
This sequence, in the instance which we are considering, exhausts the whole
system of points. We observe that all sequences begin with P, and that se-
quences of an uneven order end with PQ, sequences of an even order with QP.
4. In general, when the continued fraction is given, and it is required to
obtain the arrangement of the points P and Q, we denote a sequence of order i
by S it and we then find successively S 1 = P M & S S = S^P, S a = 8^ S lf ..., the
final sequence (which exhausts the whole series of points) being S s = 8^ S,_ t .
Vice versd, when the arrangement of the points is given, and it is required
to infer from it the development in a continued fraction, we count the points P
till we come to the first point Q ; if there are MI of them, MI is the first quotient,
and (Sj = PI*I Q. If we can repeat this sequence / 2 times, without departing from
the given arrangement, the second quotient is n 2 , and the sequence of order 2 is
S 2 = Si*P. This sequence we now repeat as often as we can do so without de-
parting from the given arrangement, observing, however, that the last repetition
of $ 2 is to be followed by a sequence 8^ If, subject to this condition, we can
repeat S 2 M 3 times, the third quotient is MS, and the sequence of order 3 is S 3 S lf
The subsequent quotients and sequences are to be determined in the same man-
11
ner ; and, if - - is the convergent M,- + - - , p t and q f are respectively the
<?, M 2 + Mi
numbers of points P and points Q in the sequence S { .
rn
5. Since n 1 < <n 1 + l, or n 1 P< Q< (fJL 1 + l)P, it is evident that the ar-
rangement of the first MI + 1 points of the series is represented correctly by the
sequence S 1 = P^i Q. We therefore proceed to show that the arrangement of the
first MI M 2 + 1 points P, and the first n 2 points Q is correctly represented by the
sequence S 2 = 8? P. Since
1 P 1
/*! T * ** Mi T ; 7 >
M 2 q /" 2 + !
we have (MI k + 1 P > JcQ, for all values of k< M 2 , but (^ Jc + 1) P < hQ, if k > M Z -
VOL. II T
138 NOTE ON CONTINUED FRACTIONS. [Art. 6.
If we write down the sequence P^ Q, 1 + M* times over, so as to obtain the
aeries
IP, 2P, ... MiP, Q,
(l+Mi)P, (2 + M^P, Zt^P, 2<?,
(1+2MOP, (2 + 2 Ml )P, ... 3 Ml P,
(l + M 2 Mi)P,
P,
the inequalities k^ P < IcQ < (&MI+ 1) P, which hold as long as &^M 2 , show that
all these points, with the exception of the last of them (1 + M 2 ) Q, succeed one
another in the proper order. But the last is in error, for, putting & = 1+M 2 ,
(1 +Mi + MiM 2 ) P < (1 + M 2 ) Q, and consequently (1 + M 2 ) Q does not follow immedi-
ately after (1 + M 2 ) Mi P. We conclude, therefore, that we can repeat the sequence
P^i Q M 2 times, but that we cannot repeat it 1 + M 2 times. And, since two points
Q cannot come together, the series (P^i QY* is necessarily followed by a point P,
so that the sequence S 2 = S P correctly represents, as far as it goes, the arrange-
ment of the points.
6. We have thus shown that the relation between the continued fraction
and the sequences Si, S 2 , S 3 , ... holds as far as S 2 . Assuming, therefore, that it
holds as far as S f , where *>2, we have to prove that it holds as far as S i+l .
The proof depends on an elementary theorem relating to continued fractions,
which was first established by Lagrange.
' If --^ , are consecutive convergents to the same rational or irrational
ft-i to
quantity 6, Pi-i qi-i0 is less in absolute magnitude than any quantity of the
form y xQ, where x and y are positive integers, of which x is less than q^
Supposing, for brevity, that i is uneven, we infer from this principle that
the least segment in the sequence S ( is its last segment q { Q p { P, and that the
next least segment in $, is the last segment of S t _i, viz. jp,-_iP <?,-_i Q- We
have to add that pi-iP qi-iQ is also less than the segment P(l +Pi)-qt Q
which immediately follows S ( . For if
a condition which is certainly satisfied when i> 1, we have f-^- - >
-
Art. 6.] NOTE ON CONTINUED FRACTIONS. 139
i.e. > , because i+ 1 is even. Let us write down the sequence $,- 1 + k times
over, and let yQ xP be any segment of S { contained between two consecutive
points P and Q, of which Q is to the right of P ; the corresponding segment in
. will be
i.e. Q will be still further to the right of P, and the distance between P and Q
be increased. Next, let xP yQ be a segment of S{, contained between two
consecutive points P and Q, of which P lies to the right of Q ; or, again, let
xP yQ represent the segment (1 +p}) P <7,- Q, which immediately follows S { .
The corresponding segment in (& + !)$,, or immediately following (& + !)$,, will
be (kp i + x)P-(kq i + y)Q = xP-yQ-k(q i Q-p i P);
so that, if k be not too great, the two new points P and Q will lie in the same
relative position with regard to one another as the two points originally con-
sidered, the distance between them being diminished ; but, for values of k which
surpass a certain limit, the point Q will be shifted to the right of P, and the
segment QP will be replaced by a segment PQ. As long as this interchange of
places between two consecutive points Q and P does not occur, so long the suc-
cessive repetitions of , will represent with accuracy the arrangement of the points
P and Q. Now the least of the segments xP yQ is pi-iP qt-iQ, and
p4-iP-q t -iQ~*+i(q i 4-p { F)
is still positive ; therefore we may repeat S { 1 + M< + i times, but we cannot repeat
it 2 +M.--H times, for
is negative. The sequence S? M $,-_! will therefore truly represent, as far as it
goes, the arrangement of the points P and Q ; but the sequence Sl +>lM $,-_! would
fail to do so. We should in fact come to an error in the last two points of $,-_!,
which, according to the law of that sequence, we should have to write down as
QP, whereas the true arrangement of these points is PQ. This suffices to
establish the general theorem of Art. 4 ; but it is of interest to add, that the
error which we have just shown must occur in the last two points of the sequence
Sl** M $,-_!, is the only error that can occur in that sequence. And this is
certain ; for, in the first place, we have seen that there is no error in $ 1 1+M>+1 ; and,
in the second place, if xP yQ be any segment of >S',-_ 1 of the same positive sign
as pi-iP-qt-iQ, xP-yQ-(l+t*i + i) (ViQ-piP) is necessarily positive ; for, by
T 2
140 NOTE ON CONTINUED FRACTIONS. [Art. 10-
the theorem of Lagrange, xP-yQ> <fr_ 2 Q -Pi-a P ; and
xP-yQ+p i P-q i Q>(p i -p i .,)P-(q i -q i . i )Q>p i . l P-q i . 1 Q
by the same theorem ; whence
is positive, because
pi-i-to-i-n+ii-pi
is positive.
7. It will be noticed that the sequence ^ can only be repeated /u 2 times,
whereas any subsequent sequence S { can be repeated l+/u,. +1 times. The ex-
ception in the case of $ x is apparent rather than real, and arises from the fact,
that the period S consists of only one term. If we were to attempt to repeat
the sequence S 1 2 + // 2 times, the sequence S , which commences the last repe-
tition of S l} ought, according to the general theory, to be in error; viz. its last
point must be interchanged with the preceding point ; and, as S contains but
one point, this interchange vitiates the sequence S^ immediately preceding.
8. Any finite continued fraction may be written either with an even or with
an uneven number of quotients, because the last quotient may be made either
equal to unity or greater than unity. If the number of quotients be even, the
two extreme points P and Q, which coincide with 1, must be written in the order
QP ; if the number of quotients be uneven these points must be written in the
order PQ.
9. If we omit these two last coincident points, the remaining p 1 points P
and q 1 points Q evidently form a symmetric series, being similarly distributed
on either side of the middle point of the line. And, similarly, if we remove from
any sequence whatever its two final points, we obtain a symmetrical series,
because the sequence S { corresponds to the division of a line into p ( equal parts
and also into q t equal parts.
10. If we wish, from the arrangement of the points P and Q, to infer the
arrangement corresponding to the fraction /u,- + 1 +- - + ..., obtained from the
fraction -- by omitting its first i quotients, we have only to replace the se-
quences S { and $,._! by single points. Thus, in the example of Art. 3, if we put
S t = A, S = B, we find
A,2A,3A,B\iA,5A, GA,2B\7A\\SA,9A, WA, 3B\
HA, 12A, ISA, 4 1 144 1| 15,4, 16 A, 17 A, 5B\\\,
Art. 12.] NOTE ON CONTINUED PEACTIONS. 141
corresponding to ^ = 3 + ^ - . And, again, if we wish to obtain the arrange-
ment corresponding to the fraction /u. { H -- - , where i +j < s, we first
replace S t , S i _ l by single points, and then consider in the resulting arrangement
the sequence of order j. Thus the arrangement
A, 2 A, 3 A, B 1 4A, 5 A, 6 A, 2B \ 7 A \\
corresponds to the fraction 3 + ^.
Addition to the preceding note.
11. The theorem of Lagrange, on which the demonstration in the preceding
note depends, will be found in the second paragraph of his ' Additions to Euler's
Algebra.' But as this theorem is no longer included in elementary treatises, we
shall here place Lagrange's demonstration of it.
If </>, is the complete quotient of order i in the development of 6, we have
&&-! + ?<- Pi-i-li-*
But </>,- is positive and greater than unity; hence, ^_ 2 g,-_ 2 0, and p i _ l q i , l Q
are of opposite signs, and p i _ l q i _ l 6 is less in absolute magnitude than
pi- 2 -qi- 2 0.
Again, since p { q {-1 jp,--! <7,- = ( 1)*, we can always find, whatever the given
integral numbers x and y may be, two integral numbers X and M satisfying the
equations
x = \q i _ l + nq i , y
whence we obtain
As p i ^ l q i _ 1 6 and pi qiQ are of opposite signs, if y xQ is less than
Pi-i <?, _i &, X and n must be of the same sign; that is to say, x and y are
either respectively equal to q { and p it or else they are respectively greater than
q { and Pi .
12. In the same place Lagrange has also established the converse theorem,
that if b a 9 is a minimum difference, i.e. if 6 ad is less in absolute magnitude
than any difference y xQ, in which x is less than a, - is a convergent to 6.
Cv
Writing p t-l for 6, and q i _ l for a, we first determine the positive numbers p f _ z
and q ( _2> respectively less than ^>,-_ 1 and q { _i, which satisfy the equation
Pt-i ( li-t~Pi-2^i-i = e > e denoting an unit of the same sign as >,-_! <7,-_] 0. If
142 NOTE ON CONTINUED FRACTIONS. [Art. 14.
we write p i -i-q i -iO = u i _ 1) p,-_ 2 -g f ,-_2^ = w,--i, we find, on eliminating 0,
In this equation ,-_ 2 is greater, by hypothesis, than <_!, because g f ,-_ 2 <Q',--i 5
?/ *?/
d fortiori - - is greater than - -. But M,-_J is of the same sign as e ; there-
?i-2 <?-!
fore, u i _ l and ,-_ 2 must have contrary signs. Consequently the quotient
is positive, and greater than unity ; and if ^ - be developed in a continued
fraction having ^^ for its last convergent (which is always possible), we obtain
i.e. *^l and ^-^ = - are successive convergents to 0.
2<-2 <7<-i
13. Combining the two theorems of Lagrange, we see that if we have ascer-
tained, by observation, that p i , 1 q i _ l 6 is less than any difference y xO in
which x is less than <?,_!, we can at once infer that pi-\ <?,_! is also less than
any difference y xQ in which q i ^<x<q i .
14. The two theorems of Lagrange serve to define the successive minima of
the expression y xQ. The theory of the successive minima of the expression
aj
- 6 is perhaps less complete. Thus we have the elementary theorem, that the
C
difference ^1 6 is less than any difference - 6 in which x does not surpass
qt-i x
qt-i, and is also less than any difference of the same sign with itself, in which x
does not surpass q { ; but there may be differences of a contrary sign to - - 0,
qi-i
in which x does not surpass q { , and which are less in absolute magnitude than
^-^ 0. And again, if 9 be a minimum difference (i.e. if 6 be less in
</<_i a a
absolute magnitude than any difference - 0, in which x is less than a), we can-
7
not in general infer that - is a convergent to Q. We shall attempt, in what fol-
C
lows, to define accurately the successive minima of the expression - 6, and thus
x
Art. 15.] NOTE ON CONTINUED FRACTIONS. 143
to give a greater amount of precision to this part of the theory of continued
fractions.
15. We still consider a rational or irrational quantity 6, of which the de-
velopment is
~P JcT) -4" *W
and, adopting the designation of Lagrange, we term the fractions r~- = jp-
where > k < n it intermediate fractions. These fractions are evidently interme-
diate between LL=* and ; hence 6 lies between any one of them and ^-^
b q t -a q f ?.--i
If is a minimum difference, we can, by reasoning as in Art. 12, arrive at an
equation of the form
1 1 1 1 1
where
and we can prove that in this equation >// is positive. But we cannot prove that
^ is greater than unity; i.e. instead of the equation X = MJ, we have the in-
equalities < X < /jj ; ; and thus from the hypothesis that 6 is a minimum
difference, we cannot infer that - is a convergent to 0, but only that - is either a
convergent or an intermediate fraction. But not every intermediate fraction can
p
give a minimum difference ; for in order that TT should be a minimum differ-
p p__
ence, ^ - 9 must (at any rate) be less than - 0, because <?,-_!<<?*. The
absolute value of
6 is r-
and the absolute value of
<?>-" -
whence, if - ^ 6 is a minimum difference, we must have
v*
144 NOTE ON CONTINUED FRACTIONS. [Art. 16.
or yu< + - + ...<
And this necessary condition is also sufficient. For, since 6 lies between - p and
Vi
p
*-^ , and since (if the condition (A) be satisfied) 6 also lies nearer to -. - than
<?-! Vfc
P P
to ^- ir:J , any fraction which is nearer to 6 than - must lie between -^ and
?,-! ft ft
, and must therefore have a denominator greater than Q k , because
_
ft ?,--l ft-1 Qk
We are thus led to divide the fractions, intermediate between -^^ and
fc-2 ffi
into two sets, according as they do or do not satisfy the condition (^4). "We may
call those fractions which do not satisfy that condition the inferior, and those
which do satisfy it the superior intermediate fractions. We then have the
theorem :
tf
' The complete series of successive minima of the expression - is obtained
00
by taking in succession for - the convergents, and the superior intermediate
v
fractions in their natural order.'
16. If k> \fjti, the condition (A) is satisfied; if & = ijM;, the condition is
satisfied if n i _ l <MJ + I| if ^ = i/ t -, /*,-_i = /*,+!> the condition is satisfied if
ju,-_ 2 >/,. + 2 , and so on continually. If the continued fraction be finite, sym-
metrical, and of an uneven number of quotients, fj. t = 2k being the middle
P-- Pk
quotient, we have a singular case in which the errors of - - and ~=r ar e
p " i v t
exactly equal ; we may in this case regard -^ as an inferior fraction. It will be
Vfc
seen that, as nearly as possible, one-half of the fractions intermediate between
*2=i and are superior. Thus, if /*. = 2 h -1*1 is uneven, there are h inferior and
?<-2 &
h superior intermediate fractions ; if /,- = 2 h is even, there are certainly h - 1
p
inferior and h 1 superior intermediate fractions ; but whether -^ is inferior or
Art. 18.] NOTE ON CONTINUED FRACTIONS. 145
superior, can only be decided (as we have just seen) by comparing the quotients
which precede p { with those which follow it.
D. P
17. The difference -i=l -0 is of the sign (- 1) 1 '- 1 ; the differences -^ 6,
ft-l V
which, in forming the complete series of minima of - 6, we have to intercalate
so
between ^^ 6, and 6 are of the same sign as the latter of these differ-
ft-i ft
ences, i.e. they are of the sign ( - l) f . Thus, after every convergent there is a
change of sign in the series of minimum differences, and the minimum differences
formed with convergents are distinguished by this criterion from the minimum
differences formed with superior intermediate fractions.
18. Again, if -- 8 be any minimum difference, and if q i _ 1 ^a<q { , the
Oi
y b
only differences - 6, which are less than -- 0, and which have denominators
J x a
/yj f
x less than q { . are the minimum differences which lie between -- 6 and 0.
a q,
p
It is sufficient to prove this for the case in which a = q i _ 1 , &=p,-_ t . Let *~ !
Vx-i
be the last of the inferior fractions, intermediate between --=1 and - * : then
P *>._ ?t '- 1 q <
0, which lies between 7^* and -- JL - - , is nearer to the latter than to the former
^X-l ft-l
of those fractions. If then - be nearer to 6 than *-*=? is, - must itself lie
p
x ft-
between =^ 1 - 1 and -:- . But, if x <q t , - cannot lie between *-=* and :
ft-i &-i x ft., ft
hence, - must lie between and x-1 . But the only fractions between these
x ft Vx-i
limits, which have denominators less than q { , and lie nearer to & than -fcz! f are
the superior fractions intermediate between and . For all such fractions
q { _ 2 q {
are of the type ~ * '~ 2 , the relatively prime numbers <r and r satisfying
<7 ft-lT" T ft-2
the inequalities a-
X-l<-</x i , .......... (1)
1 <5',., whence <J-<M,-, ....... (2)
VOL. II. U
146 NOTE ON CONTINUED FRACTIONS. [Art. 20.
ff
Now if /a,. = 2A + 1, we have X l=h, and the inequalities h<-, and
T
tr<2h+l (from which equality is excluded), show that unity is the only
p
admissible value for r. Again, if n t = 2h and - - is an inferior fraction, we have
Vi
h < - , a < 2h, and unity is the only admissible value for T. In both these cases,
therefore, the only fractions having denominators less than q iy which lie between
P >
- and , are the superior intermediate fractions. If, however, n^Zh, and
Vx-i <?,
p
- is a superior fraction, the inequalities (1) and (2) are satisfied by the values
VA
<r = 2h1, r = 2, so that, besides the superior intermediate fractions, the fraction
i+2j>-i jj between the limits P*=l and Pf m But t hi s fraction is
more remote from 6 than *-^ is, because the equation - = s is inconsistent
2,--i T
with the inequality (3).
p
19. The inferior intermediate fractions j t k^\ l, do not give minimum
Vt
fV\ f
differences, because <?,- _!<$*, and 6 is less in absolute magnitude than
7^ 0. But, with the single exception of -1 6, all other differences - 6, in
Qk <?i-i
P 91 P
which x is less than Q k , are greater than -. 6. For, if - lie between - * and
Vfc x v t
*-=!, x must be greater than Q h ; if -^ lie between - and ^S=l, the difference
<?,-! Vfc X <?i-l
- 6 is certainly greater than -^ 6, because 6 lies between -^ and ' ;
x y Q k Q k &_,
lastly, if lies between - and -.^ , we find (taking the case in which i is
?<-!
uneven)
.- __ _
x x 5,..! a^.i ftg,.! y,-.! Q k Q*
20. The theorem of Lagrange admits of an important geometrical interpre-
tation. If with a pair of rectangular axes in a plane we construct a system of
unit points (i.e. a system of points of which the coordinates are integral numbers),
and draw the line y = Qx, we learn from that theorem that if (x, y) be an unit
point lying nearer to that line than any other unit point having a less abscissa
Art. 20.] NOTE ON CONTINUED FRACTIONS. 147
(or, which comes to the same thing, lying at a less distance from the origin),
- is a convergent to 6 ; and, vice versd, if - is a convergent, (x, y) is one of the
X 3C
' nearest points.' Thus the ' nearest points ' lie alternately on opposite sides of
the line, and the double area of the triangle, formed by the origin and any two
consecutive ' nearest points,' is unity.
fin
In particular, if = , p and q being relatively prime integers, the coor-
dinates of the two ' nearest points ' above and below the finite line joining the
origin to the unit point (q, p) satisfy respectively the equations px qy = l, and
px qy= 1. We thus obtain a simple geometrical method of finding the least
solution in integral numbers of either of those indeterminate equations.
IT 2
XXIX.
NOTE ON THE THEORY OF THE PELLIAN EQUATION,
AND OF BINARY QUADRATIC FORMS OF A
POSITIVE DETERMINANT.*
[Proceedings of the London Mathematical Society, vol. vii. pp. 199-208. Read May 11, 1876.]
1. -L/ET 6 = fjL + - be any continued fraction, of which Q 1} 2 ,... are
MI + Ma T
the complete quotients ; , , . . . the successive convergents ; so that
Also, let pi _j - eq^i = ( - 1)* *<_!, so that #,- = ;
e -i
we have #1 = , and hence e ; _ l =
, _ l
e tfj 6*2 . . .
* The following summary of the contents of this Note may be of use to the reader :
Art. 1. The relation, in a continued fraction, being the quantities e and 0.
Art. 2. The theorem that T+ U*/D is equal to the product of the complete quotients in the
development of -v/Z).
Art. 3. The same theorem for the period of complete quotients in the development of any
quadratic surd.
Art. 4. Theorems as to the number of different periods of complete quotients ; viz., equations
OH*).
Art. 5. Theorems as to the number of non-equivalent classes of quadratic forms ; viz., equations
(5) and (6).
Art. 6. Equations arising from a comparison of the formulae (5) and (6) with those of Dirichlet.
Arts. 7-13. Discussion of the nature of the periods in the more important special cases.
Art. 14. On the symmetry of any periodic series.
Art. 15. On the arithmetical conditions under which the various special cases present themselves.
(It would be difficult to say that anything in the Addition (Arts. 7-15) is new: the discussion
there attempted has never been given completely (see Art. 7) ; but this may have been because no one
has thought it worth giving.)
Art. 3.] NOTE ON THE THEORY OP THE PELLIAN EQUATION, ETC. 149
This expression, which supplies a measure of the rate of decrease of the differ-
ence e,._j, admits of an interesting application to the theory of the Pellian
equation, and of binary quadratic forms of a positive determinant.
2. Let D be any positive integer, not a perfect square. In the development
of VD in a continued fraction, let
p > p > > p
*i *i *<
be the period of complete quotients ; so that, if a is the integral number
next inferior to
VD+Q l ^D + a VD+Qi
-PT "D=*' ~P~
Let Tand U be the least integral numbers satisfying the equation
and let VD = a-\
we have T= Pi _,, U= qi _
whence, by the preceding theorem,
Example. The continued fraction equivalent to \/13 is
and the period of complete quotients is
-v/13 + 3 V13 + 1 V13 + 2
giving (V13 + 3) 2 (V13 + 1) 2 (v/13 + 2) x = 18 + 5 VI 3,
and 18 2 - 25x13= - 1.
3. Again, let $2 = a + 2Zw + cw 2 =
represent any properly primitive equation of determinant D (i. e. any quadratic
equation whatever, in which a, b, c are integral numbers satisfying the equation
b 2 ac = D, and a, 26, c have no common divisor). If
^t&z
p > >
*i
150 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 3.
is the period of complete quotients obtained by the development of either
root of Q, we shall have, as before,
For the equations of the period are of the type
a + 2ft w - 0,^ = 0,
- HI - 2ft M! + a a U\ = 0,
(Q)
so that, if /*o> Mi, /2> > /".--I, is the period of integral quotients, we have
Ml
where
Hence
,,,
n, =1
+ Q.
*
But, by a known theorem (see the ' Report on the Theory of Numbers,' in the
' Report of the British Association for 1861,' Art. 96 (i.), p. 315*), we have
r_Pt-* = ff.--i-g.--2 = gi-i .
whence
= r-
or
T4.
If the given equation (Q) is improperly primitive (i.e., if the numbers
a, 26, c have 2 for their greatest common divisor), we have to replace the
* Vol. i. p. 195.
Art. 4.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 151
numbers T and U in the equations (Q), by ^ 7\ and \ U lt where 7\ and U^ are
the least numbers which satisfy the equation T\ D U\ = ( 1)' 4 ; and we find
4. Every primitive quadratic equation of determinant D, of which one root
is positive and greater than unity, and the other negative and less in absolute
magnitude than unity, occurs in one, and only in one, of the periods of equations
of determinant D (see the Report cited, p. 309, Art. 93*). Hence, every ex-
pression of the form - p - , in which, P and Q being positive, Q is less than
JD, P is a divisor of DQ 2 intermediate between \/D Q and </D + Q, and
D-Q*
the three numbers P, Q, p are relatively prime, is the root of an equation
contained in a period of equations of determinant D ; and, for any given
determinant, the number of periods of complete quotients is equal to the
number of periods of quadratic equations, if we regard two quadratic equations
8uchas
which differ only in sign, as identical with one another.
DQ 2
Let P"= - , and let k be the number of periods of properly primitive
complete quotients of determinant D ; we have evidently
......... (1)
VD+Q
the sign of multiplication II, and the sign of summation 2, extending to all
positive numbers Q which do not surpass \/D, and to all divisors P of D Q*,
which are intermediate between <JD + Q and VD Q, and are such that the three
numbers P, 2 Q, P" admit of no common divisor other than unity. Let ^ (Q)
be the number of such divisors of D Q 2 ; observing that, if - p ' is a com-
plete quotient in a period, = -ry- ^ is also a complete quotient in the
Mr \' AJ (^/
* Vol. i. p. 186.
152 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 5.
same or in a different period, we may write
where it will be noticed that, if P = P', D = P 2 + Q 2 , the two identical complete
VD+Q P rV
quotients -- ^ - and rj=. ^ are each 01 them equal to .
Jr / D (J Lv-
When D = \, mod 4, let ^ be the number of periods of improperly primitive
complete quotients of determinant D, we find as before
(3)
the symbols II and 2 extending to all positive uneven numbers Q which are less
than */D, and to all divisors P of D Q 2 which are intermediate between
VD + Q and <JD Q, and are such that the three numbers P, 2 Q, P f have 2 for
their greatest common divisor. If ^ (Q) be the number of such divisors
of D QP, we may also write
It will be observed that, if Q is even, we have always \f--j (Q) = 0.
5. Let h and h t respectively denote the numbers of properly and improperly
primitive classes of quadratic forms of determinant D, and let [T, w], \r lt wj be the
least numbers which satisfy the equations T 2 Dv 2 = +1, rj Dv\ = + 4 ; so that,
when the equations x* Dy 2 =l x 2 Di/ 2 =4: . (T)
are resoluble [they are either both resoluble or both irresoluble],
and when the equations (T) are irresoluble,
r + v^D=T+UVD, r l +
We can now establish the equations
(
(6)
For this purpose it is only necessary to show that, when the equations (T) are
not resoluble, we have h = 2k, A, = 2jfc,; .......... (a)
Art. 6.] QUADRATIC FORMS OF A POSITIVE DETERMINANT. 153
but when these equations are resoluble, we have, instead,
h = k, *! = *, ........... (b)
To the single period of complete quotients
(7)
"1 "2 "3
there correspond two periods of reduced quadratic forms, viz.,
(,&,-<!), (-a lt &,<*,), (,,&, - 3 ), (8)
and (-o,A>, i)> Oi,ft, - 2 ), (-a,&,a), (9)
For the complete quotients (7) are the positive roots of the equations (Q) ; they
are also the positive roots of the same equations with their signs changed ; and
the period (9) is related to the period ( 0) exactly as the period (8) is related
to the period (Q) ; viz., the coefficients of the forms are the same as the co-
efficients of the equations, except that in the periods of equations the middle
coefficients are alternately positive and negative, whereas in the periods of forms
these coefficients are all positive. The two periods of forms are, in general, but
not always, distinct ; and we shall now prove (what is indeed well known) that
these two periods are, or are not, identical, according as the equations (T) are,
or are not, resoluble. We may observe that the form (a, b, c) is termed the
opposite of the form (a, 6, c), and ( a, b, c) the negative of (a, b, c) : thus
( a, b, c) is the negative of the opposite of (a, b, c).
(i.) If the equations (T) are resoluble, any form (a, b, c) of determinant
Z> is properly equivalent to the negative of its opposite ; viz., (a, b, c) is
transformed into ( a, b, c) by
bU-T, -cU
-aU, bU+T'
Hence the reduced forms (a , /3 , aj), ( a c , /3 , Oj) are properly equivalent;
either of them is therefore contained in the period of the other ; i. e. the two
periods are identical.
(ii.) If the two periods (8) and (9) are identical, the form ( a , /3 , aj) must
occur in 'the period of (a , j3 , aj) ; and because its first coefficient is negative,
it must occupy an even place in that period. Hence the period of complete
quotients (7) consists of an uneven number of terms ; and we infer from the
formulae (Q) of Art. 3 that the equations (T) are resoluble..
6. If D= 1, mod 8, we have h = h 1 ; if D = 5, we have h = 3^ when T X and
VOL. II. X
15-4 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 7.
! are even, but h = h^ when T, and i>j are uneven, in which case
(lr 1 + v
We thus find, in both cases alike,
_ .
where <r = 1, or = 3, according as D = 1, or =5, mod 8.
Again, since, by the formulae of Lejeune Dirichlet, we have (see the Report
cited, Art. 1.1.) ilog[T+ . VB] _ s ^ 2( ^l,
where the sign of summation extends to all numbers prime to 2D, and ( j
is the generalised symbol of quadratic reciprocity, we obtain
<fl()i-*Hfl>fcf.
S () = ^ W) + + + ... ..... (13)
Similarly, from the formulas (see ibid.)
where a- has the same meaning as before, we infer
It is probable that a direct demonstration of the equations (12), (13), (14), of
which any two involve the third, would offer considerable difficulties.
Addition to the preceding Note.
1. As the preceding determination (equations 5 and 6) of the number of
non-equivalent classes for a positive determinant depends on the equations (a)
and (b), which assign the relation between the number of periods of complete
quotients and the number of periods of reduced forms, it is worth while, for the
sake of distinctness, to describe fully the characteristic appearances presented
by these periods in certain special cases which are of some importance.
Every form, or class of forms, is, of course, properly equivalent to itself, and
improperly equivalent to its opposite. But a form, or class of forms, may be
(i.) Properly equivalent to its opposite, and improperly equivalent to itself
(in this case the class is ambiguous) ;
* Vol. i. p. 217.
Art. 8.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 155
(ii.) Properly equivalent to its negative, and improperly equivalent to the
negative of its opposite ;
(iii.) Properly equivalent to the negative of its opposite, and improperly
equivalent to its negative.
Since, if any two of these specialities coexist, they necessarily involve the
third, there are four cases to be considered, viz., the cases (i.), (ii.), (iii.), in which
the specialities (i.), (ii.), (iii.) exist singly, and the case (iv.) in which they all
exist simultaneously. We shall briefly refer to each of these cases in succession.
We may observe, however, that the case (i.), which is that of an ambiguous class,
has been fully considered by Gauss (' Disq. Arith.', Art. 187, Obs. 6, 7, 8) ; of the
rest, the case (ii.) has, perhaps, attracted less attention than might have been
expected.
8. If the period of reduced forms equivalent to (a, b, c) is
(o> A -i), (-i. A, "2), -, (- 2 *-i, &*-i, ), .... (8)
the periods of reduced forms equivalent to (a, b, c), ( a, b, c),( a,b, c)
are respectively
(0> /&!*-l) a 2*-l)> ( a 2*-l> Ai*-2> Oat-a), > ( 1 A a o)>
( a O 02k--l> 2*-l) ( a 2*-l, &*-2> 2*-2)> (1, A), o)>
As in Art. 3, we designate the period (7) of complete quotients, or, which is
the same thing, the period formed by the positive roots of the equation (Q), by
u , u lt ..., UM-I-
The negative roots of the same equations we represent by
JL JL i
Vo' Vi ' VM-I '
so that, if /u , Mi, , M2*-i is the period of integral quotients, we have, by a well-
known theorem, ,- T
the symbol Ix denoting the greatest integral number not surpassing x. The four
periods of forms (8), (10), (11), (9) we represent for brevity by the symbols
<t>0> <f>l> #2, ', <t>2k-l, ........ *
X 2
156 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OP [Art. 10.
Two such forms as (a, b, c), (c, b, a) are said to be associated ; thus, <, and
4",, or again (f>, and \J/-,, are associated forms. The periods 4> and i% and again
the periods 4> and ^, are themselves termed associated periods.
The period of complete quotients corresponding to the periods $ and 4> is
(Art. 5) , ML ..., ,_,;
and similarly the period of complete quotients corresponding to the periods Sk
and ^ is 4 , 4 .
V 2*-U V 2*-2> > V l> V 0-
9. Case (i.) If (a, b, c) is properly equivalent to (a, b, c), the periods <J
and ^ must coincide; i.e. we must have, for some value of a-, ^ = v l'2<r + i) the
suffix 2cr + l being uneven, because the extreme coefficients of ^a + i must have
the same signs as the extreme coefficients of < . Hence <^ = ^ 2<r , < 2 = ^ 2( r-i) ;
and finally ^,7 = ^ + 1, ^0 + 1 = ^^, or there occur in the period two consecutive
forms of which each is the associate of the other. As we may begin the period
with any form we please, we may suppose that ^> and c^ are these two con-
secutive forms, so that /3i = /3 , a 2 = a , 2/3 = 0, mod ctj. It will be seen that, if
we have not only <f> a = ^ a + 1 , $0 + 1 = ^0, but also
Thus a sequence of two associated forms occurs twice in the period ; and, assum-
ing (as we have done) that a- = 0, the period of forms is of the type
^1.^1 ; ^>2> <t>3> > <t>k-\\ <*, "ta; ^k-i, "f*-2> , 4-2.
Where ^>, + 1 = 4 2 *-8 f r every value of s. 'The period of integral quotients is of
the type x , n lt /< 2 , ..., /x t _j ; \, n k-l , ^_ 2 , ..., // 1(
where X = . ** =
i a *+i
The period of complete quotients is of the type
v lt ,; M 15 M 3 , ..., M*_X; W*,V A ; ^_ 1; v t _ t , ..., v f ,
where / w, tl = / y 2 *_, for every value of s. The periods of the coefficients a and /8
are respectively of the types
a i ' a 2>'"> a *; a k + i'j *> a z 5
A. A; A, > ft-iJ A,^ A ; ft.!, ..., /3 2 .
The two ambiguous forms are < x and -v^.
10. Case (ii.) If (a, 6, c) is properly equivalent to ( a, 6, c), the
periods <1> and ^ must coincide. Hence we must have (for some even value of
the suffix 2cr) ^> = v i'2ff; whence </> (T = ^ <7 , and also ^> <r+ i t = ^ < T_* = 1 f (M .*. The
Art. 12.] BINARY QUADRATIC FORMS OP A POSITIVE DETERMINANT. 157
equation (f) a = ^ ff is equivalent to the equation a a = a a + l ; we thus see that the
period contains two forms, in each of which the extreme coefficients are equal in
absolute magnitude ; so that (supposing, as before, that o- = 0), we have
The period of reduced forms is of the type
^o=0o; 0i, 02, -, 0*-i; 0* = ^*; ^*-i, ^*-2, > ^i
where <p t = ^ 2 *_, ; the period of complete quotients is of the type
U = v ; u l ,u 2 ,...,u k _ l ; u k = v k ; v t _ l} v k _ t , ..., v lt
where u t = v., k _ t .
Lastly, the periods of integral quotients and of the coefficients a and /3 are
of the types
MO, MI, ..., M*-I; M*-i, M*_I, , MI, M O ;
1> "2, ", *-!> a *, a *> a *-l, > 2> a l 5
^o5 A> &! &-i> ft; ft-i, A-2, , ft-
11. Case (iii.) If (a, &, c) is properly equivalent to ( a, b, c), the periods
$ and 4> coincide, and we must have ^>o = ^2<r+i, <o = ( 2<r+i where 2<r + l is less
than 2k. From these equations we infer <o = < / > 2(2<r + i)> or 2<r + l = ^, since if
#0 = < m is a multiple of 2^. The period of forms is therefore of the type
<o, 0i, , 0*-i, 0o, 0i, , 0*-i,
i being an uneven number ; the periods of complete quotients, of integral quo-
tients, and of the coefficients a and /3, consist each of a period of k terms, twice
repeated. The period of equations (Q) in like manner consists of a period of k
equations, twice repeated ; but each equation appears in the second half of the
period with its sign changed.
12. Case (iv.) If (a, 6, c) is properly equivalent to any two of the forms
(a, 6, c), ( a, b, c), ( a, 6, c), and therefore to all three of them, the
nature of the periods is most readily ascertained by considering the series of
integral quotients. Since the conditions characteristic of the cases (i.) and (iii.)
must be united, the semi-period
^o, MI, MZ, > M*-i
must be term for term identical with the semi -period
\, M*_i, M 2 , , Mi,
k being an uneven number 2i+ 1. Hence \ = \, and the period is of the type
X, /uj, n 2 , ..., // <} /*,., ..., p 2 , (*!,
158 NOTE ON THE THEORY OF THE PELLIAN EQUATION, AND OF [Art. 13.
twice repeated ; which combines the characters of the periods of integral quo-
tients in the cases (i.), (ii.), (iii.).
The period of forms is of the type
the period of complete quotients is of the type
i\, %; 2 , u 3 , ..., w,. +1 ; Vi,
twice repeated, where u, = v 2i+2 _, ; and the periods of the coefficients a and j3 are
respectively O .
each of them twice repeated.
13. If therefore we develope the two roots of a given primitive quadratic
equation, we obtain, in the general case, two distinct associated periods of com-
plete quotients and four distinct periods of reduced forms. In the special cases
(i.) and (ii.), we have but one period of complete quotients and two periods of
reduced forms ; in case (i.) the two associated periods of each pair combine ; in
case (ii.) each period of reduced forms becomes identical with the negative of the
opposite of its associated period. In case (iii.) we have two distinct periods of
complete quotients ; but only two distinct periods of reduced forms ; the period
of any form being identical with the period of the opposite of its negative, and
consisting of an uneven number of forms followed by the opposites of the
negatives of the same forms ; the period of complete quotients contains only half
as many terms as the period of reduced forms. Lastly, in case (iv.) we have but
one period of complete quotients, and but one period of reduced forms, the four
periods, which in the general case are distinct, being all identical with one
another.
We may observe that, if the equation
V ,4 /
can be satisfied by three numbers X, M, ", which also satisfy the condition
the form (a, 6, c) is transformed into (a, 6, c) by the substitution
A, /z
and consequently has a period of the type (i.). If, instead of the condition (q),
the condition u 2 X>/= 1 (r)
Art. 14.] BINARY QUADRATIC FORMS OF A POSITIVE DETERMINANT. 159
is satisfied by the three numbers (X, ft., v), (a, b, c] is transformed into ( a, b, c)
by
and the period of (a, b, c) is of the type (ii.). Lastly, if the equa-
tion ( p) can be satisfied by two different sets of numbers, of which one set satis-
fies the condition (q) and the other the condition (r), the period of (a, b, c) is of
the type (iv.).
14. The periods which we have to consider in the cases (i), (ii.) and (iv.)
afford examples of each of the three kinds of symmetry which can exist in a
periodic series. Let ... c , c l5 c 2 , ... c n _ l , ... be a period of n terms repeated indefi-
nitely in both directions ; it will be found that the series thus formed may be
symmetrical (i.e. may be the same whether we follow it forwards or backwards)
in three, and only in three, different ways.
(i.) Let n be even ; and let the series continued from c forwards coincide
with the series continued from c 2k+l backwards, so that c = c n+1 , c 1 = c ik) ...: the
period then is Co , Cl , ..., c k ; c k , ..., c , c 2k+2 , ..., Cfl+1 ;
or, if n = 2i>,
where there are two centres of symmetry, one falling between the two terms c k ,
the other falling between the two terms c_ v + k+l .
Of this type is the period of the coefficients /3 in case (i.) ; and the periods
of the integral quotients and of the coefficients a in case (ii.).
(ii.) Let n still be even, but let the series continued forwards from c coin-
cide with the series continued backwards from c 2Jt ; the period is
CD> Cj, ..., Cfc_ 1( C k , Cj_j, ..., CQ, C 2k + i, ..., C n _i,
or, if n = 2 v,
where there are two centres of symmetry falling on the terms c_ v+k and c k
respectively. The symmetry of the integral quotients, and of the coefficients a
in case (i.), and of the coefficients /3 in case (ii.) is of this type.
(iii.) Let w = 2i> + l be uneven; and let the series continued forward from c
coincide with the series continued backward from c. 2k + 1 . The period is of the
type r r r f r r r
* ^0^l '>*> ^fcj ^0? v 2fc + 2'*M 7* 1'
where again there are two centres of symmetry, one falling on the term c_ v+kt
the other between the two terms c k . If we had supposed c = c. ik we should have
obtained a period of the same form. It is evident that, if this period be doubled,
160 NOTE ON THE THEORY OF THE PELLIAN EQUATION, ETC. [Art. 15.
it combines the symmetries of the periods (A) and (B). Of this type are the
periods of integral quotients and of the coefficients a and /3 in case (iv.) In case
(iii.) there is no symmetry; but an unsymmetrical uneven period is twice repeated.
15. Every determinant has ambiguous classes ; and every ambiguous class
has a period of the type (i.). But developments of the types (ii.), (iii ), (iv.) can
only present themselves in the case of determinants of the form P or 2 P, where
P is a product of uneven prime numbers of the form 4, + 1. For in case (ii.) D
must, as we have seen, be the sum of two square numbers prime to one another,
and in case (iii.) the equation T* DU*= 1 must be resoluble, whence again
D is the sum of two squares prime to one another.
If ft. is the number of different prunes dividing P, the number of ways in
which P can be decomposed into the sum of an even and uneven square prime to
one another is 2' x ~ 1 . Let D = P, and let D = A* + B 2 be one of these decomposi-
tions, A being uneven and B even ; the forms ( A, B, A), (A, B, A),( B,A, B),
(B, A,B) are all reduced, and the first two are properly, the last two im-
properly, primitive. We thus have 2, fl ~ 1 properly primitive periods of reduced
forms of the type (ii.) ; and as many improperly primitive periods of the same
type ; i.e. since there are 2''" 1 properly and as many improperly primitive am-
biguous classes, there are as many classes having periods of the type (ii.) as there
are classes having periods of the type (i.).
If D = 2P, we have a similar result; viz., there are 2* 1 " 1 equations of the
form D = 2P = A* + B\ in which A and B are both uneven. We thus obtain 2^
properly primitive periods of reduced forms of the type (ii.) ; i.e. as many as
there are of type (i.) There are, of course, no improperly primitive classes of a
determinant of the form 2 P.
When the equations (T) are not resoluble, but the determinant is of either
of the forms P or 2P, the developments of the type (i.) and those of the type (ii.)
are entirely distinct from one another. On the other hand, when the equations
(T) are resoluble, the developments of the types (i.) and (ii.) coincide, giving rise
to developments of the type (iv.), and all the remaining developments are of the
type (iii.).
It is known that, when D is an uneven power of an uneven prime of the
form 4n + 1, the equations (T) are always resoluble. But when D has any other
value of either of the forms P or 2 P, there is no known criterion for deciding
whether these equations are or are not resoluble.
XXX.
ON THE VALUE OF A CERTAIN" ARITHMETICAL
DETERMINANT.
[Proceedings of the London Mathematical Society, vol. vii. pp. 208-212. Read May 11, 1876.]
JjET (m, n) denote the greatest common divisor of the integral numbers m and
w ; and let ^ (m) be the number of numbers not surpassing m and prime to m ;
the symmetrical determinant
A m = Z(l, 1)(2, 2)...(m,m)
is equal to \|<- (1) x -^ (2) x . . . x \f<- (m).
This theorem may be established as follows. Let p lt p 2 , ... be all the
different primes dividing m, and consider the columns (P) of which the indices
are
m m m m
T/i'. . * * j ? )
Pi Pi PiP* PiPzPz
Take these columns with the signs of the corresponding terms in the product
and, attending to these signs, replace the terms of the last column of A m by the
sum of the corresponding terms in the columns (P). The value of A m is not
changed : the term (m, m) is evidently replaced by ^ (m) ; and we shall now
show that every other term (m, K) in the last column is replaced by zero ; i.e.
that A m = 4- (m) x A m _j, which is the theorem to be proved.
First, let k be prime to m ; then
VOL. II.
162 ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT.
and (m, k) has to be replaced by a sum of units, of which as many are negative
as positive ; i.e. by zero.
Secondly, let k be a divisor of m, other than unity or m itself; and let
us separate the primes p into two classes, q and r, in such a manner that k
9M
does not divide any quotient of the form , but does divide every quotient of
/yn
the form . There may or may not be any primes q, but there must be at least
one prime r, or we should have k = m : we further observe that
/m 7 \ k / m ,\ k
I >*)"-, ( -- ,k) = - ,
\q / q \ qi q 2 / q t q 2
Thus, if we were to attend only to those columns of which the indices are
m, -,... , -,..., we should have to replace (m, k), or k, by k Il(l - -V just
as before we replaced TO by -^ (m). But we have to attend to the complete
series of columns (P) ; and thus we have to replace (TO, k), not by HIM. )
-t i
taken once, but by Hl/1 ) taken as often as there are terms in the product
Ilf 1 V and taken each time with the sign proper to the corresponding term
of that product ; i. e. (TO, k) is replaced by zero.
Lastly, let k = hS, S being the greatest common divisor of k and TO; so
that (TO, k) = (TO, S) = S. If d is any divisor of TO, we have the elementary theorem
\77 ' / = \tf' ^/' ^ or ' ^ (TT~ ' fy = ^'' we nave a ^ so ( m > ^) = c ^' > anc ^ hence
S, which is a common divisor of TO and dtf, divides dX, which is the greatest
TO
common divisor of those two numbers. But h is prime to -j- ; therefore, a
,....,. . TO./TO,<K TO . . 8
jortiori, h is prime to -r& ; i. e. ( j V , h T . ) = 1, for -5-57 is prime to - > as well as
do vrfo o ' do o
to h, or, which is the same thing, (-, , h} = S' = (-j , <SY It appears from this
that in the columns (P) the terms which lie in the row of which the index is hS,
are precisely the same as the terms which lie in the row of which the index is
S; and hence (TO, hS) is replaced by terms of which the sum is zero, because (m, S)
is replaced by terms of which the sum is zero.
ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. 163
The following remarks are suggested by the preceding theorem, or by its
demonstration :
(1) If we denote by la the greatest integer not surpassing the positive
quantity a, the theorem may be expressed by the equation
1 . 2 . 3 ...m
the sign of multiplication extending to all primes not surpassing m.
(2) Instead of the greatest common divisors themselves, we may consider
their powers of exponent s ; writing ^ (m) = m'Tl(l j , and following the same
course of demonstration, we obtain the theorem
from which we infer, as a particular result,
f^- 2 = 1 . 2 . 3 . . . m x n(l + -) Z P.
A m , i v y
(3) The equation (J?) is an identity with respect to the exponent s, which
may have any value whatever : the case in which s = 1 is especially interesting.
Let [m, n\ be the least common multiple of m and n, so that [m, ri\ = -. - ^ :
(m, n)
we find V w = 2 [1, 1] [2, 2] ... [m, m]
V i-
whence = II . f
A
and, in general, if V m> , = 2 [1, 1]' ... [m, m]',
J-
, >
S
the sign being that of ( 1) p.
(4) If, for the greatest common divisor S of m and n, we substitute any func-
tion whatever <p (S) of S, and denote by <E> (m) the function
Y 2
164 ON THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT.
we arrive at the identity
2 $ (1 , 1) $ (2, 2) . . . $ (m, m) = $ (1) x 0> (2) x <I> (3) x . . . x * (m).
Two particular cases are worth attention.
(a) Let s be an integral number, and let
<k(m) = l' + 2* + 3+...+ (m-l)' + m ! ,
so that, when s > 1,
t, (m ),^+l m . + S ,^ m .-,-B^-^-V m --* + ...,
B lt B 3i ... being the fractions of Bernoulli, and the last term being
t-3 g-2
(-l)^B._ 2 |m 2 , or (-1) 2 B t _ im>
according as s is uneven or even. Let also -^f, (m) be the sum of the powers * of
the numbers prime to m and not surpassing m ; we shall have
2^(l,l)</> 8 (2,2)...^(m ) 7^) = ^.(l)x^ g ( 2 )xv|'(3)x...x^ 8 (m ) 74
The forms of the functions ^ 8 ( m ) are deducible from the expression for
<j>,(m) (see a paper by Mr. Thacker in ' Crelle,' Vol. XL., p. 89) ; we thus find
^(m)=im^n(i-i), ft()-i.<*n(i-|)+iii(i-),
the general rule being that, in order to obtain ^,(m), we are to substitute
m fc ll(l p n - k ) for m k in the expression for < g (m).
(^) Let a- e (m) be the sum of the powers s of the divisors of m ; it will be
found that m , m
For, if m=pl 1 p? ..., and if we put
we have o"(w) = P! x P 2 x . . . ,
We thus obtain the equation
2o- i (l j l)<r.(2,2)<r,(3 ) 3)... < r 8 (m,m) = (1.2.3...m)',
OX THE VALUE OF A CERTAIN ARITHMETICAL DETERMINANT. 165
in which s may be any quantity whatever: the cases in which s = 0, s= 1,
s = + 1, are equally remarkable.
(5) Returning to the equation
we may observe that it is by no means necessary that the numbers 1, 2, 3, ..., m
should be the natural series of numbers. We may, in fact, take any different
numbers MI, M 2 > Mth*t we please, subject only to the condition that, if M be any
one of these numbers, every divisor of ft, must also appear among them, a con-
dition which implies that unity is always one of the numbers M. Subject to this
condition, we have always
2 (M I? MJ) 0* 2 , M 2 ) ... (M M , M m ) "= ^ (MI) x -^(MZ) ...-f (M),
or, more generally (see 4 supra),
2< (MI, Mi) < (M 2 , M 2 ) <}>(t*m, Mm) = $ (% , MJ) $ (M 2 , M 2 ) * (Mm > Mm)-
The most obvious cases are (a) when we reject the multiples of given primes ;
e.g., when the numbers M are the uneven numbers in their natural order ; (/3)
when we consider only numbers composed with given primes, e.g. when the
numbers are all the divisors of one of them M ; (7) when we consider only linear
numbers, i.e. numbers not divisible by any square. In all these cases the results
are immediately obtained by the methods which we have already used, and
which it is unnecessary to exemplify further.
(6) Lastly, the symbols M need not represent integral numbers at all, but may
be any quantities which admit of resolution into factors in a definite manner.
If, for example, a'' = x l ~ l or x>~ 1 according as i < j, orj < i, we have
XXXI.
ON THE PRESENT STATE AND PROSPECTS OF SOME
BRANCHES OF PURE MATHEMATICS.
[Proceedings of the London Mathematical Society, vol. viii. pp. 6-29. Bead November 9, 1876.]
_1_ HAVE been led to believe that the Society may not be unwilling to allow
a certain latitude in the scope of the remarks which they permit their Presidents
to address to them upon retiring from the Chair. Relying upon this belief,
I propose, on the present occasion, to invite your attention to some considerations
relating to the present state of Mathematical Science, with especial reference
to its cultivation in this country, and to our own position as representing a
great number of those who are interested in its advancement. The subject is
so extensive that I am sure you will excuse me if I endeavour to limit it in
every way I can. I propose, therefore, to exclude from what I have to say
all that relates to Applied Mathematics, and to ask you to confine your attention
to questions of Pure Mathematics only. I am well aware how much by this
exclusion I restrict the field before me ; but the restriction is forced upon me,
not only by the limit of time, but by the far narrower limits of my own know-
ledge. And I cannot help adding that I shall regard it as a fortunate circum-
stance, if the attention of my successor, when he in his turn is looking round
him for a subject for his own Presidential Address, should be attracted by a
domain, upon which I must myself decline to enter, but of which he, better
perhaps than anyone among us, is fitted to take a clear and comprehensive view.
The restriction which I have mentioned is far from being the only one
which I must impose upom myself. I can only presume to offer fragmentary
remarks upon great subjects, in the hope that even such casual and hasty notices
may not be without their use, if they serve to remind us of the vastness of our
science, and yet of its unity ; of its unceasing development, rapid at the present
ON SOME BRANCHES OF PURE MATHEMATICS. 167
time, and promising to be no less rapid in the immediate future ; of its marvellous
power of assimilating to itself the accessions which each year brings to our know-
ledge of external nature, while yet it derives strength and vitality from roots
which strike far back into the past, so that the organic continuity of its gigantic
growth has been preserved throughout.
In every science there is a time and place for general contemplations, as
well as for minute investigations. And it is a rule of sound philosophy that
neither should be neglected in its proper season. ' Itaque alternandae sunt istae
contemplationes,' says Lord Bacon, ' et vicissim sumendse, ut intellectus reddatur
simul penetrans et capax.' * Perhaps it is the besetting sin of mathematicians
to concentrate the mental vision upon as narrow and definite a field as possible.
And there is much to be said in excuse for our indulgence of this tendency. If
we are to find anything worth finding in the mines of mathematical research,
we must dig deep ; and if we want to dig deep, we must, if we are not gifted
with Herculean force, confine our efforts to a narrow superficial area. But the
tendency is not without its peril. The illustrious mathematician under whose
auspices this Society was founded, felt it right in his opening address to warn
us against the danger. ' Our subject,' said Augustus De Morgan, on the 15th of
January, 1865, witha characteristic irony of expression, ' Our subject is really rather
a wide one. But there are mathematical publications in which it is contracted ;
and it is often treated as a narrow subject.' He cautioned us against falling
into ' a line which may be useful, but which is still confined and partial ' ; and,
while exhorting us to do our part in the additions to the more rapidly developing
' branches of the science,' he bid us at the same time take care ' not to let any
one particular branch overgrow us.' It would not have been Augustus de Morgan
if he had not added some pointed criticisms upon examinations in general,
and on Cambridge examinations in particular, and if he had not cautioned
us against any excessive admiration for that part of mathematical ingenuity
which devotes itself to the narrowest of all the narrow fields ever chosen by
a mathematician, the invention and solution of ' ten-minute conundrums.'
It is now nearly twelve years since these warnings were given to the infant
Society by its first President ; and perhaps the time may have arrived when
we might put to ourselves the question whether its subsequent history has
shown that we have profited by the lessons of that eminent and large-minded
teacher. It would ill become me to attempt to answer such a question. I
* Novum Organon, Lib i., Aph.
168 ON THE PRESENT STATE AND PEOSPECTS OP
would only venture to express, and that with great diffidence, the double
opinion that, on the one hand, the mathematical world will wholly acquit the
Society of having devoted its energies to little or trivial subjects ; but that,
on the other hand, while it would be universally conceded that the volumes
of our Proceedings contain memorable additions to mathematical knowledge, it
might be alleged by an ' advocatus diaboli ' (if such a character should be assumed
by some severe critic) that we, in this respect resembling the other mathe-
maticians of our country, have shown, and still continue to show, a certain
partiality in favour of one or two great branches of the science, to the com-
parative neglect and possible disparagement of others. Perhaps it would be
well to begin our reply by denying the charge ; but, having done so, if we should
be advised to urge a second and somewhat contradictory plea, we might with
great plaiisibility rejoin that ours is not a blameable partiality, but a well-
grounded preference. So great (we might contend) have been the triumphs
achieved in recent times by that combination of the newer algebra with the
direct contemplation of space which constitutes the modern geometry so large
has been the portion of these triumphs which is due to the genius of a few great
English mathematicians so vast and so inviting has been the field thus thrown
open to research, that we do well to spend our time and our labour upon a
country which has, we might say, been ' prospected ' for us, and in which we
know beforehand that we cannot fail to obtain results which will repay our
trouble, rather than adventure ourselves into regions where, soon after the first
step, we should have no beaten tracks to guide us to the lucky spots, and where
the daily earnings of the searcher for mathematical treasure are (at the best) but
small, and do not always make a great show even after long years of work.
Such regions, however, there are in the domain of pure mathematics, and it
cannot be for the interest of science that they should be altogether neglected by
the rising generation of English mathematicians.
I propose therefore, in the first instance, to direct your attention to some
few of these by us comparatively neglected regions ; and foremost among them
I must name the Theory of Numbers. Of all branches of mathematical enquiry
this is the most remote from practical applications ; and yet, more perhaps than
any other, it has kindled an extraordinary enthusiasm in the minds of some of the
greatest mathematicians. We have the examples of Fermat, Euler, Lagrange,
Legendre, and Gauss, of Cauchy, Jacobi, Lejeune Dirichlet, and Eisenstein,
without mentioning the names of others who have passed away, and of a few
who are still living. But, somehow, the practical genius of the English mathe-
SOME BRANCHES OP PURE MATHEMATICS. 169
matician has in general given a different direction to his pursuits ; and it would
sometimes seem as if we in England measured the importance of the subject
by what we find of it in our text-books of Algebra, or as if we regarded its
enquiries as problems of mere curiosity, without a wider scope, and without
direct bearing on other branches of mathematics. I might endeavour to remove
this impression if indeed it exists in the minds of any of those who hear me by
enumerating instances in which the advancement of Algebra and of the Integral
Calculus appears to depend on the progress of the arithmetic of whole numbers.
But, instead of wearying you with the details which would be necessary to make
such an enumeration intelligible, I would rather ask you to listen to what is
recorded of the most eminent master of this branch of science. ' Gauss,' we are
told by his biographer, ' held Mathematics to be the Queen of the Sciences, and
Arithmetic to be the Queen of Mathematics.' ' She sometimes condescends ' so
spoke the great Astronomer and Physicist 'to render services to Astronomy
and the other natural sciences, but under all circumstances the first place is her
due.' * In a more serious mood he wrote, ' The higher arithmetic presents us
with an inexhaustible storehouse of interesting truths of truths, too, which
are not isolated, but stand in the closest relation to one another, and between
which, with each successive advance of the science, we continually discover new
and sometimes wholly unexpected points of contact. A great part of the theories
of Arithmetic derive an additional charm from the peculiarity that we easily
arrive by induction at important propositions, which have the stamp of simplicity
upon them, but the demonstration of which lies so deep as not to be discovered
until after many fruitless efforts ; and even then it is obtained by some tedious
and artificial process, while the simpler methods of proof long remain hidden
from us.' f Or, again, let the young mathematician, who feels an instinctive
liking for arithmetical enquiry, be encouraged by the observation which has been
put on record by Jacobi in his brief notice of the life of Gopel, that many of
those who have a natural turn for mathematical speculation find themselves
in the first instance attracted by the Theory of Numbers. J
There are three great departments of arithmetic, not, it must be admitted,
wholly separable from one another, which seem to me at the present time to
offer a very inviting field to the researches of the mathematician. Of these
* 'Gauss. Zum Gedachtniss.' Von W. Sartorius v. "Walter shausen. Leipzig, 1856.
t Preface to Eisenstein's ' Mathematische Abhandlungen.' Berlin, 1849.
J Notiz iiber A. Gopel, ' Crelle's Journal,' vol. xxxv. p. 313.
VOL II. Z
170 ON THE PRESENT STATE AND PROSPECTS OF
I will name first the arithmetical theory of homogeneous forms, or quantics,
as we in England have now learned to call them. It is worthy of remembrance
that some of the most fruitful conceptions of modern algebra had their origin in
arithmetic, and not in geometry or even in the theory of equations. The charac-
teristic properties of an invariant, and of a contravariant, appear with distinct-
ness for the first time in the ' Disquisitiones Arithmeticae ' ; * and in that treatise
also the attention of mathematicians was for the first time directed to the study
of quantics of any order and of any number of indeterminates. t Again, Eisen-
stein in the course of his researches on the arithmetical theory of binary cubic
forms was led to the discovery of the first covariant ever considered in analysis
the Hessian of the cubic form.J But the progress of modern algebra and of
modern geometry has far outstripped the progress of arithmetic, and one great
problem which arithmeticians have before them at the present time is to endea-
vour to turn to account for their own science the great results which have been
obtained in the sister sciences. How difficult this problem may prove is perhaps
best attested by the little progress that has been made towards its complete
solution. One or two instances may serve to illustrate the actual position of the
enquiry. The algebraical problem of the automorphics of a quadratic form,
containing any number of indeterminates, may be regarded as completely solved
by a formula due to M. Hermite and Professor Cayley. The arithmetical formula
* ' Disq. Arith.,' Arts. 157, 267, 268, where binary and ternary quadratic forms are considered.
t ' Sed manifesto hoc argumentum ' [the theory of binary quadratic forms] ' tanquam sectionem
maxinie particularem disquisitionis generalissimse de functionibus algebraicis rationalibus iutegris
homogeneis plurium indeterminatarum et plurium dimensionum considerare . . . possumus.' ' Sufficiat
hunc campum vastissimum geometrarum attentioni commendavisse, in quo materiem ingentem vires
suas exercendi, arithmeticamque sublimiorem egregiis incrernentis augendi invenient.' (' Disq. Arith.,'
Art. 266.)
J See ' Crelle's Journal,' vol. xxvii. p. 89. It is remarkable that the cubic covariaut does not
explicitly appear in this paper (December, 1843) or in the note (p. 105) which immediately follows it.
This omission is supplied in a subsequent note, dated March 3, 1844 (ibid. p. 319). The earliest
papers of Boole, who approached the study of linear transformations from a geometrical point of view,
belong to the years 1841 and 1843 (Cambridge Mathematical Journal, vol. ii. p. 64 and vol. iii. p. 1
and p. 106). In these papers (of which perhaps only the first should be cited here) covariants do not
appear, but the first general theorem of iuvariance ever enunciated, the theorem of the invariauce of the
discriminant of any quartic, is distinctly stated and proved.
M. Hermite, in ' Crelle's Journal,' vol. xlvii. p. 309, appears to have considered forms of three
indeterminates only ; his solution was subsequently generalised by Professor Cayley (ibid. vol. i. p. 288).
See also a later memoir by Professor Cayley ' On the Automorphic Linear Transformation of a Bipartite
Quadric Function,' in the 'Philosophical Transactions' for 1858.
SOME BRANCHES OF PURE MATHEMATICS. 171
which gives the automorphics of a binary quadratic form has long formed a part
of the elements of the Theory of Numbers ; and the corresponding investigation
for an indefinite ternary quadratic form may be now regarded as completed by
the memoirs in which M. Paul Bachmann has followed up the earlier researches
of M. Hermite.* Again, the problem of the equivalence of two positive or definite
ternary quadratic forms was completely solved by Seeber ; and the problem of
the arithmetical automorphics of such forms, by Eisenstein. f The corresponding
but far more difficult problem of equivalence for indefinite ternary forms has
received its first solution only in very recent times from M. Eduard Selling ; J
and perhaps it is not too much to hope that these profound researches may receive
some further development from their distinguished author, and may be brought
into closer relation with other parts of arithmetical and algebraical theory. So
far, then, as binary and ternary quadratic forms are concerned, we have not much
reason to complain of the slowness of the advances made by arithmetic. But if
we pass to quadratic forms of four or more indeterminates, we shall find that the
limits within which our arithmetical knowledge is confined are indeed restricted.
The fundamental theorem of M. Hermite, that the number of non-equivalent
classes of quadratic forms having integral coefficients and a given discriminant is
finite, and the recent researches of M. Zolotareff and Korkine on the minima of
positive quadratic forms, mark the extremest limit to which enquiry has been
pressed in this direction. $
' The solution of the problem is made to depend on the solution in integral numbers of the
indeterminate equation 2>' + f(q l , q^, q^)=l, where F is the contravariant of the given ternary form.
No general method, however, of obtaining the solutions of this equation has as yet been given. (See
M. Hermite in the memoir already cited, ' Crelle,' vol. xlvii. p. 307 sqq. ; M. Bachmann, ' Borchardt,'
vol. Ixxi. p. 296, and vol. Ixxvi. p. 331, together with the note by M. Hermite completing his former
solution, ibid. vol. Ixxviii. p. 325.)
t See L. Seeber, ' Untersuchungen ueber die Eigenschaften der positiven ternaren quadratischen
Formen,' Freiburg, 1831 ; and, in connection with this work, the review of it by Gauss (in the Got-
tingen 'Gelehrte Anzeige' for 1831; or in 'Crelle,' vol. xx. p. 312; or in the collected edition of
Gauss' Works, vol. ii. p. 188), and the subsequent and simpler investigations of Dirichlet ('Crelle/
vol. xl. p. 209), and M. Hermite (ibid. p. 173). The theory of the automorphics of positive ternary
quadratic forms is given by Eisenstein in the Appendix to his ' Table of Reduced Positive Ternary
Quadratic Forms.' (' Crelle/ vol. xli. p. 227.) He observes (see the note at p. 230) that Seeber, without
actually solving the problem, had come extremely near to its solution.
\ ' Borchardt's Journal/ vol. Ixxvii. p. 143.
See the letters of M. Hermite to Jacobi ('Crelle/ vol. xl. p. 261, sqq.), and the papers of M.
Zolotareff and Korkine (' Clebsch/ vol. v. p. 581, and vol. vi. p. 366). I may perhaps also be allowed
to refer to my own papers ' On the Orders and Genera of Quadratic Forms containing more than three
Indeterminates/ in the Proceedings of the Koyal Society, vol. xiii. p. 199, and vol. xv. p. 387.
Z 2
172 ON THE PRESENT STATE AND PROSPECTS OF
As a second and much simpler instance of the difficulties which remain for
arithmetic after the work of algebra is done, let us consider the system of two
binary quadratic forms. The first question that we naturally ask, is, what is the
arithmetical meaning of the evanescence of their joint invariant ? I gave myself
an answer to this question some years ago in the following theorem, which for
brevity I express in the proper technical language.
' If the joint invariant of two properly primitive forms vanishes, the deter-
minant of either of them is represented primitively by the duplicate of the
other.' *
This theorem is very far from exhausting the subject to which it refers. But
it may serve as a fair illustration of the class of enquiries which I wish to propose
to the attention of the Society as likely to be not unfruitful. The geometrical
interpretation of the invariantive character to which the theorem relates is (as
we all know) that the two pairs of elements, represented by the two quantics,
are harmonically conjugate ; and I think it especially deserving of notice that
the same invariantive character has an important meaning both in arithmetic
and in geometry, but that neither of the two interpretations seems in the least
likely to suggest the other. If we pass on to the case of two ternary quadratic
forms, the geometrical signification of the evanescence of either of their joint
invariants is now embodied in well-known elementary theorems ; but I do not
think that any answer has been given to the corresponding arithmetical question,
nor indeed do I know that anyone has occupied himself with it. I would, how-
ever, venture to hazard a conjecture that the arithmetical interpretation of these
invariantive conditions may have an important bearing on the researches of M.
Selling, to which I have already referred.
I do not wish to weary the Society with too many particular examples ; but
I will venture to point to one more instance from which it would appear that
modern geometrical and analytical conceptions may help us a little, if only a
little, on our way in the trackless wilds of arithmetic. Let us take a question
which has some relation to the familiar notions of ' unicursality ' and ' one-to-one
correspondence.' It is an old theorem, that if the homogeneous indeterminate
equation of the second degree containing three variables admits of one solution,
it admits of an infinite number ; and there is a Memoir of Cauchy f showing how
* Report on the Theory of Numbers in the Keports of the British Association for 1863, p. 783,
Art. 123 [vol. i. p. 284].
t 'Exercices de Math6matiques,' vol. i. p. 233. Cauchy considers the ternary cubic as well as the
ternary quadratic equation in this memoir.
SOME BRANCHES OF PURE MATHEMATICS. 173
from one given solution all the solutions are to be derived. But here two things
deserve our notice (1) that no geometry (so far as I am -aware) helps us in any
way to decide whether the given equation does or does not admit of solution.
The criterion turns on the definition (first given by Eisenstein) of the generic
characters of ternary quadratic forms a definition which itself depends on a
simple arithmetical inference from the algebra of such forms.* But (2), in strong
contrast to what I have just stated, when once we have a single solution, the rest
is a matter of intuitive geometry. Our equation represents a conic ; the conic is
real, because we have a single rational point on it ; every rational line drawn
through this point meets the conic again in a rational point ; and thus the uni-
cursality of the conic enables us to deduce from any one solution all the solutions
which exist, and, what is more, to obtain them all in a natural sequence. To pass
to a question of a somewhat higher order of difficulty, there is no known criterion
(so far as I am aware) for deciding whether a ternary homogeneous cubic equa-
tion does or does not admit of solution in integral numbers. Such a criterion
would be of great interest, and ought not, one would suppose, to lie beyond the
present scope of analysis, f But here again geometry shows us at once, that if
we have one solution, we have, in general, an infinite number. For the tangential
of a rational point on a rational cubic curve is itself a rational point, and the line
joining two given rational points meets the cubic in a third rational point.
(There are, of course, cases of exception to this mode of derivation of one integral
solution from another, but I need not advert to them here.) Advancing a little
further, we have not to look very far into the connexion which modern algebra
has established between ternary cubic and binary quadratic forms in order to
satisfy ourselves that the Diophantine problem of rendering a biquadratic expres-
sion a perfect square (a problem which has been the subject of numerous researches
ever since the time of Euler) is the same as the problem of finding rational points
upon a cubic curve of which the equation is rational ; and that, in particular, the
tangential method to which I have just referred enables us in general to deduce an
infinite number of solutions of the Diophantine problem from any given solution.
A second department of Arithmetic which, as it seems to me, has in quite
recent times received less attention than it deserves, is the Theory of Congru-
* See Eisenstein, in ' Crelle,' vol. xxxv. p. 117 ; and a note of my own in the Proceedings of the
Royal Society for 1864, vol. xiii. p. 110 [No. XIII. vol. i. p. 410].
t Special cases of the equation here considered have attracted much attention. It will be suf-
ficient to mention Fermat's theorem of the impossibility of the equation a 3 + y* + = 0.
174 ON THE PRESENT STATE AND PROSPECTS OF
ences. Some time before the notation of congruences had been introduced into
the Theory of Numbers by Gauss, Lagrange, to whom the conception of a con-
gruence (apart from any special notation) was perfectly familiar, had established
the elementary theorem : *
f (x\
' If an expression of the form 44-r , where f(x) and < (x) are rational func-
9 ( x )
tions of x having integral coefficients, acquires an integral value for any given
integral value of x, the value of <p (x) must be a divisor of the resultant of f(x)
and (f> (x).'
This theorem naturally suggests another which was subsequently given by
Cauchy : f
' If two congruences which have the same modulus admit of a common solu-
tion, the modulus is a divisor of their resultant.'
These propositions suggest the possibility of transferring to the Theory of
Numbers some at least of the results which have been obtained by modern
researches in the theory of algebraical elimination. For example, we are led to
consider the problem :
' Given two congruences, to find the number of their common roots when we
take in succession for modulus each divisor of their resultant.'
In all such inquiries we shall find that the considerations which suffice for
the solution of the algebraical problem enter as indispensable elements into the
arithmetical investigation, but that this investigation compels us to take notice
of other elements also, with which, in algebra, we were not concerned. Thus, in
the solution of the problem which I have mentioned, and which I hope at some
future day to bring more fully under the notice of the Society, we should have
not only to consider in a general manner the system of the divisors of the result-
ant itself, but we should also have to distinguish, in that system of numbers,
those which are at the same time common divisors of certain systems of minors
in the dialytic matrix of which the resultant is the determinant.
If I may be allowed to regard the subject of complex numbers as belonging
to the theory of congruences, I must also be allowed to modify to a certain
' The ' Disquisitiones Arithmetic ' were published in 1801. The Memoir of Lagrange, entitled
' Nouvelle m6thode pour r^soudre les Problemes indetermins,' appeared in the Transactions of the
Academy of Berlin for 1768. See also paragraph 4 (p. 528) of his Additions to the French translation
of the Algebra of Euler (Lyons, 1774, and often reprinted since) ; the reference here is to the edition
of the ' an iii de 1'ere r^publicaine.'
t 'Exercices de Mathcmatiques,' vol. i. p. 164.
SOME BRANCHES OP PURE MATHEMATICS. 175
extent what I have said as to the indifference with which that theory has been
looked upon in very recent times ; for there is no reason to complain that com-
plex numbers have received insufficient attention, at least from the mathema-
ticians of the Continent. Without referring here to the results obtained by
Lejeune Dirichlet, or to the splendid series of researches upon the complex num-
bers formed with roots of unity which we owe to M. Kummer, we may notice
that the general theory has attracted the attention of M. Kronecker, whose
investigations relating to it have unfortunately not as yet been published, and
are only known from the application which he has made of them to the equations
which present themselves in the Theory of Elliptic Functions."* In a Supplement,
added to the second edition of Lejeune Dirichlet's Lectures on the Theory of
Numbers, M. Dedekind has given the outlines of a complete and very original
theory of complex numbers, in which he has to a certain extent deviated from
the course pursued by M. Kummer, and has avoided the introduction (at least in
a formal manner) of ideal numbers, t I may remind my hearers that the inap-
plicability, in general, of Euclid's theory of the greatest common divisor to com-
plex numbers formed with the roots of equations having integral coefficients
renders it impossible to define the prime factors of such numbers in the same
way in which we can define the prime factors of common integers, or of complex
numbers of the form a + &^/ 1 ; and that, in the effort to overcome the difficulty
thus arising, M. Kummer was led to introduce into arithmetic an entirely new
and very important conception that of ideal numbers. I shall ask leave to
mention a very recent and very interesting application of this conception which
has been made by M. Zolotareff to the solution of a problem of the Integral
Calculus, which was first attempted by Abel, and has since attracted the atten-
tion of MM. Teh eby chef and Weierstrass. We owe to Abel the remarkable
theorem that the differential expression
du= (x + \)dx
V (x* + ax 3 + bx* + ex + d)
can or cannot be integrated by logarithms, for some value of the parameter X,
according as the radical can or cannot be developed in a periodic continued
* See the ' Monatsberichte ' of the Academy of Berlin for June 26, 1862, p. 370. M. Dedekind
also refers to the investigations of M. Kronecker at p. viii. of his Preface to the second edition of
Dirichlet's Lectures, presently to be noticed.
t M. Dedekind has also commenced the publication of a resume of his theory in the ' Bulletin des
Sciences Muth&natiques et Afctronomiques ' for December, 1876.
176 ON THE PRESENT STATE AND PROSPECTS OF
fraction. But Abel gave no criterion for deciding whether the development is
or is not periodic a question which obviously cannot be decided by mere trial.
The problem was first solved by M. Tche'bychef for the case in which the coeffi-
cients a, b, c, d are rational ; and the complete solution has at last been obtained
by M. Zolotareff, with the help of a new theory of complex ideal numbers.*
The last part of arithmetical theory to which I would wish to direct the
attention of some of the younger mathematicians of this country is the determin-
ation of the mean or asymptotic values of arithmetical functions. This is a field
of enquiry which presents great difficulties of its own ; and it is certainly one in
which the investigator will not find himself incommoded by the number of his
fellow-workers. 'Vix ullus reperietur geometra,' wrote Eulerf in the last
century, 'qui non, ordinem numerorum primorum investigando, haud parum
temporis inutiliter consumpserit.' But I do not think that, as a rule, the
* The Memoir of Abel, ' Sur 1'integration de la fonnule differentielle -- -^ , R et p elant des
V K
fonctions entieres,' will be found in ' Crelle,' vol. i. p. 1 85, or ' (Euvres Completes," vol. i. p. 33. In
this memoir Abel demonstrates the general theorem that, if R be a rational and integral function of x of
any order, such that V R can be expressed by a symmetrical and periodic continued fraction of the form
'
it is always possible to find a rational and integral function p of x such that
y being a rational function of x; and that, conversely, if the equation (2) can be satisfied, the
development of V R is of the form (1). Abel further shows that y is always the convergent
[r, 2/u, 2/x,, ..., 2/Xj, 2/x]; and he gives the application of his theorem to the case in which S is a
biquadratic function. The memoir of M. Tchebychef was first published in the ' Bulletin de 1' Academic
de St. Petersbourg' for 1860 (torn. iii. pp. 1-12), and has been reprinted in the 'Melanges Mathc'ma-
tiques et Astronomiques de St. Petersbourg' (torn. iii. pp. 182-192), and also in 'Liouville's Journal'
(Second Series, 1864), vol. ix. p. 225. M. Tchtbychef has given his method without demonstration;
this was first supplied by M. Zolotareff in a paper, ' Sur la methode d'integration de M. Tchebychef '
(' Mathematische Annalen,' vol. v. p. 560). The complete solution of the problem is contained in a
work by M. Zolotareff (' Theorie des Nombres entiers complexes,' St. Petersbourg,) which I have not
yet seen. The result obtained by M. Weierstrass (however remarkable in itself), that the integrability
by logarithms of the differential du depends on the possibility of expressing a certain constant in the
form aK + ifiK', where a and /3 are rational numbers, and K and K' are certain complete elliptic
integrals, does not supply an available criterion. (See the ' Fortschritte der Mathematik.' vol. vi.
p. 118.)
t Dec. 1, 1760, ' Novi Commentarii Petropolitani,' vol. ix. p. 99; or, ' Commentationes Arith-
metic Collects' ('Petropoli, 1849), vol. i. p. 356.
SOME BRANCHES OP PURE MATHEMATICS. 177
mathematicians of the present day have any reason to reproach themselves on
this score, or stand in any need of the apology which Euler proceeds to deliver.
Nevertheless, much has been done in this direction since the days of Euler ;
enough, certainly, to give abundant encouragement to further inquiry. The first
asymptotic results that were obtained are due to Gauss, and are given without
demonstration in the ' Disquisitiones Arithmetics ' ; * they relate to the average
number of classes of binary quadratic forms of a positive or negative determinant.
The general principles on which such inquiries depend were laid down by Lejeune
Dirichlet, forty-eight years later, in a memoir entitled ' Ueber die Bestimmung
der mittleren Werthe in der Zahlentheorie,' and inserted in the Transactions of
the Berlin Academy for 1849. The subject has recently been resumed by M.
Mertens of Cracow, who, in an interesting memoir (' Borchardt's Journal,' vol.
Ixxvii. p. 289), has determined the asymptotic values of several numerical
functions. In particular, he has demonstrated the expression given by Gauss
for the mean value of the number of quadratic forms of a negative determinant ;
this mean value being, in the vicinity of n, when n is a considerable number, the
quotient obtained by dividing f Vn by the sum of the cubes of the reciprocals of
the natural numbers.
As to our knowledge of the series of the prime numbers themselves, the
advance since the time of Euler has been great, if we think of the difficulty
of the problem ; but very small if we compare what has been done with what
still remains to do. We may mention, in the first place, the undemonstrated,
and indeed conjectural, theorems of Gauss and Legendre as to the asymptotic
value of the number of primes inferior to a given limit x (the former theorem
r x dx
assigns for this value the integral logarithm Li (x) = I r , the latter an ex-
pression of the form y ).f But these theorems are only approximately
* See Arts. 302, 304, and the Aclditamenta to Art. 306, X; also various passages in the post-
humous fragments, ' De nexu inter multitudinem classium,' &c. (Gauss, ' Werke,' vol. ii. pp. 269-303.)
t See Legendre, 'Theorie des Nombres' (Paris, 1830), vol. ii. p. 65 : Gauss to Encke (Dec. '24,
1849), in Gauss's Collected Works, vol. ii. p. 444. Legendre assigns to a the conjectural value 1 '08366.
Gauss compares his own formula and that of Legendre with the results of actual enumerations of the
prime numbers, and finds that, as far as the end of the third million, the comparison is in favour of the
formula of Legendre, but that the error of that formula shows a tendency to increase more rapidly than
the error of his own. He also observes that the mean value of the constant a tends to decrease, and
that its true limit may possibly be 1, or may differ from 1 by a quantity of the order - . Enckc
VOL. II. A a
178 NOTE ON THE PRESENT STATE AND PROSPECTS OF
consistent with one another, and are perhaps still less approximately true. The
memoir of Bernhard Riemann, 'Ueber die Anzahl der Primzahlen unter einer
gegebenen Grosse,' contains (so far as I am aware) the only investigation of the
asymptotic frequency of the primes which can be regarded as rigorous.* He
shows that, if F (x) be the number of primes inferior to x, there exists an ana-
lytical expression for the series
which consists (1) of a term which does not increase without limit with x ; (2) of
a non-periodic term Li (a;) ; (3) of an infinite series of periodic terms of the type
IA(xl~ ai ) Li(o^ + '), the constants a being the roots (infinite in number) of a
certain transcendental equation. It follows that the non-periodic part of the
expression for F(x) is of the type
Li (x) - 1 Li (scl) - 1 Li (a;*) - Li (a; *) + Li (*) - } Li (z*) + . . . ;
and thus the equation of Gauss, F(x) =Li(a;), can, roughly speaking, be correct
only as far as quantities of the order x' 2 .
No less important than the investigation of Rlemann, but approaching the
problem of the asymptotic law of the series of primes from a different side, is the
celebrated memoir, ' Sur les Nombres Premiers,' by M. Tche'bychef,t in which he
has established the existence of limits within which the sum of the logarithms
of the primes P inferior to a given number x must be comprised. The limits
assigned by M. Tche"bychef are not very close, not even close enough to determine
^ 1 D
the asymptotic value of the quotient - , to which the value 1 has been
fiC
conjecturally assigned. But the theorem of M. Tche"bychef may be, perhaps, said
to mark the furthest point to which our knowledge of the series of prime num-
bers has yet been carried ; and while it is truly remarkable that, in a matter of
so much difficulty, a process so apparently simple as that which he has employed
X
had communicated to Gauss a formula of his own, r x 10 21ugl , which, as Gauss observes, may, for
log a;
very great values of x. be regarded as coinciding with . . . . (See the letter cited, and also
log x J log 10
the note of M. Schering, ibid. p. 521.)
* See the ' Monatsberichte ' of the Academy of Berlin for November, 1859; or Riemann's .
' Mathematische Werke,' p. 137. In the ' Annali di Matematica,' torn. iii. p. 52 (1860), M. Genocchi
has given a very interesting account of the method of Riemann, and has arrived at a result differing in
one respect (see p. 58) from that of Riemann.
t 'Liouville,' First Series, vol. xvii. p. 366. The memoir was presented to the Academy of St.
Petersburg in 1850.
SOME BRANCHES OP PURE MATHEMATICS. 179
should be capable of leading to a result of so much interest and importance, it is
somewhat disappointing to find that this method, even in the hands of its emi-
nent inventor, should seem incapable of being pursued further, and unlikely to
furnish any nearer approximation to the truth. The method of M. Tchebychef,
profound and inimitable as it is, is in point of fact of a very elementary character,
and in this respect contrasts strongly with that of Eiemann, which depends
throughout on very abstruse theorems of the Integral Calculus.
I do not know that the great achievements of such men as Tchebychef and
Eiemann can fairly be cited to encourage less highly gifted investigators ; but at
least they may serve to show two things first, that nature has placed no insu-
perable barrier against the further advance of mathematical science in this
direction ; and, secondly, that the boundaries of our present knowledge lie so
close at hand that the inquirer has no very long journey to take before he finds
himself in the unknown land. It is this peculiarity, perhaps, which gives such
perpetual freshness to the higher arithmetic. It is one of the oldest branches,
perhaps the very oldest branch, of human knowledge ; and yet some of its most
abstruse secrets lie close to its tritest truths. I do not know that a more strik-
ing example of this could be found than that which is furnished by the theorem
of M. Tche"bychef. To understand his demonstration requires only such algebra
and arithmetic as are at the command of many a schoolboy ; and the method
itself might have been invented by a schoolboy, if there were again a schoolboy
with such an early maturity of genius as characterised Pascal, Gauss, or Evariste
Galois.*
* In addition to the memoirs to which reference has been made in the text, we may also mention
the following: (1) M. Tchebychef, 'Sur la totality des nombres premiers inf6rieurs a une limite
donneV (' Liouville,' 1st Series, vol. xvii. p. 341). In this paper (which was presented to the Academy
of St. Petersburg in 1848) M. Tchebychef proves (among other things) that, if the expression
-p r log a; has a limit at all, the value of that limit must be 1. This result shows that in the
approximate formula of Legendre, F(x) = , we ought to take a = + 1. (2) A paper by the
late Judge Hargreave, in the ' Philosophical Magazine ' for 1849 (vol. xxxv. p. 36), in which it is shown
(but not by any very rigorous demonstration) that the average interval between two consecutive primes
in the vicinity of any very great number x is log x ; a result at which Gauss had arrived while still a
boy, as may be inferred from his letter to Encke, quoted above. (3) A paper in the ' Mathematische
Annalen,' vol. ii. p. 636, in which the author, M. Meissel, by the aid of a method suggested by those
employed by Legendre (' Theorie des Nombres,' vol. ii. p. 86), obtains a formula which greatly facilitates
the determination of the number of prime numbers contained between given limits ; he has thus found
that the number of primes in the first ten millions is 664579: and in a later note ('Mathematische
Annalen,' vol. iii. p. 523) that the number of primes in the first hundred millions is 5,761,460. (4) A
A a 2
180 NOTE ON THE PRESENT STATE AND PROSPECTS OF
I pass on to speak of some other branches of analysis which appear to me at
the present moment to promise much in the immediate future.
I will first refer to one or two points to which the transition from the arith-
metic of whole numbers is easy and natural.
We owe to Jacobi the first suggestion of a method of approximation which
forms a natural extension of the theory of continued fractions, but which still
remains in an incomplete condition. In the memoir ' De functionibus duorum
variabilium quadrupliciter periodicis,' (' Crelle,' vol. xiii. p. 55,) Jacobi demonstrated
the theorem that an uniform function of a single variable can at most be doubly
periodic ; and that, if it be doubly periodic, the ratio of the two periods is neces-
sarily imaginary. He effected this by proving that, if aa'a", W b" are inde-
pendent irrational quantities, it is always possible to find integral numbers
mm'm" such that the value of each of the two expressions
ma + ma + m"a",
mb +m'b' + m"b",
shall be less than any quantity that can be assigned.
This idea of Jacobi was subsequently further developed by M. Hermite, who
showed its connexion with the theory of the reduction of quadratic forms (see
his letters to Jacobi in ' Crelle's Journal,' vol. xl. p. 261 sqq.). The same con-
ception lies at the basis of Lejeune-Dirichlet's researches on complex units, and
led him to his celebrated generalisation of the theory of the Pellian Equation.*
Since the death of Jacobi, a memoir of his (apparently left incomplete) has
been published in ' Borchardt's Journal ' (vol. Ixix. p. 29), in which he examines
the relations between the successive sets of integral numbers x , x ly x 2 , which
render an expression such as
X + X l W l + X 2 ft> 2
(where Wj and 2 are irrational quantities) approximately equal to zero. He
note by Mr. J. W. L. Glaisher in the Report of the British Association for 1872 (' Transactions of the
Sections," p. 19), in which the results of some enumerations of the primes are given, and are compared
with Mr. Hargreave's theorem as to their average frequency. (5) A paper by M. Mertens (' Borchardt's
Journal,' vol. Ixxviii. p. 46), in which he determines the asymptotic values of the functions 2 -p-
Ft (l p-) . These functions had been already considered by Legendre (' Th6orie des Norabres,' vol. ii.
p. 67), and by M. Tcheliychef in the memoir already cited in this note : but M. Mertens obtains a more
precise result by more rigorous reasoning. (6) A preliminary note on an enumeration of primes, by
Mr. Glaisher, in the Proceedings of the Cambridge Philosophical Society, Dec. 4, 1876.
* See the ' Monatsberichte ' of the Academy of Berlin for 1842, p. 95, and for 1846, p. 105.
SOME BRANCHES OF PURE MATHEMATICS. 181
applies the theory to the examples ! = 2*, =3*, =5*; &> 2 = 2 S , =3^, = 5 5 , and
finds that in each case the development is periodic ; but he appears not to have
obtained any demonstration of the general theorem that the corresponding de-
velopment in the case of the root of any cubic equation having integral coeffi-
cients is always periodic.* These unfinished researches of Jacobi, to which M.
Borchardt has called the special attention of mathematicians (in the preface to
the 68th volume of his Journal) have been resumed by M. Bachmann, f and still
more recently, though from a somewhat different point of view, by M. Flirstenau.
The latter of these mathematicians defines a continued fraction of the second
order to be a continued fraction in which each element is itself a continued
fraction ; and, availing himself of this definition, he has succeeded in showing
that we can always approximate to the real root of an equation of the order n,
having integral coefficients, by means of a periodic continued fraction of the order
n l.J It is evident that the discovery of such a mode of approximation to the
root of an equation may lead to theoretical considerations of great interest,
though it is hardly likely that the method itself will be found practically useful.
Closely allied to the investigation of new methods of approximation is the
problem of determining the arithmetical or transcendental character of irrational
quantities. It was first shown by M. Liouville that irrational quantities exist
which cannot be the roots of any equation whatever having integral coefficients ;
a proposition which certainly required the proof which it has received from him,
although it might easily seem incredible CL priori that such irrational quantities
should not exist. We may, perhaps, be allowed to designate by the terms arith-
metical and transcendental the two classes of irrational quantities between which
the theorem of M. Liouville has taught us to distinguish ; and it becomes a
problem of great interest to decide to which of these two classes we are to assign
the irrational numbers, such as e and TT, which have acquired a fundamental im-
portance in analysis. To Lambert, the eminent Berlin mathematician of the last
century, the first great step in this direction is due. He showed that neither v
nor 7T 2 is rational ; with regard to e he was even more successful, for he was able
* This theorem had been already obtained by M. Hennite. See the Letters already cited, ' Crelle,'
vol. xl. pp. 286-289.
t ' Borchardt's Journal,' vol. Ixxv. p. 25.
\ E. Fiirstenau, ' Ueber Kettenbriicke hb'herer Ordnung,' "Wiesbaden. I regret to say that I only
know this work from the notice in the ' Jahrbuch fiber die Fortschritte der Mathematik' for 1874.
'Comptes Eendus,' vol. xviii. (1844), p. 883, and p. 910; reproduced with additions in ' Liou-
ville' s Journal ' (1st series), vol. xvi. p. 133.
182 NOTE ON THE PRESENT STATE AND PROSPECTS OF
to prove that no power of e, of which the exponent is rational, can itself be
rational. * There (with one slight exception) the question remained for more
than a century ; and it was reserved for M. Hermite, in the year 1873, to com-
plete by a singularly profound and beautiful analysis, the exponential theorem of
Lambert, and to prove that the base of the Napierian logarithms is a transcen-
dental irrational, t But, in the letter to M. Borchardt already cited, M. Hermite
declines to enter on a similar research with regard to the number IT. ' Je ne me
hasarderai point,' he says, ' a la recherche d'une demonstration de la transcendance
du nombre IT. Que d'autres tentent 1'entreprise ; nul ne sera plus heureux que
moi de leur succes ; mais croyez m'en, mon cher ami, il ne laissera pas que de
leur en couter quelques efforts.' It is a little mortifying to the pride which
mathematicians naturally feel in the advance of their science to find that no
progress should have been made for one hundred years and more toward answer-
* 'Memoire sur quelques proprieies remarquables des quantites transcendantes circulates et
logarithmiques,' in the ' Memoires de I'Acad&nie des Sciences de Berlin" for 1761, p. 265; the
demonstration depends on the continued fraction
e*- e -* x a? y? a?
" 1+ 3+ 5+ 7 + ...
A different method of proving the incommensurability of e (depending on the exponential series) has
found its way into many elementary treatises ; it would seem that this simple method cannot be applied
to prove the more general proposition that e" is incommensurable; but it has been successfully em-
ployed by M. Liouville (' Liouville's Journal,' 1st series, vol. v. p. 192) to show that neither e nor e 2 can
be the root of a rational quadratic equation. This result forms the only extension which the theorem
of Lambert had received up to the date of the memoir of M. Hermite, ' Sur la fonction exponentielle,'
to which we shall presently refer. The only elementary work in which (so far as I know) the incom-
nt
mensurability of e* is demonstrated, is Mr. Todhunter's 'Algebra,' ed. 5, p. 530. The theorems as to
the incommensurability of it and ir 2 are excluded from English text-books ; the only exception that
occurs to me being Sir David Brewster's English edition (Edinburgh, 1824) of the Geometry of
Legendre, where Legendre's reproduction of the demonstration of Lambert is given in Note iv. p. 239.
The exclusion of these theorems is a matter of regret ; for they constitute the only ' short method with
the circle- squarers ' ; and perhaps the extraordinary prevalence within the United Kingdom of the form
of delusion known as circle-squaring may partly arise from the appearance of an ' ipsi dixerunt ' on the
part of the mathematicians, which is certainly suggested by the omission in elementary works of any
rigorous demonstration of the irrationality of w. M. Hermite has given a demonstration of the irration-
ality of it and n 2 , which is very beautiful and entirely different from that of Lambert (Letter to M.
Borchardt in ' Borchardt's Journal,' vol. Ixxvi. p. 342) ; with this single exception, the theory of the
quadrature of the circle rests to-day where Lambert left it in the year 1761.
* See the Memoir ' Sur la fonction exponentielle,' already cited in the preceding note, ' Comptes
Rendus,' vol. Ixxvii. pp. 18 etc.; and also published separately by Gauthier-Villars, 1874.
SOME BRANCHES OF PUEE MATHEMATICS. 183
ing the last question which still remains to be answered with regard to the
quadrature and rectification of the circle. But mathematical discovery is like
electricity ; it follows the lines of least resistance ; and an adherence to the rule
which this analogy suggests is certainly conducive to the comfort of the indi-
vidual mathematician, and is probably also, in the long run, conducive to the
progress of mathematics themselves. It has often happened in mathematical
history that a difficulty, which had for ages resisted all direct attempts to over-
come it, has yielded at last to the gradual advance of science ; just as in the
operations of strategy a strong position, which cannot be carried by a front
attack, may nevertheless be turned and taken in the rear by an enemy who has
possessed himself of the country round it.*
I may, perhaps, mention yet one more class of questions lying on the border
land of arithmetic and algebraic analysis ; I mean the questions which relate to
the transcendental or algebraic character of developments in the form of infinite
series, infinite products, or infinite continued fractions. The theorems of Eisen-
stein and M. Heine, of which a simple and beautiful demonstration has lately
been laid before us by our colleague M. Hermite, are amply sufficient to awaken
the expectation of great future discoveries in this almost unexplored field of
enquiry, f
I have detained you so long over arithmetical and quasi-arithmetical sub-
jects that I can only venture to glance hastily at some topics on which I could
have wished to have dwelt much longer. I am afraid that I have only given
you an additional instance of that one-sidedness against which, as I have re-
minded you, we were cautioned by our first President. In the hope of convincing
you that I have not wholly forgotten the claims of other parts of our science, I
will now hazard the assertion, that (after all) the advancement of the Integral
Calculus is at once the most arduous and the most important task to which a
mathematician can address himself. In the applications of mathematics to physics
' I find that I am here closely following (haud passibus aequis) some observations of the late Dr.
Hermann Hankel in his inaugural address, ' Die Entwickelung der Mathematik in der letzten Jahrhun-
derten,' Tubingen, 1869. This discourse, by one who was at once a learned scholar and an original
investigator, contains much which deserves the attention of all who are interested in the progress of
mathematical science, or who wish to see a higher spirit infused into the mathematical teaching given
in the schools and universities of this country.
t Eisenstein, ' Monatsberichte ' of the Berlin Academy for 1852, p. 441 ; M. Heine in 'Crelle's
Journal,' vol. xlv. p. 285, and vol. xlviii. p. 267. The note of Eisenstein is reproduced in the first of
these memoirs.
184 NOTE ON THE PRESENT STATE AND PROSPECTS OP
the Integral Calculus is confessedly of ever increasing importance ; and it is
especially interesting to observe that some of the most recent developments
which it has received have had their origin in considerations of pure analysis,
and yet have come just in time to furnish us with the most appropriate instru-
ments for dealing with the problems which at the present moment are the most
prominent in physical enquiries. But I must not dwell on the prospects of great
future extension which are thus opened up for the various branches of mathe-
matical physics ; I can only advert (and that very hastily) to some of those parts
of the integral calculus which, even from the point of view of the pure mathe-
matician, seem to promise an abundant and immediate harvest.
Let me first mention the theory of ordinary differential equations. This is
a subject which ought to have a special interest for ourselves, as one of the latest
advances that have been made in it the introduction of symbolical methods is
due in great measure to English mathematicians, and above all others to George
Boole. Nor have Englishmen been behindhand in the cultivation of another
branch of the subject (intimately connected with the use of symbolical methods)
the representation of the solutions of differential equations by means of definite
integrals. But, simultaneously with these investigations, a line of research has
been pursued on the Continent to which we in England have not paid equal
attention. I refer to the endeavours which have been made to determine the
nature of the function defined by a differential equation, from the differential
equation itself, and not from any analytical expression of the function, obtained
by first ' solving ' the differential equation. The generality and importance of
such an enquiry (whatever be its difficulty) cannot be overrated ; for, just as we
long since learned to regard integration in a finite form (or, more properly, inte-
gration by means of algebraic or exponential and logarithmic functions) as only a
very small part of the problem which the Integral Calculus has to solve with
regard to differential expressions containing a single variable, so also, when we
come to differential equations, we are forced to remember that the variety and
complexity of the functional relations expressed by them may altogether tran-
scend any other means of expression at our disposal. Perhaps we m,y regard as
the fundamental theorem in the whole subject the proposition of Caiichy, that
every differential equation admits (in the vicinity of any non-singular point) of an
integral which is synectic within a certain circle of convergence, and which is
consequently (within that circle) developable by the series of Taylor. Various
applications of this theorem (together with a demonstration somewhat simpler
than that given by Cauchy) will be found in the classical treatise of MM. Briot
SOME BRANCHES OF PURE MATHEMATICS. 185
and Bouquet.* Closely allied to the point of view indicated by the theorem of
Cauchy is that adopted by Riemann, who regards a function of a single variable
as defined by the position and nature of its singularities, and who has applied
this conception to the linear differential equation of the second order which is
satisfied by the hyper-geometrical series. In the memoir, ' Beitrage zur Theorie
der durch die Gauss'sche Reihe F(a,fi,y,x) darstellbaren Functionem,' f Rie-
mann sets out with the conception of a function which possesses three discrimin-
antal points (I venture to propose this word as the most natural English render-
ing of ' Verzweigungs-punkte '), and which is further characterised by the
property that any three of the values which it admits at any point are connected
by a linear and homogeneous equation with constant coefficients. Such a function
Riemann shows, by reasoning of great beauty and originality, necessarily satisfies
the linear differential equation of the hypergeometric series ; and thus the nature
and mode of existence of the functions defined by that equation are put before
us with a precision and clearness which could not, perhaps, have been attained
by any application of the ordinary methods of analysis to the discussion or inte-
gration of the equation.
The collected works of Riemann include another, but unfortunately unfinished
memoir (' Zwei allgemeine Lehrsiitze ueber lineare Differential-gleichungen mit
algebraischen Coefficienten'), relating to the case in which the number of inde-
pendent functional values is any whatever instead of only three. And the
fertility of the conceptions of Cauchy and of Riemann is further attested by the
researches to which they have given rise, and are still giving rise, in Germany
researches among which I must especially mention those of L. Fuchs, whose
papers on linear differential equations, in the 66th and subsequent volumes of
' Borchardt's Journal,' must form, it seems to me, the basis of all future inquiries
on this part of the subject.
There is one celebrated problem connected with differential equations which,
after all that has been written and said about it, remains a problem still ; I mean
the problem of Singular Solutions. If it were not for the papers of M. Darboux
(' Bulletin des Sciences Mathematiques et Astronomiques,' torn. iv. p. 158), and
* 'Th6orie des Fonctions Elliptiques,' ed. 2, Paris, 1875, p. 325. See also a memoir by the same
authors in the 'Journal de 1'Ecole Polytechnique,' cahier 36, p. 137. For an enumeration of Cauchy's
Memoirs on Differential Equations, see his Life by M. Valson, Paris, 1868, vol. ii. cap. 9, pp. 104-117.
t ' Transactions of the Academy of Gcittingen ' for 1875, vol. vii., or in Eiemunn's Collected Works
(Leipzig, 1876).
VOL. II. B b
186 NOTE ON THE PRESENT STATE AND PROSPECTS OF
of Professor Cayley (' Messenger of Mathematics,' vol. ii. p. 6, and vol. vi. p. 23),
I do not know where I should advise a student to turn to acquire any distinct-
ness of insight into this important question. These papers have at any rate
rendered one great service ; they clearly show that there is a difficulty, and a
difficulty not yet surmounted. The point of the difficulty I presume to be, that
whereas a singular solution, from the point of view of the integrated equation,
ought to be a phenomenon of universal, or at least of general, occurrence, it is, on
the other hand, a very special and exceptional phenomenon from the point of
view of the differential equation. The explanation suggested by M. Darboux is
(to say the least) deserving of very careful consideration. He says, at p. 167 of
the memoir just cited, ' Since differential equations are formed by the elimination
of constants from an equation in finite terms and its derived equations, writers
have supposed (and it would seem erroneously) that, when we are given a differ-
ential equation of the first order (for example), it always possesses an integral of
the first order expressible in the form f(x, y,c) = 0, where f is a function having in
the whole extent of the plane the properties generally recognised in analytical
functions. This function f might be more or less difficult to find, but it was
conceived of in every case as existing. Now this is just the disputable point,
and we think that recent researches on the theory of functions ought to lead us
to adopt a different view.' It is evident that, if the observations of M. Darboux
are well founded, an important series of questions arises as to the nature of the
integral equation answering to a given differential equation ; and, further, that
some of the elementary considerations with which it is usual to introduce the
subject of differential equations must be abandoned as untenable. The rules that
can be given in aid of mathematical discovery are, I suppose, very few, and I
have already ventured to call your attention to one of them the rule that bids
us follow up any opening that may present itself, rather than try to force a way
against obstacles which may prove insurmountable. In the case before us, I
think we come upon an illustration of another rule, which is of less general ap-
plication, but nevertheless often useful. The rule is, that an apparent contra-
diction (as distinct from a mere misunderstanding) is always to be regarded as
an indication of some undiscovered truth. Yet it is remarkable what a tendency-
there is in the minds of men to ignore or soften down such apparent contra-
dictions, instead of looking for the reality which lies at the bottom of them. The
/X (%
= log x must have been familiar to mathematicians for a century
_ *c
at least before they set themselves seriously to examine the apparent contra-
SOME BRANCHES OF PURE MATHEMATICS. 187
diction presented by the equation of a single- valued to a multiple- valued expres-
sion. And yet what a flood of light was thrown on the whole theory of functions
by the researches to which Cauchy and others were led when the endeavour was
at last made to account for this and similar apparently inexplicable phenomena.
If the clue offered by such familiar instances as the equations
* dx r x dx
ax , r x ax . .
- = log x, I = sm- 1 x,
x -A) ./I ar
had been seized and followed up, it is difficult to believe that the main outlines
of the Theory of Elliptic Functions would not have been discovered much sooner
than they actually were. If we look back on the history of the past, the dis-
covery of the 'principle of double periodicity,' and, with it, of the essential
characteristic of an elliptic function, cannot but appear to us as one of the most
extraordinary efforts of mathematical genius. The integration of the differential
equation of elliptic integrals by Euler an integration obtained by a sort of
divination, which has deservedly remained celebrated in the history of science ;
the systematisation of the calculus of elliptic integrals by Legendre ; the simul-
taneous discovery by Jacobi and Abel of the double periodicity latent in the
equation of Euler, these were the successive steps and each one a gigantic
step by which those great mathematicians arrived at the theory of elliptic
functions in the form in which we now possess it. But if we compare the actual
history of the discovery with the outlines of the theory, as we find them, for
example, in the work of MM. Briot and Bouquet, it is impossible not to be struck
with the contrast. Each step in the theory, as exhibited in that work, appears
to follow from those that precede it in such a natural and necessary order that
we are inclined to wonder why those who discovered the great results them-
selves should have failed to find the easiest path of access to them.
If I had had the honour of addressing the Mathematical Society ten years
ago, I think I should have had to complain of the neglect in England of the study
of elliptic functions. But I cannot do so now. The University of Cambridge has
given this subject a place in its Mathematical Tripos ; the University of London
in its examination for the Doctorate of Science. The British Association has
supplied the funds requisite to defray the cost of printing Tables of the Theta
function Tables of which the mathematicians of this country may justly be
proud, and which will form an enduring memorial of the great ability and inde-
fatigable industry of our colleague, Mr. Glaisher. We further owe to Professor
Cayley an introductory treatise on elliptic functions, the first which has appeared
B b 2
188 NOTE ON THE PRESENT STATE AND PROSPECTS OF
in our language. I consider that the service which he has thus rendered to
students is an important one, and one for which we ought to be very grateful.
I am convinced that nothing so hinders the progress of mathematical science in
England as the want of advanced treatises on mathematical subjects. We yield
the palm to no European nation for the number and excellence of our text-books
of the second grade I mean, such text-books as are intended to guide the studies
of the undergraduate within the courses prescribed by our University examina-
tions in honours. But we want works adapted to the requirements of the stu-
dent when his examinations are over works which will carry him to the fron-
tiers of knowledge in various directions, which will direct him to the problems
which he ought to select as the objects of his own researches, and which will free
his mind from the narrow views he is too apt to contract while ' getting up '
subjects with a view to passing an examination, or, a little later in his life, while
preparing others for examination. Can we doubt that much of the preference
for geometrical and algebraical speculation which we notice among our younger
mathematicians is due to the admirable works of Dr. Salmon ; and can we also
doubt that, if other parts of mathematical science had been equally fortunate in
finding an expositor, we should observe a wider interest in, and a juster appre-
ciation of, the progress which has been achieved ? There are, of course, other
treatises besides those of Professor Cayley and Dr. Salmon to which I might
refer ; there is, for example, the work of Boole on Differential Equations ; and
there are the great historical treatises of Mr. Todhunter, so suggestive of research
and so full of its spirit ; we have also a recent work by the same author on the
functions of Laplace, Lame, and Bessel. But the field is not nearly covered,
though, indeed, my enumeration is not complete ; and, even without leaving the
domain of the Integral Calculus, I might point out that there are at least three
treatises which we greatly need one on Definite Integrals, one on the Theory
of Functions in the sense in which that phrase is understood by the school of
Cauchy and of Biemann, and one (though he should be a bold man who would
undertake the task) on the Hyperelliptic and Abelian Integrals. I fear that our
colleague, Professor Clifford, would hardly listen to us if we were to appeal to
him to undertake this task ; but at least we may express the hope that he may
be able to continue the profound researches which he has commenced on this
great branch of analysis.
I feel I must now bring these somewhat desultory remarks to a conclusion ;
though, if your time and patience were unlimited, there are many things I could
wish to say. Among other matters I should have adverted to the great efforts
SOME BRANCHES OF PURE MATHEMATICS. 189
which have been made in very recent times, in Germany, in Russia, and in Nor-
way, to advance the theory of partial differential equations ; and I should have
noted with pleasure that our own Society has received important communications
on this subject from Professor Tanner. And again, leaving the field of the Inte-
gral Calculus, I think I should have hazarded some references to the Theory of
Substitutions, and its applications to the Theory of Equations ; and though I
should have been relapsing into a region dangerously near to the Theory of Num-
bers, I should have exhorted the younger mathematicians of our time not to turn
away from a subject which, if forbidding at the first aspect, contains so much
promise of future development, and lies so near the very centre and fountain of
much that is important in Algebra. Lastly, I should have endeavoured to make
my peace with Geometry, which all this time I have been treating with such
marked neglect, and would have invited your attention, though it were but for a
few moments, to some of those questions of geometry which the natural advance
of science has brought to the forefront at the present time. Even here, I might
have found one example more of a study which we in England too much neglect ;
and I might perhaps have reminded you of the great hopes which Gauss enter-
tained of the Geometry of Situation, containing, according to him, vast and as
yet quite uncultivated regions over which our present analytical methods can
pretend to no dominion. I certainly should not have forgotten to congratulate
the Society on the part which its members, under the guidance and inspiration
of Professor Sylvester, have taken in the development of the great geometrical
theory of link-work movements.
' Verum hsec ipse equidem spatiis exclusus iniquis
Praetereo, atque aliis post commemoranda relinquo.'
I will not sit down without again offering my excuses for the fragmentary
and disconnected nature of the reflexions I have laid before you this evening.
But, indeed, over so wide a field I could only take a wandering course. My
object has been to impress upon those who have been indulgent enough to listen
to me that the vast increase which this century has witnessed in the extent of
the ground already covered by mathematical science has been accompanied by a
proportionate increase in the number and variety of the objects of interest to
which the mathematician may turn his attention, and by an even more than pro-
portionate increase in the opportunities of discovering new truths which have
been brought within his reach. Our border on every side is the unknown ; and
the further our boundary line is extended, the more multitudinous become the
points at which we may hope to penetrate beyond it. In these days when so
190 NOTE ON SOME BRANCHES OF PURE MATHEMATICS.
much is said of original research and of the advancement of scientific knowledge,
I feel that it is the business of our Society to see that, so far as our own country
is concerned, mathematical science should still be in the vanguard of progress.
I should not wish to use words which may seem to reach too far, but I often find
the conviction forced upon me that the increase of mathematical knowledge is a
necessary condition for the advancement of science, and, if so, a no less necessary
condition for the improvement of mankind. I could not augur well for the
enduring intellectual strength of any nation of men, whose education was not
based on a solid foundation of mathematical learning, and whose scientific con-
ceptions, or, in other words, whose notions of the world and of the things in it,
were not braced and girt together with a strong framework of mathematical
reasoning. It is something for men to learn what proof is, and what it is not ;
and I do not know where this lesson can be better learned than in the schools of
a science which has never had to take one footstep backward, which has never
asserted without proof, or retracted a proved assertion ; a science which, while
ever advancing with human civilization, is as unchangeable in its principles as
human reason ; the same at all times and in all places ; so that the work done at
Alexandria or Syracuse two thousand years ago (whatever may have been added
to it since) is as perfect in its kind, and as direct and unerring in its appeal to
our intelligence, as if it had been done yesterday at Berlin or Gottingen by one
of our own contemporaries. Perhaps also it might not be impossible to show,
and even from instances within our own time, that a decline in the mathematical
productiveness of a people implies a decline in intellectual force along the whole
line ; and it might not be absurd to contend that on this ground the maintenance
of a high standard of mathematical attainment among the scientific men of a
country is an object of almost national concern. But I need not ask your assent
to such wide assertions ; I shall be more than satisfied if anything that may have
fallen from me may induce any one of us to think more highly than he has
hitherto done of the first and greatest of the sciences, and more hopefully of the
part which he himself may bear in its advancement.
XXXII.
ON THE CONDITIONS OF PERPENDICULARITY IN
A PARALLELEPIPEDAL SYSTEM.
[Proceedings of the London Mathematical Society, vol. viii. pp. 83-103. Read December 14, 1876.]
1. 1 HE conception of a parallelepipedal system (i.e., of a space divided by three
systems of equidistant parallel planes into similar and equal parallelepipeds)
which forms the basis of the usually received theory of crystallography,* has
also led to important researches in a domain belonging partly to arithmetic and
partly to geometry, f
It is the object of the present paper to discuss, more completely than perhaps
has been done as yet, the conditions for the perpendicularity of lines and planes
in a parallelepipedal system. The arithmetical principles required in the dis-
cussion are very simple, and are to be found, for the most part, in articles 1-7
of a memoir on ' Linear Systems of Indeterminate Equations and Congruences '
in the 'Philosophical Transactions' for 1861 J. The geometrical considerations
involved, although well known, are less elementary, and relate to the theory of
* See Dr. Leonhard Sohncke ' Die unbegrenzten regelmassigen Punktsysteme als Grundlage einer
Theorie der Krystallstruktur,' Karlsruhe, 1876, for a different and more general hypothesis as to the
ultimate structure of a crystal. See also a memoir by Dr. Sohncke, ' Die regelmassigen ebenen Punkt-
systeme von unbegrenzter Ausdehnung' in ' Borchardt's Journal,' vol. Ixxvii. p. 47, and a memoir by
M. Camille Jordan ' Sur les groupes de mouvements ' in the ' Annali di Matematica ' (Brioschi e
Cremona), vol. ii. p. 167.
t See (1) Gauss's Review of Seeber's ' Untersuchungen ueber die Eigenschaften der positiven
ternaren quadratischen Formen ' in the Gb'ttingen ' Gelehrte Anzeige ' for 1 83 1, or in ' Crelle's Journal,'
vol. xx. p. 312; (2) Lejeune Dirichlet, ' Ueber die Keductiou der positiven quadratischen Formen mit
diei unbestimmten ganzen Zahlen/ 'Crelle's Journal,' vol. xl. p. 209; (3) the 'Etudes Crystallogra-
phiques' of Auguste Bravais, Paris, 1866; (4) A Letter of Eisenstein to M. Charles Hermite ('Journal
de Math^matiques,' vol. xvii. p. 473). The researches to which this letter refers are purely arithmetical,
but they have an important bearing on a class of questions which are naturally suggested by the present
enquiry, and of which the following may be taken as an example, ' In what cases can a parallelepipedal
system, possessing a spherical symmetry, be regarded as containing a system in which the parallel-
epipeds are cubes ? '
* No. XII. vol. l.p. 367.
192 ON THE CONDITIONS OF PERPENDICULARITY [Art. 2.
contravariant systems of cones. For the definition and some of the principal
properties of such systems, we may refer to a paper ' On some Geometrical
Constructions ' in the ' Proceedings of the London Mathematical Society,' vol.
ii. p. 85* ; the most important case, that of two-fold systems, forms an essential
part of the theory of cubic cones or curves ; the considerations relating to that
theory which we shall have occasion to employ, will be found in the ' Intro-
duzione alia Teorica delle Curve Piane ' of Professor Cremona, or in the ' Treatise
on the Higher Plane Curves ' of Dr. Salmon.
The results of the present enquiry (which has been undertaken at the request
of Professor N. S. Maskelyne, and owes much to his suggestions) are submitted
to the Mathematical Society with great diffidence ; because, while they do not
seem likely to admit of any direct application to the measurement of crystals,
there is also some uncertainty as to their exact relation to other parts of
crystallographic theory.
2. It is, perhaps, hardly necessary to explain that by a ' line of the system '
we understand a line joining any two points of a given parallelepipedal system ;
by a ' plane of the system,' a plane containing any three points of the system ;
the ' points of the system ' being the points of intersection of the three sets of
equidistant parallel planes by which the system is defined. For our present
purpose it will be sufficient to consider exclusively origin lines and planes ; i.e.,
lines and planes passing through a fixed point of the system taken as origin.
Whenever a line of the system is perpendicular to a plane of the system, the
system has a certain ' symmetry of aspect ' with regard to that plane. Let be
the plane, and let O be any point of the system lying in it. The planes and
lines of the system which pass through are symmetrically distributed with
regard to ; but the points of the system are not (in general) symmetrically
distributed with regard to Q ; thus, if OP is any line of the system, and OQ
is the reflexion of OP in the plane 12, OQ is a line of the system, but the
points of the system which lie on OQ are not, in general, the reflexions of the
points which lie on OP. Hence, while the points of the system are not them-
selves symmetrically distributed with regard to 12, the directions in which they
would be viewed by an eye situated at O, are symmetrically distributed ; and
this we may express by saying that the system has a ' symmetry of aspect ' with
regard to 12. It will be seen (1) that ii is, or is not, a plane of absolute
symmetry (i.e., a plane of symmetry with regard to the points of the system),
* No. XIX. vol. i. p. 524.
Art. 2.] IN A PARALLELEPIPED AL SYSTEM. 193
according as the point of the system, that lies the nearest to upon the normal
at O, does, or does not, lie upon the nearest plane of the system parallel to Q ;
(2) that two parallelepipedal systems may have the same aspect without coinciding.
Suppose, for example, that the given parallelepipedal system has a symmetry of
aspect with regard to three rectangular planes intersecting in (this is the case of
an ' ellipsoidal symmetry ' to which we shall presently refer). Let OA, OB, OC be
the three lines of intersection of these planes, A, B, C being the points of the
system nearest to on these lines respectively. The four points 0, A, B, C
determine a new parallelepipedal system (OABC) composed of rectangular paral-
lelepideds. This system is contained in the given system, because every point
of (OABC) is a point of the given system ; but the two systems are not equivalent,
because every point of the given system is not a point of (OABC). In general,
a certain number of points of the given system lie in each rectangular parallelepiped
of (OABC), and are similarly distributed in each of them, though not symmetrically
in any one of them. But the ' aspect ' of the two systems from the point (or
from any point common to both of them) is the same ; and the question whether
the given system has an ' ellipsoidal symmetry of aspect ' is the same as the question
whether it contains a rectangular parallelepipedal system. Similar considerations
apply to every case of symmetry of aspect ; and indeed, whenever one of two
parallelepipedal systems contains the other, the two systems have the same
aspect at any one of their common points.*
As we shall have no occasion to consider planes of absolute symmetry, we
shall, henceforward, for the sake of brevity, use the word symmetry in the sense
of 'symmetry of aspect.' Thus any line, and any plane of the system, which
are at right angles to one another, are an axis and a plane of symmetry.
The cases of symmetry, as thus defined, which can present themselves in
a parallelepipedal system, are four in number. There is (1) the case of simple
symmetry, where there is only one axis and one plane of symmetry ; and there
are three cases of triple symmetry, which may be characterized as (2) the ellip-
soidal, (3) the spheroidal, and (4) the spherical. In an ellipsoidal system there
are three rectangular planes of the system which are planes of symmetry ; in a
* The relation between a contained and a containing parallelepipedal system is the same as the
relation between a contained and a containing ternary quadratic form (see Gauss, loc. cit.). The problem,
' To find the conditions that a given parallelepipedal system should have an ellipsoidal symmetry of
aspect,' may be stated in a purely arithmetical form as follows : ' To find the conditions that a definite
ternary quadratic form (of which the coefficients may be rational or irrational) should contain a form of
the type Ax'+By' + Cz'.
VOL. II. C C
194 ON THE CONDITIONS OF PERPENDICULARITY [Art. 3.
spheroidal system there is one equatorial plane of symmetry, but every plane of
the system at right angles to this plane is also a plane of symmetry ; in a system
having spherical symmetry, every plane of the system is a plane of symmetry, and
every line of the system an axis of symmetry. Two simple symmetries cannot
coexist without forming a triple symmetry, which is ellipsoidal, if the axis of
one of the symmetries lies in the plane of the other, but is spheroidal in every
other case ; three simple symmetries form an ellipsoidal symmetry, if the three
axes are at right angles to one another ; a spheroidal symmetry, if one of the
axes is at right angles to the plane of the other two, which are not at right
angles to one another ; a spherical symmetry in every other case. These asser-
tions are, in part, simple consequences of the well-known theorem that if a plane
contains two pair of lines of the system at right angles to another, every line
of the system in that plane is at right angles to a line of the system ; in part,
however, their demonstration requires a fuller discussion of the conditions of
symmetry, to which we now proceed.
3. General Conditions of Perpendicularity.
We begin by recording, in the form which is most suitable for our present
purpose, the conditions for the perpendicularity of lines and planes in space
referred to oblique axes.
The coordinates of any point being x, y, z, and X, Y, Z being the angles
between the positive directions of the axes, the formula
F (x, y,z) = x 2 + y* + z 2 + 2yz cos X + 2xz cos Y+ 2 xy cos Z,
which denotes the square of the distance of the point x, y,z from the origin, supplies
a convenient mode of expressing the condition for the perpendicularity of any two
lines x _ y__ z
I m n'
x y z
I' = m' = rf'
viz., this condition is v dF ,dF ,dF /1X
I -jj- + m -j h n -j = 0, (1)
dl dm an
. dF' dF' dF'
or I jjr + m-jr + n -j-r =
dl dm an
where F=F(l, m, n), F = F (l r , m, n),
Art. 3.] IN A PABALLELEPIPEDAL SYSTEM. 195
Similarly the condition for the perpendicularity of the line
x _ y _ z
I m n'
and the plane \x + r*.y + vz = Q, .
dF dF dF
dl dm dn /o\
is -7~ = = .......... ' '
A fJL V
To obtain, in a form similar to (1), the condition for the perpendicularity of
two planes, we introduce the contravariant of F, i. e., the function
* ( v> D = 2 sin 2 Jr+ >? 2 8in 2 Y+ 2 sin 2 Z + 2^(cos YcosZ- cos X)
+ 2 (cos Zcos X- cos Y) + 2& (cos X cos F- cos Z),
or, as it may be written, if X 1} Y 1 ,Z l are the angles between the normals to the
coordinate planes,
*(& i> ) = 2 sin 2 X+y 2 sin 2 F + 2 sin 2 Z'+2 7 sin FsinZcos X t
+ 2 sin Zsin J cos Fj + 2^ sin X sin F cos Z^ .
The condition for the perpendicularity of the two planes
may now be expressed in either of the two equivalent forms
/o\
-r
dv
and the condition (2) for the perpendicularity of a plane and a line may be
expressed in a second, and equally convenient form,
d\ d/j, dv /A\
I m n
The formulae (1), (2), (3), (4) are well-known, and are only given here for
convenience of reference ; their demonstration by the ordinary methods of
analytical geometry presents no difficulty. We may, however, call attention to
the following points :
(i.) The equation F (x, y,z) = represents a sphere of evanescent radius having
its centre at the origin ; or, which is the same thing, a certain imaginary cone
having its vertex at the origin, and asymptotic to any sphere concentric with
the origin. Such an imaginary sphere-cone possesses the characteristic properties,
c c 2
196 ON THE CONDITIONS OP PERPENDICULARITY [Art. 4.
that (a) any two lines, (/3) any line and plane, (7) any two planes, which are har-
monically conjugate with regard to it, are at right angles to one another. The
conditions (1) and (2) are the analytical expression of the properties (a) and (0).
(ii.) Again, the tangential equation of the imaginary sphere-cone, i. e., the
condition that the plane x + >iy + ?z =
should be one of its tangent planes, is
*(UD-0;
and the conditions of perpendicularity (3) and (4) are the analytical expressions,
by means of the tangential equation $, of the properties (7) and (/3).
(iii.) As the formula F(x, y, z) expresses the square of any rectilineal segment
in terms of its oblique projections upon the three coordinate axes ; so the formula
4> (, 17, ) expresses the square of any plane area in terms of its oblique projections
sin X, n sin Y, sin Z, upon the three coordinate planes. The imaginary
sphere-cone possesses, it will be noticed, the two paradoxical properties, that the
square of any segment lying on one of its lines, and the square of any area lying
on one of its tangent planes, are each of them equal to zero.
4. Conditions of Perpendicularity in a Parallelepipedal System.
Adopting the notation of the classical treatise of Professor Miller, we
designate by a, b, c the parameters of a parallelepipedal system ; we thus have,
for the square of the distance between any two points of the system, the
expression
f(x, y, z) = F(ax, by, cz)
= a 2 ce 2 + 6 2 y 2 + c 2 z 2 + 2bcyz cos X + 2cazx cos Y + 2abxy cos Z,
where xyz now denote integral numbers. Again, if
<t> ( i, = * ( bc & ca > a& ) = b2 2 ? sin2 x + c * aV sin2 Y+ a2&2 2 sin2 z
+ 2 a 2 6c n% sin Fsin Z cos X l + 2 6 2 ca sin Z sin X cos Y l
+ 2c 2 a&>; sin X sin Y cos Z lt
the form <, which is the contravariant of/, characterises (in the manner indicated
by Gauss) a new parallelepipedal system (the polar system of Bravais) in which
every line is perpendicular to a plane of the given system, and in which the
parameter corresponding to any line is the elementary parallelelogram in the
plane to which the line is perpendicular.
Writing, for brevity, .
/= A x 2 + S f + Cz* + 2 A'yz + 2 Rzx + 2 C 'xy,
Art. 4.]
IN A PABALLELEPIPEDAL SYSTEM.
197
C^AB-C' 2 ,
C( = A'B'-CC',
(so that
A = a 2 , ...,A' = be cos X, ..., A l = b 2 c* sin 2 X, ...,A1 = a 2 be sinFsin Zcos X 1} ...,)
and denoting the determinant
A , (J,
C', B, A
R, A, C
by A, we have the well known relations
A^BC-A' 2 , B 1 =CA-B' 2 ,
Al = BC' - AA, Bl = C'A - BB',
, Cl, S,
, S lt A[
B\,
A4 =B 1 C 1 ~.
LA -Bid-.
which serve to show that, if we suppose A = 1, or the volume of the elementary
parallelepiped of the system f to be unity (a supposition which is admissible,
because, if A be not unity, we may alter the parameters in the ratio of /A : 1),
the relation between the two parallelepipedal systems is reciprocal ; i.e., the
system f is derived from the system <f> in the same way in which the system <f>
has been derived from the systemy.
The five quantities upon which the nature of the parallelepipedal system
ultimately depends are unquestionably the angles X, Y, Z, and the ratios of the
parameters a, b, c. But the combinations of these quantities which enter into the
conditions of perpendicularity and symmetry are precisely the six covariant
coefficients ^ ^ c, A', B', C,
and the six contravariant coefficients
A C =
! - C?,
Thus, if
x y z
au bv cw
x y z
(5)
are any two lines of the system, the condition for their perpendicularity is
df df df
ni _ ^ I A _ / _ I at _!!_ == 1 1
198 ON THE CONDITIONS OF PERPENDICULARITY [Art. 4.
i i \
or u -r-+v-f^ + w~^ = 0,
cWj CM?! dw 1
i. e., the condition that the lines (5) should be perpendicular is the same as the
condition that the lines
? = I? = 5. = y~ = (6)
u = v w' u v " v t w^
should be harmonically conjugate with regard to the cone f=Q. Again, the
condition that the two planes of the system
a
+ i =()
a b c ~ '
(7)
should be perpendicular to one another is the same as the condition that the
planes hx + ki/ + lz = 0. ) /Q v
I (o)
n O* ml^ L* tl I / W ~~ C\ [
&1 IAS 1i "*\ tf 1^ ^1 ** ^~ ^ J I
should be harmonically conjugate with regard to the same cone f= 0, of which
the tangential equation is <f> = ; viz., this condition is
dl
h
And similarly the condition that one of the lines (5) should be perpendicular to
one of the planes (7), is the same as the condition that the corresponding line (6)
and plane (8) should be a polar line and polar plane with regard to the cone
/= o or <p = 0.
We shall, in what follows, adopt the mode of expressing the conditions of
perpendicularity which has just been explained. By rational lines, planes, or
cones, we shall understand lines, planes, or cones, of which the equations have
integral numbers for their coefficients. We shall term the lines and planes (6)
and (8), the rational lines and planes corresponding to the lines and planes of the
system (5) and (7). Except in the case of a spherical symmetry, the cone f= 0,
or </> = 0, is not itself a rational cone : and thus (in other cases) the problem, ' to
determine all the lines and planes which are perpendicular to one another in a
given parallelepipedal system,' is the same as the problem, ' to find all the rational
lines and planes which are harmonically conjugate to one another with regard to
a given irrational cone.'
Art. 5.] IN A PABALLELEPIPEDAL SYSTEM. 199
5. Linear Relations connecting the Coefficients.
Let the parallelepipedal system contain a pair of perpendicular lines (5) ; the
condition of perpendicularity gives immediately the relation
A MMj + Bvv^ + CWW-L + A' (vw^ + wvt) + R (wu^ + mv^ + C' (uv l + u^v) = 0. . (9)
Unless therefore the six covariant coefficients are connected by a linear homo-
geneous equation having integral coefficients, no two lines of the system can be
perpendicular to one another ; and, correlatively, unless the six contravariant
coefficients are connected by a similar relation, no two planes of the system can
be perpendicular to one another. But the existence of such a relation connecting
the six coefficients (or the six contravariant coefficients), though a necessary con-
dition, is not a sufficient condition for the existence in the system of a pair
of perpendicular lines (or planes). We proceed, therefore, to examine more
closely the circumstances which present themselves, when the coefficients are
connected by one, two, three, four, or five relations. By a relation connecting
the coefficients we understand a linear homogeneous equation of the type
pA + qB + rC+2pA' + 2q'B'+2rC' = 0, (10)
in which p, q, r, p', q', r are integral numbers ; in connection with such a
relation we shall have to consider the quadratic form
and its reciprocal form
* = (p' 2 - q r ) x * + (q' 2 - T) y 2 + ( r/2 -
+ 2 (pp f - q'r") yz + 2 (qq f - r'p'} zx + 2 (rr - p'q') xy ;
these we shall term the quadratic form and the reciprocal quadratic form apper-
taining to the given relation.
For brevity, we shall attend only to the cases in which given relations exist
between the six covariant coefficients A, B, C, . . . ; the cases, in which given relations
exist between the six contravariant coefficients, are simply the correlatives of
these ; conditions for the perpendicularity of two planes answering to conditions
for the perpendicularity of two lines, and every condition of symmetry admitting
of a two-fold expression according as we employ the covariant, or the contravariant
coefficients.
It is remarkable that, in every case, the conditions of perpendicularity and
symmetry depend solely on the coefficients of the linear relations connecting the
crystallographic coefficients; so that two parallelepipedal systems in which
the crystallographic coefficients have different ratios, but satisfy the same linear
200
ON THE CONDITIONS OF PERPENDICULARITY
[Art. 6.
relations, would resemble one another exactly in respect of symmetry and
perpendicularity.
6. Case of one Linear Relation between the Coefficients.
Here we have the theorem :
'The system contains a single pair of perpendicular lines, or contains no
such pair whatever, according as the reciprocal form appertaining to the given
relation is, or is not, a perfect square.'
For (i.) if the system contain a pair of perpendicular lines, the coefficients are
connected by a relation of the type (9) ; and the reciprocal form appertaining to
this relation is the perfect square
a, y,
U, V,
z
w
Conversely, (ii.) if the coefficients are connected by a relation of the type (10),
which satisfies the condition Sk = n
the system contains a single pair of perpendicular lines. Let
and consider the matrix
P, r-y, <
/ + % 2> P'~ a
j'-P, p' + a, r
in which it is easily proved that every first minor is equal to zero, and in which
we may suppose that the nine elements are freed by division from any common
divisor which they may possess. If in this matrix we represent by u, v, w the
greatest common divisors of the elements in the first, second, and third rows
respectively, we find immediately
2p' = vw l + wv 1 , 2 q e wu-i + uw 1 , 2r' = uv l + vu^ ;
whence it follows that the equation (10) assumes the form of the equation (9)
and that the two lines
JL = H- =
era bv cw'
x y z
an.
cw.
are perpendicular to one another.
Art. 7.] IN A PARALLELEPIPEDAL SYSTEM. 201
For the condition that ^r must be a perfect square, we may, if we please,
substitute the condition that the form \^ appertaining to the given relation
should resolve itself into two linear factors having integral coefficients. If
these factors are
the lines (5) are two perpendicular lines.
It is important to observe that the reciprocal form ^ cannot be identically
equal to zero. If it were so, the form \J<- would itself be a perfect square, and the
cone f=0, which is certainly imaginary, would contain a real line. Again,
when we say that ^ is to be a perfect square, we understand that ^ itself
is to be the square of a linear function of x, y, z (and not merely a multiple of such
a square by a number which is not itself a square). Thus the condition ^ = Q
may be replaced by the two conditions, (1) that the discriminant of \J/- is to be
zero, (2) that the greatest common divisor of the first minors of this discriminant
is to be a perfect square ; and of these the second may, if we please, be replaced
by the condition that, of the three principal minors, one, which is not zero, is
to be a perfect square. (These three minors cannot all be zero, and if one of them
is a perfect square different from zero, the other two are either zero or perfect
squares.)
7, Since the formy is positive and definite, the coefficients A, B, C, A', B', C'
are subject to certain inequalities. But, subject to these, they may be any
quantities whatever, rational or irrational. Hence no relation of the form (10)
can in general subsist between them ; or, which is the same thing, a parallel-
epipedal system cannot, in the most general case, contain a single right angle.
The problem, ' Given any number of irrational quantities to determine the
rational linear relations (if any) which subsist between them,' presents great
difficulties, and, in the present state of indeterminate analysis, is perhaps in-
soluble. We observe, however, that if m + n quantities Y ly Y z , ..., Y m+n are
connected by m independent equations of the form
9i,\ YI + 9i,2 Y a +.,.+ g i>m + Y m + = 0,
where i = l,2, 3, ...,m, and the numbers g are integral, we can always (and
indeed, unless n= 1, in an infinite number of ways) assign n quantities Z lt Z z , Z n
such that the m + n quantities Y can be expressed in terms of these by means
of a system of equations of the type
VOL. II. D d
202 ON THE CONDITIONS OF PERPENDICULARITY [Art. 8.
t < t Z n ,
where the numbers h are integral, andy = 1, 2, 3, ..., m + n.*
It follows from this theorem that, if the coefficients are connected by one,
and only one, relation, there is an equation of the type
f(x,y, z) = a> 1 2 1 + tt 2 Z 2 + ...+ft> 6 2 5 ,
where 2^, Z 2 , ..., 2 5 are quadratic forms in x, y, z having integral coefficients, and
<a lt 2 > ."'s ar e irrational quantities between which no linear rational homo-
geneous equation can exist. In general, if i +j = 6, and if there are i relations
between the coefficients, we have an equation of the type
/=w 1 Z 1 + ...+o> j 2 J ..
The rational cones of which the equations are
2^0, 2 2 = 0, ..., 2 j = 0,
and again the rational cones of which the tangential equations are
*i(fc *,) = <>. *.(& * = > ..., *<(fc * = o,
determine two systems of contravariant cones ; every cone of the system (2)
circumscribing harmonically every cone of the system (\^), and, vice versd, every
cone of the system (\^) being harmonically inscribed in every cone of the
system (2).f
We have seen that the existence of a pair of perpendicular lines in a parallel-
epipedal system implies the existence of a relation between the coefficients ; but
that not every such relation implies the existence of a pair of perpendicular
lines. When the reciprocal form ^ appertaining to a given relation is not a
perfect square, we can only assert that the cone f harmonically circumscribes
the cone ^. But this statement seems to have no crystallographic meaning.
8. Case of two linear relations between the coefficients.
Let vf'i, ^ 2 ; i'!, S^ be the forms appertaining to the two given linear
relations ; and let us at first suppose that
*i=D, * 2 =D;
so that the system contains two pairs of perpendicular lines.
Three particular cases require especial notice.
* Seethe memoir to which we have already referred, 'Phil. Trans.,' 1861, p. 299, sqq. [vol.i. p. 375].
t 'Proceedings of the London Mathematical Society,' vol. ii. pp. 90, 91 [vol. i. p. 528].
Art. 8.] IN A PARELLELEPIPEDAL SYSTEM. 203
(1) If the squares Skj and ^ are identical, or differ only by a numerical
factor, so that, retaining the notation of Art. 6, we have the equations,
i = A_ = .7i
* fo 72 '
the planes of the two right angles coincide ; and every line of the system
that lies in this plane has a line of the system at right angles to it in the
same plane.
(2) The plane of one of the right angles may contain one of the rays of the
other right angle.
(3) The two right angles may have one ray in common ; their common ray
is then at right angles to the plane of the other two rays, and this plane becomes
a plane of simple symmetry. The condition that this should happen is that
vj/i and \^ 2 should have a linear factor in common ; viz., if these two forms have
a common linear factor, this factor is of necessity rational, the two reciprocal
forms are perfect squares, and the common factor is
6 i,
*i, #1. 7i
2, Ai> 72
Still retaining the hypothesis that ^ and ^ are perfect squares, but ex-
cluding the particular cases to which we have just referred, we observe that,
besides the two pairs of perpendicular lines which we obtain immediately from
the two given relations, the system contains one, and only one, other pair of
perpendicular lines. This is a consequence of the elementary theorem that if
AOB, COD are two right angles in different planes, the intersection of the
planes AOD, BOC\& at right angles to the intersection of the planes A OC, SOD ;
a theorem which is itself a particular case of a general property of two pairs
of conjugate lines with regard to any cone (or conjugate points with regard
to any conic), first given by Otto Hesse.
We now quit the hypothesis that ^ and ^ 2 are squares. When only one
linear relation subsists between the coefficients, we are certain that, if the
reciprocal form appertaining to it is not a perfect square, the parallelepipedal
system cannot contain a single pair of perpendicular lines. But, when we have
two such relations, we have still to inquire whether, by combining them linearly
with one another, we cannot obtain a new relation which shall satisfy the
condition
Let 6, O lt 6 2 be the three roots of the discriminantal cubic of ^i + O^. If
D d 2
204 ON THE CONDITIONS OF PERPENDICULARITY [Art. 8.
these roots are irrational, the system contains not a single pair of perpendicular
lines. If one of them, for example 0, is rational, we have still to examine
whether the factors of ^1 + 6^2 &re rational; if they are, we have a pair of
perpendicular lines. If all the three roots are rational, we have to examine the
factors of the three forms ^ + 0^ 2 , ^ + 6^, ^ + 6^ ; according as these factors
are or are not rational (if the factors of two of the forms are rational, the factors
of the third are also rational), we obtain one or three pairs of perpendicular
lines, or no pair at all of such lines.
If two of the roots of the discriminating cubic are equal, the cones ^ and \^ a
touch one another ; if the three roots are all equal, these cones osculate (the
cases of double contact and of super-osculation cannot present themselves,
as they would imply the existence of a real line on the cone/=0). When two
of the roots are equal, we have either the special case (2) considered above, or
we have one, and only one, pair of perpendicular lines ; when the three roots are
all equal, we have a single pair of perpendicular lines.
Lastly, the coefficients of the discriminating cubic may all vanish. If this
happens, either (a) ^ and S^ 2 differ (if at all) only by a numerical factor, or (/3) \f^
and ^ 2 have a common linear factor. The case (a) is the particular case (1) to
which we have already referred. For each of the equations ^ = 0, \J/- 2 =
represents a pair of lines in the plane V^i = or V^ 2 = ; these two pairs of
lines may themselves be irrational, but, even if they are, every rational line
lying in their plane has a rational conjugate in the involution determined by
them, and the pairs of this involution are conjugate pairs with regard to the
cone/=0.
The case (/S), when ^ and ^ have a common linear factor, is the case of
simple symmetry (the particular case (3) already mentioned).
We may therefore enunciate the theorem :
' The conditions that a parallelepipedal system should possess a simple
symmetry are
' (a) That the coefficients should be connected by two linear relations ;
' (6) That the two quadratic forms appertaining to these relations should
have a linear factor in common.'
For the condition (6) we may substitute the following :
' The four invariants of \^ and >^ 2 must vanish ; the first minors of the
discriminants of ^ and ^ not being proportional to one another.'
This is only an explicit statement of the conditions which are necessary and
sufficient to ensure the presence of a common factor in ^ and \k 2 .
Art. 9.]
IN A PARALLELEPIPEDAL SYSTEM.
205
9. Case of three linear relations between the coefficients.
If the parallelepipedal system includes three pairs of lines at right angles to
one another (these pairs being asyzygetic, i.e., such that the existence of one of
them is not a necessary consequence, in the manner already explained, of the
existence of the other two), the six coefficients are connected by three inde-
pendent linear relations of the type (10), and the reciprocal forms appertaining
to these relations are perfect squares. Conversely, when these conditions are
satisfied, the system contains not merely three pairs of lines at right angles to
one another, but (as we shall presently see) an infinite number of such pairs 5
indeed, this will happen even if only one of the three reciprocal forms is a perfect
square. And, even when none of these forms is a perfect square, the system
may contain rectangular pairs of lines, because (as in the last case) it may be
possible to combine linearly the given relations in such a manner as to obtain
a new relation satisfying the condition
Such a combination, as we now shall prove, is, or is not, possible, according as
the intermediate equation
C=
d n
d ' d n ' d%
=
does, or does not, admit of resolution in integral numbers.
In accordance with the theorem of Art. 7, we represent/ by an equation of
the type /_ y 4. y -t- 2
and since no linear relation can exist between o> 1 , u> z , 3 , we infer that any pair
of rational lines which are conjugate with regard to f are also conjugate with
regard to 2 T , 2 2 , and 2 3 ; the problem before us, therefore, is, ' To determine all
the rational lines which are conjugate with regard to 2 19 2 2 , 2 3 .'
With reference to this problem, we consider the system of contravariant
cones vj'j, \^ 2 , \f/3, and 2 n 2 2 , 2 3 ; the equations -^ = being equations in plane
coordinates, and the equations 2 = being equations in point coordinates. The
Cay ley an cone of the system is the cone of the third class C= 0, of which the
equation has been already given. This cone is the Jocobian of the cones
206
ON THE CONDITIONS OF PERPENDICULARITY
[Art. 9.
4"i> ^2. 4s 5 the Jacobian of the cones 2,, 2 2 , 2 3 is the Hessian cone of the
system ; its equation is
H=
c^Zj rfZj rfZj
dx ' dy ' dz
dx ' dy' dz
= 0.
dx' dy ' dz
The Cayleyan C is a combinantive covariant of the three forms ^ lt \J<- 2 , \^ 3 ;
the Hessian H is a combinantive contravariant of the same three forms ; the
developed expressions of C and H have been given by Dr. Salmon,* and it is
unnecessary to repeat them here.
All the pairs of lines harmonically conjugate with regard to the system of
cones (Z) lie on the cubic cone H=0; and vice versd, every line lying on this
cubic cone has a corresponding line lying on the same cone which is its conjugate
with regard to all the cones of (Z) ; it only remains, therefore, to ascertain
whether any of these pairs of conjugate lines are rational. The problem, ' To
determine whether a homogeneous cubic equation, containing three indeterminates,
and having integral coefficients, does, or does not, admit of solution in integral
numbers,' is one of which, in its general form, no complete discussion has as yet
been given. It is obvious, however, that, if the equation _ff=0 admits of a
single solution in integral numbers, it admits of an infinite number of such
solutions. In general this is true for any homogeneous cubic equation containing
three indeterminates, and having integral coefficients. For we may regard the
cubic equation as defining a rational cubic cone ; if there is a rational line ap-
pertaining to this cone, the successive tangentials of this rational line are
themselves rational lines ; and the plane meeting the cone in any two rational
lines meets it in a third rational line. This mode of construction is subject to
certain exceptions ; if, for example, the given rational line is a line of inflexion,
it is its own tangential, and gives rise to no second rational line. But in the
case before us (when the cubic cone appears as the Hessian of a rational system
of cones) the line conjugate to any given rational line of the Hessian cone is
always rational, and distinct from the given line ; and the plane joining the
two lines meets the Hessian cone in a third line distinct from either of them, of
which we may again take the conjugate line, and so on continually. The line
* ' Conic Sections,' Ed. 5, pp. 346, 347.
Art. 10.] IN A PAEALLELEPIPEDAL SYSTEM. * 207
conjugate to a given line of the Hessian cone is most readily found by taking the
polar planes of the given line with regard to the cones (2) ; these polar planes
are all rational, and intersect in a rational line, which is the line required. We
may also note (1) That if AA ', BB' are two pairs of rational conjugate lines,
the lines (AB', A'B), (AB, A'B'} are themselves a pair of rational conjugate
lines ; (2) that if C is any rational line of the Hessian, the planes CA and CA'
meet the Hessian again in a pair of rational conjugate lines ; (3) that A, A' have
the same rational tangential D, which is the conjugate of the line in which the
plane A A' meets the Hessian again.
From what precedes, it appears that, if the equation H =0 admits of a
single solution in integral numbers, the parallelepipedal system contains an
infinite number of pairs of perpendicular lines all lying on the cubic cone
H(xa, yb, zc) = 0. And the same thing is true (as has been already said) if the
equation C = admits of a single solution in integral numbers ; for the two
equations (7=0 and H=0 are simultaneously resoluble or irresoluble ; or, which
is the same thing, if the Cayleyan cone (7=0 has rational tangent planes, the
Hessian cone H=Q has rational lines, and vice versd ; this, indeed, is evident,
because the plane containing any two conjugate lines of the Hessian is a tangent
plane of the Cayleyan ; and the intersection of any two conjugate planes tangent
to the Cayleyan is a line of the Hessian.
10. Three Relations between the Coefficients Conditions of Symmetry.
The system may have a simple symmetry, or an ellipsoidal symmetry, or
none at all. But it cannot have a spheroidal or a spherical symmetry.
If there is a simple symmetry, the cones (2) must have in common a
rational polar line and polar plane ; but must not have in common a rational
self-conjugate system of diametral planes. On the other hand, when there
is an ellipsoidal symmetry, the cones (2) must have such a system of diametral
planes.
(A) Conditions that the cones of the system (2) should have in common a
single polar line and polar plane (the line and plane are necessarily rational).
We take for the axis of z the polar line, the polar plane for the plane of xy,
and for the axes of x and y the lines in which this polar plane is intersected by
the tangent planes drawn from the axis of z to touch any one of the cones (^).
(Except in a special case, which we shall notice presently, these cones all touch
the same pair of planes intersecting in the axis of z.) We may then take for the
representatives of the system (2) the three loci
208 ON THE CONDITIONS OF PERPENDICULARITY [Art. 10.
x* = 0, y 2 = 0, z 2 -2Xiry = 0;
and for the representatives of the system (x^) the three envelopes
X being a constant different from zero. Thus the Hessian and the Cayleyan
respectively become xyz and ( 2 ,-x*),
the former resolving itself into a product of three linear factors, the latter into
a product of a linear by a quadratic factor. The axis of z is conjugate (with regard
to the cones 2) to every line in the plane xy ; every line in either of the planes
o;z or yz is conjugate to a line hi the other of those two planes ; and the
planes containing these pairs of conjugate lines envelope the Cayleyan cone
2,-X 2 = 0.
The conditions that a ternary cubic should resolve itself into three linear
factors is that the Hessian of the form should coincide with the form itself;
and the condition that a ternary cubic should resolve itself into a linear and
a quadratic factor is that the evectants of its two principal invariants should
coincide.*.
The conditions, therefore, for a simple symmetry are that the Hessian of H
should coincide with H, and that the two evectants of C should coincide with
one another ; but that the Hessian of C should not coincide with C.
The rational line corresponding to the axis of symmetry is the line, and the
only line, common to all the quadratic cones represented by the first minors of
the Hessian ; the linear factor of C represents it tangentially. The rational
plane corresponding to the plane of symmetry is the polar plane of this rational
line with regard to the quadratic Cayleyan cone. One of the linear factors of
the Hessian represents this polar plane, and is certainly rational ; the other two
factors (x and y) may be irrational, or even imaginary. When they are rational,
there exist in the parallelepipedal system two planes, not at right angles to one
another, intersecting in the axis of symmetry, and each containing an infinite
number of lines of the system at right angles to one another.
These two planes may coincide ; if they do, the plane of coincidence is
certainly rational. The special case at which we thus arrive has been excluded
from the preceding discussion ; for we have supposed that the cones -^ all touch
two distinct tangent planes intersecting in the common polar line of the cones 2.
* Dr. Salmon's 'Higher Plane Curves,' pp. 190 aud 202, sqq.
Art. 10.] IN A PAEALLELEPIPEDAL SYSTEM. 209
But the cones ^ may instead touch two coincident planes ; i. e., they may all
pass through the polar line, and touch a fixed plane along that line. If we take
this plane for the plane of yz, and any other plane through the polar line for the
plane of xz, we may represent the systems (2) and (\[/-) by the loci
x 2 = 0, x
and the envelopes
=<>, =0, -\,*
The Hessian and the Cayleyan are respectively
x 2 z and
the former being a product of a linear by a square factor ; the latter being a
product of three linear factors, of which two are certainly imaginary, because, if
X were negative, the cones 2 would have two real lines in common, which would
consequently appertain to the cone f. The conditions for this special case of
simple symmetry, which is characterised by the presence of a single plane passing
through the axis of symmetry, and containing an infinite number of pairs of
perpendicular lines, are that the Hessian of H should vanish identically, and
that the Hessian of (7 should coincide with C.
(B.) Conditions that the system (2) should have in common a self-conjugate
system of diametral planes, of which one at least is rational.
The cones ty) must have three tangent planes in common (viz., the three
self-conjugate planes of the system 2). Thus the Cayleyan cone resolves itself
into three lines (the three lines of intersection of the diametral planes), and the
Hessian resolves itself into those three diametral planes themselves.
For this the necessary and sufficient conditions are that the Hessian of the
Hessian should coincide with the Hessian, and the Hessian of the Cayleyan with
the Cayleyan.
The three linear factors of the Cayleyan (or the Hessian) may, however, be
all irrational, or one of them only may be rational, or they may all three be
rational ; and accordingly there may be either no symmetry at all, or a simple
symmetry, or an ellipsoidal symmetry.
To distinguish between these three cases it is sufficient to examine the
discriminantal cubic of any two of the cones 2, or of any two of the cones ^ ; for
it is not difficult to verify that the linear factors of the Cayleyan, or Hessian,
are rational or irrational according as the roots of any one of these equations
are rational or irrational.
VOL. II. E e
210 ON THE CONDITIONS OF PERPENDICULARITY [Art. 12.
11. Three Relations between the Coefficients Special Cases.
Some special cases, in which there is no symmetry, have (nevertheless) a
certain interest. Of these we may briefly mention two.
(1) The cones -^ have one, and only one, tangent plane A in common. The
Hessian in this case consists of A, and of a quadratic cone which is the reciprocal
cone of A with regard to the cones of 2. Thus, in the parallelepipedal system,
there is a cone and plane such that every line lying in the plane has a line at
right angles to it lying in the cone, and vice versd ; the lines of intersection of
the plane and cone are at right angles, and may be lines of the system.
(2) The cones ^ have a polar line and polar plane in common.
The result is as follows : The parallelepipedal system has an infinite number
of pairs of lines at right angles to one another, all lying in the same plane ; and
it may have a second set of such pairs lying on the surface of a quadratic cone,
the plane of each such pair passing through the polar line of the first-named
plane with regard to the cone.
12. Case of Four Linear Relations between the Coefficients.
In this case the form /may be expressed by an equation of the type
/=l 2 l + <>2 2 2,
where the ratio of w, and > 2 is irrational, and where the cones 2j , 2 2 cannot have
any real line in common.
Every rational line has a rational line conjugate to it with regard to each of
the cones 2 X = 0, 2 2 = 0, and consequently with regard to the cone f= 0. Hence
every line of the parallelepipedal system has a line at right angles to it ; and
this distribution of pairs of perpendicular lines may exist without the presence of
any symmetry whatever.
Conditions of Symmetry.
The symmetry (if any) may be simple, or ellipsoidal, or spheroidal, but
cannot be spherical.
There is a simple symmetry when the discriminantal cubic of the system
(2 1( 2 2 ) has one rational root, and an ellipsoidal symmetry when this cubic
has three rational and unequal roots. When two of the roots are equal,
the cones 2j and 2 2 have an imaginary double contact, and the axis and
plane of the double contact are the axis and equatorial plane of a spheroidal
symmetry.
Art. 13.]
IN A PARALLELEPIPEDAL SYSTEM.
211
If we represent by S the harmonic co variant of Sj and 2 2 , the equation
dS dS dS^
dx' dy ' dz
=
dx ' dy ' dz
dx ' dy ' dz
represents the three diametral planes which are self-conjugate with regard to the
cones 2j and 2 2 ; and its left-hand member / vanishes identically when 2 X and
2 2 have double contact.
The Jacobian J is a combinantive covariant of 2 X and 2 2 ; its coefficients
therefore involve only the determinants (of the second order) which can be formed
with the coefficients of 2 X and 2 2 . But these determinants* are proportional
to the reciprocal determinants (of the fourth order) formed with the coefficients
f ^i4'4''^ r *! i" e -> the coefficients of the Jacobian can be expressed in terms of
these reciprocal determinants. To avoid introducing a new notation, we abstain
from giving the developed expression of J, which indeed is not required here.
It will be observed, however, that, in the case which we are now considering, the
conditions for the various kinds of symmetry, and the determination of the
planes of symmetry, depend on a single ternary cubic form (resoluble into three
linear factors) which is a combinantive contravariant of the four forms >//. (See
the concluding remark in Art. 5.)
13. Case of Five Linear Relations between the Coefficients.
In this case the ratios of the coefficients are themselves evidently rational ;
and the parallelepipedal system has a spherical symmetry.
For, the coefficients of f being rational (after division, if necessary, by a
common irrational factor), every rational line has a rational polar plane ; i.e.,
every line of the parallelepipedal system has a plane of the system at right
angles to it.
And, conversely, if a parallelepipedal system has a spherical symmetry, the
ratios of the coefficients are rational ; for, if we consider any three lines of the
system at right angles to one another, we obtain three independent relations
(see Art. 5) between the coefficients ; and if we consider any line of the system
* See 'Philosophical Transactions,' 1861, p. 301 [vol. i. p. 376].
E 6 2
212 ON THE CONDITIONS OF PERPENDICULARITY, ETC. [Art. 13.
(not lying in the same plane with two of the first three), and the plane per-
pendicular to it, we obtain two more relations between the coefficients, inde-
pendent of one another and of the former three ; i.e., the ratios of the coefficients
are rational.*
* The question of the rationality or irrationality of the ratios of the crystallographic coefficients
had already attracted the attention of Gauss, who, as appears from his Life (' Gauss. Zum Gedachtniss ; '
by W. Sartorius von Waltershausen ; Leipzig, 1856), had in the year 1831 devoted himself with great
ardour to the study of crystallography. See the concluding paragraphs of the Review of Seeber's
Untersuchungen ' already cited.
XXXIII.
ON THE CONDITIONS OF PERPENDICULARITY IN
A PARALLELEPIPED AL SYSTEM.
[Philosophical Magazine, Ser. v., vol. iv. pp. 18-25. Read before the Crystallological Society,
June 14, 1876.
1. JL HE conception of a parallelepipedal system (i.e. of a space divided by three
systems of equidistant parallel planes into similar and equal parallelepipeds) may
be regarded as forming the basis of the usually received theory of crystallography.
It is the object of the present note to state some of the conditions for the per-
pendicularity of lines and planes in such a system. The results of this enquiry
(which has been undertaken at the request of Professor N. S. Maskelyne, and
owes much to his suggestions) are submitted to the Crystallological Society with
great diffidence, because they do not seem likely to admit of any direct ap-
plication to the practical work of the crystallographer. Such interest as they
possess belongs to a domain which borders on the one hand on pure arithmetic,
and on the other hand on pure geometry.
2. It is perhaps hardly necessary to explain that by a ' line of the system ' we
understand a line joining any two points of the given parallelepipedal system, by
' a plane of the system ' a plane containing three points of the system, the points
of the system being the points of intersection of the three sets of equidistant
parallel planes by which the system is defined. It will be sufficient to consider
origin-lines and planes, i.e. lines and planes passing through a fixed point of
the system taken as origin.
3. Whenever a line of the system is perpendicular to a plane of the system,
the system has a certain ' symmetry of aspect ' with regard to that plane. Let
Q be the plane, and let O be any point of the system lying in it. The planes
and lines of the system which pass through are symmetrically distributed with
regard to fi ; but the points of the system are not (in general) symmetrically
214 ON THE CONDITIONS OP PERPENDICULARITY [Art. 5.
distributed with regard to Q : thus, if OP is any line of the system not lying in
the plane Q, and if OQ is the reflection of Q with regard to the plane ft, OQ is a
line of the system as well as OP, but the points of the system which lie on OQ
are not (in general) the reflections of the points of the system which lie on OP.
Hence, while the points of the system are not themselves symmetrically dis-
tributed with regard to ft, the directions in which they would be viewed by an
eye situated at O are symmetrically distributed ; and this is what we intend
to express by saying that the system has a ' symmetry of aspect ' with regard to
the plane ft.
As we shall have no occasion in what follows to consider planes of absolute
symmetry, we shall for the sake of brevity use the word symmetry in the sense
of ' symmetry of aspect.' Thus any line and any plane of the system which
are at right angles to one another are an axis and a plane of symmetry.
4. The cases of symmetry, as thus denned, which can present themselves in
a parallelepipedal system are four in number. There is (1) the case of simple
symmetry, when there is only one axis and one plane of symmetry ; and there
are three cases of triple symmetry, which may be characterised as (2) the ellip-
soidal, (3) the spheroidal, and (4) the spherical. In an ellipsoidal system there
are three mutually rectangular planes, which are planes of symmetry ; in a
spheroidal system there is one equatorial plane of symmetry, but every plane
of the system at right angles to this plane is also a plane of symmetry ; in
a system having spherical symmetry every plane of the system is a plane of
symmetry, and every line of the system an axis of symmetry. Two simple
symmetries cannot coexist without forming a triple symmetry, which is ellip-
soidal if the axis of one of the symmetries lies in the plane of the other, but
is spheroidal in every other case : three simple symmetries form an ellipsoidal
symmetry if the three axes are at right angles to one another, a spheroidal
symmetry if one of the axes is at right angles to the plane of the other two which
are not at right angles to one another, a spherical symmetry in every other case.
5. Adopting the notation of the classical treatise of Professor W. H. Miller,
we designate by a, b, c the parameters appertaining to the three lines of the
system taken for the coordinate axes ; we also denote by X, Y, Z the angles
between the coordinate axes, and by X lt Y l} Z the angles between the normals
to the coordinates planes. We thus have for the square of the distance between
any two points of the system the expression
f(x,y,z) = a*x 2 + b* y 2 + c 2 ? 2 + 2 bcyz cos X + 2 cazx cos Y+ 2 abxy cos Z,
Art. 6.] IN A PARALLELEPIPEDAL SYSTEM. 215
where x, y, z denote any positive or negative integral numbers ; and this ternary
quadratic form may be regarded as characterising the given parallelepipedal
system. Again, if
<t> ( i> D = &2 C T sin2 x + 2 a * i 2 sin2 Y + 2 &2 1 2 sin2 z
+ 2a 2 be ^ sin Fsin Z cos X 1 + 2b 2 ca sin Z sin X cos Y l
+ 2c 2 ab^rj sin X sin YcosZ 1}
the form <, which is the contravariant of f, characterises (in the same way in
which y characterises the given system) a new parallelepipedal system (the polar
system of Auguste Bravais) in which every line is perpendicular to a plane of
the given system, and in which the parameter corresponding to any line is the
elementary parallelogram of the given system lying in the plane to which the
line is perpendicular.
6. We write for brevity
/= Ax 2 + By* + Cz* + 2 A'yz + 2 B'zx + 2 C'xy,
(so that
A = a 2 ,..., A' = bc cosX,..., A l = b 2 c 2 8iaX,..., ^4j = a 2 6csin J'sin.ZcosJLi,...);
and we observe that although the five quantities upon which the nature of the
parallelepipedal system ultimately depends are the ratios of the parameters
a, b, c, and the three angles X, Y, Z, yet the combinations of these quantities
which it is most convenient to consider in discussing the conditions of per-
pendicularity are precisely the six coefficients
A, B, C, A', B', C,
and the six contravariant coefficients
Thus the condition that the lines of the system
= y. =
au bv cw '
' fyl CW 1
should be perpendicular to one another is
df df df
U + V r njg J -
l du l dv 1 dw
or
df df df
u J+ V --+w-!-
216 ON THE CONDITIONS OF PERPENDICULARITY [Art. 7.
the condition that the planes of the system
mi
iV + iZ =
_
a b c
should be perpendicular to one another is
l dl
and the conditions that the first of the lines (i) should he perpendicular to the
first of the planes (ii) may be written in one or other of the equivalent forms
(df\ f df
\du'
h
\dh' \dk/ \dl'
U V W
7. Let us now suppose that the given parallelepipedal system contains
a pair of perpendicular lines (i) ; the condition of perpendicularity gives im-
mediately
Auu^ + Bvv l + Cww 1 + A' (vw + wv^ + B' (uni^ + u w^ + C' (uv-^ + vu^ = 0.
Unless, therefore, the six covariant coefficients are connected by a linear homo-
geneous relation having integral coefficients, no two lines of the system can be
perpendicular to one another; and correlatively, unless the six contravariant
coefficients are connected by a similar relation, no two planes of the system
can be perpendicular to one another. But the existence of such a relation con-
necting the six covariant coefficients (or the six contravariant coefficients),
though a necessary condition, is not a sufficient condition for the existence of
a pair of perpendicular lines or planes. We proceed, therefore, very briefly
to describe the principal cases which present themselves when the coefficients
are connected by one, two, three, four, or five linear relations. By a linear
relation connecting the coefficients we understand a linear homogeneous equa-
tion of the type p A + qB + rC+2p'A f + 2q'' + 2i f C' = 0,
where p, q, r, p', q', r' are integral numbers which we may suppose free from
Art. 9.] IN A PARALLELEPIPED AL SYSTEM. 217
any common divisor. In connexion with such a relation we shall have to
consider the quadratic form
and its contravariant or reciprocal form
Sk = (p' 2 - qr) x z + (q' a - rp) y z + (r 2 -pq) z 2 + 2 ( pp - q' /) yz
+ 2 to?' - r/ y) zx + 2 ( rr/ - p'<f) x y-
These we shall term the quadratic form and the reciprocal quadratic form
appertaining to the given relation. For brevity we shall attend only to the
cases in which given relations exist between the six covariant coefficients
A, B, C, A', If, C', the cases in which given relations exist between the six
contravariant conditions being simply the correlatives of these. It is remarkable
that hi every case the conditions of perpendicularity and symmetry depend
solely on the coefficients of the linear relations connecting the crystallographic
coefficients ; so that two parallelepipedal systems, in which the crystallographic
coefficients have different ratios but satisfy the same linear relations, would
resemble one another exactly in respect of symmetry and perpendicularity.
8. Case of one linear relation between the coefficients.
Here we have the theorem, ' The system contains a single pair of per-
pendicular lines, or contains no such pair whatever, according as the reciprocal
form appertaining to the given relation is or is not a perfect square.'
For the condition that the reciprocal form ^ should be a perfect square,
we may if we please substitute the condition that the quadratic form \^ ap-
pertaining to the given relation should resolve itself into two rational factors.
Or, again, we may replace this condition by the two conditions, (1) that the
discriminant of ^ is to be zero, (2) that the greatest common divisor of the first
minors of this discriminant is to be a perfect square.
9. Case of two linear relations between the coefficients.
We represent the quadratic forms and the reciprocal quadratic forms
appertaining to these relations by ^ lt ^ r l , -vJ/- 2 , ^ 2 , and by 9, 0', 6" the roots of
the discriminantal cubic of ^ + 0\J/- 2 . If these roots are irrational, the system
contains not a single pair of perpendicular lines. If one of them, for example 6,
is rational, we still have to examine whether the factors of ^ + 0\^ 2 are rational ;
if they are, we have a pair of perpendicular lines. If all the three roots 6, 0', 6"
are rational, we have to examine the factors of each of the three forms
vf'j-f 0\J/ 2) ^ l + &'^ 2 , ^1 + 0" "ta; according as these factors are or are not rational
VOL. II. F f
218
ON THE CONDITIONS OF PERPENDICULARITY
[Art. 10.
(if the factors of two of them are rational the factors of the third are so too), we
obtain one or three pairs of perpendicular lines, or no pair at all of such lines.
When two of the roots 0, 6', 0" are equal, we have either one, and only one,
pair of perpendicular lines ; or we may have two pairs, the plane of one of the
right angles containing one of the rays of the other right angle. When the
three roots are all equal we have a single pair of perpendicular lines.
Lastly, the coefficients of the discriminating cubic may all vanish. If this
happens, either (a) ^ and ^ 2 differ, if at all, by a numerical factor, and every
line of the system that lies in a certain plane has a line of the system at right
angles to it in the same plane ; or (/3) x^ and ^ 2 have a common linear factor,
and the system possesses a simple symmetry.
We may thus enunciate the theorem :
' The conditions that a parallelepipedal system should possess a simple
symmetry are (a) that the coefficients should be connected by two linear re-
lations, (&) that the two quadratic forms appertaining to these relations should
have a linear factor in common.'
10. Case of three linear relations between the coefficients.
We represent by ^ lf \f' 2 , ^ 3 the quadratic forms appertaining to the given
relations, and we obtain the following theorem :
' The system contains no right angle, or an infinite number, according as
the indeterminate cubic equation
C =
d
=
!>" ' 7 ' "~ 7 <S
at, drj a
does or does not admit of solution in integral numbers.'
By virtue of the three given relations the characteristic expression f(x, y, z)
of Art. 5 assumes the form
the ratios of the quantities w l , 2 , w 3 being irrational, but the coefficients of the
quadratic forms f ly f 2 , f 3 being integral numbers. If II (x, y, z) denote the
Jacobian of these three forms, we have the theorem :
' When the indeterminate equation C= admits of solution, the infinite
number of right angles which the system contains all lie on the cubic cone
Art. 11.] IN A PARALLELEPIPED AX SYSTEM. 219
H(xa, yb, zc) = ; viz. an infinite number of lines of the system lie on this cone,
and every line of the system which lies on it has a line at right angles to it,
also lying on the cone.'
The system may have a simple symmetry or an ellipsoidal symmetry, or
none at all ; but it cannot have a spheroidal or a spherical symmetry.
The conditions for a simple symmetry are that the ternary cubic form
(7(, n, ) should resolve itself into a rational linear factor and a rational
quadratic factor, and that the ternary cubic form H(x, y, z) should resolve
itself into three linear factors. These conditions admit of being further de-
veloped (see Dr. Salmon's ' Higher Plane Curves,' pp. 190 and 202 seqq.) ; it
is sufficient for our purpose to observe that the coefficients of the Jacobian
H(x, y, z), no less than those of (7(, ;, ), depend solely on the coefficients
of the forms ^, -^ 2 , "^3, *& on the integral numbers entering into the given
linear relations.
The conditions for an ellipsoidal symmetry are that C(, i, ) should
resolve itself into three rational linear factors, and that H(x, y, z) should
resolve itself into three factors.
Two special cases of the general theory (which, however, are not cases
of symmetry) deserve attention.
(1) There may exist in the parallelepipepal system a quadratic cone and
a plane, such that every line of the system lying in the plane has a line of
the system at right angles to it lying in the cone.
(2) Or, again, the parallelepipedal system may have an infinite number
of pairs of perpendicular lines all lying in the same plane ; and it may also have
at the same time a second set of such pairs lying on the surface of a quadratic
cone, the plane of each pair of this second set passing through the polar line
of the first-named pair with regard to the cone.
11. Case of four linear relations between the coefficients.
Here every line, without exception, of the parallelepipedal system has a line
at right angles to it ; and this distribution of pairs of perpendicular lines may
exist without the presence of any symmetry whatever. The symmetry (if any)
may be simple, or ellipsoidal, or spheroidal, but cannot be spherical.
The characteristic form f(x, y, z) may be expressed by an equation of
the type f=> l f l + < 2 f 2 ,
the ratio of o>, and w 2 being irrational, but the coefficients of the quadratic forms
/, and f a being integral numbers. There is a simple symmetry when the
F f 2
220 ON PERPENDICULARITY IN A PARALLELEPIPEDAL SYSTEM. [Art. 12.
discriminantal cubic of fi + Of 2 has one rational root, and an ellipsoidal symmetry
when it has three rational and unequal roots, a spheroidal symmetry when it
has two equal roots. (It cannot have its three roots equal, because the cone
f(x, y, z) = is imaginary.)
We suppress the further discussion of these conditions', only observing that
they may be so expressed as to show that they depend only on the coefficients
of the four given relations, and not on the six coefficients A, B, C, A',R, C'
themselves.
12. Case of Jive linear relations between the coefficients.
In this case the ratios of the coefficients are themselves evidently rational,
and the parallelepipedal system has a spherical symmetry. It is also true,
conversely, that when there is a spherical symmetry the ratios of the coefficients
are rational.
We may mention that the question of the rationality or irrationality of the
ratios of the crystallographic coefficients had attracted the attention of Gauss,
who, as appears from the memoir of his life (' Gauss. Zum Gedachtniss,' von
W. Sartorius v. Waltershausen : Leipzig, 1856), had in 1831 devoted himself
with great ardour to the study of crystallography.*
* Some of the demonstrations, which have been omitted in the present note, will be found in a
paper inserted in the ' Proceedings of the London Mathematical Society,' vol. vii. p. 83 [No. XXXII.
vol. ii. p. 191.]
XXXIV.
SUR LES INTEGRALES ELLIPTIQUES COMPLETES.
[Atti della R. Accademia del Lincei. Transunti Ser. iii. vol. i. pp. 42-44. Bead January 7, 1877.]
suivant la notation usuelle, posons w = x + iy, la quantite y etant positive,
[1 -1- /?
r
de'signons aussi par < (& 2 ) et * (k 2 ) les inte'grales rectilignes
dx
la valeur initials de chacun des deux radicaux etant + 1. Lorsque & 2 est reel,
positif et plus grand que I'umte', la definition de la premiere mte"grale prdsente
une ambiguitd ; celle de la seconde se trouve pareillement en defaut, lorsque k z
est re'el et n^gatif. Nous conviendrons done, dans ces deux cas, de prendre pour
$ (k 2 ) celle des deux valeurs admissibles dans laquelle le coefficient de i est
positif ; et pour Sk (k' 2 ) celle des deux valeurs de I'inte'grale correspondante dans
laquelle le coefficient de i est ne'gatif.
On salt que, si la partie re"elle de w s'eVanouit, c'est &, dire, si k 2 et k' 2 sont
re'els, positifs et moindres que 1'unit^, on a les deux equations
=$(#), K' = *(k 2 ) ......... (1)
Mais il a dtd de'montre' par M. Hermite, que ces equations ne peuvent pas avoir
222 SUE LES INTEGRALES ELLIPTIQUES COMPLETES.
lieu pour toutes les valeurs de o>*. II est vrai qu'en doit avoir, dans tous
les cas,
dx (2}
'
mais ces integrates ne sauraient 6tre, en gdndral, rectilignes ; et la determination
du chemin de 1'tntegration a paru ofirir quelque difficulte. On y parvient de la
maniere suivante.
Soient a, /3, y, des nombres entiers, positifs ou ne*gatifs, qui satisfont a
1'equation aS l3y = l, et aux congruences a = S = l, mod 4; /Q = 7 = 0, mod 2 ;
soit aussi Q = X+iY, Y etant positif, et les quantity's rdelles X et Y etant
assujetties & verifier les ine'galites
-\<X<1, -X<X*+Y*2iX. ...... (3)
On conclut facilement de la theorie de la reduction les formes binaires h, ddtermi-
nant ndgatif, qu'on peut toujours satisfaire, et cela d'une maniere unique,
a 1'equation
= 00.
Les nombres a, /3, 7, 8, et la quantite complexe O e"tant ainsi determines, la
theorie des transformations lineaires donne aussitfit
de plus Ton ve"rifie sans peine que les Equations (1) subsistent, tant que a> satis-
fait aux ine'galites (3). En effet, tant qu'on a
1 < x < I, x<x z + y z <x,
les fonctions P et k' 2 ne peuvent pas atteindre aucune des valeurs pour lesquelles
les integrales rectilignes cessent d'etre completement determinees. Et, quant
aux cas limites qui se prdsentent lorsqu'on a
x = 1 , ou bien x 2 + y z = x,
les conventions, que nous avons adoptdes pour lever les ambiguitds dans les
definitions des integrales $ et "&, ont ete choisies de maniere h, faire accorder dans
* Voyez la Note sur la theorie des foncticms elliptiques ajout^e au Calcul diffire-ntiel de Lacroix,
torn. ii. pp. 420-425, Paris, 1862.
SUR LES INTEGRALES ELLIPTIQUES COMPLETES. 223
ces memes cas limites, les valeurs des integrates < et ^ avec celles des fonctions
K et K'. On a done
ou bien, en observant que <p s (fi) = <p s (&>) = k 2 ,
.fiT (Q) =
d'ou Ton tire finalement
Ces equations remplacent les equations (1), et font connaitre le chemin que 1'inte-
gration doit suivre dans les Equations (2).
Nous ferons remarquer, en terminant, que la transformation de la quantite
complexe w, qui donne la solution du probleme, est la meme que Jacobi avait
employee pour la reduction des fonctions 9, sauf la difference qui provient des
congruences auxquelles nous avons assujetti les nombres a, /3, -y, S.
XXXV.
MEMOIRE SUR LES EQUATIONS MODULAIRES.
[Atti della R. Accademia dei Lincei. Memorie della classe di Scienze fisiche, matematiche e naturali.
Ser. iii. vol. i. pp. 136-149. Read February 4, 1877.]
1. UN connait les beaux resultats auxquels sont parvenus MM. Kronecker et
Hermite, en etudiant les rapports qui existent entre les Equations modulaires et
les formes quadratiques binaires k determinant n^gatif. Mais les points de rap-
prochement, qu'on a trouve's jusqu'ici entre la thdorie des equations modulaires et
celle des formes quadratiques & determinant positif, ont dte peu nombreux ; et, a
cet egard, nous ne saurions citer que le Me"moire si remarquable de M. Kronecker,
'sur la solution de liquation de Pell par le moyen des fonctions elliptiques.'
Oependant, nous avons ete conduits h, reconnaitre qu'il existe entre ces deux
theories des liens tres intimes. C'est ce que nous nous proposons de faire voir
dans ce Memoire, en demontrant que, si Ton designe par N un nombre entier
quelconque, et par <j> ^ X 2 ) = $ (X 2 , k-) =
une des equations symdtriques, qui definissent les transformations modulaires du
yieme O rdre, la courbe represented par liquation cartesienne
aura la propriety singuliere de presenter une veritable image geometrique du
systeme complet des formes quadratiques reduites appartenantes au determinant
positif N. C'est en suivant la route tracee par les illustres geometres que nous
venons de nommer, que nous avons ete conduits k ce resultat, qui nous a paru
offrir une interessante application de 1'arithmetique ^, la geometric, aussi bien
qu'h, la theorie des fonctions elliptiques.
2. Soit <a = x + iy une quantite complexe, la valeur de y etant positive.
l J? (~\
Posons w = - ; a, /3, 7, S etant des nombres entiers qui satisfont ^ 1'equation
Art. 3.]
MEMOIRE STIR LES EQUATIONS MODULAIRES.
225
et aux congruences
a, /3
1,0
0,1
, mod 2 ; a = S = 1, mod 4.
En representant, comme on fait ordinairement, les quantites complexes par les
points d'un espace de deux dimensions, nous exprimerons la relation qui subsiste
entre to et ii, en disant que les deux points correspondants sont equivalents.
Cette definition de 1' equivalence est plus restreinte, et par consequent, moins
naturelle que celle qu'on emploie ordinairement en arithmetique ; et c'est uni-
quement pour abre*ger le discours que nous 1'admettons ici. Dans le plan xy,
dont toutefois nous ne conside*rons que la partie situe"e au dessus de 1'axe des
abscisses, trafons les deux droites P = x 1 = 0, P~ l = x + 1 = 0, et les deux demi-
cercles =
Soit Z 1'espace compris entre les deux droites, mais exterieur aux deux cercles ;
P et Q dtant censes appartenir k cet espace, mais P" 1 et Q~* en e*tant exclus.
Cela pose, on aura les propositions suivantes, qu'on deduit sans peine de la the'orie
de la reduction des formes quadratiques binaires & determinant negatif.
' Etant donne" un point quelconque to, il existe toujours un point reduit (c'est
a dire, un point appartenant k 1'espace reduit 2), qui est equivalent aw; et il
n'en existe qu'un seul.' ' La substitution reduisante est aussi unique.'
Pour abre"ger, nous conviendrons de nommer normales les substitutions telles
a, ft
3. Soit N = 6 2 ac ; a, b, c dtant des nombres entiers. Liquation
que
appartient & un cercle re*el ; nous representerons ce cercle par [a, b, c], et la forme
quadratique correspondante par (a, b, c). Nous conviendrons d'appeler cercle ra-
tionnel tout cercle tel que [a, b, c] ; mais nous ne considererons toujours que les
demi-circonfe'rences situe'es au-dessus de 1'axe des x. Lorsque N est un carre", on
peut avoir c = ; dans ce cas le cercle rationnel devient une droite.
Soit (A, B, O) une forme e"quivalente k (a, b, c) par la substitution normale
; les cercles correspondants [A, B, (7] et [a, b, c] seront aussi Equivalents
i J? O
par la m^rne substitution. En effet, liquation to = -^ etablit, entre les points
w et & des deux cercles [a, b, c\, [A, B, C], cette espece de correspondance ge'ome'-
trique qui a dte appelde affinitu circulaire par Moebius, et qui, & la ve'rite', ne
differe point essentiellement de la relation si connue de 1'inversion. II est bon
VOL. II. G g
226 MEMOIRE SUE LES EQUATIONS MODULAJRES. [Art. 4.
de remarquer que la transformation par affinitd circulaire du cercle [a, 6, c] dans
le cercle [A, B, (7], est en me'me temps une transformation homographique.
Ainsi, lorsque N est non-carrd, les substitutions automorphiques de la forme
quadratique (a, 6, c) sont repre'sente'es gdome'triquement par des transformations
homographiques du cercle [a, b, c] dans lui-me'me. Et de meme que les substi-
tutions automorphiques normales sont les puissances, positives ou negatives,
d'une seule d'entr'elles ; de me'me les transformations homographiques, que nous
avons a conside'rer par rapport au cercle [a, b, c], proviennent de la repetition,
dans les deux sens, d'une seule transformation fondamentale.
En supposant que [a, b, c] soit un cercle primitif, et en designant par t, u les
moindres nombres positifs qui satisfont a 1'equation t 2 Nu 2 = l, u etant pair, t
impair, on trouve que 1'ellipse, rdgulatrice de cette transformation fondamentale,
est repre'sente'e par liquation
L'excentricite de cette ellipse est donnde par 1'equation A/ - =
V JL ^ C?
d'ou Ton voit que les transformations homographiques, qui correspondent aux
substitutions automorphiques normales, sont semblables pour tous les cercles
primitifs du me'me determinant.
4. Soit a- = 2 ou = l, selon que le plus petit nombre u, qui satisfait a
1'dquation tfNu? = 1, est pair ou impair. II est facile de verifier que la restric-
tion, que nous avons du apporter h, la definition de 1'equivalence, entraine la
repartition des formes de chaque classe proprement primitive en 3 a- classes
subalternes, qui satisfont, en nombre e"gal, aux conditions exprimees par les
congruences ( A ) a=c = l, mod 2,
(B) a = 0, c = l, mod 2,
(C) a = l, c = 0, mod 2.
Pareillement, lorsque N = l, mod 8, chaque classe de formes improprement primi-
tives se partage en six classes subalternes, dont il y a toujours deux qui satisfont
a chacun des systemes (Af) a = c = 0, mod 4,
(B'} a = 0, c = 2, mod 4,
(C") a = 2, c = 0, mod 4.
Enfin, lorsque N=5, mod 8, chaque classe improprement primitive contient 2 a-'
classes subalternes, <S etant = 1, ou = 3, selon que 1'equation t 2 Nu* = 4 est reso-
luble ou irresoluble en nombres impairs ; de plus, chacune des congruences
(A"), 6 = 1, mod 4, (B"), 6 = - 1, mod 4,
est satisfaite par a-' de ces classes subalternes.
Art. 5.] MEMOIRE SUR LES EQUATIONS MODULAIRES. 227
Soient h, Ji les nombres des classes proprement et improprement primitives,
qui appartiennent au determinant N ', soient H, H' les nombres correspondants
des classes subalternes de cercles. En observant que les deux fermes (a, b, c),
( a, b, c) ne correspondent qu'a un seul cercle [a, 6, c], et que ces deux formes
appartiennent toujours a des classes subalternes diffdrentes, bien que, dans
certains cas, elles peuvent e~tre comprises dans la mSnae classe, on parvient a
etablir les deux Equations
H=%<rh, H=<r'h'.
5. Maintenant, aux points du cercle primitif [a, b, c] substituons les points
rdduits correspondants. Ce cercle se changera en un assemblage d'arcs circulaires
reduits, dont la totalite" formera une ligne L, qui en apparence sera brise'e, mais
dont on mettra en evidence la continuity, en repliant sur lui-me'me Fespace 2, de
maniere a former une surface ferme"e tricuspide, les droites P, P~ l e"tant reunies
ensemble, et aussi les cercles Q, Q~ l . II importe surtout de savoir quels sont
les cercles Equivalents a [a, b, c] qui traversent 2 ; et quelle est la loi qui, dans
la ligne composed L, gouverne la succession des arcs rdduits. Voici la solution
de ce probleme pour le cas d'un determinant non-carrd.
Soit 6 1'une ou 1'autre des racines de 1'equation a + 2bQ + c6 2 = 0. Ddvelop-
pons 6 en fraction continue, en prenant toujours pour quotient integral le nombre
pair, positif ou negatif, qui est le plus rapproche" du quotient complet correspondant.
Soit . . ..
la periode de la fraction continue ; les nombres entiers MI , /2 > /"as e"tant positifs ;
e,, e 2 , ...,e 2 de"signant des unites positives ou negatives ; et le quotient complet
0j etant de rang impair dans le developpement de 6. Posons aussi
et conside'rons les 2/x quantitds in-ationnelles
3 ,
' e,,- 2e 2 ,' 6> 28 -
Qg2
228 MEMOIRE SUR LES AQUATIONS MODULALRES. [Art. 7.
Chacune de ces MI + ^2+ quantitEs est la racine d'une Equation quadratique
Equivalente aa + 260 + c0 2 = 0; par consequent, les cercles correspondants sont
Equivalents h, [a, 6, c]. Or, tous ces cercles traversent 1'espace 2, et la ligne L
est composEe des parties de leur circonferences qui se trouvent dans 1'intErieur
de cet espace, ces parties Etant prises dans 1'ordre indiquE par le developpement.
6. Les formes quadratiques correspondantes aux cercles dont nous venons
de parler, difterent des formes rEduites de Gauss, (1) parceque nous nous servons
ici d'une fraction continue avec des quotients pairs ; (2) parceque nous admettons
parmi les formes re'duites, non seulement les formes re'duites principales qui
correspondent aux quotients complets 6 lt , ... , mais aussi les formes interme-
2
diaires, correspondantes aux racines
fi 9*
u l~ " e l> > 2 o~ i
2 -2e 2
(3) parceque nous prenons pour les racines des Equations qui correspondent aux
quotients de rang pair, les quantitEs
.1.1 1
a > ~a~ ' n~ '
V 2 V t t 2 ,
au lieu des quantitEs 2 > ^> > ~^2- H est presque inutile d'aj outer que,
pour avoir la suite des arcs rdduits, on pourrait se servir, au lieu du deVeloppe-
ment en fraction continue, de 1'algorithme des formes contigties de Gauss, en y
apportant une legere modification.
7. Lorsque le determinant est carre, les arcs reduits, Equivalents h, un cercle
donne", forment toujours une suite continue ; mais cette suite, au lieu d'etre perio-
dique, commence avec un arc passant par un des points singuliers de la surface
tricuspide, et se termine de la meme maniere. Designons les points (0, GO), (0, 0)
par p et q; et 1'un ou 1'autre des deux points Equivalents (1, 0), (1, 0) par r.
La suite des cercles rEduits, Equivalents & un cercle proprement primitif donnE,
aura pour ses points extremes rr, qq, pp, selon que le cercle donnE satisfait aux
congruences (A), (B), (C) de 1'article 4. Pareillement, dans 1'ordre improprement
primitif, les points extremes de la suite des cercles rEduits, Equivalents h, un
cercle donnE, seront pq, qr, rp, selon que 1'Equation de ce cercle satisfait aux
congruences (A'), (B^, (C") de 1'article citE. Pour dEterminer completement la
ligne L qui correspond h, un cercle donnE [a, b, c], dont le dEterminant est un
nombre carrd, il suffira de connaitre les Equations des deux cercles extremes de
L, et d'en dEduire la substitution normale unique qui transforme 1'un d'eux dans
1'autre. Soit, en effet,
Art. 8.]
MEMOIRE SUE LES EQUATIONS MODULAIRES.
229
cette substitution ; on en de'duira le deVeloppement fini
w=2 ^> + 277 + + 27V + S'
ou il faut remarquer qu'on peut avoir Mi = 0, / 28 = 0. Ce developpement rempla-
cera la fraction continue pEriodique de 1'art. 6, et fera connaitre tous les cercles
de L dans leur ordre naturel de succession. Tout se rEduit done a trouver les
Equations des deux cercles extremes de L. Pour cela, soit
(a, b, c) (x, y) 2 = m (px +p'y) (qx + q'y),
m dtant le plus grand diviseur commun de a, 2 b, c. En designant toujours par
une substitution normale, et en regardant comme inconnus les nombres
entiers a, /3, <y, S, tj, /, \, X', 1'Equation indetermine'e
P>P
a, 13
\, \'
admet une solution unique, dans laquelle les valeurs absolues de ?, n ne surpassent
2
pas runite", et celles de X, X' ne surpassent pas + (pq p'q) = - \fN. Le cercle
m
'] est 1'un des deux cercles cherches; 1'autre peut s'ob-
tenir en dchangeant entr'eux dans la solution pre'ce'dente, les deux facteurs de
(a, b, c). Mais, un des deux cercles extremes dtant trouve', il vaut mieux partir
de 1'dquation nouvelle
X, X'
a,/3
/
li> 1 1
i, i'
X
7> S
>
puisque ainsi on est conduit immEdiatement jl la substitution
a,/3
7> $
qui transforme
le premier cercle dans le second. On tire aussi de cette Equation la conclusion,
tres importante pour notre but actuel, que si Ton connait la substitution ' .
% o
et les deux points extremes de L, on a tout ce qu'il faut pour pouvoir determiner
les Equations des deux cercles extremes et, par consequent, les Equations de tous
les cercles rEduits, pris dans leur ordre naturel.
8. Le nombre .ZV Etant quelconque, les arcs rEduits Equivalents k un cercle
donnE sont de six especes diffErentes, qu'on peut distinguer entr'elles par les sym-
boles (PP- 1 ), (QQ- 1 ), (PQ), (PQ- 1 ), (P~ 1 Q), (P-^Q- 1 ), qui indiquent les
diffErentes parties du contour de 2, sur lesquelles se trouvent les points d'entrEe
et de sortie de 1'arc que Ton considbre. Lorsque N est carrE, les cercles
230
M^MOIKE SUR LES AQUATIONS MODULAIKES.
[Art. 8.
extremes restent exclus de cette classification; on pourrait, au besoin, les
exprimer par les symboles (p Q), (p Q~ l ), (q P), (qP~ l ), (r P- 1 ), (rQ- 1 ), (r'P),
(r 'Q). II est aussi convenable de distinguer entre deux symboles tels que (P, Q)
et (Q, P), pour pouvoir indiquer le sens dans lequel 1'arc reduit est censd d'etre
parcouru. Cela pose", la table suivante fera connaitre 1'espece de 1'arc rdduit qui,
dans une fraction continue quelconque, correspond h, un quotient donnd.
r impair; s = 1, 2, ..., p. r 1.
r pair; a= 1, 2, ..., fi r 1.
f r-l
fr
&r
r -2se r
+ 1
+ 1
(Q- 1 P)
(P-' P)
+ 1
-1
(Q- 1 P- 1 )
(/>-' P)
-1
+ 1
(QP)
(P-> P)
-1
-1
(Q P- 1 )
(P P-')
( r-i
tr
Or
r -2af r
+ 1
+ 1
(P- 1 Q)
(Q- 1 Q)
+ 1
-1
(P- 1 Q- 1 )
(QQ- 1 )
1
+ 1
(PQ)
(Q- 1 Q)
-1
-1
(P Q- 1 )
(Q <?-')
Re"ciproquement, e"tant donnd le tracd des arcs re"duits, Equivalents ^, un
cercle quelconque, la table servira pour retrouver la fraction continue, et, par
consequent, les Equations des cercles re*duits. Mais on peut obtenir le meme
rdsultat sans faire usage de la table. Posons
peut se mettre, et cela d'une
les exposants e^, e z n z , ..., e 2t /m 2i , (dont le premier et le dernier peuvent s'evanouir)
etant les memes nombres qui se pre'sentent dans le deVeloppement
1 P \
1,
Q -
1,2
\ * \
2,
1
'
0,1
et observons que toute
substitution
normale
a, /3
7 ,S
P
maniere unique, sous la forme
P
'
= 2e
D'un autre c6td, un arc r^duit qui se termine en P, P~ l , Q, Q' 1 est toujours suivi
par un arc re"duit qui commence en P-*, P, Q- 1 , Q; et les substitutions qui corre-
spondent it ces quatre cas sont respectivement
II suit de Ik que, pour avoir les exposants e 1 n 1) e 2 /u 2 , ... , il suflira de compter
les arcs rdduits, en faisant attention a. leurs points d'entrde et de sortie. Ainsi,
Art. 9.] MEMOIRE SUR LES EQUATIONS MODULAIEES. 231
par exemple, dans le cas d'un determinant non-carre, supposons qu'on commence
la pe"riode avec un arc reduit (6) qui prend son origine en Q ou Q' 1 et qui se
termine en P i ; /Uj sera le nombre des arcs reduits, y compris (0) lui meme, qu'on
aura a parcourir avant de venir a un arc qui se termine en Q ou Q~ l ; supposons
encore que le premier arc reduit, qui n'aboutit pas en P\ se termine en Q ( * ; / 2
sera le nombre des arcs reduits qui se terminent en Q f %, avant qu'on arrive k un
arc qui aboutit en P ou P~ l ; et ainsi de suite.
On remarquera que les arcs des deux premieres especes correspondent aux
quotients interme'diaires
tandis que ceux des quatre dernieres correspondent aux quotientes complets 6 l ,
-g- .... On peut aj outer que tout cercle qui coupe le cercle x 2 + y* = l est un
cercle reduit d'un de ces quatre especes, et re"ciproquement.
9. Pour faire 1'application de ce qui precede aux fonctions modulaires, soit
toujours <a = x + iy, et posons, avec M. Hermite, k 2 = (/> 8 (&>), H* = \p (w) ; faisons
aussi $ 8 (>) = ^ + X+iY, >p (to) = ^ X iY, X et Y dtant des quantity's rdel-
les. A chaque point w du plan xy ou, si Ton veut, & chaque point seulement de
1'espace reduit 2, faisons correspondre le point X+ i Y du plan illimite' X Y ; c'est
pour mettre en Evidence la syme'trie des figures, que nous avons choisi pour
origine des axes rectangulaires OX, OF le point ( 8 (w) = \p (w) = -|, qui a pour
correspondant le point ta = i. On sait que, dans une telle transformation, les
parties infinitesimales correspondantes des deux figures sont, en gdn^ral, sem-
blables. Dans le cas actuel, il n'y a exception que pour les trois points singuliers
p, q, r, auxquels correspondent respectivement les points ( -g, 0), (^, 0), et 1'infini
du plan X Y. De plus, si 1'on ne considers que 1'espace reduit, la correspondance
sera parfaitement de'termine'e ; de sorte qu'a chaque point reel de 2 il correspondra
toujours un seul point reel de XY, et re"ciproquement.
Maintenant il est facile de voir que, si Ton suppose re"elles les quantite"s a, b,
c, d, x, y, liquation
c + d(x + iy)
= -- -
entralne les deux suivantes
a d,
De cette seule observation on tire imme'diatement le tbdoreme que voici :
232
MEMOIRE SUB LES AQUATIONS MODULAIRES.
[Art. 11.
' Tous les cercles rationnels du plan xy sont repr^sent^s, dans le plan X Y,
par des courbes algebriques.'
'Les Equations des ces courbes se deduisent des Equations modulaires en
posant k* = \ + X + i Y, X 2 = + X - i Y.'
10. Considerons d'abord quelques cas particuliers, et designons par A lt A 2
les points (^, 0), ( ^, 0). Le cercle x 2 + y* = x, et les droites x = Q, x = l sont
represented dans le plan XY par les trois parties de 1'axe des X, depuis + oo
jusqu'h, A 1} depuis A^ jusqu'a A 2 , et depuis A 2 jusqu'a oo. L'axe des Y depuis
GO jusqu'a + oo, correspond au cercle a; 2 + y 2 = 1, qui est le seul dont la demi-
circonfdrence soit entierement comprise dans 1'espace 2. Enfin, les deux cercles
equivalents x z + y 2 = +2x, et les deux droites e'quivalentes 2x= +1, sont repre"-
sentes respectivement par les cercles (X + ^) 2 + Y 2 = 1, (X J) 2 + Y 2 = 1.
On voit que les cercles rationnels de determinant + 1 sont repr^sentes par
des droites et des cercles dans le plan XY; ce sont les seuls cercles rationnels
qui jouissent de cette propri^te". Ces cercles divisent 1'espace 2 en douze
parties distinctes ; la consideration de cette division, et de la division correspon-
dante du plan XY, est tres importante pour la theorie des transformations du
premier ordre. On peut aussi remarquer que deux points, syrndtriques par
rapport a une droite de determinant + 1, ou inverses par rapport a un cercle de
determinant +1, sont remplaces dans le plan XY par deux points qui ont la
me"me propriete" par rapport k la ligne correspondante.
11. Ceci suffit pour les cercles de determinant + 1, tant proprement qu'im-
proprement primitifs. La table suivante, dans laquelle nous avons posd
donne les resultats correspondants pour les determinants 2, 3, 4, 5.
n = 2
Fraction continue
Cercles reduita
Courbe modulaire
[2,2]
(x- 1) 2 4Y = 2
(x+l) 2 +y 2 = 2
^sj^-r*)
[4, -2]
(a:-2) 2 + 2/ 2 =2
af + yi^ 2
(x+2Y + y"'=2
R[= 16jfJ|
[-2,4]
2(x + l) z +2y 2 =l
2(a5-l) 2 + 2y 2 = 1
2*' + 2y*= 1
J%= 16 Rl
Art. 11.]
MEMOIRE SUB LES EQUATIONS MODULAIRES.
n = 3
233
[4, -4]
2? + y>= 3
a; 2 + f = 3
=
[2, 2, -2, 2]
(a5+l) 2
3(o;+l) 2
3(a;-l) 2 +32/ 2 =l
-^-l)-(/f 2 -l) 4 =
[2, -2, 2, 2]
-l =0
= 4
co = 2 + 11
1
ti
3(x' l +y ! )-2x~l =
= 2" ( F 2 -X 2 ) ^ 4 + 2 8 . 3 . 7 .
=
_
( as _ 2 ) 2 + 2 /> =
=
8a; + 4 =
-2 + 12
=
4a:-l =
= o
VOL. II.
4a; 3 =
+3 =
8a; + 3 =
4x+3 =
Hh
234
MMOIRE SUB LES EQUATIONS MODTTLAIKES.
[Art. 12.
n = 5
[4,4]
[(1 + 4 K*)*- 16 X*]
= 5
[6, -2, 2, -2]
=
6a;+l=0
10a;+5=0
[-2, 2, -2, 6]
63+1 =
2-l=0
X [(1 + 5J) (2 5 1%- 7
-(2.
=
-^) 2 + 3 . 2 8 7 2 )
2*-2 =
[2, -2, -2, 2]
2(c 2 +y 2 )+2a;-2 =
=
12. Maintenant, pour dclaircir le th^or^me gdn^ral de 1'article 9, il faut
rappeler quelques rdstdtats relatifs aux Equations modulaires.
(i) Soit N un nombre impair ; ddsignons par F (k 2 , X 2 , N) = 1'equation
modulaire normale pour les transformations du N ikme ordre, dans laquelle, lorsque
N admet des diviseurs carrds, nous supposerons qu'on ait supprimd les facteurs
correspondants aux transformations d'un ordre inferieur. En posant k- = 4> 8 (w),
on sait que les racines de cette dquation sont comprises dans la formule
X 2 = $ 8 (^ ; V dans laquelle y, y' sont des diviseurs conjugue's de N; k de"signant
un terme quelconque d'un systeme de rdsidus pour le module 7', et les nombres
y, y', k dtant assujettis h, ne pas avoir un diviseur commun. Cette dquation est
symetrique par rapport b- k 2 et X 2 ; en outre, elle ne change pas lorsqu'on substitue
k 2 et X 2 deux fonctions semblables, prises parmi les six fonctions anharmoniques
que voici
Art. 12.]
MEMOIRS STTR LES AQUATIONS MODULAIRES.
235
_X 2
A -v 2
X 2
X 2 -l
X 2 ' 1-X 2 ' X 2 -!' X 2
En substituant des fonctions dissemblables, on obtient le systeme complet des
dquations modulaires du N~ lism<i ordre. Nous dcrirons ces Equations, comme il suit :
(i)
(iv)
elles correspondent, comme on salt, aux six formes diffdrentes
0, 1 1,1 1,0 1,0 1,1 0, 1
1, ' 0, 1 ' 1, 1 ' 0, 1 ' 1, ' 1,1
que peut avoir une substitution de determinant impair par rapport au module 2.
Les Equations (v) et (vi) s'dchangent entr'elles lorsqu'on permute k 2 et X 2 ; les
quatre premieres sont symdtriques par rapport a k 2 et X 2 . Done en dcrivant
& 2 =<3+ X + iY, \ 2 = ^ + XiY, dans les Equations (i), (ii), (iii), (iv) on aura les
Equations rdelles de quatre courbes gdomdtriques, que nous appellerons ddsormais
la premiere, la seconde, la troisieme et la quatrieme courbe modulaire ; nous
observons toutefois que lorsque AT =3, mod 4, la quatrieme courbe se rdduit aux
deux points conjugues ( + ^, 0). Des Equations (v) et (vi) on de"duit les equations
de deux courbes imaginaires conjugue'es, dont nous ne nous occuperons pas
dans ce Memoire.
(ii) Soit N= 2 1 *. Dans ce cas, on a 1'dquation modulaire normale/(^ 2 , X 2 , 2^) = 0,
dont les racines sont donnees par la formule & 2 = < 8 (a>), X 2 = ^> 8 (53 ), en de"-
signant par h un terme quelconque d'un systeme de rdsidus pour le module 2^.
Cette Equation n'est pas symdtrique par rapport a k 2 , X 2 ; mais elle jouit des
deux propridtds de ne pas se changer (1) lorsqu'on dcrit rj pour k 2 , (2 C ) lorsqu'on
X 2
e*crit r- ^ pour X 2 . II suit de la que les trente-six substitutions anharmoniques
ne donnent que neuf Equations diffdrentes :
H h 2
236
MMOIBE SUB LES EQUATIONS MODULAIRES.
F
[Art. 12.
(ii) /(*, 1-X)=0,
(iv)
(viii)
(v) /(!-*, 1-\*) = 0,
(vii)
qui correspondent respectivement aux neuf formes diffdrentes
1,1
0,1
0,0
1,0
0,0
0,0
1,0
1,1
0,1
1,1
,
0,0
,
1,0
9
0,0
1
0,1
>
1,1
J
1,0
9
0,0
0,1
,
que peut avoir une substitution de determinant pair (la forme
0,0
0,0
e"tant exclue)
par rapport au module 2. En effet, les racines de ces Equations sont donne"es par
les formules :
(iv)
(viii) X 2 =
(ix)
Les Equations (i), (ii), (iii) sont syme'triques par rapport k k 2 et X 2 ; en y dcrivant
k* = \ + X+i Y, \ 2 = ^ + X iY, comme ci-dessus, on obtient les Equations de la
premiere, de la seconde, et de la troisieme courbe modulaire. Les equations (iv)
et (v), (vi) et (vii), (viii) et (ix) s'e"changent entr'elles, lorsqu'on ^change k 2 et X 2 ;
par consequent, elles ne fournissent que des courbes imaginaires conju^ees.
(iii) Soit enfin N=2> i n > n dtant impair. Dans ce cas encore il y a neuf
equations modulaires ; on les obtient successivement, en eliminant z de 1'^quation
F(z, X 2 , w) = 0, et de chacune des neuf Equations modulaires de 1'ordre 2^, dans
lesquelles on remplace X 2 par z. En effet, en ddsignant par N t et N 2 deux
nombres premiers entr'eux, et par J\ (k 2 , X 2 , NJ = 0, / 2 (k 2 , X 2 , N 2 ) = deux
equations modulaires appartenantes aux ordres N t et N a respectivement, le resultat
Art. 13.] MEMOIKE SUE, LES EQUATIONS MODULAIRES. 237
de 1'Elimination de z des deux Equations f t (k 2 , z, ^V,) = 0, ei>f 2 (z, X 2 , -AT 2 ) = 0, est
toujours une des Equations modulaires de 1'ordre .ZVi x N 2 . Si N l} N 2 avaient un
diviseur commun, cette proposition serait encore vraie ; mais le rEsultat serait
encombre de facteurs etrangers, qu'on peut assignor d, priori, et qui appar-
tiennent a des transformations d'ordre inferieur. Dans le cas actuel, on vErifie
facilement que les Eliminations indiquees fournissent le systeme complet des
Equations modulaires pour les transformations de 1'ordre 2' l xw. Les trois
premieres de ces equations sont les seules symEtriques, les autres Etant conjuguEes
par couples. On en deduit (comme dans le cas precedent) les Equations de trois
courbes modulaires rEelles, et de trois couples de courbes imaginaires conjuguEes.
Le theoreme suivant rEsulte de cette discussion :
' En dEsignant par N un nombre quelconque positif, les cercles proprement
primitifs de dEterminant N, qui appartiennent aux classes subalternes (^4), (B), (C),
sont reprEsentes respectivement dans le plan (XY] par la premiere, la seconde, et
la troisieme courbe modulaire.'
'Lorsque N = \, mod 4, les cercles improprement primitifs sont reprEsentes
par la quatrieme courbe.'
13. Soit, en premier lieu, ^un nombre non-carrE. Les courbes modulaires
ne peuvent rencontrer la ligne droit & 1'infini que dans les deux points cycliques
imaginaires ; de plus, aucune de ces courbes ne peut avoir une branche reelle pas-
sant par A! ou A 2 : toutefois le point A l peut appartenir, comme point conjuguE,
k la seconde courbe, le point A 2 & la troisieme, et tous les deux a la quatrieme.
Chacune des trois premieres courbes est composEe de \H =^<rh parties fermEes,
entierement distinctes entr'elles, et dont la forme gEnErale est celle d'une spirale
lemniscatique entrelacEe, qui s'entortille alternativement autour des deux points
A ly A 2 . Chaque spirale peut etre considErEe k volontE comme representant, soit
une classe subalterne de cercles proprement primitifs, soit un cercle unique choisi
arbitrairement dans cette classe subalterne. De m6me, lorsque N=l, mod 4, la
quatrieme courbe modulaire est composEe de //' = </ h' spirales, qui reprEsentent
respectivement les H ' classes subalternes des cercles improprement primitifs.
Soit, en second lieu, N=n 2 un nombre carrE. DEsignons par A le nombre
des solutions de la congruence a 2 + 1=0, mod 2 n, de sorte qu'on ait A = 0, si n
est pair, ou divisible par un nombre premier de la forme 4 m + 3. Comme dans
le cas prEcEdent, chaque courbe modulaire est composEe d'un certain nombre de
spirales distinctes ; mais ici chaque spirale doit passer soit par le point A lt soit
par le point A 2 ; ou bien elle doit avoir un point a 1'infini ; de plus une spirale
peut 6tre simple, ou multiple ; c'est u dire qu'elle peut satisfaire a ces conditions
238 MEMOIRE sun LES EQUATIONS MODULATRES. [Art. 14
soit une fois, soit plusieurs fois. Dans chacune des trois premieres courbes il y a
A spirales simples, et ^ (// 3 A) = | (h A) spirales doubles; chaque spirale
ayant soit un point, soit deux points, a 1'infini dans la premiere courbe, et passant
soit une fois, soit deux fois, par le point A^ dans la seconde courbe, et par le point
A 2 dans la troisieme courbe. Enfin, dans la quatrieme courbe, qui n'existe que
lorsque n est impair, il y a A spirales triples, ^ (jH 7 3A) = ^(h' A) spirales
sextuples ; chaque spirale ayant, soit un point, soit deux points a 1'infini, et
passant, en outre, soit une fois, soit deux fois, par chacun des deux points A ly A 2 .
Une spirale simple repre'sente une seule classe subalterne de cercles ; les
deux arcs re'duits, qui appartiennent aux cercles extremes, e'tant repre'sente's par
les parties de la spirale les plus voisines, de part et d'autre, soit du point a Tinfini,
soit de celui des points A v A z qui appartient a la spirale. Une spirale double re-
pre'sente, au contraire, deux classes subalternes, comprises dans la mdme classe, mais
non e'quivalentes entr'elles ; les deux parties de la spirale, qui correspondent a ces
deux classes subalternes, e'tant se'pare'es Tun de 1'autre par le point double A l ou
A 2 , ou bien par les deux points a 1'infini. On remarquera que lorsqu'une spirale
double .a deux points a 1'infini, ces deux points sont toujours distincts, ayant
chacun une asymptote a distance finie ; et de meme, lorsqu'une spirale double
passe deux fois par le mme point A, les points extremes de chacune des deux
parties dans lesquelles elle est divise'e, bien que re'unis au point A, appartiennent
a deux branches qui s'y croisent a un angle fini. Les spirales triples et sextuples
de la quatrieme courbe modulaire donnent lieu a des observations tout-a-fait
semblables, que nous pouvons passer sous silence.
Quelque soit le nombre N, toutes les courbes modulaires sont syme'triques
par rapport a Faxe des X ; la premiere et la quatrieme sont aussi symetriques
par rapport a 1'axe des Y; la seconde et la troisieme sont syme'triques entr'elles
par rapport a ce me'me axe; de plus, ces deux courbes sont les inverses de la
premiere par rapport aux cercles Rl = l, Rl = l; enfin, lorsque N=l, mod 4, la
premiere courbe se change dans la quatrieme par la substitution X = i Y, Y iX.
Ajoutons que dans chaque courbe modulaire, quelques unes des spirales sont elles-
mSmes syme'triques par rapport a 1'axe des X ; pareillement, dans la premiere et
dans la quatrieme courbe, il y a certaines spirales qui sont syme'triques par
rapport a 1'axe des Y, ou bien qui ont le point pour centre. Mais pour abre"ger,
nous supprimons la discussion de ces particularite's, qui dependent de 1'ambiguite'
des classes et de la re'solubilite de 1'dquation t 2 - Nu z = - 1 en nombres entiers.
14. Imaginons que dans le plan XY on ait ope're' une coupure, suivant 1'axe
des X, depuis A z jusqu'a - oo, et depuis A^ jusqu'h, + oo, en y comprenant les
Art. 14.] MEMOIRS SUR LES EQUATIONS MODULAIRES. 239
deux points A 1} A z eux-mdmes. Chaque spirale modulaire sera divise"e dans un
certaine nombre de parties, que nous nommerons les spires de la spirale, et que
nous considererons se"paremment. II est Evident que, si Ton remplace le cercle
rationnel, qui est represents" par la spirale, par 1'assemblage des arcs re"duits qui
lui sont equivalents, chacun de ces arcs reduits aura pour image ge"ome"trique
dans le plan XY une certaine spire de la spirale. Done, en ne"gligeant, pour
abreger, les spires qui ont un point a 1'infini, et celles qui passent par un des
points A 1} A z , on pourra en distinguer six especes differentes, dont voici la
description generale.
Une spire de premiere espece (P~ l P) prend son origine dans un point du
segment (mA 2 ) et se dirige vers la partie inferieure du plan; ensuite elle
traverse le segment A 2 pour passer dans la partie supe"rieure du plan, et aboutit
a un point de ( oo A^, different en ge"ne"ral du point d'origine. Les spires de
seconde espece (Q~ 1 Q) sont Byrne triques, par rapport a 1'axe des Y, aux spires
de premiere espece.
Une spire de troisieme espece (PQ) commence dans un point de ( oo A^) et
aboutit & un point de A-^, en restant toujours au-dessus de 1'axe des X. Une
spire de quatrieme espece (PQ~ l ) prend son origine dans un point de ( oo A 2 ) et
se dirige vers la partie supe"rieure du plan, mais ensuite elle traverse A 2 A l} pour
continuer son chemin au-dessous de 1'axe des X, et se termine a un point de
(A^).
Enfin les spires de la cinquieme et de la sixieme espece, (P^ 1 ^) et (P~ 1 Q~ 1 ),
sont respectivement symetriques par rapport a 1'axe des X, aux spires de la qua-
trieme et de la troisieme espece.
On voit que les differentes especes se distinguent entr'elles par des caracteres
qui se rapportent a la ge'ome'trie de situation. Et il r^sulte de ce qui a e'te' dit
dans les articles 5, 6, 7, que, si la courbe modulaire e"tait une fois de'crite, on
n'aurait qu'a suivre des yeux le trace" d'une spirale quelconque pour avoir, en
premier lieu, la fraction continue qui correspond a cette spirale, et pour retrouver
ensuite le systeme complet des formes quadratiques re"duites, repre"sentdes re-
spectivement par les diffe"rentes spires dont la spirale se compose. Voici le prin-
cipal re"sultat auquel nous voulions parvenir dans ce Me"moire. Nous nous pro-
posons, dans une autre occasion, d'exposer quelques applications, que nous avons
faites des principes prdce"dents, et qui nous semblent propres a montrer le parti
qu'on peut espdrer d'en tirer en diverses questions d'arithmetique et de la thdorie
des fonctions elliptiques.
240 MEMOIKE SUR LES EQUATIONS MODULAIRES.
[The following abstract of the preceding paper was published in the Transunti,
Ser. iii. vol. i. pp. 68-69.]
II Socio L. Cremona presenta una Memoria : Sur les equations modulaires
del signer Henry J. Stephen Smith professore aH'Universitk di Oxford.
L'auteur se propose dans ce Mdmoire d'dtudier la liaison qui existe entre les
Equations modulaires, et les formes quadratiques binaires & determinant positif.
Cette liaison, qui peut-etre n'avait pas dte remarqude jusqu'ici, est a la verite"
purement analytique ; mais on parvient h, la mettre en Evidence, en regardant
d'une part les Equations modulaires comme ddfinissant des courbes gdomdtriques,
et en se servant, de 1'autre part, d'une nouvelle representation geometrique des
formes quadratiques, dont voici le principe. A chaque forme (a, b, c) du determi-
nant positif D = 6 2 ac, on fait correspondre un cercle a + 2 bx + c (x 2 + y 2 ) = 0, trace
dans un plan (xy) dont toutefois on ne considere que la partie situde au dessus
de 1'axe des x. On appelle espace reduit la partie de ce plan comprise entre les
deux' droites x= 1, mais extdrieure aux cercles x 2 + y 2 =+x; et Ton regarde
comme arc reduit tout arc de cercle qui se trouve en dedans de 1'espace rdduite.
Cela pose", la pdriode des formes rdduites appartenantes k une classe donnde est
reprdsentde par les arcs rdduits correspondants, dont 1'assemblage forme une
ligne en apparence brisde, mais qui peut 6tre envisagee comme continue. En
iK'
suivant la notation usuelle des fonctions elliptiques, qu'on pose ta = x + iy = -^r,
k~ = < 8 (o>) = | + X + i Y, et qu'on fasse correspondre aux points x + iy de 1'espace
reduit les points X+iYd'un nouveau plan illimitd (XY), par le moyen de liqua-
tion | + X + i Y=<p*(x + iy), dans laquelle on suppose reelles les quantitds x, y,
X, Y. Les lignes brisdes correspondantes aux diffdrentes classes de formes
quadratiques de determinant D se trouveront reprdsentees dans le plan (XY) par
autant de spirales formdes distinctes, dont 1'ensemble formera une courbe gdomd-
trique complete. L'dquation cartesienne de cette courbe sera simplement
F( + X+iY, + X-iY) = 0, en ddsignant par F(k 2 ,X 2 ) = une des dquations
modulaires pour les transformations elliptiques de 1'ordre D. On tire de Ih, ce
rdsultat remarquable que si, sans penser aux formes quadratiques de determinant
D, on trace la courbe modulaire, on aura sous les yeux une image exacte du
systeme complet de ces formes ; de sorte que, par un simple ddnombrement des
spirales, et des diffdrentes spires dont chaque spirale se compose, on pourra ddter-
MEMOIRE SUE LES EQUATIONS MODTJL AIRES. 241
miner (1) le nombre des classes non equivalentes ; (2) le developpement en fraction
continue, propre & chaque classe, developpement qui donne, comme on sait, le
systeme complet de formes re"duites, representantes de la classe. Le cas d'un
determinant carre' est compris (sauf quelques particularite's) dans la the"orie
gene"rale.
II est bon de remarquer que les notions arithmetiques de classe, d' 'equivalence,
et de forme reduite, doivent subir une legere modification pour les adapter & la
theorie des fonctions modulaires. On donne dans le Mdmoire toutes les expli-
cations necessaires a cet egard.
Au lieu des equations modulaires entre k 2 et X 2 , on aurait pu introduire les
equations plus simples de Jacobi entre fk et ^/\ ; mais la thdorie arithmdtique
en deviendrait un peu plus compliquee.
Les mdmes principes peuvent s'appliquer avantageusement a d'autres ques-
tions de la the'orie des fonctions elliptiques, parmi lesquelles on peut signaler le
probleme pose par Jacobi dans la note ajoutee ^ la page 75 des ' Fundamental
L'auteur espere de soumettre prochainement h, 1' Academic la solution qu'il pense
avoir trouve de ce probleme difficile.
VOL. IL II
XXXVI.
ON THE SINGULARITIES OF THE MODULAR
EQUATIONS AND CURVES.
[Proceedings of the London Mathematical Society, vol. ix. pp. 242-272. Head February 14
and April 11, 1878.]
1. IT is proposed, in this paper, to examine the characteristic singularities
of the modular equations and curves. The method employed is applicable to
all the modular equations hitherto considered by geometers ;* but, for brevity,
the discussion is confined to the equations containing the squares of the modules,
and to the case in which the order of the transformation is uneven.
* The modular equations considered by Jacobi in the ' Fundamenta Nova ' are (2) the equation
between u = <$> (<o), and v = < (li), (see Art. 4 of this paper, equations (3) and (4),) and (5) the equation
between 8 and 8 , of which the characteristics are discussed here. M. Kronecker, in his researches on
the modules which admit of complex multiplication, would seem to have also employed (3) the equation
between w 2 and *, and (4) the equation between u* and v 4 . (See the account of these researches in the
Reports of the British Association for 1865, pp. 332 and 358 [vol. i. pp. 301 and 336] ; see also Pro-
fessor Cayley, ' Phil. Trans.,' vol. clxiv. p. 450.) M. Joubert (' Comptes Rendus,' vol. xlviii. pp. 290-
294) was the first to consider (6) the equation between 8 (1 u*) and w 8 (1 v 8 ). Dr. Felix Miiller,
in his Inaugural Dissertation (Berlin, 1867), drew attention to the equation (7) between
and the discussion of this equation has recently been resumed by Professor Klein \' Mathematische
Annalen,' vol. xiv. p. 112, May 1878). These geometers have expressed T(a>) and T(Q) rationally in
terms of a third indeterminate, in the cases N 2, 4 ; 3, 5, 7, 13 ; the deficiency of the equation (7)
being zero in these six cases ; but neither of them has given any example of the equation in its explicit
form. Writing x for T(u>) and y for T(H), I find, when N=3,
x (x + 2 7 . 3 . 5'Y+y (y + 2 7 . 3 . 5 s ) 3 - 2 w ary + 2". 3 2 . 31 tfy 1 (x + y)- 2 2 . 3 s . Q907xy (* 2 + y 1 )
+ 2 . 3*. 13 . 193 . 6367*y + 2 8 . 3 6 . 5 5 . 4471s;?/ (x + y)- 2 18 . 5". 22973xy = 0.
The equations (2) ... (7) are symmetrical with regard to the two indeterminates, and, the number N
Art. 2.] SINGULARITIES OF MODULAR EQUATIONS AND CURVES. 243
2. We represent by q, the square of the modulus of a given elliptic function ;
by p, the square of the transformed modulus, the transformation being primary,*
and of the uneven order N; by
F(p,q,l) = . ......... (1)
the modular equation subsisting between p and q ; in connection with this
equation, we consider the modular curve C, of which the trilinear equation
F(a, P, y) = (2)
is obtained by writing = - , q =- We denote by P, Q, E the vertices of
7 7
the triangle afiy ; and by S the point a = /3 = y, or p = q = l ; the three inter-
sections (PS, QE), (QS, EP), (ES, PQ) we represent by a, I, c. We regard
p and q as the parameters of two pencils of lines a py,
fi qy, of which the centres are Q and P, and between the
rays of which a correspondence is established by the equation
(1) ; we observe that, to the values 0,1, oo of either para-
meter, there answer in the two pencils respectively the rays
QE, QS, QP ; PE, PS, PQ ; and that, by a known property
of the modular equation, the pairs of rays (QE, PE), (QS, PS), (QP,
PQ) are corresponding rays in the two pencils, either ray of any of these
being uneven, they are of the order A +5 (Art. 3) in the indeterminates separately, and of the order 2 A
in the indeterminates jointly. I have recently found that the Eulerian functions x (<">) an( l X (^)
defined by the equation *m
x() = 4/ 2 - e " x nru:- (- i) '""),
satisfy an equation (1) having the same properties ; for the cube x (<) this had already been shown by
M. Keenigsberger (' Borchardt's Journal,' vol. Ixxii. p. 182 sqq). Writing M X = X (<">)> v \ X (^) we
have, in the cases N= 5, 7, 11,
u\ + $u\ v{ 23 u l v l + i> 6 = 0,
r " = 0,
The function \(<o) is a twenty-fourth root of w s (l M 8 ) ; the formulae relating to its linear trans-
formation have been given by M. Hermite ('Comptes Rendus,' 1858, vol. xlvi. p. 721). In respect of
simplicity of form, the equations (1), (2) ... (7) appear to arrange themselves in this numerical order ;
but, in respect of simplicity of algebraical theory, the order is reversed, as the deficiency decreases from
(1) to (7).
A transformation
a, b
c, d
1.0
0, 1
a,
c, d
of the uneven order N is primary when it satisfies the congruence
, mod 2. For the theory of the elliptic multiplier it is convenient to fix the signs of
a, b, c, d by tlie additional condition a = 1, mod 4; but for our present purpose this restriction is
unnecessary.
I i 2
244 ON THE SINGULARITIES OF THE [Art. 4.
pairs being the only ray answering to the other ray of the same pair ; the
modular curve C is the locus of the intersections of corresponding rays in
the two pencils.*
We denote by ra and n the order and class of C ; by H its deficiency ;
by K and / its cuspidal and inflexional indices ; by D and T its discriminantal
order and class; by F(p) = E(q) the highest, by E'(p) = E'(q) the lowest,
exponents of p and q which present themselves in the equation (1).
ARTS. 3-8. N not divisible by a square.
3. We confine ourselves, in the first instance, to the case in which AT is
not divisible by any square ; and we represent, in this case, by A the sum
of the divisors of N which surpass */N ; by B the sum of the divisors of N which
are less than VN; by v the number of divisors of either sort, so that Zv is the
whole number of divisors of N. We then have the formulae
(i.) m = 2A,
(ii.) n = 3 A -B,
(iii.)
(iv.)
(v.) I-K=3(A-B),
(vi.)
(vii.) T-D = (A-B)(5A-B).
To these we may add the equations
(viii)
'
Of these formulae (i.) and (viii.) are well-known ; of the remainder, it will
suffice to attend to (iii.) and (iv), because, when the values of m, H, and K are
given, the values of n, I, D, and T are known from the equations of Pliicker.
4. The Deficiency. The demonstration of the formulae (iii.) and (iv.) de-
pends on the simultaneous expression of the modular parameters p and q as
* Iii a paper which I hope shortly to lay before the Society, I have discussed, with some fulness
of detail, the relation of the algebraical singularities of the parametric equation F(p, q, 1)=0 to the
characteristic singularities of the curves of which the equations are included in the formula
^\S' 2)' = 0> ^> B, C, D being the equations of straight lines. The discussion comprises an
examination of the effect of any quadric transformation on the singularities of a curve.
Art. 4.] MODULAR EQUATIONS AND CURVES. 245
transcendental functions of the quotient of the periods of the given elliptic
function. As we have already developed these considerations elsewhere,* we
* In an Introduction (now in the press) to Mr. J. ~W. L. Glaisher's Tables of the Theta Functions
[No. XLHI. of the present volume]. The method indicated in this article has been already employed by
Professor Klein, in the paper to which reference has already been made (' Math. Ann.,' vol. xiv. p. Ill;
see especially 6-8, 7-13), and by Professor Dedekind, in a letter addressed to M. Borchardt (June
1877, see 'Borchardt's Journal,' vol. Ixxxiii. p. 265, especially 14 and 7). In the year 1873, I sub-
mitted to the Academy of Sciences in Paris a Me"moire ' Sur les Equations Modulaires ' (' Comptes Eendus,'
August 1873, vol. Ixxvii. p. 472). In this Memoire (which was ultimately presented, without altera-
tion, to the Accademia dei Lincei, and was printed in their ' Atti,' vol. i. series 3, p. 136, February
1877) [vol. ii. pp. 224, etc.], I had employed the same method (see Arts, ii, ix, and x of the
Memoire) to establish the relation which exists between the modular equations of order N and the
binary quadratic forms of the positive determinant N. The M6moire was devoted to that theory alone,
as I attached more importance to it than to any other result relating to the modular equations at
which I had then arrived. But I had already in the year 1873 obtained : (i) a proof of the existence
of the modular equations, simpler perhaps than that of M. Dedekind, and based solely on those elementary
properties of the function <j)(ca), which were deduced from the theorem of Fourier by Cauchy and
Poisson, without employing any elliptic formulae ; (ii) a determination in the simpler cases of the
Pliickerian characteristics of the modular curves ; (iii) a solution of one part at least of the problem
relating to complex modules, proposed by Jacobi in Art. 32 of the 'Fundamenta Nova.' I communicated
to Professor Cayley, in 1873, the formulae for the deficiency of the equations (2) ... (5) when N is an
uneven prime (see his Memoir on the Transformation of Elliptic Functions, presented to the Royal
Society in that year) ; the formulas for the cuspidal index I obtained by transforming into normal
developments the parametric developments which give the deficiency (see Art. 5 of this paper) : thus,
the order of the curves being known, all their Pliickerian characteristics were determined. The case
when N is a product of uneven primes presents no greater difficulty than the case when N is a prime ;
and I had (in fact) obtained the formula for this more general case as early as 1873. The case when N
is divisible by a square, and still more the case when N is itself a square, appeared to involve some
difficulty ; and these I left untouched till the spring of the present year, when I found that the intro-
duction of the arithmetical function f ( see Art. 9 of this paper) caused the supposed difficulty to dis-
appear. To the more exact determination of the indices characteristic of each special singularity of the
modular curves, I was guided by the methods employed in a former paper on the Higher Singularities
of Plane Curves.
A complete system of formulae, analogous to that given in the present paper for the modular
equation (5), I have already obtained for the equations (2), (3), and (4) ; with the equation (7), and
with the eight equations between corresponding powers of x (<") and x (&)> I have not advanced equally
far, but I have not found that they offer any peculiar difficulty.
In the Memoire ' Sur les Equations Modulaires," I have confined myself (as in the present paper) to
the equation (5) between the squares of the modules. At the time when the Memoire was written, I
was well acquainted with the characteristic property of the function T (to) ; viz. that it is unchanged
by any linear transformation of the elliptic functions ; and I even thought of employing it in the
Memoire instead of the function <i> 8 (to). I had conjectured (erroneously however) that the modular
curves T derived from the equation (7) would represent ordinary periodic continued fractions with
positive integral quotients, in the same way in which the modular curves G derived from the equation
(5) represent periodic continued fractions with even quotients. But I was deterred from employing
246 ON THE SINGULARITIES OF THE [Art. 4.
shall in this place assume the results of the discussion as known, and shall
confine ourselves to their application to the formulae (iii.) and (iv.).
Denoting by x and y real quantities, of which y is essentially positive, and
by o> the complex variable x + iy, which is thus subject to the restriction
that the coefficient of i in its imaginary part is positive, we define the function
(p (o>) by the equation
'
and we consider the A + B quantities O, of which the values are given by the
equation '
y, g being any two conjugate divisors of N, and k being any term of a complete
system of residues for the modulus g. We then have the fundamental theorem,
'If q = <j> B (w), the A + B corresponding values of p are included in the
formula ^ = ^ 8 ^ ;
or, which is the same thing,
the sign of multiplication extending to the A+B values of Q.'
It results from the discussion to which we have referred, that, if we regard
q as an independent complex variable, and represent its values in the usual
the equation (7) by the consideration that there was not a single calculated example of it ; indeed, at
that time I was not acquainted with the researches of Dr. Felix Milller, and did not know that the
equation had attracted any attention. I have since found that the curves T do not precisely represent
the reduced forms of Gauss, but instead a system of forms determined by a different regulative
principle. I am disposed to think (notwithstanding the considerations mentioned in the note on Art. 1),
that there is some advantage in continuing to regard the equation between the squares of the modules,
as the principal modular equation, rather than either the equation (1) or (7). At least, as far
as concerns the arithmetical theory to which the Memoirs relates, and which I have since extended to
the equations (1), (2), and (7), (the theory of the equations (3), (4), and (6) hardly requiring a separate
discussion), the modular curves (5) present phenomena in some respects simpler than those presented
by the curves (7), and in all respects simpler than those presented by the curves (1) ... (4).
Both in the M6moire and in this paper, I have given especial attention to the case in which N is a
square, because the solution of the problem of Jacobi for the transformations of order N depends on a
consideration of the spaces into which modular curves of order N* divide a plane. A note published
in the 'Transunti' of the Accademia dei Lincei (vol. i. p. 42, 7 Jan. 1877) [vol. ii. pp. 221 etc.],
contains what is in fact a solution of Jacobi's problem for the case N= 1 ; to this particular case of the
general problem, the attention of geometers had been called by M. Hermite in the note appended to
the second volume of M. Serret's edition (1862) of the Differential Calculus of Lacroix, pp. 421-425.
Art. 4.] MODULAR EQUATIONS AND CURVES. 247
manner by the points of a plane, p, considered as a function q, has no spiral
points other than the three q = 0, q = 1, q oo (it will be remembered that in the
plane of double algebra the infinitely distant is a point). Thus, if we cause
q to describe any closed contour, not including one of the three points 0,1, oc,
the values of p will undergo no interchange ; but each root of the equation
F (p, q, 1) = will return, when the contour is closed, to the same value which
it had at the beginning ; although the contour may include points (other than
0, 1, oo) at which two of the values of p become equal. But the case is different
if we cause q to describe a closed contour round one of the three exceptional
points. At each of these points all the values of^> become equal to one another,
and to the value of q indicated by the point. As the general result is the same
for each of the three points, it will suffice to attend to one of them only ; for
example, to the point q = 0. If, then, we cause q to describe a closed contour
round 0, the g values of p, or of the expression <p s (- -1 , which contain the
i/
divisor g in the denominator, change into one another cyclically, and thus the
~Zg = A + B roots of the equation arrange themselves in 2v cycles, corresponding
to the different divisors of N; or, which is the same thing, the developments
obtained by expanding the different values of p in series proceeding by ascending
powers of q are singular ; being, in fact, of the type
/
p = *q' + .............. (5)
Similarly, at the points + 1 and oo, we have singular developments of the types
p-l=\(q-iy> + ..., ......... (6)
1 = X(1)% .............. (7)
p \qs
As these are all the singular developments that can exist, we infer that, if
W(p) represent the number of the spiral points of p, each point being reckoned
with its proper multiplicity,
Substituting, in the equation of Biemann,
2H=W(p)-2E(p) + 2, ........ (8)
the value of E(p) (equation viii., Art. 3), and the value just obtained for W(p),
we find H
which is the equation (iii.).
248 ON THE SINGULAEITIES OF THE [Art. 5.
5. The Cuspidal Index. To determine the cuspidal index of C, we first
consider the developments (5) which appertain to the point R. Since N is
not a square, we cannot have g=g' ; ifg'>g, we have the normal development*
r* (/+< ..........
if g > g, we have the normal development
r
The branches corresponding to the developments (9) and (10) have, for their
cuspidal indices, g-1 and g' l respectively. Hence each of the lines QR, PR
is touched at the point R by a set of branches of which the aggregate cuspidal
index is B v.
In the same manner, it will be found that at the point S each of the lines
QS, PS is touched by a set of branches having the same cuspidal indices as the
branches which touch QR or PR at R.
Lastly, from the developments (7), we deduce normal developments of the
types g'
and from these we infer that the line PQ is touched at each of the points
P and Q by a set of branches of which the aggregate cuspidal index isAJB v.
Since C can have no other cuspidal branches, we find
which is the formula (iv.).
* If A, B, C represent straight lines forming a triangle, a development of the type
iu which a,, a t , ... are positive and increasing, and a, is greater than unity, is termed a normal
development ; A is, of course, the tangent to the branch, B is any line passing through the point to
which the development refers.
Art. 6.] MODULAR EQUATIONS AND CURVES. 249
6. The discriminant of F(p, q, 1). The values of m, E(p), E(q), E' (p),
E' (q) are inferred from the equation (1), by a method, due (as it would seem)
to M. Kronecker, of which examples are given in the Report on the Theory of
Numbers ('Reports of the British Association' for 1865, p. 349 sqq.).* This method
is also applicable to the discriminant of the equation (1) ;f i.e. to the expression
v(?) = n[^(Q 1 )-<m)] 2 , ........ (is)
where the sign of multiplication extends to every pair of values of Q. Let
(A + BY = A 2 + B 2 , ......... (14)
where A 2 comprises all the terms in (A + E) 2 which are greater than N, and B 2
all the terms which are less than N', as for the terms which are equal to .ZV
(of which the sum is evidently 2/jV), we divide them equally between A 2 and
2 ; thus, if g 1} g 2 , ...g t are all the divisors of N (unity and N included), we have
'iK={g r g e , g r g s >N;
K:(K:\g r g s , g r g,<N-
A 2 + B 2 =X: t 1 K:l ri
Applying the method of M. Kronecker, we find that the highest power of q in
V (q) is 2 A 2 A B, and that the lowest power of q is 2 B 2 A B. By a
known property of the modular equation,
hence V (q) must be divisible by 1 q as often as by q ; we have, therefore,
V(2) = (?-5 2 ) 2 *-^xx(<7); ....... (16)
where x (?) is a rational and integral function of q, not divisible by q or 1 - q,
and of the order 2A 2 -4B 2 + (A + B}.
By another property of the modular equation we have the identity
* [Vol. i. p. 325.]
t It has been applied by M. Koenigsberger (' Vorlesungen iiber die Theorie der Elliptischen
Funktionen,' vol. ii. p. 154) to the discriminant of the modular equations between u and v, in the case
in which N is not divisible by any square ; the result had already been given by M. Hermite in his
Memoire sur la Theorie des Equations Modulaires (' Comptes Rendus,' vol. xlviii. p. 1079). The discrimi-
nant of the modular equation between x 3 (<o) and x 3 (&) lias been similarly treated by M. Krause (' Math.
Annalen,' vol. sii. pp. 1-3).
VOL. II. K k
250 ON THE SINGULARITIES OF THE [Art. 7.
If therefore we write p+p' for p, and q 4- q' for q, the dialytic discriminant
of the bipartite binary quantic
(PW^XJ^, ^ l)
is symmetric with regard to q and g' ; i.e. it is of the form
[?<?'(?' -<Z)] 2B '-^x x fo<A ....... (17)
where x (?, <?') is symmetric and of the order 2A 2 1B 2 + A+B.
It will be observed that the order of the discriminant (17) is
as it ought to be ; and that the equation V (q) = is to be regarded as having
2B 2 AB infinite roots, and as having lost the same number of dimensions.
We may add that x (?) ia a perfect square. For, if q q , be any factor
of x (q), all the developments of p, proceeding by powers of q q , have integral
exponents ; the exponent of q q in x (<?) is the sum of the discriminantal
indices of these developments taken in pairs ; and this sum is always even.
7. The discriminantal indices of P, Q, R, 8. We next examine the
discriminantal indices in the curve C of the points P, Q, R, S. Representing
these indices by D (P), D (Q), D (R), D (S), we have
To establish these formulae, it will suffice to consider the points P and R.
At P we have v superlinear branches of the aggregate order A B touching
PQ we may symbolize the branch of which (11) is the normal development
by (g', g), where gg' = N, and g'>VN>g. The discriminantal index of the
branch (g', g) taken by itself is g' (g' g 1) ; the joint discriminantal index of
the two branches (g', g) and (g\, g^) is 2g' (g\ g^, if g' > g\, and consequently
g<gi- Hence we have
the summations extending to all values of g' and g\ which satisfy the inequalities
g>VN, g\<g', g\>VN. ........ (20)
Attending to these inequalities, we find
Art. 8.] MODULAR EQUATIONS AND CURVES. 251
and substituting these values in (19), we obtain the value of D (P), given by
the formula (18).
Again, at the point R, we first consider the branches touching PR, the
normal developments of which are of the type (10). For the discriminantal
index of these branches, taken singly and in pairs, we have the expression
the summations extending to all divisors g and g 1 which satisfy the inequalities
g>VN, g l<g , g L >VN. ....... (21)
The discriminantal index of the branches touching QR is evidently the
same as that of the branches touching PR and as the aggregate order of
each of the two sets of branches is B, they intersect one another in B 2 points,
and the part of D (R) which arises from their crossing one another at R is 2B 2 .
We have, therefore,
but, attending to the inequalities (21), we also have
?g = A,
whence, in accordance with (18),
8. The intersections with C of the polar curves of P and Q. Since the
branches which touch PR at R are of the aggregate class A B, the line PR,
considered as a tangent drawn from P, counts A B times as a tangent at R.
Similarly PS counts A B times as a tangent at S. Again, since the branches
which touch PQ at P and at Q are of the aggregate order A B, and of the
aggregate class B, PQ, considered as a tangent drawn from P, counts
A B + B = A tunes as a tangent at P, and B times as a tangent at Q. Thus
the three lines PR, PS, PQ count as
tangents from P ; i.e. no other tangents can be drawn to C from P. Again,
the polar curve of P intersects C at P, Q, R, S, in
D(P) + A = 2A*-A 2 ,
D(Q) + B
points respectively; or, in all, in 4.4 2 + 4f? 2 2A 2 3A B points. The whole
number of intersections of C by any one of its first polars is 2 A (2 A 1) ; hence
K k 2
252 ON THE SINGULARITIES OF THE [Art. 9.
the polar of P intersects C in
2A (24-1) -(44- + 4^-2^-34 -B) = 2A 2 -4B 2 + A + B
points, other than P, Q, R, S. These 2A 2 -4B 2 + A + B intersections correspond
in the discriminant of F(p, q, 1) to the factors of x(?)> of which the aggregate
order is the same.
As the intersections other than those at P, Q, R, S correspond to the
factor x (q), so also the intersections at R correspond to the factor <f B *- A - B , and
the intersections at S to the factor (1 qY s t~ A - s . But it is proper to observe
that the remaining intersections at P and Q (of which the aggregate number
is 4: A 2 2A 2 + B A) surpass the order of the remaining factor q' 2 S t -A-s O f
complete dialytic discriminant ; the difference
being, as it ought to be, equal to the difference between the number of inter-
sections of C by any one of its polars, and the order of the dialytic discriminant.
We reserve, for a future communication to the Society, a complete discussion
of the relations which subsist between the exponents of the factors of the dialytic
discriminant of any parametric equation, and the corresponding intersections
of the locus curve by the polars of the centres of the generating pencils.
The points, other than P, Q, R, S, in which C is intersected by the polar
of P, are all ordinary double (or it may be multiple) points, free from super-
linearity, and having tangents which do not pass through P. For C has no
super-linear branches beside those at P, Q, R, S, and the only tangents which
pass through P are PQ, PR, PS. The same thing is also evident from what
has been said in Art. 6 of the exponents of the factors of x (q)-
ARTS. 9-11. N divisible by a square.
9. Definition of certain Arithmetical Functions. We now pass to the
general case in which N is any uneven number whatever. Let
AT" rt a i nv n a s
J.1 U.j U>2 1*3 ">
a lt a 2 , ... being different uneven primes; let g, g', as before, be two conjugate
divisors of N, and let *\ be the greatest common divisor of g and g'. We resolve
g into the product of two factors 7^ and y 2 , of which y^ contains only those prime
divisors of g which do not occur in >j and g / ; and y 2 contains only those prime
divisors of g which do occur in i? and g'. Representing by f(z) the number
of numbers prime to any given number z and not surpassing it, we write
Art. 9.] MODULAR EQUATIONS AND CURVES. 253
and we observe that we have the equations
i 9 9'
each of these quotients being equal to
n(i-l),
if e denote any prime divisor of 17. We still retain the symbols v, A, B,
A z , B 2 ; but with extended significations, which we proceed to explain.
We define v by the equation
2 Z/(,), (22)
the sign of summation extending to every divisor g of N. We observe that,
in general, each term/(>/) occurs twice in ^f(i), because >j is the same for each
of any two conjugate divisors ; but that, if N= 6 2 is a perfect square, the term
f(ff) =f (&) occurs only once in 2/(>/).
Again, we define A and B by the equations
(23)
the summations extending to all divisors g of N which satisfy the inequalities
(23) respectively ; when N = 6* is a perfect square, we divide the term/' (&) =/(#)
equally between A and B. We thus have in every case
the summation extending to every divisor g of N.
Lastly, we define A 2 and B 2 by the equations
(24)
in which g l and g 2 are any two divisors of N (the same or different) which satisfy
the inequalities specified ; so that, if g t and <jr 2 are different, the t&crs\f'(g^f'(g^
occurs twice in A 2 , or in B s , as the case may be. If g i g :i = N', we divide the
double term Zf'(gi)f'(g2), corresponding to these two divisors, equally between
A t and B 2 ; if, in particular, N=6* is a perfect square, the single term [/(0)] a
is to be divided equally between A 2 and B 2 . It is evident that we have, in
every case, A 2 + B 2 = (A + Bf.
The sums 2/ = 2/(^), and A + B = 'Zf'(g), may be conveniently expressed
in terms of the prime divisors of N. Observing (1) that the terms of the product
+...a a ]
254 ON THE SINGULARITIES OF THE [Art. 10.
represent, after development, all the divisors of N, and (2) that
if g be a product of two relatively prime factors hi and h 2i we find
whence A+B = Nx Ill + . ........ (25)
Again, if we write f"(g) for f(i), and give to h lt h. 2 the same signification
as before, f"(g) satisfies the equation
whence we infer that
s
First, let a = 2/* + 1 ; then
') = 2
Secondly, let a = 2^ ; then
If therefore -ZV=n& 2/3+1 x lie 2 ?, where &,..., c, ..., are different prune numbers,
we have
(26)
It will be observed that the definitions which we have now given of the symbols
v, A, B, A 2 , B 2 , coincide, in the case in which N is not divisible by any square,
with the definitions of Arts. 3 and 6.
10. Case when N is not a square.- Excluding, for the present, the case in
which N is a perfect square, we have to show that, in all other cases, the
formulae of Arts. 3-8 hold without further modification. For brevity, vve shall
establish only a few of the assertions included in this general statement, as the
method to be pursued with regard to all of them is the same.
(i.) If <&(N) is the sum of the divisors of N, and e 1( e 2 ,... are the primes of
which the squares divide N, the order of the irreducible modular equation of
order N is
-l^Z
Art. 10.] MODULAR EQUATIONS AND CURVES. 255
(See M. Joubert, ' Comptes Rendus,' vol. i. p. 1041 ; Report on the Theory of
Numbers, loc. cit., p. 332*). But this expression has for its value NH ( 1 + - ) ;
V CL'
i.e. the order of the modular equation in p or in q is A + B (equation 25).
(ii.) If we write q = <f> s ((a), the A+B corresponding values of p are given
by the equation
ft \
in which k is any one of the ^g =f'(g) residues of g which are prime to n,
i
the greatest common divisor of g and g'. If, as in Art. 4, we cause q to describe
a closed contour round Q,f'(g) values of p, which answer to any given divisor
g, arrange themselves in f(n) cycles each containing roots ; and thus the
developments (5), which appertain to the simultaneous values p = Q, # = 0,
assume, in the general case which we are now considering, the form
^*?
|> -A <g} *+.;.; ......... (27)
the least common denominator of the exponents being . It will be observed
that there are f(n) developments, in which g and g' have the same values ; the
coefficients \ having different values in these /(?) developments. Similarly,
there are 2/(ij) developments of each of the types
(p-l) = \(q-l)^+..., ........ (28)
; + ............. (29)
p \q
Hence W = 3^r,[S.- 1]
and, consequently, as in Art. 4,
H=
(in.) From the developments (27), (28), (29), we can deduce the normal
developments of the six sets of branches which touch PR and QR at R, PS and
QS at S, PQ at P and Q. Each set comprises v branches ; if A 2 is the greatest
square dividing N, h of these are linear in each of the first four sets ; all
* [Vol. i. p. 302.]
256 ON THE SINGULARITIES OF THE [Art. 10.
the rest are super-linear. It will suffice here to determine the cuspidal and
discriminantal indices of the branches touching PQ at P. The normal develop-
ments of these branches are of the type
g'-g
!f>9-
Hence their aggregate cuspidal index is
or, which is the same thing,
= A - B - "
To obtain the discriminantal index, we first consider a single group of f(n)
branches corresponding to given values of g, g. The discriminantal index of one
of these branches, taken by itself, is
n i 'n
the joint discriminantal index of two different branches of the group is
1]
so that the aggregate discriminantal index of the group (g', g) is
We next consider the two groups (g', g) and (g{, g^ consisting respectively
of f(n) and /(?i) branches. If g' > g^, or, which is the same thing, if g'g 1 >N,
the joint discriminantal index of the two groups is
Thus the aggregate discriminantal index of the branches touching PQ at P is
given by the equation
D (P) = Zf'(g') [/'(<$ -/'(</) - 1] + 22Z/' (/) [/'(</;) -/'to
the summations extending to all values of g f and ^ which satisfy the inequalities
g'>VN, g\<g', g{
Art. 11.] MODULAR EQUATIONS AND CURVES. 257
But, as in Art. 7,
whence, as before, J5 (P) = 2A 2 A 2 A.
11. Case wAen JV is a square. The case in which N=6* is a perfect
square requires separate consideration, because the modular curve of order 2
meets the line PQ wf(6) points distinct from one another and from P and Q ;
and again, at each of the points R and S, it has/(0) linear branches, of which
the tangents are different from one another, and from the lines RP, RQ, SP, SQ.
Thus some of the characteristics of the singularities at PQRS are changed ; and
with them some of the characteristic indices of the curve.
We write & for /(#)=/' (0). It will be found, on referring to Art. 10,
(i.) and (ii.), that E(p) = E (q) = A+B, m = 2A,
as in the case when N is not a square. Again, the cuspidal index of each of the
four sets of branches which touch PR and QR at R, PS and QS at S, is,
as before, B v; but the cuspidal index of the branches at P and Q is
A B v + ,6' instead of A B v. For this index is
(see Art. 10, iii.) ; and
To find the discriminantal indices of PQES, we denote by A, B, A 2 , B. 2
the numbers obtained by omitting in ABA 2 B 2 the terms depending on 6 ; we
thus have
Using these expressions, we find, as in Art. 10, (iii.),
D (P) = D (Q) = 2A* - A, - A;
VOL. II. L 1
258 ON THE SINGULARITIES OF THE [Art. 13.
or, substituting for A and A 2 their values,
To determine D (R), we have : (i.) For the discriminantal index of the set of
branches touching either PE or QR, 2 AI} 2 ; (ii.) for the joint discriminantal
index of these two sets of branches, 2B* ; (iii.) for the joint discriminantal index
of the & linear branches, 6' (Q' 1) ; (iv.) for the joint discriminantal index of the
linear branches taken with the branches touching either PR or QR, 2ff B.
Hence D (R) = D(S) = 2\B z -A-B*] + 2B* + 6'(e'-\) + &B
= 2(B 2 -A),
as in the case when N is not a square.
There is no change in the expressions for the order of V (q), and for the
exponents of the factors q and 1 q in V (q) (see Art. 6) ; and these expressions
agree with the values which we have obtained for D (R) = D (S), and for
D(P) = D(Q). For PR, touching the curve at R, counts as A B tangents
drawn from P ; and hence the order of q in the discriminant ought to be, what
hi fact it is, 2(B,-A) + (A-B) = 2B 2 -A -R
And again, PQ, considered as drawn from P, counts as A \& tangents at P,
and as B ^6' tangents at Q. Thus, the number of intersections of C by the
polar of P, which lie on the line PQ, is
and this number is, as it ought to be, the excess of 2 A (2 A 1) above the order
of V (q) ; i.e., the excess of the whole number of intersections above the inter-
sections lying on PQ.
ARTS. 12-14. Formulae applicable to all values of N.
12. If, in the formulae relating to the case when AT is a square, we omit the
terms containing the symbol 6' defined by the equation
we obtain the corresponding formulae for the case when N is not a square. We
shall henceforward denote by 6' a number which is equal to zero when N is not
a square, and which is equal tof(VN) when N is a square ; and we shall treat
the two cases simultaneously, except when it is necessary to call attention to
the difference between them.
13. The discriminantal class of the superlinear branches. In the paper on
the Higher Singularities of Plane Curves* (Arts. 12 and 13), it has been shown
* Proceedings 'of the Society, vol. vi. p. 153 [vol. ii. p. 112].
Art. 14.] MODULAR EQUATIONS AND CURVES. 259
that, if d and t are the order and class of a superlinear branch, D and T its
discriminantal order and class, we have the equation
And again, that if there be a second superlinear branch of the order d' and class t'
touching the first, and if we represent by T and D the joint discriminantal indices
(of order and class) appertaining to the two branches, we have the equation
T-D = 2(tt'-dd').
Combining these two results, we obtain the theorem
' If any number of branches touch one another at the same point, the
difference between the discriminantal order and class of the singularity is equal
to the difference between the squares of its order and of its class.'
Employing a notation explained in Art. 14, we apply this theorem to
determine the discriminantal class of the branches (PPQ), (PQQ), (PER),
(QBE), (PSS), (QSS). We thus find
T (PPQ) - D (PPQ) = T(PQQ) - D (PQQ)
= (B-ff)*-(A-B)> ...... (31)
T(PEE) - D (PEE) = T(QEE) - D (QEE)
= T(PSS) - D (PSS) = T(QSS) - D (QSS)
= (A-BY-(B-^J- ......... (32)
so that
= D (PER) = D (QEE) = D (PSS) = D (QSS),
and T(PEE) = T (QEE) = T(PSS) = T(QSS)
= 2A*-A 2 -A=D(PQQ) =
14. Summary of the results. For convenience of reference, we exhibit the
preceding results in a tabular form.
Characteristics and Singularities of the Modular Curve C.
I. Explanation of the symbols :
(1) The order of the transformation is the uneven number N.
(2) fj and (/ are conjugate divisors of N; h 2 is the greatest square
dividing N.
Lla
260 ON THE SINGULARITIES OF THE [Art. 14.
(3) 77 is the greatest common divisor of g and g.
(4) f(rf) is the number of numbers not surpassing ? and prime to it.
(5) f'(fj) and f'(g') are defined by the equation
gig
(6) 2, = Z/(,), A + B = -2f'(g), A 2 + B>= (A + B)*.
In these equations the summations 2 extend to all divisors g of N ; A compre-
hends all the terms f'(g) in which g>*/N, and also, if N=Q Z , the term
l0' = i/(0); A 2 comprehends all the terms of 2/'(gr 1 ) x 2/'(</ 2 ),| in which
<j 1 g 2 >N, and one-half of every term in which g l g z = N, g l and <7 2 denoting any
two divisors of JV, the same or different. The definitions of B and B 2 follow from
those of A and ^4 2 .
(7) ra, n, K, I, D, T, H denote respectively the order, the class, the cuspidal
index, the inflexional index, the discriminantal order, the discriminantal class,
and the deficiency of the curve.
(8) The symbol (XX Y) or (YXX) denotes a branch, or an aggregate of
branches, touching the line XY at the point X.
(9) The symbols 0(XXY), C(XXY), K(XXY), I(XXY), D(XXY),
T(XXY) denote the order, class, cuspidal index, inflexional index, discriminantal
order, discriminantal class of the branches (XX Y). The symbols 0(X), K(X\
D(X\ C(XY), I(XY), T(XY) are to be similarly interpreted with regard to
the branches which pass through a given point X or touch a given line XY.
Lastly, the symbols D (XXY, XXZ] and T(XXY, XYY) denote respectively
twice the number of points common to the branches (XXY), (XXZ), and twice
the number of tangents common to the branches (XXY), (XYY).
II. Characteristics of the Curve.*
T=(SA-B-6'Y-5A+B + e',
* Several of the formulee which follow may be more simply expressed by using the symbols
Z, B, ~A V J3 2 of Art. 1 1, and by writing v = v $0*.
Art. 14.] MODULAR EQUATIONS AND CURVES. 261
III. Characteristics of the Special Singularities.
(i.) Characteristics of (PPQ) and (PQQ).
0(PPQ) = A-B; C(PPQ) = B-i6',
K(PPQ) = A-B-v + \tf, I(PPQ) = B-v,
D (PPQ) = 2 A* -At- A + i0',
The number of distinct branches is v ^Q'. They are all superlinear ; viz.
corresponding to every divisor g of N, which is less than VN, there are in (PPQ)
/(>?) superlinear branches, each of the order - - , and of the class -; (PQQ) is
of the same type as (PPQ).
(ii.) Characteristics of (PRR), (QRR), (PSS), (QSS).
All these singularities are of the same type.
C(PRR) = A-B,
D (PRR) = B 2 -B 2 -A-8'
2 - .
The number of distinct branches in (PRR) is v \Qf ; of these, h 6' are
linear (Art. 10, iii.) ; the characteristics of (PRR) and (PPQ) are reciprocal ; viz.
corresponding to any divisor g of N, which is less than VN, there are in (PRR)
f(n) branches, each of the order - and of the class - -
i n
(iii.) Characteristics of (PQ).
C (PQ) = 2 C(PPQ) = 2JB-y;
I(PQ) = 27 (PPQ) = 2B- 2v,
T(PQ) = T(PPQ) + T(PQQ) + T(PPQ, PQQ)
(iv.) Characteristics of(R) and (S).
These are the same for the two points.
(1) 0(R) = 0(PRR) + 0(QRR)
= 2B.
(2) K(R) = K(PRR)+K(QRR)
262 ON THE SINGULARITIES OF THE [Art. 14.
(3) D(R) = D (PER) + D (QRR) + D (PRR, QRR)
+ D(6) + D (6, PRR) + D (6, QRR)
The symbol (6) is used to represent the & linear branches which pass through
R, having tangents distinct from one another and from PR, QR.
(v.) Tangents to the Curve from PQRS.
(1) PQ, considered as drawn from P, counts as A ^0' tangents at P, and
as B \6' tangents at Q ; PR counts as A B tangents at R ; thus PQ, PR, PS
count as 3 A B 6' = n tangents drawn from P.
(2) The tangents to the branches (PRR), (QRR), and (6) count as
2 (A - -| B'] + 2 6' = 2 A + 6' tangents drawn from R. Thus, there are A-B-26'
other tangents which can be drawn to the curve from R.*
(vi.) Intersections with the Curve of the sides of the quadrangle PQRS.
(1) PQ meets the curve in A 1 6' points at P, and in as many at Q ; and
in & non-singular points distinct from either P or Q.
(2) PR meets the curve in A B points at P ; at R it meets the branches
(PRR) in A \Q' points; the branches (QRR) in B \Q' points; the branches
(0) in 0' points ; in all in 2 A points. The same statements hold, mutatis mu-
tandis, for the lines QR, PS, QS.
(3) RS meets the curve 2 (B \ ,6') + 6' = 2 B points at R, and in as many at
S; and also in 2 (A - 2B) other points, f
IV. Residual singularities of the Curve.
Designating by K l ,I 1 ,D 1 , T the parts of the indices K, I, D, T which arise
from the singularities connected with the points and lines of the quadrangle
* If x ( 77, 2 )= represent the equation of the multiplier, which is of the order A 4- B in , and
of the order \(A B) in q, the values of q appertaining to the points of contact of these tangents are
determined by the equations \(Vn, - - ) = 0, x( -v^w", )=0. When N=0'*, the first
of these equations has & roots equal to zero, and O' infinite roots ; both these sets of roots are to be
rejected.
t At each of these points we have p=q. The equation F(p, p, 1)=0 is divisible by [p (p 1)] 2B ;
the remaining roots, which are 2A 4B in number, give the intersections of the curve by US at points
other than E and S. These roots may be determined by the method (due to M. Kronecker) described
in the Eeport on the Theory of Numbers, Arts. 131-133 [vol. i. p. 325].
Art. 15.] MODULAR EQUATIONS AND CURVES. 263
PQRS, we find, from the preceding formulae,
and for the residual singularities we have
ARTS. 15, 16. Case when N is a square. The Linear Branches (6).
15. The developments appertaining to the 6' linear branches U, which inter-
sect PQ at points other than P and Q, are
1 e' ' e iu (1 + <?'") e'"(l + e') (21 + 11 e'")
~ ~ ' - o o - a~A 2 - + > . .
p q 2q 2
21 + 1
where = - -IT, 2Z + 1 beuig any term of a system of residues prime to 2 6.
We hence obtain the normal developments
S 1 /v -i- ?' (a - a ^ ^
I32;.6(l-e*^+... l ..... (34)
so that the tangents of the 6' branches are the lines
which meet one another in the point a = /3 = J-y ; i.e. in the point in which RS
intersects ab.
The developments appertaining to the Q' linear branches at R are
p = e iv q + ie i (l-e i <>)q 2 + ..., ...... (35)
where v = v, h being any term of a system of residues prime to ; so that
the tangents are a e v /3 = 0,
none of the branches being inflected at R.
Similarly the developments appertaining to the 6' linear branches at S are
p-l=e i "(q-l) + ^e i "(l-e <v )(q-l) z +..., ..... (36)
and the tangents are a y = e iv (/? 7),
there being no inflexion.
264 ON THE SINGULAEITIES OF THE [Art. 16.
The two sets of tangents at R and S meet PQ in the same points in which
it is intersected by the linear branches U ; for, if
we have u + v = (2k + l)-}r, whence e iv = e~ iu .
16. The developments (33-36) may be obtained as follows, with the help of
formulae established in the Report on the Theory of Numbers already cited.
n
If ) = 1 + - , where or is positive and increases without limit,
g-$()-l-^-(tV)
increases without limit ; and the limit of q -r 16e* ff is unity. The corresponding
values of p are comprised in the formula
x 9 '
where g and g' are conjugate divisors of N, and g, g', k have no common divisor.
But this expression may be exhibited in a form from which the dimensions of
<p (Q), as compared with < (), may be inferred ; viz., we have (Report, loc. cit.,
p. 350)
A ;
N
where d' is the greatest common divisor of g' + 2k and g, d = -^ , and 21 + 1 is
determined by a certain congruence for the modulus d. In order that the de-
velopment of -, in a series proceeding by powers of -, should correspond to a
branch intersecting PQ elsewhere than at P or Q, - and - must be of the same
dimensions. But #
Lim. p -r e d =1, Lim. g-f e* a = 1;
hence d' = d, or N is necessarily a square, and d = d' = 0. Since 6 = d' is the
A
greatest common divisor of g' + 2k and g, let or = A0, a' = -; then 6 divides
6 6
- + 2k; i.e., - divides g, g', and 2k, which are relatively prime. Hence X = 0,
g = 6 2 = JV, g' = 1. Now there are just 6' values of 2 k for which & is the greatest
common divisor of 2 and 1 + 2 k ; viz., if 2/j. + 1 be any number less than 2 6 and
prime to 6, the 6' values of 2k are included in the formula 2k =
Art. 17.] MODULAR EQUATIONS AND CURVES. 265
and it will be found that the congruence determining 2Z-J-1 is
(2/x + l)(2Z + l)= -1, mod0.
Hence we have, for the ff values of 2k which we are considering,
- +
Expanding the values of
1 <b B (i(r) j P 1 , R / , u
- - and of - = 8 -
- ., , . / x
q I-<p s (icr) p TT
by means of the formula
$ s (< a ) = 16e il ' a (l-8e i " 1 + Ue* i -...), ..... (37)
which arises from the expansion of (3) ; and equating the coefficients of like
powers of e~* ff in the series
l,fi + 5 + C .
p q q* g 3
we obtain the developments (33).
Similarly the developments (35), which appertain to the linear branches at
(nj -^
i<r + -} and
q = <f>* (iar) in the assumed series
ARTS. 17-19. TJie Six Modular Curves.
17. If we represent by e(z) any one of the six anharmonic functions
1 1 x x-l /QQ x
x, 1-x, -, - - , ^ , , ...... (38)
x 1-x x1 x
the modular equation (1) is unchanged by the simultaneous substitution of e (p)
for p and e(q) for q. Hence, if e 1 (x), e 2 (x) denote any two, the same or different,
of the functions (38), the thirty-six substitutions
f{fi(p), *fe). 1] ......... (88)
give only six different equations. As representatives of these, we take the
following:
(ii.)
VOL. II. M m
266 ON THE SINGULARITIES OF THE [Art. 18.
(iii.)
(iv.) F(p, q, 1) = 0,
= 0,
The equation F(p, q, 1) = is symmetric with regard to p and <? ; and it
will be found that the equations (L), (ii.), (iii.) possess the same property ; thus,
for example, the equations
F(p, , l) = 0, and F(q, |, l) = 0,
are the same, because
F(x, y, l) =
The fifth and sixth equations, on the other hand, are changed, each into the
other, by the interchange of p and q.
18. Denoting by X and Y rectangular Cartesian coordinates, and writing in
the equations (L), ..., (vL), p = i + X -iY \
q = l + X + iY,\
we obtain the equations of six curves, which, in the Mdmoire ' Sur les Equations
Modulaires,' * we have called the first, second, third, fourth, fifth, and sixth
modular curves. The equations of the first four of these curves are real, as
appears from the symmetry of the equations (i.) (iv.) with regard to p and q ; the
equations of the fifth and sixth curves are imaginary and conjugate to one another.
The first and fourth curves are each of them symmetric with regard to both
axes ; the fourth curve is its own inverse (anallagmatic) with regard to each of
the two real circles
. Y 2 1
and the first curve with regard to each of the two imaginary circles
The second and third curves are symmetric with regard to the axis of X, and
symmetric to one another with regard to the axis of Y '; the fifth and sixth
(imaginary) curves are symmetric with regard to the axis of Y, and symmetric
* [VoL ii. p. 224.]
Art. 19.] MODULAR EQUATIONS AND CURVES. 267
to one another with regard to the axis of X . The second and third curves are
the inverses of the first, with regard to the circles
respectively ; similarly the two imaginary curves are the inverses of the fourth
curve with regard to the two imaginary circles
Lastly, the substitution X = iY', Y= iX'
changes the first curve into the fourth, the second into the fifth, the third into
the sixth, and vice versd.
These assertions are the geometrical equivalents of the properties of the
modular equation stated in Art. 17 ; it will suffice to verify one of them. The
equation of the first modular curve is
its inverse with regard to the circle (X + J) 2 + Y 2 = l is obtained by writing
so that _
The equation of the inverse curve is therefore
and this is identical with the equation
i.e. with the equation of the second modular curve, because
-y 1-x
is identical with F(X, -. l) = 0.
\ y /
19. The equations of the first and fourth modular curves are included in the
general equation f ^ ^^ = Q]
M m 2
268 ON THE SINGULARITIES OF THE [Art. 19.
viz. to obtain the first curve, we write
and, to obtain the fourth curve, we write
a = i + X-iY, /3 = i + X+iY, 7 = 1.
Thus the theory of the singularities of these two curves is implicitly contained in
the preceding discussion of the singularities of C.
In both curves the points P, Q are the cyclic points, and (ab, RS) or O is
the origin : in the first curve ab and RS are the axes of X and Y ; a, b being
the points ( + 5, 0), and R, S the points (0, +|i) : c is the point at an infinite
distance on the axis of Y; in the fourth curve R, 8 are the points ( + 3, 0), and
a, b the points (0, +5*), c being the point at an infinite distance on the axis of
X. Both equations (as has been already said) are real ; and it follows, from the
theory explained in the Me"moire cited, that both of them represent real curves,
except when N=3, mod 4 ; in which case the fourth curve reduces itself to the
pair of conjugate points ( + \, 0).
When N is not a square, both curves are completely and parabolically cyclic,
having at each cyclic point / branches, of the aggregate order AB and class B,
touching the line at an infinite distance.
When N is a square, each of the two curves has 6' real infinite branches.
The fourth curve has also 6' real branches passing through each of the points
( 2> 0) ; (these two points always belong to the curve, though, when N is not a
square, only as isolated points :) the tangents to the & branches are parallel to
the asymptotes of the curve. Similarly the first modular curve acquires & linear,
but imaginary, branches at each of the points (0, + \i) ; the tangents to these
branches being imaginary lines parallel to the real asymptotes.
The equations of the asymptotes of the first and fourth curves are re-
spectively
Y cos \u X sin \u = 0, j
Y sin \u + X cos \u = ; j
2Z + 1
u denoting ^ TT, as in Art. 15. And it may be inferred from the develop-
ments (33) and (33), given in that Article, that the rectangular hyperbola
( Y cos ^u X sin \u) ( Y sin \u + X cos |w) = 5 sin u
osculates at an infinite distance the branches asymptotic to the two lines (41).
Art. 19.] MODULAR EQUATIONS AND CURVES. 269
Lastly, if v = v, as in Art. 15, the tangents of the fourth curve at the
points ( + |, 0) are
Y cos^v + (X l) sin \v = ;
the tangents of the first curve at the imaginary points (0, + ^i) are
( ^ 2 *) sm 2 v ~ X cos iv = :
and these tangents are parallel to the asymptotes of the curves to which they
respectively appertain ; because, if
The points ( + |, 0) and (0, \i) are foci, and indeed the only foci, of hoth
curves: of these, the points (0, \i) lie on the first curve, and the two real
points ( + 1, 0) are its two foci (properly so called) ; the axis of Y being the only
corresponding cyclic axis, or directrix. The points ( + \, 0) belong to the fourth
curve (only as isolated points, when N is not a square), and this curve has, pro-
perly speaking, only the pair of imaginary foci (0, + \i).
19. The second and third modular curves may be regarded as derived from
the equation (1), by the substitution
(42)
7 = 1.
X and Y being rectangular Cartesian coordinates, and the upper signs relating
to the second curve, the lower to the third.
Thus the theory of each of these curves is comprehended in that of the curve
C' ', of which the trilinear equation is
(43)
Or (ay) B ~ A X F(a/3, y 2 , ay) = 0.
The singularities of C' may be examined by the method already employed
in the case of (7. Attending, for brevity, only to the case in which N is not
divisible by any square, we write p = - , q ^ , in the parametric developments
of Art. 4, and we deduce, as follows, the normal developments of the singular
branches of C'.
270 ON THE SINGULARITIES OF THE [Art. 19.
(i.) From (6) we obtain
or, multiplying by = 1 +
ft
(44)
which is itself a normal development, if g' > g, and gives rise, by reversion, to
such a development, if g <g. Hence C' has a singularity at S, having the same
characteristics as the corresponding singularity of C.
(ii.) From (5) we infer
Here, when p and q are small, a must be small compared with 7, and 7 compared
with ; i.e. the coordinates of the point (p = 0, 5 = 0) are o = 0, 7 = ; and the
normal development is
S+jf
(iii.) Similarly from the development (7) we deduce
Thus the coordinates of the point (p = oo, q = eo) are /3 = 0, 7 = ; and we find,
after reversion and multiplication by - , the normal development
which is of the same type as (45).
Thus the curve has at P and Q singularities of one and the same self-reci-
procal type, not resembling the singularities which C has at the same points.
The point R does not lie on C'.
Art. 20.] MODULAR EQUATIONS AND CURVES. 271
ART. 20. Characteristics and Singularities of the Modular Curve C'.
I. Characteristics of the Curve.
, n = 3A + B,
-5A-3B,
It will be noticed (1) that these formulae do not contain 0', although the case
when N is a square is included in them ; (2) that, when N is not a square, / K
has the same value for C' as for C.
II. Characteristics of the Special Singularities.
(i.) Characteristics of (PPR) and (QQR).
(PPR) =A+B=C (PPR),
K(PPR) = A + B-2 V = I (PPR),
D (PPR) = A 2 + 3B 2 - 2A -2B = T(PPR).
The number of distinct branches at each of the points P and Q is 2v ; viz., cor-
responding to every divisor g of N, there are in (PPR), f(rj) branches of the order
- and of the class - ; of the 2 v branches, h are linear, and, in particular, when N
i i
is a square, 0' of these are also non-inflexional.
(ii.) Characteristics of (PSS), (QSS), and (S).
These are the same for C" as for C (see ^4rt. 15, III., (ii.) and (iv.)). When
N is a square the equations of the tangents to the linear branches are
a 7 = e iv (7 18)
(see Art. 15, equation (36)).
(iii.) Tangents to the Curve from PQRS.
(1) PR, considered as a tangent drawn from P, counts as 2 A + 2B tangents
at P ; and PS counts as A B tangents ; hence PR and PS are the only
tangents from P.
272 ON THE SINGULARITIES OF THE [Art. 20.
(2) RP, considered as a tangent drawn from R, counts as A + B tangents at
P ; and so does RQ at Q ; thus there are A B other tangents which can be
drawn to C' from R.
(3) Besides the tangents at S, there are A B 2 & other tangents which
can be drawn to the curve from S (see Art. 14, III., (v.) (2)).
(iv.) Intersections with the Curve of the sides of the quadrangle PQRS.
(1) PQ meets the curve in A + B points at P, and in A+B points at Q.
Thus it never meets the curve again, and touches it nowhere.
(2) PR and QR each meet the curve in 2A + 2B points, touching it at P
and Q respectively, and meeting it nowhere else.
(3) PS and QS meet the curve in A + B points at S, and in A + B points at
P and Q respectively ; thus they never meet the curve again.
(4) RS meets the curve in 2B points at S, and in 2 A other points.*
III. Residual Singularities of the Curve.
Employing the notation of Art. 14, IV., we have
D l = 2A 2 + 8B 2 - 6 A -
The indices K 2 , D 2 have the same values for C and C', because these indices refer
to singularities which do not lie, in either figure, upon the fundamental triangle
of the quadric transformation by which the curves are changed into one another.
The equality of the indices I 2 , when N is not a square, implies the theorem :
' Each of the first three modular curves has as many non-singular inflexional
tangents as it has osculating circles, which pass through the point (|, 0) ; or
again, through the point ( ^, 0).'
* The points of contact of the AB tangents, (iii.) (2), and of the A 2? 20' tangents, (iii.) (3), and
the 24 points of intersection, (iv.) (4), can be determined by methods similar to those indicated in the
Notes on Art. 14.
Art. 20.] MODULAB EQUATIONS AND CURVES. 273
Of the formulae contained in the preceding enumeration, we shall demon-
strate only one ; viz., the expression for D (PPR) or D (P).
We have, as in Art. 10, (iii.),
where g is any divisor of N, and g' is the conjugate divisor of g ; g l is any divisor
less than g, so that g( > g'; the summations 2 X and 2 extend to every value of g
and g respectively. Hence we find
- - (47)
But we have, evidently,
V W(0) + 2 sz./'foo/'fo) - (^ + ^) 2 = 4, + A ;
and, observing that g'g^ < N, we also find
W WW + zs,
Introducing these values into the equation (47), we obtain
in accordance with the formula II. (i.) supra.
VOL. n. N n
XXXVII.
NOTE ON A MODULAR EQUATION
OF THE
TRANSFORMATION OP THE THIRD ORDER.
[Proceedings of the London Mathematical Society, vol. x. pp. 87-91. Read February 13, 1879.]
1 HAVE given elsewhere* the modular equation for the transformation of the
third order between
(1-^ + fc 4 ) 3 2 * 3
and =
x (i-xy
viz., F(x,y) = x(x + V.3. 5 s ) 3 + y (y + 2. 3 . 5 3 ) 3
- 2 16 x 3 y a + 2 11 . 3 2 . 31 x 2 y 2 (x + y}
- 2 2 . 3 3 . 9907 xy (x 2 + y 2 ) + 2 . 3 4 . 13 . 193 . 6367 x*y*
+ 2" . 3 5 . 5 3 . 4471 xy (x + y) - 2 16 . 5 6 . 22973 xy = 0.
The following is the process by which the coefficients were determined :
It follows from the general theory of modular equations that F (x, y) is sym-
metrical with respect to x and y ; and that it is of the order 4 in x and y
separately, and of the order 6 in x and y jointly. Hence F (x, y) is of the form
A 3t 3 a-y + A^ x 2 y* (x* + y*) + A tt , xy (x 3 + y 3 )
>0 (x 3 + y 3 ) + A 2i2 x*y* + A 2tl xy (x + y)
Several of the coefficients may be conveniently found by employing the
method of Sohncke ; i.e. by substituting for x and y their expressions as series
* ' Proceedings of the London Mathematical Society,' vol. ix. p. 243, note [vol. ii. p. 242].
NOTE ON A MODULAR EQUATION. 275
proceeding by powers of q, and equating the coefficients of the powers of q to
zero. We have, by a known formula,
(1 _ Z \3
calling this quantity z, we have x = - ~ ;
%
we also write Z= 2 *g l" (1 + 2 6 "- 3 )- 2 *,
(1 Z) 3
so that 2/ = 72
Substituting these values, and equating to zero the coefficients of q~ 2S , q~ K ,
and q~**, we find successively
^ = 0, ^4,1 = 0, 4y,= -2".
To determine ^ 3j2 , we observe that q~ zz presents itself only in the terms
Its coefficient in x z y 3 is 2~ 40 ; its coefficient in x s y 3 is 3 2 . 31 x 2~* 5 ; viz. this is
the coefficient of q 2 in
Hence A 3t2 x2- io + 2~ . 3 2 . 31 . A 3>3 = 0,
Again, to determine A 3tl we consider the coefficient of q~ i0 ; this power of q
presents itself only in the terms
y [A 3i3 x -r^L 3i 2x T-"3,i'j>
It will be found that in the development of y 3 there is no power of q intermediate
between q~ ia and 2~ 12 ; hence, the coefficient of q~ 2 in
A /y3 i A rt.2 _i_ A ff
"3,3 "- ~T -"-3,2 "^ i- a -3,l' c i
or the coefficient of q 1 in
'(1 + g 3 ) 1 "-^, 2(1 + 2)'
is equal to zero. Substituting the values of A 3>3 , A 3>t> and reducing, we find
N n 2
276 NOTE ON A MODULAR EQUATION OF THE
All the coefficients may successively be determined by this method, but the work
becomes very laborious for the later coefficients. They may be more easily ob-
tained by the consideration that, if , n are any two corresponding values of k 2
and X 2 ,./() and y (i?) are corresponding values of x and y. Availing ourselves of
this principle, we find
(i.)
(ii.)
(iii.) F( y , y) = -2 y (y-) (y-
From (1) we infer that F (x, y) is of the form
..... (A)
= 0. )
From (iii.) we find ^ i= _ 216 . 5 . 22973,
J 4 2il = 2 8 .3 6 .5 3 .4471,
A 2s2 = 2" . 1262587 + 2 3 . 3 3 . 9907 - 2
= 2. 3*. 15974803
= 2. 3 4 . 13. 193. 6367.
This completes the determination of the coefficients ; a verification is supplied
by the equation (ii.) ; and it only remains to show how the equations (i.), (ii.),
(iii.) are inferred from the modular equation between = k 2 and >j = X 2 ; viz. :
(i.) Let p denote either root of the equation p 2 p + 1 = ; if k 2 = p, we have
x =f(p) = ; we also find
3>(p,ri) = ( n - P ) (>,* - [128 - 253,0] , - [128 + 253^] n + 1).
If w b to 2 , w 3 are the roots of the cubic factor, the four roots of F (0, y) are
But the cubic factor is one of the two conjugate factors of the expression
Hence /(,) =/(o, 2 ) =/(o, 3 ) = - V . 3 . 6,
and
TRANSFORMATION OP THE THIRD ORDER. 277
(ii.) Let k 2 = %, so that/(& 2 ) = ^ ; we find
If n\, la, la> ii are the roots of $ (^, >?), the four roots of F(^-, y) are /(?i),
(iz)y fM> f(i*) *- e - ^ z i> Z 2 are the roots of z 2 48^ z+ ^, these four roots are
* (1 - z,) 3 , zj 2 (1 - z 2 ) 3 , each taken twice. Hence
and by the ordinary methods we find
32 33 113 23 3
[y - zf 2 (1 - z,) 3 ] [y -^(l-z^J = f--^ 133283 y-^-
(iii.) Since the only roots of the equation
are
the roots of the equation F(y,y) = Q are all of the form y (5), where is a root of
one of the six equations :
iy(e*-6 + l) = 0, ............. (1)
(3)
1 _0) 2 ] := 0;. . (4)
-l) 2 ] = 0,. ... (5)
(6)
These equations are of order 8, the first and second having each two infinite
roots. The 48 roots give, in all, five distinct values for/(0) according to the
following scheme :
A. f (0) = oo ; 12 roots, viz. :
d = oo,2 roots in (1) and in (2) ;
6 = 0, 2 roots in (1) and in (6) ;
6 = 1, 2 roots in (1) and in (3).
278 NOTE ON A MODULAE EQUATION.
B. f(&) = ; 6 roots, viz. :
02 _ Q + 1 = o, in (1), (4), and (5).
C. /(0)=^A 3 ; 6 roots, viz.:
160 2 -160 + 1 =0, in (2);
= 0, in (3);
= 0, in (6).
D. f(&) = - 128 ; 12 roots, viz. :
1280 2 (0-l) 2 = 0, in (4) and (5).
=; 12 roots, viz. :
(40* -40- 1) 2 = 0, in (2);
= 0, in (3);
3 s 5 3 5 s
Hence the roots of F(y,y) = are 0, ' , each once; o>, -2 7 , , each
twice ; i.e. :
The multiplicity of the roots of F (y, y) = may be otherwise, and more
simply, determined as follows. The form of F (x, y) (see the equation (A), supra)
shows that F(y,y) has one root equal to zero and two roots equal to infinity);
the multiplicities of the finite roots are determined by the equation
53
- -2x
XXXVIII.
NOTE ON THE FORMULA FOR THE MULTIPLICATION
OF FOUR THETA FUNCTIONS.
[Proceedings of the London Mathematical Society, vol. x. pp. 91-100.* Head February 13, 1879.]
_L HE normal formula for the multiplication of four Theta Functions is (' Pro-
ceedings of the London Mathematical Society,' vol. I., part viii., p. 4 1)
2 * w fa) 0,*, * (**) 6 *.
In this formula we have
^^(x) = 2":!:(-l) m '''2i< 2ro+ ^e''( 2m+ '')^=2^:!:(-l) m A'e^(2'+/') 2 e^ 2m+ ''^; (2)
2s = x 1 + x 2 + x 3 -\-x t ,\
2a- =Mi + M 2 + Ms + A l 4 f (3)
2o-' = MI + /2 + MS + M! ; J
q = e iva> being a constant of which the analytical modulus is inferior to unity,
and the indices n, // being integral numbers, which render the sums 2o-,
2o-' even.
I. The Eleven Cases of the Formula.
Since M+2> , (x) = (- I)' M> ^ (as), | .
H,^I(I)-OH
we need only attribute the values 1, to the indices /*> M'. We may also
* [This paper is referred to on p. 75 of vol. x., and appears in the Contents of the volume under
the title, 'On the formula for four Abeliau functions answering to the formula for four Theta
functions.']
t [Vol. i. p. 443.]
2 SO
NOTE ON THE FORMULA FOR THE
permute the four arguments x 1 , x t , x 3 , x t w any way we please ; thus the matrix
0i,*i,* l iH mav have e i even different values, which are enumerated in the
following table :
TABLE I.
A. <r = 0, o-'=0, mod 2 : Cases i. iv.
1111
0000'
B. a- = 0, <r' = 1 , mod 2 : Cases v., vi.
1111
1100
C. <r = l, or'=0, mod 2 : Cases vii., viii.
1100
1111
D. a- = 1 , er = 1 , mod 2 : Cases ix. xi.
1111
1111
0000
1111
0000
1100
1100
0000
0000
0000
1100
1100
1100
1100
0011
1010
The formulae appertaining to these eleven cases are all different from one
another; i.e., none of them can be derived from any other by a permutation
of the four arguments x ; we can, however, pass from any one of them to any
other by means of the formula which connects any two different Theta func-
tions ; viz.,
~
Using a notation with a single suffix, and writing
i , o = * *> o , o = s ,
we may exhibit the eleven formulae as follows.
TABLE II. Formula for the Multiplication of Four Theta Functions.
(i.)
T ~M) ^ ~ ^ ^ X ~ " ~ 3 ^ ~ 3 ^ ~ 3 ^ ~ 3'
(ii) 25 2 x5 a x5 a x5 2 =
H~ <C/3 X rJs X (vT3 X (Js " fiTfl X <C/o
MULTIPLICATION OF FOUR THETA FUNCTIONS.
281
(iii.)
(iv.)
"I" rJl X rJl X ^j X rjj - rj% X rj 2 X ~2 X ~2'
"f" rC/2 ^ ^2 ^ ~2 ^ ^ 2 ~~ ^ 1 ^ ^1 ^ ^1 ^ ""'I*
B.
rjj} X rjj X fwTo X rjfl
3 s x5 s x3 x3o
,J Z X Si X ^7j X /!
c.
X ^Jfl X rJ 3 X
X ^7j X
X rj%*
*^ IWJ X <WJ X fJ~Q X r*/Q "^ i>/2 X r*/2 X rj"3 X nJ*<J*
(vi.)
(vii.)
(viii.)
-f 3 2 x 3 2 x 3 3 x 3 3 - 3j x ,3-j x 3 x 3 .
D.
(ix.) 23 1 x3 1 x3 3 x3 3 = ^X^X^iX^ + ^oX^oxSaX.^
^ fCTg X o/g X rwT0 X rJ"Q T~ rJi X rCTj X o/g X rCTg.
(x.) 2^x^x30x33= ^sX^oX^.x^ + ^oX^sX^x^a
"|" rJg X rC/J X rJ\J X rCT() "~ rvTj X fs/^ X rCT|) X rj"g.
(xi.) 23 2 x3 2 x3 x3 = 3 x3 x3 2 x3 3 +3 3 x3 3 x3 1 x3 1
^ rj"l X rjj X fCA 3 X rjjj ~!~ rj" 2 X nj" 2 X rjQ X r-TQ*
For brevity, the arguments x lt x 2 , x s , x^, and the arguments s x l , s x z , s x t ,
s x t are omitted in the left and right-hand members respectively.
It will be observed that" of the 256 sets of values which may be attributed
to the constituents of the matrix
"l "2 "2 "4
192 are excluded by the condition that 2<r and 2c/ are even ; of the remaining
64, 4 are represented by the formulae (i.) (iv.), which are symmetrical with respect
to all the arguments ; 24 by the formula (x.), which is entirely unsymmetrical ;
and 6 by each of the formulas (v.) (ix.) and (xi.), which are symmetrical with
respect to the arguments taken in pairs.
VOL. II. O
2*2
NOTE ON THE FORMULA FOR THE
II. Application of the Formula to the Abelian Functions.
The Abelian functions are defined by the equations
o /TT X \
-Kg)' ^(|z) N
i = e x > '
Al 3 (x) = i
> /
(6)
the constants ^ and ^ being determined by the equations
\Y=
4 *~l-
(7)
(8)
The formulae of Table II. give rise to a corresponding system of formulas for the
multiplication of four Abelian functions. To obtain this second system, we
have only to express the Theta functions as Abelian functions, and to attend
to the equations
o)
'
TABLE III. Formula for the Multiplication of four Abelian Functions.
A.
(i.)
x Alj = Al t x Alj x Al! x Alj + -r^ A1 2 x A1 2 x AL x A1 2
+ -TJ A1 x A1 x A1 x A1 - rA-lj x A1 3 x Al s x A1 3 .
(ii.) 2Al 2 x Al 2 x Al 2 x A1 2 = Al 2 x Al 2 x Al 2 x A1 2 + Je' 2 Al t x Al t x A^ x A^
1 k' 2
+ -rjAl^x Al s x Al 3 x A 3 - - A1 x A1 x A1 x A1 .
MULTIPLICATION OF FOUR THETA FUNCTIONS. 283
(iii.) 2A1 x A1 x A1 x Alj = A1 x A1 x A1 x A1 + ^ A1 3 x A1 3 x Al s x A1 3
k 2
+ k 2 Al l xA.\ l xA\ l -xA\ 1 - TKJ A1 2 x A1 2 x A1 2 x A1 2 .
(iv.) 2A1 S x Al s x A1 3 x A1 3 = A1 3 x A1 3 x Al s x A1 3 + k' 2 A1 x A1 x A1 x A1
+ k 2 Alj x A1 2 x Alg x A1 2 - k 2 k' 2 A.I, x M l x Al x x Al^
B.
(v.) 2Al t x AljX Al 2 x A1 2 = A1 2 x A1 2 x Al x x Alj + A^ x M l x A1 2 x A1 2
+ pA! 3 x Al 3 x Al x A1 - -r z Al x Al x Al 3 x A1 3 .
(vi.) 2A1 x Ala x A1 3 x A1 3 = A1 3 x A1 3 x A1 x A1 + A1 x A1 x A1 3 x AL,
+ k 2 A1 2 x A1 2 x Alj x Alj - k 2 Al x x A^x Al 2 x A1 2 .
C.
(vii.) 2 Al t x Al x x A1 x Alo = A1 x A1 x Al x x Al x T/J A1 3 x A1 3 x A1 2 x Al
+ Alj x A^ x A1 x A1 + TTJ Ala x A1 2 x A1 3 x A1 3 .
(viii.) 2 A1 2 x A1 2 x A1 3 x A1 3 = A1 3 x A1 3 x A1 2 x A1 2 k' 2 A1 x A1 x Alj x Alj
+ Al 2 x Al 2 x Al 3 x A1 3 + k' 2 Al x x Al t x AloX Alo.
D.
(ix.) 2 Alj x Alj x A1 3 x A1 3 = A1 3 x A1 3 x Al, x Al t - A1 x A1 x A1 2 x A1 2
+ A1 2 x A1 2 x A1 x A1 + Alj x Alj x A1 3 x A1 3 .
(x.) 2 Alj x A1 2 x A1 x A1 3 = A1 3 x A1 x A1 2 x Al x + A1 x A1 3 x Alj x A1 2
+ A1 2 x Alj x A1 3 x A1 Alj x Al 2 x Al x A1 3 .
(xi.) 2 A1 2 x A1 2 x Al u x A1 = A1 x A1 x A1 2 x A1 2 A1 3 x A1 3 x Alj x Al x
+ Alj x Alj x A1 3 x A1 3 + A1 2 x A1 2 x A1 x A1 .
III. Case when the Sum of the Four Arguments is zero.
Putting s = x l + x 2 + x 3 + x 4 = 0,
and attending to the equation
we find that in each of the formulae (i.) (xi.), one of the terms on the right-hand
side cancels one of the two equal terms on the left, and that the four formulae
A., and the formulae of the three pairs B., C., D., (ix.) and (xi.), become respectively
coincident ; the formula D. (x.) remains sui generis. Introducing the elliptic
functions A^ (x) A1 2 (x) Al, (x)
snx= -;, / {, cnx= .,; % dn x = ' / '
Al (x) Al (x) Al (x)
002
284 NOTE ON THE FORMULA FOR THE
into the five formulae thus obtained, we arrive at the following system :
(L) k 2 k' 2 Il.Bnx-k*n.cnx + n.dnx-k' 2 = 0,
(ii.) k 2 sn x l sn a en #3 en x 4 k 2 en x t en x% sn x 3 sn x 4
dn a!q dn x 2 + dn a% dn a; 4 = 0,
(iii. ) &' 2 sn x 1 sn x 2 k' 2 sn a? 3 sn ce 4
+ dn i dn a; 2 en ie 3 en a? 4 en x l en x 2 dn 0% dn x 4 = 0,
(iv.) sn Xj sn a; 2 dn x 3 dn a; 4 dn a^ dn o; 2 sn # 3 sn x 4
+ en x 3 en a; 4 en x l en x 2 = 0,
(v.) sn a^ en ce 2 dn a; 4 + dn x l en x 3 sn x 4
+ dn x z sn # 3 en a; 4 + en x sn x^ dn a^, = 0.
The first of these, which alone is symmetrical with respect to the four arguments,
is the formula given by Professor Cay ley (' Proceedings,' p. 43). The formulae
(ii.), (iii.), (iv.) are symmetrical with respect to the two pairs xx z , x 3 x 4 , and with
respect to the two arguments of each pair ; thus each of these formulas represents
a set of three. Lastly, the formula (v.) remains unchanged when any two of
the arguments are interchanged, provided that the other two are interchanged
at the same time ; i.e. there are six formulas of this type.
As a verification of these formulae in a particular case, let a? 4 = ; we find
/C S
(11)
(where we have written, for brevity, 5 1 = sna; 1 , Ci = cna; 1 , c? 1 = dna; 1 ; s a c 2 d 2 and
SOD having similar meanings with respect to the arguments x 2 and x l + x 2 = x 3 ) ;
and these equations are easily shown to be true by means of the formulae for the
addition of elliptic functions.
IV. Formula for the Multiplication of Four Multiple Theta Functions.
We define a double Theta function by the equation
' + \
,~ _\
/
where $ is the quadratic form (a, b, c) ; iwa, iirb, iirc being the hyperbolic
MULTIPLICATION OF FOUR THETA FUNCTIONS. 285
logarithms of A, B, C ; and where the condition of convergence is that the real
part of i<$> must be a negative form of negative determinant.
Considering four different Theta functions, and writing
2s ^x^ + Xt + Xs + Xt, 2t =
2 a- =
we have the formula
the signs of multiplication IT referring to the four values 1, 2, 3, 4 of the index y,
and the sign of summation 2 in the right-hand member referring to the symbols
a, a', /8, ft, each of which is to have the values 0, 1 ; so that the right-hand
member contains sixteen terms. The formula may be demonstrated in the same
manner as the corresponding formula for the single Theta functions (see the note
already cited, 'Proceedings,' vol. I., part viii., Art. 2).*
If, instead of a double Theta function, we consider a multiple Theta
function containing X arguments x, y, ..., and depending on X pairs of indices
MM', vv', pp', ,.., we have the equation of definition
a f I 'I I I \ ^?m= + co ^=+ ^>r=+oo
*(|A*,M|; ||) = 2 11I= _. 2. = _ 2 r ._....
t, [, (15)
+ ... 7* (2 f " 2r + "
where <|) is a quadratic form, containing X indeterminates, and such that the real
part of i<p is definite and negative ; and we obtain a formula similar to (14) ;
viz. taking four Theta functions, such that the sums of the homologous indices
are all even, we have
(16)
where the symbols | MJ , M/ | , \<r MJ + , a-' ^ + a' |
are placed by abbreviation for the X pairs
MJ, M>; "j, v'j; ...
and a-Mj + a, er'-Mj + a'; T- i/,- + /3, T'- ^-f-/^; ...,
respectively ; and the symbol a</ + aa'| represents the sum of the X terms
* [Vol. i. p. 444.]
286 FORMULA FOR THE MULTIPLICATION OF FOUR THETA FUNCTIONS.
Each of the 2X indices aa, /3/3', ... is to receive successively the values 0, 1 ; and
the sign of summation 2 extends to every combination of these values, so that
the right-hand member consists of 2 2X products, each affected with its proper
sign, of four Theta functions.
The multiple Theta function (15) satisfies the equations
, ......, , .
.;...); }
, /*';...; ...) ; . . . . (18)
0,0; 0,
dd> deb
"
,. .(19)
which correspond to the equations (4), (10), (5), relating to a Theta function of
one argument.
The equations (17) show that we need only attribute to the indices the
values 0, 1 ; thus, the formula (16) may be regarded as comprehending 11 A
different equations, if we regard two equations as identical which may be de-
duced from one another by permutation of the four indices j ; or, as compre-
11.12.13...10 + X ,.. ,. . ., A . . .
hendmg r -^ - r - dinerent equations, 11 we also regard as identical two
1 2 O A
equations which may be deduced from one another by a simultaneous permutation
of the \ arguments, and corresponding pairs of indices, in each of the Theta
functions; viz. counting the different equations on this principle, we have as
many of them as there are X-combinations, with repetition, of the eleven matrices
of Table T. But we can always pass from any one of the equations (16) to any
other by means of the formula (19), which may be employed to express any one
of the 4 X Theta functions as a product of any other by an exponential factor ;
although (for brevity) we have supposed that the indices of one of the Theta
functions compared in that formula are all equal to zero.
XXXIX.
DE FRACTIONIBUS QUIBUSDAM CONTINUIS.
[Collectanea Mathematica (in memoriam Dominici Chelini), Milan 1881, pp. 117-143.
The paper is dated 1879.]
1. ROPOSITA aequatione
oliin demonstravimus fore*
* 'Journal fur die reine und angewandte Mathematik,' vol. 1. p. 91 [No. III. vol. i. p. 33].
Verum hoc theorema ante nos invenerat vir clarissimus Serret.
Numeratorem fractionis continues
per formulam (y,, q 2 , q,, ...,}) denotamus ; cujua proprietates a forma determinantali
q lt -I, 0, 0,
1, &, 1, 0,.
0, 1,9-j, -1,
0,0,1, q t ,
,
.0
.0
0, 0, 0, 0,...,1, ?,,_ - 1
0, 0, 0, 0, .,<), 1, q n
pendere in commentatiuncula supra memorata observavimua. Ipsam autem fractionem continuam ita
significamus ut quotientium series uncinis quadratis includatur. Itaque habetur sequatio
Coeterum in fractione periodica utimur punctis superpositis ad distinguendos primos atque ultiraos
periodorum quotientes. Sic exempli gratia erit
[a, 6, c, ...,?,,, ..... q n ]
fractio contiiiua constans e quotientibus heteroclitis a, b, c, ..., et periodo q v q v ...,? B infinities repetita.
288 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 2.
atque hinc haberi partitiones numerorum P l et P 2 in summas duorum quadratorum.
Eadem fere methodo sequationes
tractari possunt ope lemmatis quod viro clarissimo J. J. Sylvester acceptum
referimus ; < Sit q , q z , .... q., q ., ..., q 2 , q,
quotientium series ordine symmetrico disposita, e quibus alterni per numerum /x
multiplicentur ; determinans
erit formse A 2 + nB 2 , designantibus A et B numeros inter se primes.'
Fit enim, si numerus i par est,
(?i. M <?g, &> -, n
si vero idem numerus impar est,
(M^I, q z , nq z , ...,
Itaque erit aut
K=(q l ,nq*, q 3 , .... /
aut
K=(<li> MQ'z, <? 3 . -, (affi-O'+^feu^ffn A* ?,-!, g,-) 2 -
Aperte autem confitemur totam hanc disquisitionem angustis finibus con-
tineri; quam, annis ab hinc amplius viginti a nobis inchoatam, nunc demum
longo post intervallo retractatani absolutamque, ideo arithmeticorum hominum
judicio committere ausi sumus, quod elementis arithmeticse incrementa etiam
tenuissima afferre operae pretium putamus. Qusestiones autem in eodem genere
latius patentes et nos olim attigimus, et fusius tractavit vir clarissimus M.A.
Stern in egregia commentatione.*
2. Initium operis facimus ab sequatione
P,P,-
in qua fiat 7?
=
' Report on the Theory of Numhers,' Art. 123 (Report of the British Association for the
Advancement of Science for 1863) [vol. i. p. 283].
' Journal filr die reine und angewandte Mathematik,' vol. liii. pp. 1-102.
Prater autem Goepelii diasertationem (ibid. vol. xlv. pp. 1-13) maxime memorabilia est com-
mentatio clarissimi viri C. Hermite (ibid. p. 191); quse tamen a theoria continuarum fractionum
paullo longius recedit.
Art. 2.] DE FBACTIONIBUS QUIBUSDAM CONTINUIS. 289
et generaliter
R n = R n -\ ?n-i P n >
p.*l-
ubi quotientes integros q ita determinari volumus, ut fiant numeri residui R
quam minimi. Ponimus autem in aequatione data numeros P, , P 2 (atque adeo
omnes numeros P) esse positives, et non impariter pares ; R l autem numerum
esse sive positivum sive negativum, quern positive sumptum per [JRJ denota-
bimus. Prseterea statuimus, quod licet, [JF2J esse minorem quam -|Pj. Erit ergo
P! > P 2 , P 2 < | \_R^\ ', unde patet quotientem q T evanescere non posse, et fore [J? 2 ]
< |P 2 , P 3 < | [R 2 ~\ ; ideoque [P,] > [PJ, P 2 > P 3 ; sive generaliter
P 1 >P 2 >P 3 ,....
Itaque veniemus aliquando ad sequationem
in qua P {+1 = 1, qi = R t , quseque adeo omnium postrema erit. Turn vero habe-
buntur sequationes
,._ 2 , 5-,.,
P^fei. 3g 2 , 2-3, ..., 3g 3 ,
quae facile aliae ex aliis probantur, ope formularum
__ 1
* i p
* i
>
Itaque ex data sequatione P l P z = ZR\-\-\ elicimus repra3sentationes nu-
merorum Pj et P z per quadratum et triplicem quadrati.
VOL. ii. p p
290 DE FRACTIONIBUS QTJIBUSDAM CONTINUIS. [Art. 3.
Exemplum. Sit sequatio data, 1999 x 436 = 3 x 529 2 + 1 : erit
1999 = (1, 3,1, -6,1, 3, -2,3,1,3)
= (1,3,1, -6) 2 + 3(1, 5,1, -6, I) 2
= 26 2 + 3x21 2 .
436 = (3, l,-6, 1, 3, -2, 3, 1)
= (3,1, -6, I) 2 + 3(1, 3, -2)*
3. Statuamus in fractione continua
p
3^- = [?i, 3 ft, g-3, -, 3gr a , q t ,
hunc haberi quotientium completorum ordinem
6 lt 36,, 3 , ..., 6 i sive 3 6 it 3 ft sive ft, ..., ft,
ita ut fiat Pi 1
Quantitatum 6 et <^> ea est conditio ut semper evadat [$]>>
Quod ut certa demonstratione confirmetur, adnotamus primum, si inter quo-
tientes q duse signis diversis unitates juxta positse inveniantur, continuari
signum secundae unitatis in proximo quotiente. Sit enim, exempli causa, </! = !,
q z = 1 ; numeri R^ et R 2 erunt positivi, quia signa numerorum q, , R, semper
inter se conveniunt. Quum autem sit P 2 < | R 1} sequitur fore
<-R
2 ,
ideoque etiam q 3 < 0. Monemus tamen duos ultimos quotientes q^i et q { unitates
signis diversis esse posse ; neque ulla lege adstringi signum quotientis duas
unitates proxime antecedentis. His positis, apparet quantitatem <p f+ ^ satisfacere
conditioni supra memoratse, si eidem ab ipsis </>,, ^>,_i, ... satisfiat. Quum
enim sit
Art. 4.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 291
manifest um est signa ipsorum < 8+1 , <,... congruere cum signis quotientium
2*+!) 3s> Sit igitur, brevitatis causa, q s+i numerus positivus ; erit certe
quia quotiens tertius q,_^ unitas positiva esse nequit. Itaque ad extremum
habebitur
fa + i ^[1> 3 > 2], sive <i>,_i^-75-
Vo
Et similiter si est [Q e +i\ >f, [Q,] >,- erit etiam
1 1
quia post quotientes q, = 1, g 8+1 = 1, necessario obtinet proximum locum quotiens
negativus.
Sit 16 numerus integer qui omnium proxime accedit ad valorem ipsius 9 ;
nanciscimur ex antedictis sequationes sequentes
At vero est ^ = ^ -5--; quumque fieri possit ut sit [<,-]<, sequatio
q i = !B i non semper valet. Valet tamen prseterquam in eo casu, in quo est <,-_!
unitati sequalis, [^] < 2, et in quo insuper alternantur signa quantitatum
g,-_ 2 , <?,-_!, 6i', quo in casu erit [<,]<, /[0,-] = 2, q ( unitas unitati q i , l contraria.
Cujus rei exemplum prsebet sequatio 52x7 = 3xll 2 +l, ubi fit
If = [2, -3, 1,3, -1,6].
Hie enim est 3 = ^, ideoque I6 Z = 2, quum sit tamen q 3 = 1.
Quodsi in fractione continua ad postremum quotientem, qui est Sq^, accedat
qusevis ejusdem signi quantitas, valores 6 l} 6 2 ,..., 6 t ipsos quidem aliquatenus
turbatum iri manifestum est ; sed nihilo secius permansuras sequationes q 1 = Id i ,
q 2 = I0 2 ,..., q ( _i = !0 i _ 1 , atque determinationem quotientis q { modo traditam.
Quod patet ex ipsa demonstrationis forma.
4. ^Equatio P 1 P 2 = 3El 1 itafacillime tractatur si scribatur
PjP,-- 855+1,
P p 2
292 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 4.
ut P 2 et R t numeri negativi evadant. Quo facto, determinentur numeri
P a > Pi, --> Pi + i', RZ, R*, > Ri\ per eequationes hasce
p.
2 l
-3721 + 1
(in quibus numeros R quam minimos fieri intelligendum est) ... donee per-
veniatur ad sequationem in qua sit P,- + i = e, Rj = eq i} designante litera e unitatem
( 1)' ... Turn vero erit
eP 1 = (q l , -3gr 2 , ..., q a , -
eP z = (q z , -3g 3 , ..., q 3 , -
eR = (q l} -3g. 2 , ..., -3q 3 , q 2 ) ;
unde apparet ipsos P l et P 3 fore formse y 2 -3x 2 , aut formse 3 x z y-, prout
t par est aut impar.
Coeterum in sequatione penultima Ri = Ri-\ <li-iPi, si est P t +2, oritur
ambiguitas in determinatione quotientis q t _^ ; quam ita tollimus ut quotientes
^i-i> <?* (e quibus hie certe unitas erit) afficiantur signis contrariis. Facile autem
demonstratur in serie quotientium q l} q 2 , ..., q { , si duse eodem signo unitates
juxta veniant, mutationem signi fieri in proximo quotiente. Hinc in fractione
continua
si fiat deinceps P
et sic porro, erit ut supra [<]> -^> [^] > f- Quotientes autem q lt q^, ... de-
Vo
terminantur per aequationes q l = 10 l , q. 1 = I6 2 , ...; inter quas etiam extrema ilia
q ( = IQ { locum sibi vindicat. Nam, si q t = + 1, (qui solus est casus de quo dubi-
tatio oriri potest) erit necessario P ( = +2, atque discrepabunt inter se signa
quotientium q { _ } , et q { ; quo fit ut valor absolutus ipsius <, = g r ,-+ - -^j
_ l
unitate major evadat. rU-s
Quum autem numerus ePj infinitis modis per formulam y- 3x- reproesentari
Art. 4.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 293
queat, ostendere convenit earn reprsesentationem quae continetur formula
eP 1 = (q l , -3q 2 ,...,q 2 , -3q,)
per numeros minimos fieri. Et generaliter quidem, si est
inveniri possunt in una eademque repraesentationum familia totidem reprae-
sentationes reprsesentatione data magis simplices quot habentur numeri u
fractione ^ absolute minores. Etenim cum omnes indeterminati x valores in
hac formula tx uy comprehendantur, si qua est repraesentatio repraesentatione
data magis simplex, habetur insequalitas (tx uy)' 2 < x 2 ; quae, eliminate numero
t, convenit cum inaequalitate []< ^- Jam vero, si i = 2n, habemus sequa-
tionem
P 1 = y^-3x 2 = (q i , -3q z , ..., q,^, -3,? 2n ) 2 - 3( ?1 , -3q 2 , ..., q, n _^,
in qua erit
ideoque /3~P~ j~p
Sin autem i = 2 n + 1, erit
Hinc in utroque casu habetur -~- < 1 ; unde apparet numeri P l reprae-
L " i -J
sentationem magis simplicem exhiberi non posse in ea saltern familia, in qua fit
Exemplum. Sit aequatio proposita
20306 x - 5197 = - 3 ( - 5931) 2 + 1 ;
erit e= 1,
20306= -(1, 6, 1, -12, -1, 3, 4, -3, -2, -3)
= 3(1,6,1, -12, -1)2 -(1,6,1, -12)*
= 3x97 2 -89 2 ;
5197 = (6, 1, -12, -1)2 -3 (-2, -3, 4) 2
= 85 2 -3x26 3 .
294 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 6.
5. Sit X numerus primus formse 4n + 3; atque in sequatione \n =
tribuantur literae A singillatim valores 1, 2, 3, ..., 2w + l : quo facto, erit B unus
aliquis ex iisdem numeris vel positive vel negative sumptus. Hinc patet in hac
sequationum caterva binas inter se invicem respondere ; quarum numerus cum
impar sit, una super erit, quse necessario hanc formam induct XM = + 3A 2 + l.
Hinc per exclusionem nanciscimur demonstrationem non inelegantem notissimi
theorematis ; primos formse 12w + 7 esse etiam formse x* + 3y 2 ; primes autem
formse 12n + ll habere formam 3x 2 y 2 . Ac simili fere modo demonstrari
posset utramque formam x 2 + 3y 2 , x z 3y 2 , primis 12w + l competere ; neutram
vero primis 12w + 5. Sed primos formse 4n + 1 in prsesens missos facimus.
Nam reprsesentationes primorum 4, + 3 per formas x 2 + 3y 2 , 3x 2 y 2 indagari
possunt per evolutiones radicum quadratarum in fractiones continuas ; quse
proprietas ad primos formse 4 n + 1 nequaquam pertinere videtur. Quo autem
facilius inteUigantur rationes evolutionum, quibus in hac qusestione utimur,
pauca, licet aliunde nota, prsemittenda sunt.
6. Sit N integer non quadratus ; a integer ipso VN proximo minor ; quan-
titas a + <</N in fractionem continuam conversa dabit quotientium periodum
ad hoc exemplar
2 , Mi, M 2 , , M n , P, Mn, M n _i, ..., Mi J
ubi primus quotiens quasi singularis est ; coeteri ordine symmetrico circa quo-
tientem medium /8 hinc et hinc disponuntur. Cujus symmetrise eo causa est,
quodsequatio a*-N-2ax + x* = 0, ......... (A)
a qua periodus originem trahit, anceps est, cum tertius in ipsa coefficiens
secundum metiatur. Quod si evolutionem a quotiente medio incipimus, habe-
bimus hanc periodi descriptionem,
/3, M n , M_i, -.., Mi, 2, Mi, M 2 , -, Mr.,
quse est plane ejusdem formse cum periodo data. Hinc patet inter sequationes
periodicas, prseter primam illam, mediam quoque ancipitem esse. Cujus sequa-
tionis cognitio magnum momentum habet ad explorandam totius periodi naturam ;
sunt autem certi casus in quibus etiam sine evolutione innotescit. Sit, exempli
causa, iV=X 1 xX 2 , designantibus literis X t et X 2 numeros primos formse 4, + 3;
sit etiam \ > X 2 ; b integer surdo A/ proximo minor. His positis, erit 2 6
* A.,
quotiens medius in evolutione radicis x /^i x ^2, e ^ ssquatio media hanc formam
habebit
Art. 7.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 295
Cum enim X t x X 2 summa duorum quadratorum esse nequeat, duae illse
sequationes ancipites, quse in periodo inveniuntur, diversaa inter se esse debent ;
at prseter duas aequationes (^4) et (B) nulla alia existere potest, quse et anceps
sit, et characterem sequationes periodicse prse se ferat. Unde videmus evolu-
tiones radicum ^/X x X 2 et Ay ~ una eademque opera exhiberi ; idque fieri per
hujusmodi sequationes
,- /X~"
V i 2~|_ , Mi, M 2 , , M n , Ay ^
, /\ F9/ ,-
+ A/ --\_6O, M n > Mn-l, > Mi, <*> + V^l
A 2
e quarum utravis concludimus fore
X x (u. u. u } = (a u a
ita ut fiat
7. Sit primo X 2 ipsius \ residuum quadraticum ; erit ( 1)"= +1, atque
ideo e numeris (MI, M 2 , > M n ) et (MI, M 2 , , M,,, H) hie impar, ille par erit. Hinc
sequitur, sequationem 4 \ x 2 X 2 7/ 2 = 1 resolubilem esse; et, designantibus a' et
V numeros ipsis 2 ^/Xj x X 3 et 2 A/ ' proximo minores, sequationem
X 2 x 2 - 2 6'X 2 a; + X 2 6' 2 - 4Xj =
occupare medium locum in periodo aaquationis
hoc est, evolutiones quantitatum a' + 2^/Xj x X 2 , V + 2 Ay -^ eodem modo alteram
ab altera pendere, quo evolutiones quantitatum a + /^Ai x ^2, & + / \/ ^~ '
Evoluta autem quantitate a' + 2^/Xj x X 2 sit
2a', /,, 2 , ..., f-, 2&', i/-, ..., y 2 , vi
quotientium periodus ; habebimus sequationes sequentes antecedentibus (C)
consimiles x x(c,, t> 2 , ..., f-, ft 7 ) = (a!, v lt v 2 , ..., v), }
4\,x(i/ 1 , i/jj, ..., i/j) =(', i/!, j> 2 , ...,"j, &') ;l
habebimus etiam solutionem sequationis 4XJX 2 X 2 y 2 = 1, scilicet,
296 DE FRACTIONIBUS QUIBTTSDAM CONTINUIS. [Art. 8.
quse quum convenire debeat cum solutione superius inventa (utraque enim
minimos numeros exhibet qui aequationi satisfacere possunt) colligimus sequa-
tiones sequentes
(a, MI, M 2 , ..., M,) = (', v lt v 2 , ..., vj),
(a, MI, M 2 , , Mi, &) = 2( a ' "i> V 8> ' v i> ^') >
e quibus patet quomodo, ex evolutione quantitatis ^/Xj x X 2 , evolutiones quanti-
tatum 2 ^/X! x X 2 et 2 Ay -~ derivari possint per conversionem fractionis vulgaris
in fractionem continuam ; si enim est
- = [,Mi,M 2 ,..,M,6],
erit etiam
= \a! v v v. V}
jj
Sit, secundo, X 2 ipsius Xj residuum quadraticum; erit ( l) n = 1,
(MI, M 2 , , M, &) par, (M I} M 2 , , M n ) impar; atque habebitur solutio sequationis
\ x 2 4 X 2 y 2 = 1. Unde derivari poterunt evolutiones quantitatum ^ ^/X 1 x X 2 ,
et J Ay ^ , quibus tamen in praesens opus non est.
8. Jam vero sit X 2 = 3, \ = X numerus primus formse 12 n + 7 ; erit 3 ipsius
X residuum quadraticum ; atque habebitur ex antedictis sequatio
(a, MI, M 2 , -, M n , &) (MI, M 2 , -., M n ) = 3x (M!, M 2 , -, Mn, &) 2 + l ;
quse tamen, quia (MI, M 2 , , Mn) numerus est impariter par, non potest commode
tractari per methodum supra (Art. 2) traditam. Utendum est igitur evolutione
quantitatis 2 Ay K quae si statuitur esse
26', V-, V-_ 1} ..., V l} la, V l} ..., !/_], V-,
suppeditabit sequationem
in qua (v lt v 2 , ..., Vj) impar erit, (of, v lt i/ 2 , ..., fj, V} pariter par. Itaque si in hac
aequatione fiat
Art. 8.] DE FRACTIONIBUS QUIBUSDAM CONTINUES. 297
(a, vi, v 2 , ..., Vj, b') = (k , 3k 1} k 2 , ..., 3k 2 , k l} 3k ),
("i, "2 ". "j)=( 3 ^D ^2, , 3 &2, &l)-
("i> "2, > v j> b') = (k l) , 3ki, k 2 , ..., 3k 2 , kj,
evolutio quantitatis 2 A/ ;- ad sequentem formam redigetur
V o
Quod facile demonstratur, si in evolutione vulgar! quantitatis 2 A^ '- quotientes
2 a', 2 6', interpositis cifris ita discerpantur ut fiat
= = [a, v j} !/_!, ..., v 1} 6', 0, 6', i/i, ..., i/-, a', 0, a'] ;
O
et si prseterea intelligatur, quod in fractions infinita licitum est, quotientium
seriem ita terminari ut una ex cifris interpositis locum extremum obtineat.
Sit A-2Bd + C6 2 = sequatio quadratica ; numeri ABC integri ; deter-
minans B* + AC positivus. Sit item q 1 ,q 2 ,...,q n quotientium periodus per quam
sequatio ista in se ipsam transit ; ita ut fiat
habentur sequationes notissimse
(q 2 ,g 3 ,...,q n ) _ fo, q,, ..
A B C
in quibus si forte fiat aut (q l} q 2 , ..., q n -i) = n (q 2 , q 3 , , q n }> au ^
(q z , q 3 , ..., q n ) = f^(q l , q a , ..., ?_,),
determinans sequationis erit formse ic 2 + /uy 2 . Quodsi in evolutione quantitatis
k + 2 Ay ^ periodum ab illo quotiente inchoamus qui est aut triplex, aut tertia
pars quotientis proxime antecedents, habetur series aut hujus formse
3k { , <_!, ..., 3k 2 , k lt 6^,, k 1} 3k 2 , ..., k 2 , 3k 1} 2k , 3k 1} k 2 , ..., 3A 4 _ If k { ,
aut hujus
k ( , 3k { _,, ..., 3& 2 , k lt 6k , k t , 3k 2 , ..., k 2 , 3k lt 2k , 3k lt k 2 , ..., k^, 3k { ;
prout numerus i par est aut impar. Unde patet in sequatione periodica
-^ > <-n + 2^i + ia; + Pf_ 2 a; 2 = 0, e qua istam quotientium seriem originem trahere
statuimus, fore vel 3 P { + x + P { + 2 = 0, vel P { + : + 3 P f + 2 = 0, ita ut habeatur partitio
VOL. n. Q q
298 DE FRACTIONIBUS QUIBT7SDAM CONTINTJIS. [Art. 9.
numeri 12 A, ac proinde ipsius X, in quadratum et triplicem quadrati. Quo autem
in sequatione X = A 2 + 3 B* valores numerorum A et B facilius investigentur,
aperienda est via ad evolutionem quantitatis k + 2 /\J - sine operoso calculo
inveniendam. Nimis enim molestum esset earn evolutionem a vulgari evolutione
radicis 2 A/ - per transformationem supra traditam elicere.
9. Et primum quidem observamus in evolutione quantitatis k + 2 /\J - , si
alterni quotientes per ternarium dividantur, haberi periodum ejusdem plane
formae cum periodis primorum 4w + l. Patet autem ex iis quse antea (Art. 3)
disputavimus quotientes k , k lt ..., k {-1 deinceps determinari posse eadem prope
calculi forma quse vulgo usitata est ad radices quadraticas evolvendas. Habemus
enim schema hujusmodi, si incipimus ab sequatione
inquaest P
0-2kP+0
Vl &KQ-L iT Vo>
~p -- >
p "1
*>
QJ-12X
* 4 D ' 3 -
in quo videmus unumquemque numerorum Q esse ternarii multiplum, sed contra
alternos tantum e numeris P. Si ergo scribatur Q, = 3 b s , P 2 , = a 2 . P<tt + 1 = 3 2
habetur calculi forma paullo simplicior,
3&,-4\ ~ bl ~ 2 V .3
-
Art. 9.] DE FRACTIONIBT7S QUIBITSDAM CONTHSTTJIS. 299
vel, si notationem Gaussianam sequi placet
, 36?-4X
Oj = 6 9 , mod ! ; 2 = =
b 2 = b { , mod a 2 ; a a =
36 2 2 -4X
, , 3& 2 g _ 1 -4X
b l = b,_ l ,moda g _ 1 ; a s =
ubi numeros b,, k i _ l - , e congruentiis & = &_!, mod a g , ita determinari
oportet ut 6 g quam proxime accedat ad valorem radicis ( 1) 8 + 1 2 A/ -
*^ o
His in schematis determinationes numerorum
ex antedictis (Art. 3) accuratas esse apparet ; sed contra determinationem quoti-
entis k { I Of falsam fore, si
+ 1
hoc est, si in aequatione
fiat [fe,- + i]< T^'[ a + i]- Quocirca calculus eo usque producendus est donee perveni-
atur ad sequationem in qua aut a { + j_ + a { + 2 = 0, aut ^a i 3b i + 2a i + 1 = 0. Nam,
si [<^,-]>f, aequatio quse conditioni i + i + o[,- +2 = satisfacit quseque ideo in toto
schemate ultima est, calculi normam sequenti "suo loco se offeret. Si vero est
[<,.] < f , ultima ista sequatio quodammodo extra ordinem calculi erit, quia fallit
determinatio k i = !Q i , Sed hoc in casu est necessario k t = +1, unde sequitur
penultimam sequationem talem fore ut sit
cujusmodi sequatio si quando se obtulerit, ipsa certe est proxima ante ultimam,
atque, ut ad ultimam perveniatur, quotienti k { tribuendus est valor + 1 ; idque
observandum est etiam si forte fiat [5,-] > |. Calculum autem numerorum a et b
Q q 2
300 DE FRACnONIBUS QUIBUSDAM CONTINUIS. [Art. 9.
ulterius extenders omnino opus non est ; cum in horum periodis semissis prior
hanc habeat formam
&,, & 2 , & 3 , ...,&,.; 6,- +1 ; 6,-, &,-_!, ..., Z>!;
altera autem semissis eosdem numeros signis mutatis eodem ordine repreesentet.
Quae omnia animadvertere operse pretium est propter similitudinem primornm
formae
Exemplum I. Sit X = 199 ; erit
/2A/^:=16, =-28, b =-16, a, = l.
v o
^Equationes autem P, + 2Q i x + P e+1 x 2 = brevitatis causa per notas sic scri-
bimus (a g , &,, 8+ i) ; quo pacto habebimus ex calculi ordine has aequationes in
primo periodi quadrante
(-28, -16, 1), (1, 16, -28), (-28, -12, 13),
(13, 14, -16), (-16, -18, -11), (-11, 15, 11):
undent 4 x 199 = ll 2 + 3. 15 2 ;
16 + 2 x /ip = [32, 3, 2, 6, -3, -9, 2, 6, 1, 66, 1, 6, 2, -9, -3, 6, 2, 3].
Exemplum II. Sit X = 607 ; erit
J2/y / ^ = 28, =-76, 6 = -28, o, = l.
Hinc oritur periodus
( - 76, - 28, 1), (1, 28, - 76), ( - 76, - 48, - 59),
(-59, 11, 35), (35, -24, -20), (-20, 36, -73),
(_ 73 , -37, -23), (-23, 32, -28), (-28, -24, 25),
(25, -26, -16), (-16, -22, 61), (61, 39, 35),
(35, -31, 13), (13, 34, 80), (80, -46, 49),
(49, 3, -49), ...
undefit 4x607 = 49 2 + 3.3 2 ;
/MT=[56, 3, -1, -3, -3, 3, -3, 6, -2, 9, 1, -6, 5, -3,
+ 1, 3, -1, 15, -2, 3, 3, -6, 2, -9, -1, -9, - 1, -3,
1, 168, 1, -3, -1, -9, 1, -9, 2, -6, 3, 3, -2, 15, -1,
3, 1, -3, 5, -6, 1, 9, -2, 6, -3, 3, -3, -3, -1, 3].
Art. 10.] DE FRACTIONIBUS QUIBTJSDAM CONTINUIS. 301
Observandum autem est hie in penultima sequatione (80, 46, 49), sive
(u> &n> is)> haberi 40 - 3 x 46 + 2 x 49 = 0, unde sequitur esse k u = 1, quamvis sit
46 + 28
10. Quum autem numerus 4\ per formam x z + 3y 2 trifariam reprsesentari
possit, ilia quidem reprsesentatio, quse numeros x et y pares habet, cum nostra
convenire non potest. Ex duabus, quse reliquse sunt, una sola est in qua
x= +1, mod 12 ; atque hsec ilia est quam prsebet sequatio 4X = a? + 1 + 3& 2 +1 . Si
enim i impar est, habemus
3(k , 3^, ..., 3^.) 2 -4
sive
Unde fit a< + 2 = 1, mod 12. Ac similiter, si numerus i par est, erit a i + 1 = l,
mod 12.
Cseterum, ut ex inventa sequatione 4X = a? +1 + 3^ + 1 , ipsius X reprsesentatio
habeatur, fiat ,- + i + &,-+i, =0, mod 4 : quo pacto erit
contra, si data fuerit sequatio X =A 2 + 3JB 2 , fiat ^4 = 6/x + 2ei, 5 = 2v + e 2 , desig-
nantibus literis e ls e 2 unitates, quarum altera ipsa sequatione definitur, altera ita
capiatur ut n + v par evadat. Quibus rite animadversis, habebuntur sequationes
2 , =1, mod 12
M + e.-tfM-iy* 1 ^;
nam ( 1)' + 1 6 i + 1 certo numerus posit ivus est, cum debeat esse
Hinc nanciscimur theorema :
' Designante \ = A 2 + 3B 2 numerum primum formse 12w + 7, resolubilia sunt
haec tria sequationum paria
302 DE FRACTIONTBTJS QUIBTJSDAM CONTINUES. [Art. 10.
cujus veritas facile colligitur ex sequivalentia formarum ( 4X, 0, 3), ( 12 X, 0, 1),
(P + !> Qi+ll *i + V-
Tres isti numeri [3e 2 B e^A], [2^^], [3 e z S + 1 A], qui reprsesentant
totidem valores indeterminati x in aequatione
eo etiam nomine nobis memorandi sunt, quod, evoluta radice ^/SX in fractionem
continuam vulgarem, inter denominatores quotientium completorum uno tractu
veniunt. Atque, si est A>3B, erit 2 A medius, si vero A < 3B, erit idem vel
primus vel postremus. Nam, si A > 3B, colligimus sequationes
3B-A-2(3B + A)x+2Ax* = 0,
alteram alteram excipere in periodo sequationis 3 X + x 2 = ; utramque enim
aequationi 3X + x 2 = sequivalere per demonstrationem theorematis prsecedentis
evincitur ; utraque autem characterem sequationis periodicae habet propter inse-
qualitatem A>SB. Similiter, si A < 3B, sequationes
- 2^-2 (3B-A) x + (3B + A) x 2 =
deinceps occurrunt in eodem periodo. Hinc nanciscimur methodum non inele-
gantem solvendi sequationem \ = A 2 + 3B 2 . Namque in evolvenda radice ^/3\,
inveniemus tres juxta denominatores, quarum medius sequatur extremoram sum-
mse, idemque exsuperat radicem ^/SX ; ex his unus erit 2 A, reliqui duo habebunt
valores impares [3JB + A],
Exemplum. Sit X = 139 = 8 2 + 3. 5 2 ;
Hie est C], A = 8, 3e 2 B-e 1 A= -23, -3e, l B e 1 A = 7. Atque erit ex evolutione
nostra, cum sit 12 A/ - = 14,
v o
(32, -14, 1), (1, 14, 32), (32, -18, 13), (13, 8, -28),
(_28, -20, -23), (-23, 3,23),...
Unde fit 4xl39 = (-23) 2 + 3.3 2 ;
14 + 2 N /Ei = [ 28, -3, -2,3, -1, -3,1, -0, -1,84,
-1, -6, 1, -3, -1, 3, -2, -3].
Contra ex evolutione vulgari sequationis
-17-40a; + a; 2 = 0,
Art. 11.] DE FRACTIONIBUS QUIBUSDAM CONTINUIS. 303
orientur hujusmodi periodus
( - 17, - 20, 1), (1, 20, - 17), ( - 17, - 14, 13),
(13, 12, -21), (-21, -9, 16), (16, 7, -23),
(-23, -16, 7), (7, 19, -8), (-8, -13, 31),
(31,18, -3), (3, -18,31),...
in qua habemus
20 + 7417 = [40, 2, 2, 1, 1, 1, 5, 4, 1, 12, 1, 4, 5, 1, 1, 1, 2, 2];
videmus autem numeros 16, 23, 7, inter coefficientes sequationum periodicarum
comparers, idque accidere fere vergente ad finem primo periodi quadrante.
11. Numeri primi formse 12w + ll similes aliquatenus proprietates habent.
Namque ex evolutione vulgari
= [2&>My, My_i, .... Mi, 2 > Mi, -.., My], .....
elicimus sequationem
-(a, /* 1} ..., ftj, b)x(f*i, ..., My) = ~3(Mi, M 2 , -, My,
Quse cum sequatione P 1 P 2 = 3^ + 1 ita comparetur ut fiat
M 2 , , My, =
Quibas positis, facile perspicitur in evolutione (^4) quotientium seriem
&, My, My_i, -, M 2 , Mi,
cum hac serie , , , ,
A-O, OA/I, / 2 , ..., A/J,
sine fraude commutari posse. Erit itaque
^o + V 3 = [2^o> -3ft,, fta, ..., -3*j, ft,, -Bfto.ft,, -Sftj, ft,, -S.ftJ . (B)
Qua in evolutione altera ex sequationibus, quse medium locum in prinaa
period! semisse tenent, dabit aut hanc sequationem
f) ^\ _ /~\2
O A. - \J ' i i *
aut hanc
304 DE FRACTIONIBUS QUIBUSDAM CONTINUIS. [Art. 11.
prout scilicet numerus i par est aut impar. Ponimus autem eodem fere quo
supra modo )_ q;, p . n P -
V 8> * W 2> f 2
6 = -& = --f'V 3. i = l;
et calculum periodi instituimus secundum hoc schema
a.
X 36*
A O U s
vel ex notatione Gaussiana
6 2 = &D mod a 2 ; a 3 =
& = &_, mod a; a =
\-36f
numeris b e ita determinaris ut quam proximo ad valorem radicis (
accedant. Cui calculo finis faciendus est cum primum sequatio a,-.,
sese obtulerit ; eo enim usque valet formula k s = Id, , quse ulterius progredientem
destituit. Periodi autem numerorum a, b, k, eandem plane descriptionem
habent, quam supra exemplis illustravimus.
Quum autem in sequatione \ = 3b* i+1 a? i+1 numerus a <+1 par esse nequeat,
e venire potest ut repraesentatio ipsius X, quam ejus aequationis ope nanciscimur,
non Bit omnium simplicissima. Est tamen vel ipsa omnium simplicissima, vel
proxima post simplicissimam. Habemus enim (Art. 4)
Art. 11.] DE FRACTIONIBTJS QUIBUSDAM CONTINUIS. 305
hoc est,
ideoque etiam
At in sequatione t 2 3u 2 = ~L, valores numeri n sic procedunt, 1, 4, 15,...,
unde concludimus, si fuerit [&; + i]<[a; + i], unam fore eamque solam reprgesenta-
tionem quse per numeros minores fiat; quae si est X = 3x 2 y z , erit y par, x
impar. Itaque si A et B minimi numeri exstant, per quos sequationi X = 12B 2 A*
satisfieri possit, sequationes
3x 2 -\y 2 = eA, =3eA, =4(6^4
resolubiles erunt, signo unitatis e ita determinate ut fiat eA = 1, mod 3. Facile
autem perspicitur sequationi
-(3B-A)-2(GB-A)x + Ax 2 = Q
competere characterem aequationis periodicse : cum sit
Quare vulgaris quoque evolutio radicis ^/SX suppeditat solutionem simplicissimam
sequationis X = 12^ 2 A 2 ; veniemus enim in ea evolutione ad sequationem
l x 2 = in qua erit \pi = g_ t +p + \, ideoque etiam,
Exemplum. Sit X = 167; erit I f\f o = 7; atque habebimus ex evolutione
nostra hanc periodi fonnam
( - 20, 7, 1), (1,7,- 20), ( - 20, - 13, - 17),
( - 17, 4, 7), (7, - 3, - 20), ( - 20, 17, - 35),
( - 35, - 18, - 23), ( - 23, 5, 4), (4, - 7, - 5),
(5, 8, -5),...
Erititaquel67 = 3.8 2 -5 2 ;
7 + x/ I | I = [i4. - 3 - -1, 3, -1, -3, -1, 9, 3, -9, -3,
3, -1,3, -1,3, 1, -42,1, 3, -1,3, 1,3,
-3, -9, 3, 9, -1, 3, -1, 3, -1, -3].
VOL. II. B r
300 DE FRACTIONIBUS QUIBUSDAM CONTINTTIS. [Art. 12.
Evoluta autem radice ^/SOl secundum methodum vulgatam invenimus
periodum sequationum hancce
( - 17, - 22, 1), (1, 22, - 17), ( - 17, - 12, 21),
(21, 9, - 20), ( - 20, - 11, 19), (19, 8, - 23),
( - 23, - 15, 12), (12, 21, - 5), ( - 5, - 19, 28),
(28, 9, - 15), ( - 15, - 21, 4), (4, 19, 35),
(-35, -16, 7), (7, 19, -20), (-20, -21, 3),
(3,21, -20)...
e quibus nona suppeditat solutionem sequationis 167 = 3x8 2 5 2 , cum sit
^ x 28 = 19 - 5. Erit praeterea 22 +
[44, 2, 1, 1, 1, 1, 3, 8, 1, 2, 10, 1, 5, 2, 14, 2, 5, 1, 10, 2, 1, 8, 3, 1, 1, 1, 1, 2].
12. ^Equationes P 1 P 2 = 21^ + 1, PiP 2 = 2RI + 1, et reprsesentationes inde
orientes numerorum primorum per formas 2x 2 + 2/ 2 , 2x a y z , a Goepelio in com-
mentatione inaugurali* longe pulcherrima (ex qua disquisitionem nostram de-
libatam esse libenter fatemur) tanta felicitate ope fractionum continuarum
illustrates sunt, ut earn rem iterum attingere prope supervacaneum sit. Tamen
ne quis analogiam determinantium 2 et 3 a nobis prsetermissam desiderit, in
vestigiis ejus viri paullisper immorari liceat. Et primum quidem adnotamus,
ab sequatione P l P z = 2R\ + \, eodem fere quo antea modo, perveniri posse ad
sequationem P 1 = (k l , 2k,, k 3 , ..., 2k 3 , k,, 2k,);
eamque transformationem ita fieri posse ut quotientem unitati sequalem excipiat
ejusdem signi quotiens. Ex quo apparet in fractione continua
p
op = LA ' * *2 > *2 > 2 K! J ;
sese obc~,q ua tiones ^ = 10, k. 2 = I6 2 , ... eamque legem etiam in extreme quotiente
destituit. nigi forte fiant signa quotientium /:<_! et O f contraria, ac praeterea
habent, quam ,' n casu er it !Q i = +2, k { = +1. Nee secus in fractione continua
Quum autei,q U atione P X P 2 = -2^ + 1, erit
evenire potest ut i\
, -i i ^ I "'I * ~* f^9 i *vi * * " " "'Is "^2 > """ " "'I /)
non Bit omnmm sim 1 !
proxima post simplicist_ _J__rz. _9^ 91-1
nff ~\. K U K 2 K 3i-"> K '2> *ljj
w-u
line und angewandte Mathematik,' vol. xlv. pp. 1-13.
Art. 12.] DE FKACTIONIBUS QUIBTJSDAM CONTINTTIS. 307
&! = /#!, Jc 2 = I6 2 , ... atque adeo &, = /#<, cum sit utique i >^(2 + V2) : quae res
item docet in neutra formarum sequivalentium x z 2y z , 2x z y 2 , dari repraesenta-
tionem numeri P t repraesentatione nostra magis simplicem. Turn vero, designante
X numerum primum formae 4n + 3, vel duplum talis numeri, atque evoluta radice
secundum praacepta vulgata erit in media periodo hujusmodi aequatio anceps
(ea enim sola est quae exsistere potest), ubi numerus b ita determinatur ut
\ (X 6 2 ) integer atque positivus, idemque quam minimus fiat. Sit a numerus
integer surdo ^/X proxime minor; erit aut a = b, aut a = 6 + l. Atque si est
a = 6, habebimus hujusmodi evolutionem
unde nanciscimur aequationes
(a, ft lt n 2
e quibus concludimus fore
Itaque, si fiat e = ( !)", erit e= +1, vel e= 1, prout X (vel^X) est formae
8n + 3 aut formse 8n + 7; atque per transformationem jam ssepius in hac
commentatione usitatam habebimus hujusmodi evolutionem
cujus veritas facili confirmatur, si formulam ex evolutione vulgari oriundam
paullulum immutatam adhibemus,
+ v/X = [a, 0, a,//!, // 2 , ..., /*, ^a, 0, ^a, /*, M n _ 2 , , M 2 ,
Si autem est = & + !, erit
^/"x = [a, MI, M 2 , ..., M n , M a ~ !
quae sequatio, si sic scribatur,
V /X = [ + 1, 0, -I,;*,, .... M B , -1, 0,
abit in formam precedent! (.4) similem, atque eodem prorsus modo tractari
potest. Itaque habebitur evolutio ad hanc normam concinnata
a + l+J\ = [2(a + l), &!, 26^ 2 , ..., k 2 , 2 6 ^, e(a + l), Z e k lt k 2 , ..., 2tk,, jfej
in qua observandum est quotientem A^ negativum fore.
r a
308 DE FRACTIONIBUS QTJIBTTSDAM CONTINUIS. [Art. 12.
Schema autem calculi, quo ad has evolutiones utimus, hoc ferme est. Sit
a, = l, 2ea = /3j X, numero negative /3 ita determinate ut a integer atque
idem quam minimus fiat. Turn sumpto ab aequatione
initio caeteras hoc pacto derivamus
I I I I M I ' * j ,
X-|8 2 .
A,
mod 2 a% j
3 2ea a '
= &_i, mod 2 a, ;
2ea,
ubi numeros impares B, ad valorem radicis ( l)'x/X quam proximo accedere,
/8 8
quotientes autem k e per aequationes Ic 3 = s - determinari intelligendum est.
Finis autem calculo faciendus est simul ac pervenerimus ad sequationem in qua
est a i+ i + a i+2 = 0, quseque adeo suppeditat discerptionem qusesitam. Quando
autem e= +1, hoc est quando X, vel |X, est numerus primus formge 8n + 3, fieri
potest ut regula generalis in ultimo quotiente & { falsa sit ; quae exceptio turn
locum habet cum in discerptione qusesita X = ^4 2 + 2^ 2 numerus B major est
quam 2 A. Quo in casu erit in penultima sequatione a i + 2/3 j + 3a <+1 = ; atque,
ut habeatur ad extremum a,- +1 + a< +2 = 0, quotienti k t tribuendus erit valor +1,
etiamsi fiat I6 { = + 2.
Exemplis brevitatis causa supersedemus ; illud unum adjicimus, pro numeris
primis formsa Sn + 3, eorumque duplis aequationes
esse resolubiles, si quidem ponatur X = 2A 2 + S 2 , et numerus A eo signo aflficiatur,
ut fiat ( - l)*^- x > = 1, vel ( - 1)* ( ^~ 1) + ^ 42 - 1) = l, prout est ipse X numerus primus,
vel duplum numeri primi : quse quidem conditiones ex ipsa sequationum forma
oriuntur. Et similiter pro numeris primis formae 8n + 7, eorumque duplis,
sequationes _* =
resolvi poterunt, designantibus literis A et B minimos numeros qui sequationi
\ B 2 2 A 2 satisfaciunt, et de terminate ut supra ipsius A signo.
Art. 13.] DE FRACTIONIBUS QUTBUSDAM CONTINUIS. 309
13. Coronidis loco observamus eandem fere fractionum continuarum transfor-
mationem utilitate non carere in theoria numerorum complexorum. Sit enim
habebimus hujusmodi sequationem
E r _ I ' ' ' u '\
(.Mi, M 2 , , M 2 n) X (M 2n , , M2, Mi)
_L /,/ ' ,/ \ V (' ,,'\
TV*I) M 2 , , M2n V \^2 i, , "2, r*l/ ,
unde sequitur determinantem ^T summam esse duarum, quas vocant, normarum,
hoc est, quattuor quadratorum. Ponatur ergo
designantibus literis P lt P 2 numeros positivos reales, R l) R{ numeros complexes
conjugates ; atque in hac sequatione fiat
V t
et sic porro, numeris R s ita determinatis ut norma R t x R' t evadat quam minima.
Hinc erit P 1 >P 2 >P 3 , ... atque ad extremum habebitur
-Pi = (Mi, Ma, M 3 , -, Ms, M 2 , M'I) 5
quas formula, cum universi numeri primi formulam l+x 2 + y 2 metiantur, continet
demonstrationem theorematis Fermatiani omnes omnino numeros esse summas
quattuor quadratorum.
Plane autem eodem modo evincitur omnes numeros habere hanc formam
a; 2 + y z + 3 (u z + v z ), atque praeterea hanc etiam x 2 + 2 y 2 + 3 u 2 + 6 v z . Quarum
enuntiationum altera intra veritatem cadit, cum omnes numeri in alterutra
harum formularum
comprehendantur ; id quod a viro clarissimo Lejeune Dirichlet* jampridem
demonstratum est : altera ea ipsa est quam olim summus in hac disciplina
magister C. G. J. Jacobif ex notissima formula elliptica originem trahere in-
dicavit. Hoc autem loco satis erit demonstrare, proposita sequatione p 2 + p + 1 = 0,
omnem numerum realem esse summam normse sunplicis et duplicis.
Sint igitur A, B, P numeri reales, A Bp numerus complexus integer
continens radicem asquationis p z + p + 1 = ; erit
A 2 + A B + B 2 = N . (A -
' Journal fur die reine und angewandte Mathematik,' vol. xl. pp. 231-232. t Ibid. vol. xxi.
310 DE FRACTIONIBUS QTTTBUSDAM CONTTNUIS.
Quod si ponatur A-B P = nP+a-bp,
numerus complexus M ita determinari potest ut sit
[Art. 13.
Fiat enim, quod utique licet, a = A, b=B, mod P, \a\<>\P,
signa numerorum a et 6 contraria sunt, habebimus manifesto
Hie, si
Si vero hsec signa conveniunt inter se, atque est praeterea uterque numerus
P l
a et b minor quam . , erit iterum N(a bp) < \ (P I) 2 < ^ (P 2 1). Quod
si alter eorum, velut a, eum limitem superaverit, faciendum erit a = a + P,
[a*] < P, ita ut habeatur, si quidem P ^ 5,
experiendo autem invenitur eadem lege teneri numeros quinario minores.
Hinc patet calculum quo saepius jam usi sumus accommodari posse ad
aequationem
in qua si fiat
_
* ~
semper quam minimae evadant,
et sic porro, ea lege ut normse
veniemus tandem ad sequationem
quee continet demonstrationem theorematis, cum omnis numerus impar metiatur
formulam 1 + 2 a? + 6 y*, vel, si mavis, formulam
Simile fere theorema oritur ab aequatione o- 2 + o 1 + 2 = 0. Universi enim
numeri primi, praeter septenarium, metiuntur formulam l+a^ + Ty 2 , hoc est,
formulam l+x* + xy + 2y*: ac praeterea, posito
utique fieri potest
Art. 13.] DE FRACTIONIBUS QUIBUSDAM CONTINTJIS. 311
Unde patet numerum quemcumque integrum summam esse duarum normarum ;
vel, quod idem est, in hac formula
x 2 + xy + 2 y* + u* + uv + 2 v z
comprehendi. Omnes itaque numeri pariter pares in hac forma continentur
impares autem, atque impariter pares aut in ilia aut in hac certe
XL.
ON SOME DISCONTINUOUS SEMES CONSIDERED BY
EIEMANN.
[Messenger of Mathematics, Ser. ii. -vol. xi. pp. 1-11 (May 1881)].
-LlIEMANN, in a fragment published after his death in his collected works*,
has .determined the value of certain q series in the limiting case in which the
analytical modulus of q is unity. If q = p e ie , the series considered by Riemann
contain 6 only, and converge and diverge for an infinite number of values of 6
between any two limits however near to one another ; they appear, indeed, to
have attracted Eiemann's attention by this peculiarity. Two of these series,
which may be regarded as examples of the rest, are considered in the present
note, and some details relating to them are supplied which are omitted in
Riemann's brief record of his results.
Let SQ-a-. ......... (i)
n denoting any positive integral number from + 1 to co, so that (' Fundamenta
Nova,' p. 103)
Put q = p e ie , multiply each side of (i) by = , and take the rectilineal
integral from to pe ie along the vector e w ; we have
(ii)
' Mathematische Werke,' p. 427.
ON SOME DISCONTINUOUS SEBIES CONSIDERED BY RIEMANN. 313
the logarithms on the right being so taken as to vanish with p ; or, which is the
same thing,
' + ^tan-^r A . (iii)
q n 2 n 2 \l + p n cosnO'
where A n is the arithmetical square root of
l + 2p n cosn6+p zn ,
log A n is a real logarithm, and tan -1 ( - -r ) is an angle included between
\1 + p n cosnd/
the limits \ IT and +\v . So long as p < 1, the series S (q), and the integrated
series (ii) or (iii) are absolutely convergent. When p = 1, the series (i) and
(iii) respectively become
IV
tan %(nff), ........ (iv)
and 22log{4cos^-(e)}+4i2[i(n5)];. . . . (v)
where by [| (n6J] we understand the absolutely least angle which has the same
tangent as [| (n 0)], except when [| (n6)\ is an uneven multiple of |TT, in which
case [| (n&)~\ = 0. The imaginary part of (v) is always convergent ; the converg-
ence or divergence of (iv) and of the real part of (v) depends on the nature of
the value assigned to 6.
s\
First, let - be incommensurable. In this case the series (iv) is always di-
7T
vergent. For if -=-7- , T are two consecutive convergents to - of which the latter
has an uneven numerator a, and if fy is the complete quotient immediately
a
succeeding T , we have
whence, when 6 is very great, the absolute value of y- tan|&0 is approximately
_ . Thus the series (iv) contains an infinite number of terms of which the
VOL. II. 8 S
314 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN.
absolute value is finite, and it is consequently divergent. Riemann says that
Q
when - is incommensurable, the real part of the series (v) is also always diverg-
ent. But this appears to be an oversight ; for the real part of (v) is in fact
divergent or convergent according as the incommensurable quantity 6 does, or
does not, satisfy certain conditions ; and, in particular, it can be shown that,
whenever 6 is the incommensurable root of an equation of a finite order having
integral coefficients, the real part of (v) is convergent. For each successive
value of n we form the equation
a
where m is an integral number, e = + 1, and < v < 1 ; it is evident that m, e,
and v can always be determined in one way and in one way only. We then
have
log {4 cos 2 A (nS)} = log{4 sin 2 i (m)},
2 2 sin 1 (nr)
and, since - < -- "-* - < 1,
7T VTT
*
is absolutely convergent ; i.e. the series
log (4 cos 2 ! ( n6 )} ( vi )
is convergent or divergent according as the series
Q
is convergent or divergent. Let - be the root of an irreducible equation
of which the coefficients A, B, ... are integral ; it may be shown as follows by a
method due to M. Liouville, that the series (vii) is convergent.
Q
Let x lt c 2) ...be the real roots other than- of /(*), yiizi, yti%,
TT
its imaginary roots ; and let S 1 be a positive quantity greater than the greatest
ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN. 315
6 6
of the differences x, y ; we have
7T IT
/m + e\ . ev rf/m + e \ f/m + e \
n'f(- -} = A x xn'xlll- x)xll( y) +
J \ n ' n \ n / (\ n y '
But vPf\ -) is an integral number, which cannot be zero, because /"(
V ft, /
n
is irreducible ; hence
or log v < (s 1) log n + c, where c is a finite constant ; this shows that (vii) is
j i -^ 1 ft
convergent, because 2 2 is convergent.
g
The same reasoning would suffice to prove, that when - is a root of any
IT
equation such as/(o;), the series
always converges if 2 - p- is absolutely convergent.
<p(n)
On the other hand, between any two limits however near, there is always
an infinite number of values of 6, for which the series (vi), and indeed any series
(VU1)
becomes divergent, however rapidly the positive function <p(n) may increase
n
with n. If, for example, in the development of in a continued fraction, it
happens an infinite number of times that a convergent y , having an uneven
numerator a, is immediately followed by a complete quotient which surpasses
f*
j-e* (!l) , where c is any finite positive constant, the series (viii) contains an in-
finite number of terms of which the absolute value is finite, and is therefore
divergent.
A /7
Secondly, let - = =-, a rational fraction in its lowest terms. If a is uneven,
the series (iv) and (vi) contain terms which are infinite. But when a is even
the two series are convergent, and Kiemann has succeeded in assigning their
sums in the following manner.
s s 2
316 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN.
Writing 2 a for a, and observing that b is uneven, and designating the sum
of the series (iv) by - 2 log 2 + 2iS, we have
, ^,==00 -1)* ,
tan . = 2. =1 2 (=0 _i tan
T . =1 (=0
=M
and T^s-26 2 -' 7
whence
s- ^ 2-- (-1X- z;:f ^ 2-- (-
But 2;:f(-l)' 7 ' (20ff - T ) = 2& or 0,
according as the congruence T =2a<r + l, mod 26, is or is not satisfied. We have
therefore
7 ^
o
.v-^-D/ 1W (*{
-i ff=1 (-1) I ^
/ M.+y*'a
But /* sin = [i 01
Jo 1 + 2CCCOS0 + 0; 2
r >-/ ~i
and hence, finally, = 2 iV 2^ ^ 1} ( 1)" I <r r J
. ^ff=i.-i / iv
^2^ (-1)
the symbol fo-^ 1 denoting the excess (positive or negative) of <r r above the
integral number lying absolutely nearest to o- ^ .
The method employed by Riemann to effect the preceding summation is due
to Lejeune Dirichlet, and was employed by him in his memoir on the Arithmetical
Progression (' Transactions of the Academy of Berlin for 1837,' sections 4 and
10), and in his ' Recherches sur di verses applications de 1'analyse infinitesimale
h, la the"orie des nombres (' Crelle's Journal,' vols. xix. and xxi ; see sections
1, 9 and 10 of the Memoir).
ON SOME DISCONTINUOUS SERIES CONSIDERED BY RTEMANN. 317
Again, denoting by S' the sum of the series
we have, if & = --^ 9
But, by a formula due to Euler,
r
h
i i
-s) a n " J
_ i
- 5 2 "J
-^
V 2 /S7T\ IT V6
- cosec - sec
/S7T\
COSI-^-l
V6/
whence, finally,
2 2
sm
2 COS
m'(-
S7
When the series
2 / S7P \
8m (y)
is divergent, we must regard the equation
e"*' 9
2 r-[sH)] ( ix )
iff %
as having no assignable meaning. When that series is convergent, the equation
318 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN.
always has a meaning, but some explanation is required to show clearly
what its meaning is. As to the signification of the left-hand member no
(00
< (x) dx we understand the limit of
p h
I (x) dx when h is increased without limit, so by
J o
dp
n l+p n e nie p
we are to understand the limit of
/"
Jo
- 1 )" P n e ni$ dp
nni6 '
n i + p n e n ' p
when 1 - p is positive and decreases without limit. We thus have the equation
x -
when /> = 1. But before we can pass from this equation to the equation (ix), we
have to show that the limit of
when ,0 = 1, is
And although this is true, it cannot be assumed without proof; any more than
we could assume that the limit of the series
^ (sin x) + ^ (sin 3x) + \ (sin 5x) + . . .,
when x = 0, is the same (which it certainly is not) as the value of that series
when x = 0.
It would seem that Eiemann in his fragmentary note intended to give the
required proof ; for he developes the expression
in a series proceeding by powers of q (just as Jacobi has dealt with similar
series in the ' Fundamenta'), and finds
= 2 1 + (1)H2 * (n) T,
(n) denoting the sum of the uneven divisors of n. He also enunciates and
ON SOME DISCONTINUOUS SEEIES CONSIDEEED BY RIEMANN. 319
proves the following theorem (which is really due to Abel, CEuvres, vol. i,
p. 69*) :
' Si series
eo quo scripsimus ordine summata summam habet convergentem, functio ipsius
rMcserie
expressa, convergente r versus limitem 1, convergit versus valorem eundem.'
But the fragment breaks off abruptly with the words ' ex hoc theoremate
facile deducitur,' and it is not very easy to see how he proposed to complete
the demonstration, because the series
2 + \" $ (n) q n
is certainly not always convergent when q = e i9 . The following considerations
may serve to replace Biemann's intended demonstration. Denoting e ie by q ,
( 1) 1 + Q n
we have, in fact, to show that Km 2 - f- x log - = ^, wnen P converges
n" 1 + q n
to unity, or q = pe ie to q = e io . The imaginary part of this series is
lim [tan- Y, P "f^ ,) - [| (n fl)]1 ; . . . . (x)
L \l+ n cosnQ/
,
n \l+p n
the real part is
^^)]; .... (xi)
it will suffice to consider the latter only. Let h be any very great number ; and
let A be so great that 4A/t is greater than the greatest value of sec 2 ^ (n&) for any
value of n not surpassing h (we observe that sec 2 ^(0) cannot be infinite, because,
n
in the cases under consideration, - is not a rational fraction having an uneven
numerator) ; also let p be so near to + 1 that !-/>"< , for all values of n
not surpassing h; then, for all such values of n, p n + \^(1 -p n )} 2 sec 2 ^(n6) lies
between 1 j-j , and 1 + j-j ; and, consequently,
(n^ .... (xii)
* [The reference is to Holmboe's edition, 1839. In Sylow and Lie's edition (1881) the theorem occurs in vol. i, p. 223.]
320 ON SOME DISCONTINUOUS SERIES CONSIDERED BY RIEMANN.
is evanescent, being intermediate between the two values
which are both evanescent because r is as small as we please.
AA
Again, the series
/ i\
(n6y] .... (xiii)
is evanescent, for jo"+ {i(l-jo n )} 2 sec 2 ^ \(n6) is certainly greater than and less
than sec 2 ^ (nff) ; whence this series is less in absolute magnitude than the sum
of the two series
v n= 1 v n= Iogsec 2 i(0)
log* x 2, i=fc+1 -^ , z n=M - ?-+' ;
and of these the first is evidently evanescent when h is very great ; the second
is also evanescent because, the series 2 " 2 ^ ' being convergent in the
72*
cases here considered, the remainder after h terms vanishes when h is very great.
Hence, finally, the series (xi), which is the sum of (xii) and (xiii), is evanescent.
XLL
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
[Messenger of Mathematics, Ser. II. vol. xii. pp. 49-99 (August-November, 1882).]
I. On the Formulae of Transformation, with especial reference to the case in
which the modular equation has equal roots.
1. LET y = F(x), where F (x) is a rational fraction containing x n , but no
higher power of x, be an integral of the equation
dy* J. __ dx 2 ^ . v
2 2 -*
so that, if y = F(x^),
r y _ fy _ JL r
V{i-i-x 2 } ' MJ x
the track of the integration and the initial sign of the radical being assumed
arbitrarily on the right-hand side, and being determined on the left-hand side in
accordance with these assumptions by the equation y = F(x). If (4.K,
(4 A, 2{A') are pairs of conjugate periods of the integrals*
dy
r x j r
J V{i-* 2 i-Fz'} a J
* Any value of one of these integrals extended over a closed track, at the end of which the radical
has the same sign that it had at the beginning, is a period of the integral. The periods, and the pairs
/dx
. are, in fact, the periods, and the pairs of
conjugate periods of the doubly periodic function x = sin am u defined by the equation
dx / du \
VOL. II. T t
322 NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. [.Vote I.
respectively, the equation (2) implies the equations
4
(3)
in which a, b, c, d are integral numbers ; because whenever x describes a closed
contour, at the end of which the radical under the integral sign has the same
value that it had at the beginning, y describes a like contour, and consequently
any period of the integral on the right-hand side of the equation (2), must
be a sum of multiples of the periods of the integral on the left-hand side.
The determinant ad bc = m is different from zero, because the quotient -^
K
is not real ; and is positive, if (as we may suppose to be the case) the co-
iK' iA.'
efficients of i in the complex quantities -^ , r- have the same sign.
Let us now assume that the fraction F(x) satisfies the conditions
dy 1 r dx f' ( 6 )
in this case, we have the equation
rv dy J_ r x dx
Jo V{(l-y' J )(l-X 2 2/ 2 )} = MhV{(l-x*)(l-k*x*)l' ' ' '
and as particular cases of it, the equations
r 1 dy J_ r 1 dx
Jo v\(i-y*)(i-\*y*)} == M Jo ;
7
i/O
The equation (4) implies that F(x) is an uneven function of x, and also that
tit
the limit of - , when x = 0, is + -try. ; the condition F(<x>) = oo shows that the nu-
x ~ M
merator of F(x) is of higher dimensions than its denominator. Consequently
n is an uneven number, and the equation y = F(x) assumes the form
' = x I +fi*;+f^ + - + >-i)*" ^ (6)
the coefficients a and b being connected by the relation
and the initial values of the radicals in the equations (4) and (5) being +1.
Art. 1.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 323
We shall henceforward suppose that the pairs of conjugate peri
(4 A, 2{A'), employed in the equations (3), are primary*
On this hypothesis, since Kis one of the values of the integral
r 1 dx
*-' U '
77-
it follows that -=^ is one of the values of the integral
r 1
.
similarly, %K' is one of the values of the integral
r 00 dx
Jo
and -^f- is one of the values of the integral
But all the values of the first of the integrals (7) are comprised in the
formula (4p + 1) A + 2qiA.', and all the values of the second of these integrals are
comprised in the formula 2rA + (2s + l)iA', p, q, r, s being whole numbers; we
have therefore
or, which is the same thing, the numbers a, b, d in the equations (3) satisfy the
congruences
a = 1, o = 0, mod 4 ; d = 1, mod 2,
so that m = ad bc is an uneven number.
Let u be one of the values of the integral
ny
Jo V{(1 ;
corresponding to a given value of y ; then all the values of that integral, cor-
* The definition of primary pairs of conjugate periods, and a statement of the properties relating
to them which are employed in the text, will be found in Note EL
T t 2
324 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I.
responding to the same value of y, are comprehended in the formulae
2J!fA - u - If (4g A - 2pi A')] '
where p and ^ are indeterminate integers. Writing 2m M A for 23f A in the
second formula, which is admissible, because m is uneven, and replacing 4 A and
2iA' by their values in terms of K and 2iK', we obtain the formulae
u + ; - [(pc + qd) 4JT- (pa + qb) 2 iK*]
2K- u - -- [(pc + qd) K- (pa +
(9)
in the second of which we have written 2 K for 2dK l)iK', as we may do, because
d is uneven and 6 even, and because we may omit multiples of the periods K
and 2iK'; any such omission being in fact equivalent to a change in the indeter-
minates p and q. Thus all the values of the integral
x*)(L-k*x*)}' ........
which correspond in the equation (4) to the given value of y, are comprised in
one or other of the two formulae (9). But since the congruences
, (U)
+ qd=s
are resoluble for m and only m pairs of values of r, s, incongruous to one another
for the modulus m, the values comprised in the formulae (9) group themselves in
2m sets not reducible to one another by the addition of multiples of the periods
4K and 2iK'. As representatives of these 2m sets we take the m pairs of
values
- 2riK'\
u H
m (12)
m
and we observe that to values comprised in the same set, or in two sets of the
same pair, there answers one and the same value of x. Hence in the equation
(4), and consequently also in the equation (6), to a given value of y there answer
m values of x. We conclude that m = n.
Writing a = a, 6 = 2/3, <y = 2 c, d = S, we may enunciate the theorem :
Art. 2.]
NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION.
325
' If y = F(x) is a rational integral of the equation (1), satisfying the con-
ditions
F(x) is of the form (6), the exponent n of the highest power of a; contained in it
being uneven, and any two pairs of primary periods of the elliptic integrals are
connected by the equations
where a, /3, y, $ are whole numbers satisfying the equation a fly = n, and where
the matrix
a, ft
is primary ; viz. it satisfies the congruences
a = l, mod 4 ;
a, ft
,
1,
0, 1
, mod 2.'
2. The foregoing theorem and its demonstration will be found in substance
in the ' Traite* des Fonctions Elliptiques ' of MM. Briot and Bouquet.
Both have been reproduced here, with some slight modifications, in order to
establish with precision the relation between the rational transformations
y = F(x) of the elliptic integral (10), and the arithmetical transformations (3)
of its pairs of primary periods. The following observations, though not necessary
for the immediate purpose of this note, may be useful as further illustrating
this relation.
(a) Every rational transformation must satisfy the condition F (x) = 0, as
otherwise the equation y = F (x) would indeed be a rational integral of the
differential equation (1), but would not transform the integral (10) into an-
2 K
other of the same form having the same lower limit. Since -^ must be
equal to a sum of even multiples of A and iA.', this condition implies that b
is even in the equations (3) ; or, which is the same thing, that every rational
transformation y = F(x) corresponds to an arithmetical transformation of the
periods which may be expressed in the form
<>
the coefficient 7 being an even number.
326
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
[Note I.
Again, when the condition .F(0) = is satisfied, a in the equations (3), or
a in the equations (13) is uneven. For we can cause x to travel from a; to a;
by a route such that the right-hand member of the equation (4) shall be in-
creased by -=j- ; y will travel at the same time from y to y, and the left-hand
member of the equation will be increased by 2A together with certain multiples
2K
of the periods 4 A and 2z'A' ; -=-= is therefore equal to an uneven multiple of 2 A,
together with a multiple of 2iA'.
The condition F (0) = being satisfied, F (x) is (as we have already seen)
a fraction of which the denominator is an even, and the numerator an uneven
function of x. The order of the numerator may be either higher or lower
by an unit than the order of the denominator ; in the former case we have
n uneven, and .F(oc) = oo, in the latter, n even, and .F(oo) = 0. Further, if
iK'
F (oo) = GO, -=rjr must be the sum of an even multiple of A and an uneven
iK'
multiple of i&' ; if -F(oo) = 0, -TF must be the sum of even multiples of both A
and iA'; hence in the former case, d in the equations (3), or S in the equations
(13) is uneven ; in the latter case, this number is even. Thus the matrix
is of one of the four types, mod 2,
1,0 1,1
0, 1 ' 0, 1
1,0 1,1
0, ' 0,
(c) The demonstration that m = n, given in Art. 1, applies only to the case
when m is uneven, and b even ; but in all the cases alike the second of the
formulae (8) may be written in the form
u M[4: (q 1)A 2pi'A / ] ,
so that the second of the formulae (9) becomes
m uneven,
m even,
In discussing this expression we have to consider the congruences
pa + qb = | b + r
[,
.s j
mod m,
Art. 3.]
NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION.
327
which, like the congruences (11), are resoluble for m and only m pairs of values
of r and s, and are moreover resoluble for the same pairs of values of r and s as
those congruences, because the congruences
pa + qb = ^b \
, f 7 , L mod TO.
pc + qa = ^ d ^in }
are resoluble. Hence the values of the integral (10) answering in the equation
(4) to a given value of y are comprised in 2m, sets, which, as before, may be repre-
sented by the formulas (12) ; we have therefore m = n in all the four cases.
(d) The remaining condition, F(l) = l, renders a = l, 6 = 0, mod 4, in the
equations (3), or a = l, mod 4, /3 = 0, mod 2, in the equations (13). Thus, finally,
if F(x) satisfies the two conditions F(Q) = 0, F(i) = 1, we must have either
or
n uneven, F (x) = oo ,
n even, F(<x) =0,
,
n:
1,
7, $
0, 1
a, /3
_ __
1,
7, S
0,
, mod 2,
mod 2.
When a or a is uneven, the condition that it is to be = 1, mod 4, can always
be satisfied by changing the sign of M.
It is important to observe that, if the pairs of primary periods have been
chosen once for all in the equations (3), the rational transformation y = F (x) can
correspond to only one arithmetical transformation of the periods ; viz. an
equation of the form Q A + ^ A/ = ^ A + ft ^
implies the equations a = 1( /8 = /3 1 , because the ratio A : i\' is always ima-
ginary.
3. If in the equation (1) we regard k 2 as a given quantity having any
value except one of the three 0, 1, oo, and M, X 2 as quantities to be determined,
any rational integral y = F (x) of - that equation which satisfies the conditions
is termed a primary transformation of the elliptic integral (10)*. It results from
' The word primary was employed by Gauss in the theory of complex integers of the form a + bi,
to distinguish one of the four associated uneven numbers t* 1 (a + bi), p = 0, 1, 2, 3, from the other
three ; and the same expression has since been employed in other complex theories. The essential
character of primary numbers in these theories is that they reproduce themselves by multiplication ;
viz. the product of two primary numbers is primary. It is natural to extend the use of the term to the
328
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
[Note I.
the theorem of Art. 1 that every primary transformation of this elliptic integral
implies an equation of the form . . ,
iK'
K
i.V
(14)
where
a, /3
9
is a primary matrix. Conversely, it is known from the theory of
the transformation of the Theta functions, that every equation of the form (14)
supplies one, and only one, primary transformation of the elliptic integral ; and
further, that if two matrices
are equivalent by post-multiplication with
a primary unit matrix, the transformations arising from them are the same ; that
the transformations corresponding to matrices not so equivalent are always dif-
ferent follows from the concluding observation of Art. 2. Thus the number of
different primary transformations of the order n is the same as the number of
non-equivalent matrices* of determinant n ; i.e. it is o- (n), if a- (n) is the sum of
the divisors of n ; and there is a correspondence one to one between the primary
transformations and the non-equivalent primary matrices.
These a- (n) transformations are, however, not all primitive ;f viz, if a, /3, 7, $
have a greatest common divisor n, the transformation corresponding to
, 6
is not primitive, but is compounded of a multiplication of the argument by
fin
and of a primary and primitive transformation of the order v = , corresponding
theory of matrices ; and to characterise as primary, matrices of the type defined by
the congruences
a, b
1,
, mod 2, a = 1, mod 4,
-
c, d
0, 1
because this type reproduces itself in multiplication. Again, it is convenient to designate as primary
those pairs of conjugate periods of an elliptic integral or function which are formed from the funda-
mental pair (see note II) with primary unit matrices. Lastly, the introduction of the phrase primary
transformation is justified by the consideration that the primary periods of the elliptic integrals trans-
formed into one another by a primary transformation are connected by a primary matrix ; and that, as
a consequence of this relation, the composition of two primary transformations gives a primary
transformation.
* For brevity, matrices equivalent or non-equivalent by primary post-multiplication are here
simply termed equivalent or non-equivalent.
t The word primitive was applied by Gauss to quadratic forms of which the coefficients are
relatively prime ; it is natural to extend the use of the term to matrices of which the elements have no
common divisor other than unity ; and consequently to the rational transformations appertaining to such
matrices.
Art. 4.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION,
a ft
329
to the matrix
; in fact, in this case the transformation (6) arises from
the combination of two primary transformations
z = F 1 (x), y = F 2 (z),
of the orders /* 2 and v respectively, corresponding to the equations
dz r x dx
r
r
dy
= /. r
Jo
= W
dz
of which, however, the former represents a multiplication, and not a transfor-
mation properly so called, of the elliptic integral. The number of primitive
transformations is of course the same as the number of non-equivalent primitive
matrices
7,*
, and is given by the formula
where the sign of multiplication extends to every prime divisor p of n. The
primitive transformations are alone to be regarded as proper transformations of
the order n, and in what follows we may confine our attention to them.
4. Let = j^t ^ = r- , so that = - -^ ; and let <p (6) be the function
defined by the equations
(15)
this function is characterized by the two converse properties (i) that the
equation C + D6
where
A,B
C,D
A + B6'
is a primary unit matrix, involves the equation
(16)
and (ii) that the equation (17) involves the existence of an equation of the
form (16).
VOL. n. U u
330 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I.
It is known, from the theory of the Theta functions, that
# = 4>(). X* = <>(});
and that the </ (n) modules <^> 8 (^), which present themselves in the different
primary transformations of order n, are the roots of an equation f(k 2 , X 2 ) = 0,
termed the modular equation, which is of the order a 1 (n), and of which the
coefficients are rational and integral functions of & 2 = ( 8 (o>). In the general
case, in which the roots of this equation are unequal, the </ (n) primary trans-
formations of the elliptic integral and the transformed modules correspond to
one another one to one. We shall now show that, when the modular equation
has equal roots, the values of M 2 answering to the equal roots are different ;
so that, although in this special case two or more different primary transfor-
mations answer to one and the same transformed modulus, yet in all cases alike
the a-' (n) different primary transformations correspond one to one to & (n)
different pairs of values of X 2 and M*. To establish this assertion, the truth of
which might be inferred from the concluding observation of Art. 2, we consider
the equation of Jacobi 72/1 m 7 , 2
nfc-(L -/c ) a.\ _
M*~ X'(l-X') d.k*'
from which it appears that if for the same value of k 2 we have simultaneously
\l = X, M I = M\, we must also have
d-x; _ dAj
d.k*~ d.k*'
But this equation is inadmissible. In a memoir ' On the Singularities of
the Modular Equations and Curves/* it has been shown that the modular
curve represented by the equation
f(X, F) = 0, k* = X, X*=F,
has no super-linear branches except at the points
(x=o, r=o), (x=i, Y=I), (jc-eo, r=oo),
which may be left out of consideration ; and that it has no tangents at a finite
distance parallel to either of the axes of X or Y. Hence the only points which
need to be examined are multiple points free from any superlinearity ; and at
these it has only to be shown that two branches cannot touch one another. Let
* ' Proceedings of the London Mathematical Society,' voL ix. p. 242 [vol. ii. p. 242].
Art. 4.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
331
the matrices
i> ft
and
2> ft
7-2 >
being non-equivalent primary matrices of de-
terminant n. The equation X? = X^, or < 8 (Qj) = < 8 (Q 2 ), implies, as we have seen,
an equation of the form
0,=
C+D&,
where
C,D
is a primary unit matrix ; if then
a, |8
2, ft
A,B
7 ,S
=
72) ^2
X
C, D
we may replace the equation
by the equation
u> = -
to =
where
is a primary matrix of determinant n, equivalent te
therefore non -equivalent to
We now have
2) ft
72, 4
, and
but to calculate d.\l in the equation (19) we must employ the equation
and to calculate d . Xf, the equation
(0 =
Observing that
o+jSO,
where K(u>} is the one- valued function of e iir<a defined by the equation
- ^ =
TT
and that these derived functions are neither zero nor infinite, we infer from the
equation (19) the apparently but not really identical relation
V dta ' V d'
u u 2
332 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [.Vote I.
of which the two members have different significations. This equation gives
whence also
Vn
and finally, i = a,
contrary to the hypothesis that
and
r>
Oo
are non-equivalent.
With a little modification this demonstration may be employed to show
that no two modular curves can touch one another, except at the points [0, 0],
[1, 1], [oo, oo ]. Thus the <r'(n) primitive transformations of order n are not
only distinct from one another, but they are also distinct from any of the derived
transformations of order n.
5. The result at which we have arrived, viz. that when two roots of the
modular equation are equal the squares of the corresponding multipliers are
always different, might seem to contradict the theorem, due to M. Kcenigsberger,
that M is a rational function of k 2 and X 2 . But it is to be observed that the
demonstrations of this theorem fail (as they ought to do) when the modular
equation has equal roots. For example, M. Kcenigsberger has observed ('Theorie
der Elliptische Functionen ') that a rational expression for M may be obtained by
combining the equation (18) written in the form
/ d f \
(JT&)
" ( ]
-
\d . \*
with the equation of the multiplier ; and this is of course in general true, but
the right-hand side of the equation (20) assumes the form ^ at a multiple point.
Again, Prof. Cayley has shown (' On the Transformation of Elliptic Functions,'
hil, Trans., vol. clxiv, p. 423) how to form an
and X 2 , by considering the symmetrical functions
Phil, Trans., vol. clxiv, p. 423) how to form an expression for j,, rational in
= l, 2, ...<r'(w); r = 0,l, 2,... ;
Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 333
but the expression thus obtained must assume the form ^ when two roots of the
modular equation are equal, for its denominator is the discriminant of the
modular equation, and the multiplier M is never infinite.
"We may observe that the derived functions of X 2 and M 2 with respect to k 2
are always finite, and rational in M*, X 2 , and k 2 ; this may be inferred from the
equation of Jacobi (18) ; because in the modular curve there is no superlinearity
dY
except at the excepted points, and because -j^. is never zero and never infinite ;
hence, even at a multiple point (X , YJ), the development of each branch of the
modular curve is of the form
where /*, contains M z as well as X and Y , but /x 2 , MS,--- can contain no new
irrationality, because there is no contact of different branches.
It may also be noticed that M, which, according to the theorem of M.
Koenigsberger, is in general rational in k 2 and X 2 , is always rational in M' z , k 2 , X 2 .
This may be seen by evaluating according to the usual rule the expression for
M, obtained by the method of Professor Cayley, and substituting for the derived
functions of X 2 with regard to k 2 , the equivalent expressions, rational in M*, k 2 , X 2 ,
which we have shown to exist.
To obtain the values of M* at the multiple point (X , YJ), we have only
to form the equation, giving the directions of the tangents at that point. Sub-
stituting in this equation from the formula (18), we obtain an equation of the
form x (~TJ7> X , Y j = 0. The coefficients of this equation are rational functions
of X , Y , having rational numerical coefficients ; and, as we shall presently see,
X can always be resolved into linear factors, rational in X , Y , but having
coefficients involving an imaginary quadratic surd. The coordinates (X , F )
of the multiple point are themselves numerical quantities, which may be ir-
rational, because in the most general case the multiple points of a modular
curve present themselves in sets ; the abscissas, or the ordinates (as the case
may be) of the points of any one set, satisfying an irreducible equation.
6. In the general case in which the roots of the modular equation are
unequal, the coefficients a lt a 2 ,.,.b l , 6 2 ,... of the primary transformation are rational
in k 2 and X 2 , because the primary transformations and the values of X 2 correspond
to one another one to one ; and, when two or more roots are equal, these coeffi-
334 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I.
cients are at any rate rational in k*, X 2 , M 2 , because there is a similar corre-
spondence between the primary transformations and the pairs of values of X 2 and
M 2 . In fact the coefficients a and 6, as well as the multiplier M, can always be
expressed as rational functions of k 2 and X 2 ; but in the general case the nu-
merical coefficients entering into these expressions are rational ; in the special
case of equal roots the general expressions, or some of them, assume an indeter-
minate form, and new expressions present themselves, which may be deduced
by evaluation from the general expressions, and in which the coefficients are no
longer rational, but involve the same imaginary quadratic surd which enters into
the expression of M. It is worth while to show how these assertions are con-
sistent with the results of the algebraical theory of elliptic transformation.
It is shown in that theory (see the ' Fundamenta Nova,' or the Memoir of
Professor Cayley already cited) (i) that any primary transformation y = F(x)
satisfies an equation of .the form
where P = a + yx* + ..., Q = /3 + 8x 2 + ... , are functions of a; 2 , which are re-
spectively of the orders
i(w-l), i(n-5); ori(n-3), i(-3),
according as n = 1, or n= 3, mod 4 ; the whole number of coefficients in P and Q
being \ (n + 1) in both cases alike ; (ii) that the determination of the ratios
a : /3 : 7 ... depends on the \(n + 1) equations included in the formula
+
(fcr)- 1 [P 2 (^) + p^ P (^-5) Q
which expresses the condition of Jacobi that the equation (6) or (21) remains
unchanged when we write in it j for x, and for y. Putting U for A/ *
/ex A y * K
we may write the \(n+ 1) equations in the form
4v=E/-i- 2 ^, M = 0, l,...(n-l), . ..... (23)
where ^ and ^ M are homogenous quadratic functions of a, /3, -y, The elim-
ination of a, j8, 7, ... from the equations (23) gives rise to an equation between
U and k z , of which the order in U, according to the general theory of elimination
is (n + 1) x 2 J(-i> = (n + 1)2 fa- 3 \ but which must be capable of reduction to the
Art. 6.] NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. 335
modular equation between U and k 2 by the exclusion of extraneous factors ; of
this reduction, however, no complete account has as yet been given.* If we
assume that to each value of U satisfying that modular equation, there cor-
responds but one set of ratios a : /3 : 7 . . . , we can infer that these ratios, and
consequently the values of M and of the coefficients a and 6, are rational functions
of U and k 2 with rational numerical coefficients. And, in the general case the
assumption is justified, not indeed by anything which has been proved concerning
the algebraical nature of the system (23), but by the one to one correspondence
established in Art. 3, which suffices to show (as we have already seen in Art. 4)
that, when the roots of the modular equation between X 2 and & are unequal, only
one primary transformation can correspond to given values of k 2 and X 2 , and,
therefore, to given values of k 2 and U.
Again, to assert that M and the coefficients a and & are rational in Z7and k 2
is to assert that they are also rational in k 2 and X 2 , for, as we shall now show, U
is itself rational in k 2 and X 2 , at least when X 2 is not one of a group of equal roots
in the modular equation between k 2 and X 2 . In fact, if, as usual, u = \/k, v = v X,
/jj ni
it may be shown that - - and -'- are rational in u 8 and v 8 . For, if u r v' be any
J u n v n J
term in the modular equation between u and v, we have, by a theorem due to
Sohnke (see the Memoir '^Equationes Modulares pro Transformatione Functionum
Ellipticarum,' Crelle's Journal, vol. xvi. p. 97),
ns + r = n + 1 + 8/n,
where /m is an integer, so that
u
Hence, dividing every term of the modular equation by u n + 1 , we have
a relation of the form
where in general ./i and j^ cannot vanish simultaneously ; viz. if/! andy z vanish
for the values u = u, v = v, these functions also vanish for u = u', v = v', and
* Professor Cayley in the memoir cited has shown how this reduction takes place in the case when
n = 5 ; and Mr. Ely has given the corresponding determination of the extraneous factors for n = 7.
(' Proceedings of the London Mathematical Society,' vol. xiii. p. 1 53 ; see also Prof. Cayley. ' Phil. Trans.'
vol. clxix. p. 419.) The result in the case n = 7 possesses considerable interest, because it enables us
to conjecture in what way the reduction takes place for higher values of n : viz. by the exclusion of
factors corresponding to the modular equations of lower orders.
336 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I.
the modular equation between X 2 and k z has a pair of equal roots corresponding
V V^
to k 2 = u' s . Thus can be expressed rationally in terms of and u* ; but if
w 2 = w 2 , v* = v 2 , and w^ is any term in the modular equation between u 2 and V 2
wehave
nj . qj ^ 2 /y4
whence, as before, -f is rational in (-^ ) = and u\ = u 8 . Lastly, if Jf X" is any
term in the equation between u\ = k and v\ = X, we have
whence, finally, TJ = ^ is rational in j-^ and k z ; i.e. is rational in X 2 and k 2 .
K 16 K U
Example. Let n = 3; the modular equations between (u, v), (u 2 , v 2 ), (k, X)
may respectively be written in the forms
u\
X 4
A. /C
: F r
77 X *y
and give, by the elimination of -| and ^ , the following expression for ,
'K -i fC Iff
_.._
u 3 ~ k 2 * 32(1 - X 2 - k 2 ) + 5 (X 4 + A; 4 ) + 38 X" & 2 - 8 X 2 &- (X 2 + k 2 )
Of course, this expression is only one of an infinite number of equivalent
forms, reducible to one another by means of the modular equation between
1
k 1 and X 2 ; we have, for example,
Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 337
II tJ
To obtain similar expressions for , and , we have only to interchange
k z and X 2 in these formulae, and to change the sign.
The values of the ratios a : {$ : y ..., or of the coefficients a, b, have been ex-
pressed in terms of u and v by Jacobi (in the ' Fundamenta Nova ') for n = 3,
5, and by Professor Cayley (in the memoir already cited, and in the addition
to it, ' Phil. Trans.' vol. clxix., p. 419) for n = 7. It will be found that all these
nj ni
expressions contain only and u*, or and V B , and are consequently rational
It *U
in k 2 and X 2 , in accordance with what has been said here.
Passing to the case in which the modular equation has equal roots, let
(k 2 , X 2 ) be a point of multiplicity s on the modular curve ; we have already seen
that some at least of the expressions for the ratios a:fi:y..., and for the
coefficients M, a, b, must assume indeterminate forms, and that the s different
primary transformations may be elicited from these indeterminate forms by
d X 2
differentiating with regard to k 2 , and by substituting for ,' , 2> and for the
higher differential quotients of X 2 , their values corresponding to the s different
branches of the curve. But instead of employing a process of evaluation, we
may return to the equations (23), and, before proceeding to eliminate, we may
attribute in them to k 2 and U the values which these quantities have at the
multiple point. The system (23) will then admit of s different solutions ; and
will furnish an equation of order s for the determination of the ratio of /3 : a,
1 2/8
or, which is the same thing, for the determination of -^ = 1 H It must be
remarked however that the modular equation between U and k* may have
unequal roots, even in the case in which the modular equation between X 2 and k 2
has equal roots ; (because in this case it is possible that U may not be rational
in k z , X 2 , and that as many as four different values of U may answer to one
value of X 2 ) ; if this should happen, the equations (23) would enable us to express
the coefficients a, b rationally in U and k 2 ; but these coefficients would not be
rational in X 2 and k 2 .
Example. Let n 5 ; and, for simplicity, let us consider, instead of the
equation between k 2 and X 2 , the equation between u and v, viz.
W 6 _ ^,6 + 5 U t v 2 ( M 2 _ ^2) + 4 uv (J _ M 4 ^4) = Q
VOL. II. X X
338 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note I.
The general formulae for a primary transformation of the fifth order are
X
"M
1 V U 6
M = v-v*u'
If 6 is a primitive sixteenth root of unity, (6, 6 s ) is a double point on the
modular curve, and the equation determining the values of -*- is
whence -7- = = The expression for -^ becomes indeterminate ; evaluat-
du 5 M.
ing, we find i
]W *
f*x CiX
and we obtain two primary transformations of the integral / -., jr , included
ii ? i J T C* " * I
in the tormula ,
Jo V(l y 4 )
(2i
n/ /yi j, -A_,._
y ~ x -i .
in which either sign may be attributed to i.
Again, writing k 2 = 1, U= 1 in the equations (23), we have
whence we infer y = , 2 a 2 + 2 a/3 + /3 2 = 0,
2--I.-U ? -i.
a a
The formula (21) becomes, when n = 5,
~ a 2 + (p* + 2 a 7 + 2 a 0) a" + (y* + 2 /3 7) x* '
and gives, on substituting for the ratios a : /3 : 7 their values, the same result as
before. It will be observed that, in accordance with the foregoing theory, the
system (23) admits of two solutions, depending on a quadratic equation in /8 : a
or -v>; it is a peculiarity, with which we are not here concerned, that the
Art. 7.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 339
coefficients M, a, b are rational in u and v, though irrational in k* and X 2 ; the
circumstance that k z = X 2 is also immaterial.
The values of ^- might of course be obtained immediately by combining the
Cil)
equation (18) of Jacobi with the equation which gives the values of ~r- at the
double point. In this way we should find
the extraction of the square root introducing a new ambiguous sign. This
ambiguity may be removed by means of the equation of the multiplier, viz.
1 10 35 60^ 55_ 26-256 2 (l- 2 )
jfi ~ 3ft + M* ~ M* + M* ~ ~W~ '
of which the left-hand member, when k 2 = 1, resolves itself into the product
shewing that -r> = 1 + 2{, as we have already found.
7. The ratio of the two unequal multipliers, which correspond to two equal
values of X 2 , is always a simple imaginary quadratic surd. For, as we have seen
in Art. 4, the equality of two values of X 2 implies an equation of the form
if therefore M l and M are the multipliers corresponding to these two trans-
formations, we have
'
^ A',)
and consequently, ^ _ ^^ .,-,--- , m x
3f ~ r _L O ~ .. j^JfOf ' /
But is a quadratic surd ; viz. if
(25)
X X 2
340 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
Q is a root of the quadratic equation
[Note I.
In the equations (25) it is convenient to suppose that 2r is the greatest
common divisor of
i( aTl - l7 ), iK-V+^i-M,
which are whole numbers, because the matrices
7, *
and
!,
are primary, and which cannot vanish simultaneously because those two matrices
are not identical ; again, A and C cannot vanish, and AC IP = &, cannot vanish
or be negative, because the imaginary part of Q must be different from zero ;
thus we may attribute to T the sign of ay l -a l y or ft 1 -ft 1 S, and may suppose
that A, C and VA are positive ; we then have without ambiguity
C
Let 2 <r + 1 denote the uneven number
we find immediately
(26)
2 = (2 a- + I) 2 + 4r 2 A,
M
(27)
and, substituting hi (24) for O its value given by (26), we obtain for -~ the
expression j^ 2 <r + 1
showing that the ratio of the two multipliers is a simple quadratic surd, having
unity for its analytical modulus.
We might also arrive at the formula (28) by compounding the transformations
X to,
7, J
xQ,
(29)
and equating the product of their multipliers to the multiplier of the resulting
transformation. The multipliers of the transformations (29) are (nM^- 1 and M
respectively ; the resulting transformation
^-/3 1% ftS.-ft.S
which is in fact a complex multiplication of the argument, has for its multiplier
Art. 7.] NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. 341
The equation (27) shows that, for a given value of n, the ratio of the
multipliers corresponding to any two equal values of \ 2 can only be one out of a
finite number of imaginary quadratic surds ; for that equation can be satisfied
only by a finite number of values of a-, r, and A. We may add that, conversely,
every set of numbers 2cr + 1, 2r, A, which are all different from zero, and which
satisfy the equation (27), may be employed in the formula (28), and will serve
to express the ratio of at least one pair of multipliers corresponding to equal
values of X 2 in the modular equation of order n. The demonstration of this
converse proposition would, however, require a more complete discussion of
the multiple points of the modular curves than can be undertaken within the
limits of this note. Such a discussion would also show, that if M, M l} M 2 , ...
are the multipliers appertaining to as many equal values of \ 2 , we may have
,. M l M 3 , . M l M 2
an equation such as -j-f = -TJ- , but not an equation such as -jr-= = -jj- r or our
.. I / j I / L ,(/ ' ' I
present purpose it suffices to observe, that the ratio of two multipliers answering
to equal values of X 2 , cannot be a root of unity, because the imaginary fourth
and sixth roots of unity, which alone of all the roots of unity are simple quad-
ratic surds, cannot be represented by the formula (28).
Let
be the equation, rational in & 2 and X 2 , which determines the values of M at the
multiple point (& 2 , X 2 ). We may suppose that this equation is irreducible, i.e.
that M cannot be resolved into factors without the adjunction of some irrationality
other than those contained in k z and X 2 ; if n = is not irreducible, we must
consider successively instead of M, the irreducible factors of n.
Let M=pM l , M and M l being two of the roots of (30), and p being a
quadratic surd of the form (28). Then the equation
.......... (31)
will have some, but not all of its roots in common with the equation (30) ; viz. -^
is certainly common to the two equations ; if -ry- is also common to the two,
1 l 11
we must have M l = pM 2 , ^ being a root of (30), different from -^ and -^ ,
because M=pM 1 = p 2 M 2 ; if -=j- is common to (30) and (31), we must have
l MI
-jjj- being itself a root of (30). Since p is not a root of unity, this
342 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
process cannot terminate until we arrive at a root of (30) which is not also
a root of (31). Therefore -r? is the root of an equation which is of lower di-
mensions than (30), but of which the coefficients contain an imaginary quadratic
surd iV&. If -jTf, is any second root of this equation, we shall have M=p'M', p f
being a quantity of the form (28), different from p, but containing the same
surd tVA. We can therefore again reduce the order of the equation of which
-jrf is a root, without introducing any new irrationality, and we shall at last
arrive at an expression of the type
2 ,X'), ......... (32)
^ being a rational function, with rational coefficients, of iVA, k 2 , X 2 . It is
proper however to observe, that each time that a new ratio p is employed, the
radical i VA is introduced with an independent sign ; so that the expression (32)
(in which we regard k 2 and X 2 as given) admits not of two values only but of 2 r ,
if r is the number of different ratios p employed in the reduction.
In the example, considered in Art. 6, the quotient of the two values of
M is ^( 3 + 4i), in accordance with the foregoing theory. And in general,
if p is a prime number of the form 4m + 1, so that we may write
p = a z + l 2 ; 6 = 0, mod 2; a6 = l, mod 4 ;
the integral / -77- - , is transformed into itself by two primary transformations,
Jo V(l-z 4 )
answering to the two equations
-a + b, b ,., ,
' x(l + z).
a + b, a
The corresponding multipliers are a bi, of which the ratio is - - ,
in accordance with the formula (28).
II. On the primary periods of the elliptic functions, and on the complete
rectilinear integrals.
1. Two periods of a doubly periodic function are said to be conjugate, when
the vectors representing them are adjacent sides of an elementary parallel-
ogram in the parallelogrammic system appertaining to the function. If (P,
Art. 2.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 343
is a pair of conjugate periods of the function, all the conjugate pairs are included
in the formula Q = a P+yP 1 }
when a, /3, y, $ are integral numbers, and aS /3y = 1.
The theory of the reduction of binary quadratic forms of a negative deter-
minant is applicable in the manner explained by Gauss, in his review of Seeber's
' Untersuchungen uber die ternaren quadratischen Formen,' to the parallel-
ogrammic system appertaining to any doubly periodic function.* Thus there
is always a reduced parallelism, one side of its elementary parallelogram being
the vector which represents the absolutely least period of the function, and the
adjacent side representing the least period which is not a numerical multiple of
the absolutely least period. The sides of a reduced parallelogram cannot exceed
its diagonals, and the included angles is less than 120 and not less than 60.
If P = r + is, F^
= rr+ss, =r* + s' 2 ' '
the quadratic form (A, B, C], or
Ax? + Bxy+Cy 2 .......... (3)
represents the parallelism determined by the given pair of periods (P, P'] ; and
if this form be reduced by the method of Gauss, the equivalent reduced form
represents the reduced parallelism. The reduced parallelism as well as the
reducing substitution is unique, except when (A, B, C) is equivalent to a
multiple by any real quantity of
or 2
2. It will be convenient to restate, in a form adapted to the purpose of this
Note, some of the principal results included in the theory of the representation
of parallelogrammic systems by binary quadratic forms of a negative determinant.
This theory is well known ; but, in order to apply it to the elliptic functions, we
have to introduce two modifications into it ; viz. (i) we have to regard the sides
of the elementary parallelograms as vectors, (ii) we have to restrict the con-
ception of equivalence, confining it to primary equivalence only ; this restriction
is introduced in Art. 3 ; in this article we adhere to the Gaussian definition
of equivalence.
* Gauss, ' Werke,' vol. ii. p. 188. See also Dirichlet in ' Crelle's Journal,' vol. xl. p. 209 ; and the
'Report of the Theory of Numbers,' Part v. Art. 120 ('Report of the British Association' for 1863)
[vol. i. p. 263].
344 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
(a) When we are given the quadratic form (3), we are given the absolute
magnitudes VA and VC of the vectors P and P 1 ; we are also given the quotient
of these vectors, but with an ambiguity of sign ; viz. if
Q=j, A = AC-B*,
we have
To remove this ambiguity, we adopt the convention that the direction of
rotation from the first vector P to the second P' is to be positive*; in ac-
cordance with this convention the amplitude of Q is positive, and if we denote
by VA the positive square root of A we have, without ambiguity,
(4)
The vector defined by this equation may be termed the vector associated
to the form (A, B, C).
Thus, when the quadratic form is given, the parallelism represented by it
in any plane in which the positive direction of rotation has been assigned is
given in species, but not in position ; for, if we draw one of the two vectors in
any arbitrary direction from the zero point, and set the other at the proper in-
clination to it, we obtain a parallelism which is represented by (A, B, C), and
which by turning it about in its own plane may be made to coincide with any
other parallelism represented in that plane by (A, B, C].
If (A, B, C) is transformed into (A', B 1 , C') by an unit matrix
the vectors of the new parallelism are
Q =
Q f
and, if Q' = - , we have
* If Re 1 X + t Y is any complex quantity of which R is the analytical modulus and the ampli-
tude, we shall suppose throughout this note that it < S ft ', BO that the amplitude always has the
Fame sign as Y. The angle of rotation from one vector P to another P' is the amplitude of the quotient
P'
; thus the angle of rotation is always a positive or negative Euclidean angle, or in the limiting case
a positive angle of 1 80 ; and the direction of rotation from P to P' is positive or negative, according as
p>
the amplitude of -= is positive or negative.
Art. 2.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 345
By the convention which we have adopted improper equivalence is ex-
cluded ; for, the amplitudes of O and Q' being both positive, the equation (5)
implies that aS py is positive. Two improperly equivalent forms, such, for
example, as the opposite forms (A, B, (7) and (A, B, C) represent parallelo-
grammic systems, which are symmetric to one another, but which, except in
the cases of ambiguity to be noticed presently, cannot be made to coincide with
one another by turning either of them about in the plane in which they lie.
In the general theory of periodic functions there does not appear to be any
reason for attending to the distinction between the positive and negative
directions of rotation, or, which comes to the same thing, to the distinction
between proper and improper equivalence. But, as we shall presently see, this
distinction becomes of importance in the theory of the elliptic functions, and
especially in the connexion of that theory with the theory of the Theta functions.
(&) When the form (A, B, O) is reduced, the vector. l = X+iY is said
to be reduced, and vice versd. The conditions that the form (A, B, C) should
be reduced are (' Disq. Arith.' Arts. 171, sqq.)
(i) A^C, \E\ZA*,,
(ii) TfA = C, B^O, ; ....... (6)
hence, when (A, B, C) is reduced, O satisfies the inequalities
(i) X'+Y>zl, -\<X<\, i ,
(ii) IfZ 2 +F 2 = l, X^O,
or, which is the same thing, the extremity of the vector Q falls within the
'reduced space,' i.e. the space which lies above the real axis, between the lines
X\ = Q, and outside the circle X*+ Y 2 = l ; the bounding line JT+| = 0, and
the bounding arc, from 5 ( 1 + u/3) inclusively to i exclusively, are not con-
sidered to belong to the reduced space.
When (A, B, (7) is reduced, the imaginary part of fi has a greater ab-
solute value than the imaginary part of any equivalent vector ; for Y= .-, and
A
VA is invariant, while A in the reduced form has the least value possible.
* Square brackets are here used to denote the absolute value of the quantity included in them ;
the Eame notation may be conveniently employed in the case of complex quantities, so that [.Re* 9 ] = R.
VOL. II. Y
346 NOTES ON THE THEOEY OF ELLIPTIC TBANSFORMATION. [Note II.
When the extremity of Q falls on one of the boundaries of the reduced space,
this statement requires an unimportant modification, viz. in this case the two
vectors +X + iY, of which one only is reduced, are equivalent. The value of Y
in a reduced vector can never be less than \ V"3.
(c) A parallelogrammic system is ambiguous, when it can be brought into
coincidence with itself by rotation through an angle of 180 round one of the
lines of the system ; if (A, B, C) represent the reduced parallelism of the system,
this takes place in the following three cases :
(1) S = ; the reduced parallelogram is a rectangle ; the extremity of il
falls on the line Y= 0.
(2) A = C; the reduced parallelogram is a rhombus; the extremity of O
falls on the boundary X* + Y 2 1.
(3) A = B ; the reduced parallelogram has a side and a diagonal perpendicular
to one another ; the extremity of ii lies on the boundary X \ = 0.
In case (1) the axes of symmetry are the sides of the reduced parallelogram ;
in case (2) they are its diagonals ; in case (3) they are its least side and the line
of the system perpendicular to that side. The axes of symmetry are conjugate
lines of the system in case (1) only ; in the other two cases the vectors lying in
the axes of symmetry contain a parallelogram which is double of an elementary
parallelogram. There are two pairs of axes of symmetry, when the system
consists of squares, and three pairs when it consists of equilateral triangles ;
in all other cases of ambiguity there is one pair only.
(d) A parallelogrammic system can always be brought into coincidence with
itself by a rotation in its own plane through an angle of 180 ; this rotation
-1,
0, -1
system consists of squares, it can be brought into coincidence with itself by a
rotation through an angle of 90, and when it consists of equilateral triangles
by a rotation through an angle of 60 ; to these rotations there correspond the
two pairs of opposite automorphics appertaining to the system
(A,B, (7) = (1,0, 1), Q = t,
and the three pairs appertaining to the system
(A,B,Q = (2, 1,2), Q=i(l+iY3);
corresponds to the automorphic
of the quadratic form. When the
the identical pair
1,0
0,1
counting in each case as a pair.
Art. 3.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 347
The parallelogrammic system of sin 2 am u consists of squares when k 2 = |-,
and of equilateral triangles when k 2 is an imaginary cube root of 1.
3. We now restrict the application of the term equivalence to primary
equivalence, and henceforward when we say that two forms or two vectors are
equivalent, we shall understand that they can be transformed into one another
by a primary unit matrix ; i.e. by a matrix
satisfying the equation
y.
aS /3y = l, and the congruences a = S = 1, mod 4 ; /3 = 7 = 0, mod 2. When this
limitation is not intended, we shall say that the two forms or vectors are
absolutely equivalent.
In the general theory of doubly periodic functions we have only to deal with
absolute equivalence ; but in the case of the elliptic and modular functions,
the consideration of primary equivalence is indispensable. It will be noticed
that the two opposite parallelisms (P, -P) and (-P, P'), though represented
by the same quadratic form, are not, strictly speaking, equivalent ; for 1
is not a primary unit matrix. In the reduced parallelisms which we shall have
occasion to consider in the case of the elliptic functions, the real part of the
first vector is always positive, and the opposite parallelism need not be con-
sidered. The restriction in the definition of equivalence introduces certain
modifications into the theory of reduction, which we shall now briefly indicate.
A quadratic form (A, B, C} is said to be (primarily) reduced, when its
coefficients satisfy the inequalities
(i)
(8)
(ii) l([S] = A, or \B\=C, '
The associated vector O is reduced when the quadratic form is reduced, and
vice versd. The ' reduced space ' lies above the real axis, between the lines _5T 1 =
and outside the semicircles X 2 + Y 2 + X = ; the line X + 1 = 0, and the semi-
circle X 2 + Y 2 + X = are regarded as not appertaining to the reduced space.
The extremity of a reduced vector falls within the reduced space, so that the
vector satisfies the inequalities
-1<X1; -X<X*+Y 2 ^X. ....... (9)
Theorem. ' Every quadratic form is equivalent to one, and only one, reduced
form, and by one, and only one, reducing transformation.'
(1) Let a be the least number represented by (A, B, C) with values of the
indeterminates which satisfy the congruences (x, y) = (l, 0), mod 2 ; and, among
Y y 2
348 NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. [Note II.
all the forms equivalent to (A, B, C), which have a for their first coefficient, let
(a, b, c) be that which has the least third coefficient ; if the two opposite forms
(a, b, c) and (a, 6, c) are equivalent, we take for (a, b, c) that one of the two
in which the second coefficient is positive ; we then have
whence it follows that (a, b, c) is reduced ; i.e. a reduced form always exists
equivalent to a given form (A, B, C).
(2) Again, to show that only one reduced form can be equivalent to a given
form, it suffices to establish the following lemma, which is in substance due
to Legendre, and which serves to show that two reduced forms cannot be
equivalent without being identical.
' If (a, b, c) is a reduced form, and if the numbers primitively represented
by (a, 6, c), or by any equivalent form, are divided into three series according as
the indeterminates satisfy the congruences
the least numbers in these series are respectively
a, a 2 [li] + c, c.'
Let f(x, y) = ax 2 - 2 [b] xy + cy 2 ,
and let us suppose that x and y are positive ; the identities
/(x-2,y)=/(x,y)-4a(x-l) + 4[%,
combined with the conditions [6] a, [b] < c, give rise to the inequalities
f( x ~ 2 >y) = f( x > y)> if x > y>
f(x, y-2)< f(x, y), if y > x,
which show immediately that /(I, 0), /(I, 1), /(O, 1), or, which is the same
thing, a, a 2 [6] + c, c are the least numbers in the three series respectively.
We may add that in the second series the least number but one is a + 2 [6] + c.
(3) Lastly, it is readily seen that a reduced form cannot be transformed
1,
into itself by any primary substitution other than
0, 1
hence, the re-
ducing substitution is always unique.
To obtain a reduced form equivalent to a given form, we may employ, with
Art. 3.] NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. 349
a slight modification, the algorithm of Gauss.* Let (a , &, c) be the given
form ; if [6 ] > a > let & + 2/^ao = b l} [6J ^ a , and let ' transform ( , & , c )
1
into (a , 6 X , c t ) ; if [& 1 ]>c 1 , let 6 1 + 2 / <z 2 c 1 = &2, [&2] = c i> an d let ^ transform
(a , &j, Cj) into (a l9 6 2 , Cj) ; if [& 2 ] >a i> the process can be recommenced, and as
long as it continues, we shall have a >c 1 >a l >c. 2 > .... But a quadratic form of
negative determinant can represent only a finite number of quantities less than
any given quantity. Hence, we must at last arrive at a form (a,,, b 2fl , C M ), in
which [&J^a M , [fyj^c,,, and which, if not reduced itself, has for its opposite an
equivalent and reduced form.
The coefficient of i in a reduced vector O = X + i Y is greater than in any
equivalent vector ; for the first coefficient of a reduced form is less than the first
coefficient of any equivalent form ; the enunciation requires an unimportant
correction when the extremity of the reduced vertex lies on a boundary of the
reduced space. There is no inferior limit to the value of Y, which cannot
vanish, but may be any positive quantity however small.
From the inequalities (8), which are satisfied by the coefficients of a reduced
form (a, b, c), it appears that in the corresponding parallelism the triangle of
which the vertices are (0> 0)> (1> ), (0, 1), (10)
is acute-angled ; one of the acute angles becoming a right angle in the limiting
cases & = 0, b = a, 6 = c; in the symbol (0, il), and throughout the rest of this
article, the upper or lower sign is to be taken according as b is positive or
negative. The triangle (10), which we shall designate by UVW, and which
may be termed a reduced triangle of the system, is in fact a triangle contained
by two adjacent sides and the lesser diagonal of an absolutely reduced elementary
parallelogram. Every elementary triangle of the system, which is not obtuse
angled, is a reduced triangle ; for if UV, UW, are the least sides of such a
triangle, the elementary parallelogram contained by UV, UW, is an absolutely
reduced parallelogram, because its sides cannot exceed its diagonals ; all the
reduced triangles of the system are equal to one another in all respects. The
three sides of a reduced triangle, taken positively and negatively, can be paired
in twenty-four different ways ; in twelve of these the direction of rotation fronu
the first side to the second is positive ; we thus obtain twelve parallelisms
' Disq. Arith.,' Art. 171. See also the account of the theory of reduction in the ' Report on the
Theory of Numbers,' Art. 92 (' Report of the British Association' for 1861) [vol. i. p. 182],
350
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
[Note II.
absolutely but not primarily equivalent to one another, if, however, we confine
our attention to one of each couple of opposite parallelisms, such as P, P') and
( P, P'), it will suffice to consider the six exhibited in the following scheme :
(UV, UW), (VW, VU), (WU, WV) I
( UW, - UV), ( VU, - VW), ( WV, - WU) ) '
(11)
If
so that
/>! =
-1, +1
1, o
0, -1
1,
P\ = P\= ~^ 2 =
-I,
-1, 1
+ 1,
1,0
0,1
the three parallelisms in the upper row of the scheme (11) are transformed into
one another in cyclic order by the matrices 1, p l} pi; the three in the lower row
by 1, pi, p 2 , and any two in the same column by <p-\ The six parallelisms are
all primarily reduced ; they are in general different in species from one another ;
they are, however, of the same species in sets of two, if UVW is isosceles and
right-angled, and in sets of three if UVW is equilateral. The system has a pair
of axes of symmetry if UVW is either rectangular or isosceles ; two pair, if it is
both, and three pair if it is equilateral. But the quasi-symmetrical pairs of
parallelisms are equivalent, only when UVW is right-angled, so that in this case
only is the system ambiguous in the sense of primary equivalence.
4. Let (4Z, 2Z/) be a pair of conjugate periods of the elliptic function
sin am (u, k 2 ) = X (M), which we here regard as defined by the equations
dx
u =
the initial value of the radical being + 1, and Tf being any complex quantity
whatever, other than 0, 1, oo.
It is important in the theory of the function X (u) to consider specially the
pairs of conjugate periods, infinite in number, which satisfy the equation
X () = !, (13)
and in addition the condition that the direction of rotation from L to L' is to be
positive. Such pairs of conjugate periods of X (u) we shall term primary*
* There is a slight difference between the definition of the primary periods considered in the text
and the definition of the elliptic periods of MM. Briot and Bouquet; viz. the pair [4/f, *iiK'~\ is an
elliptic, as well as a primary, pair of periods of X (u) ; but [4Q, 2Q'] is an elliptic pair of periods, only
when /3 and y are evenly even in the equations (15) ; these authors, however, in the first edition of their
classical work (Paris, 1859) had given a slightly wider definition of elliptic periods, requiring only that
/3 should be evenly even.
Art. 4.1 NOTES ON THE THEOEY OF ELLIPTIC TRANSFORMATION. 351
One pair of primary periods of X (u) can always be assigned. Let
7T f 1
' /.:
*-:
dx
(14)
the integrations being rectilinear. The initial sign of the radical in the former
integral is taken to be positive ; the sign of the radical in the second integral
* Tf*
is so taken as to render positive the amplitude of the quotient ^- The more
accurate determination of these integrals will occupy us in Art. 5 ; for the
present either sign may be attributed to ^ in the lower limit of the second
integral. By reducing to elementary contours the different tracks along which
x may travel from the lower to the upper limit in the integral (12), it
may be shown that [4J5T, 2iA"'] is a pair of conjugate periods of the in-
tegral, or, which is the same thing, of the function X (u). But we have
iK'
evidently X (K) = 1, and the amplitude of -j=- is by hypothesis positive ; there-
fore \^K, %iK'~\ is a pair of primary periods of X (u).
Theorem I. 'If [4Z, 277] is any given primary pair of periods of X (u),
all the primary pairs [4Q, 2<7] are exhibited by the formula
in which
a, /3
is any primary unit matrix.'
It will be sufficient to demonstrate this theorem on the supposition that
(1) Let [4 Q, 2 (X] be a primary pair of periods of X (u). If X (v) = X (u), we
have, by the known properties of the function X(), either v = u, or v = 2K u,
omitting multiples of the periods of X (u) in either case. Hence the equation
X (Q) = 1 = X (K) implies an equation of the form
Again, 2 Q', which is a period of X (u), must be a sum of multiples of the
conjugate periods 4_STand 2iK' ; we have therefore
352 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
Here s' is uneven, because, [4$, 2(X] being a conjugate pair, we must have
(4r + l) s' 4/s= 1 ; the upper sign is to be taken because the amplitudes of
^- and -^- are both positive. Writing
t = 2r', 7 = 2s, S = s',
is primary, and the equations
Q=aK+ 7 iK' ]
the matrix
are satisfied.
(2) Conversely, if [Q, Q*] is a pair of quantities satisfying the equations (15),
in which the matrix
a, 18
is primary, we find immediately X (Q) = X (L) = 1 ; also
Q'
the amplitude of -~r is positive ; i.e. [4 Q, 2 Q?~\ is a primary pair of periods
ofX(tt). V
If [4 Q, 2 Q'] is any primary pair of periods of X (u) = sin am u, the pairs
[Q,Q1; [2& 2<?]; [20-20', 20+201; [20,40-];
are pairs of conjugate periods of X 2 (2w), of X 2 (w), of cos am u, and of Aamw
respectively, and may be termed primary pairs of periods of those functions.
Every primary pair [Q, Q'], in addition to the equation (13), satisfies the
equations X(2<?) = 0, \(Q') = \(2QQ') = v ]
1 , ..... (16)
\(QQ')=l )
of which the first two result from the known formulae (see Art. 7)
X (2K) = 0, X ( iK") = \(2K iK') = oo,
the last is an immediate consequence of the definitions of K and iK'. If the sign
of k be determined as in Art. 5, the upper or lower sign has to be taken in the
right-hand number of the last equation, accordingly as /3 is evenly or unevenly
even in the equations (15).
The following theorem, in the enunciation of which $(fl) is the function
defined by the equations,
, q = e< ....... (17)
may serve to show the importance of considering the primary periods of elliptic
functions.
Ait. 4.] KOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 353
(Y
Theorem II. 'If CL = -^r, [2Q, 2<7] being a primary pair of periods of
H/
X 2 (w, k 2 ), Q satisfies the equation
$*(!) = &,
and, conversely, if this equation be satisfied by a complex quantity Q of positive
amplitude, a pair of primary periods [2 Q, 2Q r ] of X 2 (u, k 2 ) can be assigned, such
Cf
that = -^-.'
V
Q'
Demonstration (1). If Q = -y, [2$, 2$'] being a primary pair of periods
ofX 2 (w, yfc 2 ), let
gwn, A=jx[2:;:: 2 "7, M =A, A-
It is known from the theory of the Theta functions that the function X[w,
is identical with the function
Q /ITU \
(18)
M \
P V
and has, therefore, [4 h, 2h f ], or [4/x^, 2/uQ'] for a pair of primary periods. The
two functions X[MM, < 8 (Q)] and X[M, A: 2 ] consequently have the same zero points
2m$ + 2nQ', and the same infinite points 2mQ + (2n + l) Q' ; that is, they
differ only by a constant factor, which may be determined by observing that
lim = 1, when u = 0. We thus obtain the equation X \jj.u, < 8 (^)] = M x X[M, & 2 ].
It
But X [Q, yfc 2 ] = 1, X [A, 8 (Q)] = 1 ; we have therefore
and, finally, differentiating with regard to , ^> 8 (Q) = & 2 .
(2) Conversely, if $ 8 (Q) = ^ 2 , Q being a complex quantity of positive
amplitude, the function X 2 (u, & 2 ) is identical with the square of the function
(18), and has [2h, 2h^ for a pair of primary periods, of which the quotient is Q.*
The primary pairs of periods of X 2 (2) are represented by a parallelogrammic
system, which we may reduce by a primary transformation. If \M, M'~\ is the
reduced primary pair, the least primary pairs of sin am u, cos am u, A am u, are
[4 M, 2 Ml [2 M- 2 M', 2M+ 1M'], [2M,
* The properties of the function < (li) employed in Note I, equations (16) and (17), are immediate
consequences of the theorems of this article.
VOL. II. Z Z
354 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
respectively. For sin am u and A am u this is evident, and it may be proved for
cos am u by observing that M+ M' are the two least values included in the
formula (2r + l) M +(2s + l) M
(see Art. 3). We shall presently show that the rectilineal integrals K, iK' are
the reduced primary pair of periods of X 2 (2w, k 2 ). In order, however, to give
a precise signification to this assertion, we have in the first place to remove
the ambiguity which attaches to the definition of iK', and in certain cases to
the definition of K. We shall give in the next article the requisite deter-
minations relating to these, and to some other elementary integrals which
present themselves in the theory.
5. By the first of the equations (14), the integral K is completely de-
termined, except when k 2 is real, positive, and greater than unity. In this case,
if k is the positive square root of k 2 , the radical vanishes when we come to the
point x = -r ; after that point its value is a pure imaginary and its sign can only
be fixed by a new convention. The convention which we adopt is that it has
a negative amplitude \ir. This is the same thing as to suppose that x,
when it arrives close to the point j, describes an infinitesimal semicircle
round that point in the negative direction, i.e. above the real axis. We, in fact,
regard k z as having an evanescent positive amplitude, so that r is a point lying
K
just below the real axis, and infinitely near to it. The effect of the convention,
in the case to which it applies, is to render positive the amplitude of every
element of K, and therefore the amplitude of K itself ; the real part of K is in
all cases positive.
To complete in the second equation (14) the determination of the integral
iK', we understand by k that square root of k 2 , of which the real part is positive,
or, if k 2 is real and negative, that square root of k 2 of which the amplitude is the
positive angle \TT. The sign of the radical is so taken that the real part of
the expression
is positive ; this real part is nowhere equal to zero in the course of the in-
tegration except at the two limits ; for if x = T + (l T) , being a real variable,
Art. 5.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 355
of which the limits are and 1, the radical V{(1 - * 2 ) (1 - k 2 x 2 )} may be written
in the form
where the first radical factor is real, and is to be taken with a positive sign, and
where it is readily seen that the real part of the second radical factor never
vanishes, because the quantity under the radical sign never becomes real and
negative. We thus have
K' =
the real part of the radical under the integral sign being positive ; whence it
follows, that the real part of every element of K', and therefore the real part
of K' itself, is positive.
The convention by which we have fixed the sign of the radical in the
expression for iK' comes to saying that at every point P in the rectilinear track
from T to 1 the direction of rotation from the vector 1 T to the vector
V{(l-x 2 )(l-k-x 2 )}
is to be negative. The rule thus indicated is equivalent to the following :
' Let a simple closed contour be drawn passing through and P, and
including +1, but not including 1, +j, --^; and let x, setting out from 0,
describe this contour in the positive direction round + 1 ; the radical, setting
out with the initial value + 1, will arrive at P with the sign which it is to have
in the integral expression for iK'.'
To verify the coincidence of the two rules, it is sufficient to show that they
agree at any one point of the rectilinear track. Let x travel from along the
real axis until it arrives at a point Q close to + 1, and let it then describe in
the positive direction a circular arc of infinitesimal radius round + 1 as centre,
until it arrives at a point P on the rectilinear track. Let a be the amplitude
of 1 T, /3 the amplitude of k' - V(l k 2 ), the real part of this radical being
supposed to be positive ; it will be seen that a and /3 are of opposite signs ; viz.
the amplitudes of k 2 and 1 T have the same sign, and the amplitudes of k z
fC
and k' have opposite signs. When x arrives at Q the amplitude of the radical
ZZ 2
356 NOTES ON THE THEOKY OF ELLIPTIC TRANSFORMATION. [Note II.
is approximately /3 ; when x arrives at P, this amplitude is /3-r ^a, or
according as the amplitude of k 2 is positive or negative ; i.e. according as
a is positive and /3 negative, or o negative and /8 positive. Subtracting the
amplitude of 1 we find that the difference j8 |a lies between IT and
and the difference Tr + fi \a between TT and 2 TT, because |a and /3 are each in
absolute magnitude less than \ir. The two rules are therefore in agreement with
one another. This conclusion holds even in the cases in which k 2 is real, and the
demonstration may be applied to them, if we regard k 2 as a quantity of positive
amplitude, evanescent if k 2 is positive, and differing from TT by an infinitesimal
if k 2 is negative. We may observe that when the amplitude of k 2 is positive the
radical arrives at the point P with its proper sign, if x travels in a straight line
from to P ; the reverse is the case when the amplitude of k 2 is negative.
6. The preceding determinations give immediately
/-
Jo '
dx if &'
V](l-x 2 )(l-k 2 x*)\ =
the integration being rectilinear, and the initial value of the radical being + 1 ;
the upper or lower sign is to be taken according as the amplitude of k 2 is positive
or negative.*
Again, writing x = ^ in the integral (19), we find
ky
I
Jo
dx r dx
y being a real variable, and the real part of the radical under the integral sign
on the right-hand side being positive ; this determination is obtained by
comparing corresponding elements, near to and near to GO in the two integrals
respectively ; two such elements must have the same amplitude, viz. that of
r
integral /
Jo
* Instead of employing the second of the equations (14), we might define 2t'A" as the elliptic
fly.
-r- rs-Tr-, > extended in the negative direction over a simple closed contour
V { (1 a; 2 ) (1 kV)}
including - and 1, but not T or 1. We should thus arrive immediately at the equation (19);
and to show that in the equation (22) infra the upper sign has always to be taken, we might employ
the principle of continuity ; viz., when k- is real, positive, and less than unity, the integrals in the
equation (22) are each of them real and positive, and as they do not liecome infinite or zero for any value
ot If, other than 0, 1 , oc , the sign cannot change when A 2 is made to pass from any one value to any
other.
Art. 6.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION, 357
Y approximately. The sign thus attributed to the radical on the right-hand side
fC
is that which it would acquire if y were to travel from along the real axis and
were to describe a semicircle round +1 in the positive or negative direction,
according as ik' or ik' has a positive real part, i.e. according as the amplitude
of P is positive or negative. Hence, combining the equation (19) with the first
of the equations (14), we find
r"
iK'= -f
Jo
dy
(20)
according as the amplitude of k z is positive or negative, the integrals being
rectilinear and the variable y being supposed to describe an infinitesimal semi-
circle in the positive direction round + 1.
We can now assign the values of all the different complete rectilineal in-
tegrals ; i.e. of all the rectilinear integrals included in the formulae
dx
(,&)=/
Ja
dx
when a, b are any two of the points 0, + 1, +-r, and Y is a point at an infinite
distance.
The following is a list of the complete integrals ; the twenty integrals (a, b)
are reduced to six by means of the formula
(a, b) = - (- a, - b) = - (6, a).
The upper or lower sign is to be taken throughout according as the amplitude
of If is positive or negative. In the symbol (a, Y,) we understand by Yj, Y 2 , Y 3 , Y 4
points lying at an infinite distance in the angles respectively contained by the
vectors drawn from a to the pairs of points
^]- ['-a- [4-']- [-'.*]=
the vector drawn from a to a is to be interpreted as the vector from to a
produced. If P is a point infinitely near to a on the track ab, or aY, the initial
sign of the radical in the integral (a, b), or (a, Y), is the sign with which the
radical arrives at P, when x travels in a straight line from to P, and when the
radical sets out with the value + 1.
358 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
A.
(0,1)-*; (Q,l)=K + iK';
,)=+iK' t (!,-!)= -2*; (l, -)=-2KiK';
0,Y 2 )
0,Y 4 )
Y,) = 3K + iK', (- 1 , Y 2 )
Y 3 ) = -K + iK', (-1,Y 4 )
(~pY 2 )= K;
In verifying the formulae (5), it is useful to notice (i) that Y,, Y 8 are
opposite points in the formulae (a, Y,), ( a, Y 3 ), and Y 2 , Y 4 in the formulae
(a, Y 2 ), (-a, Y 4 ) ; (ii) that if the tracks Y r , Y 8 are separated by the track ab,
and by that track only, we have (a, Y r ) + (a, Y 8 ) = 2 (a, &).
The formulae of this article have been constructed with reference to the
general case in which k is any complex quantity whatever ; but they retain their
validity when k is real ; we have only in this case to regard P as a quantity
of evanescent positive amplitude, and to interpret accordingly the tracks ab, aY,
and the rule by which the sign of the radical has been determined in each
integral. For example, if k 2 be real and less than unity, we have
the track of the integration lying along the real axis, above the points 1
Art. 7.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 359
and y , but below the point + 1 ; and the initial sign of the radical being such
K
that it acquires the value + 1, when x arrives at 0.
7. We shall now establish the equation
the integration being rectilinear, and the initial value of the radical being + 1 ;
if k 2 is real and negative, so that k' = V(l k z ) is positive and greater than unity,
the radical vanishes in the course of the integration ; in this case we again
suppose that F is a complex quantity, of which the amplitude is positive, and
differs from IT by an infinitesimal ; or, which comes to the same thing, we suppose
that, after we have passed the point x = -p , the amplitude of the radical, which
has now become a pure imaginary, is positive. The effect of this convention, in
the case to which it applies, is to render negative the amplitude of the integral
r l ^y
Jo *
the real part of this integral is in all cases positive.
The substitution 1 yfc'-i/ 2
-Y.
transforms any given value of the integral
dx
into a corresponding value of the integral
dx
Jo
To ascertain what value of the second integral is equal to K', we distinguish
three cases.
First, let k 2 not be real, so that j has an imaginary part different from zero,
and a real part different from zero and positive. Let -r = pe ie , x = X + iY. As
y pursues the rectilineal track from to 1, X + iY describes the finite arc of the
hyperbola 1 n 2 rnq 2 6
_^ l
p* sm 2 d
360 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
from y or (p cos 6, p sin 6), to + 1, or (1, 0) ; it will be observed that the points T
A/ **'
and +1 lie on the same branch of the hyperbola, because the real part of
j- is positive. But the curvilinear integral along the arc of the hyperbola is equal
K
to the rectilinear integral along the chord (T, +l) , because neither of the two
remaining discriminantal points 1 and j can lie in the segmental area
included between the curve and the chord. Hence
and the upper sign has to be taken because the real part of K', as well as the
real part of the integral on the right-hand side, is positive.
Secondly, let k 2 be real and negative; as y passes from to T?, X+iY
I I
describes the axis of Y from T to 0, and as y passes from p to +1, X + iY
describes the axis of X from to +1. But the integral along the broken line
(-r, 0, + l) is equal to the rectilinear integral from T to + 1. Hence the equation
(21) subsists in this case also, the sign being determined as before by the con-
sideration of the real parts of the integrals. It will be noticed that the
convention of this article by which we have fixed, between the points p and +1,
the sign of the radical in the integral expression on the right-hand side of (21)
agrees with the convention of Art. 5, according to which ik is positive ; viz.
these conventions render simultaneously negative the amplitudes of the two
integrals in the equation (21).
The remaining case, when k 2 is real and positive, presents no difficulty, as
the integrals in the equation (21) are both real and positive.
We may, henceforward, define K and K' as the rectilineal integrals
dv '
/
Jo
the initial values of the radicals being + 1, and the same conventions as before
being retained when k* is real and greater than + 1, and when k z is real and
negative.
Art. 8.] NOTES ON THE THEOEY OF ELLIPTIC TRANSFORMATION. 361
In connexion with these integrals we shall consider the equation
....... (24)
--.-
o V(k* - sin 2 v)
which is obtained from the equation (19) by writing x = j-- The initial value
K
of the radical is k, and the upper or lower sign is taken according as the
amplitude of k 2 is positive or negative. When k 2 is real, positive, and less than
unity, sinv must be supposed to describe a semicircle in the positive direction
round k ; but we shall not have occasion in what follows to employ this de-
termination.
8. Theorem I. ' The vectors K and K' make acute angles with the vector
+ 1 on opposite sides of it ; and the angle between K and K' is an acute angle.'
In this enunciation we understand by an acute angle an angle lying between
the limits inclusively and 90 exclusively.
When k 2 is real, positive, and less than unity, the vectors K and K' are real
and positive ; when k' 2 has any other real value one of them is real and positive,
and the real part of the other is different from zero and positive ; in these cases the
theorem needs no demonstration. When k 2 is not real, the vectors 1 k 2 and k-
lie on opposite sides of the real axis ; and if we produce the vector k 2 indefinitely
both ways, the vectors 1 k 2 and + 1 lie on the same side of this indefinite line ;
hence the vectors k and k' (of which the real parts are positive) lie on opposite
sides of + 1, making acute angles with it and with one another ; the same things
are consequently true for the vectors y and T> But for every value of v the
/C K
amplitude of the vector r- . . is less than the amplitude of 7 , , and is
V(i k 2 sin 2 v) k
of the same sign ; hence, the amplitude of every element of K, and therefore of
K itself, is less than the amplitude of j-, , and is of the same sign. Similarly, the
!
amplitude of K' is less than the amplitude of y, and is of the same sign ; i.e. the
K
amplitude of K and K' are of opposite signs, and each of these amplitudes,
K'
as well as the amplitude of the quotient j^-, is less in absolute magnitude
than 90.
i If
Cor. The amplitude of the quotient -^- lies between and 180, exclusively
J\.
of both limits ; it is less or greater than 90 according as the amplitude of k 2 is
positive or negative.
VOL. II. 3 A
362 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [\ote II.
Thus the ratio of the two periods of the elliptic integral
dx
r
Jo
to V{(l-x'*)(l-k*x 2 )}
is always imaginary. It is of some importance in the theory of the function
\ (u) to establish this elementary proposition by the direct consideration of the
integral expressions for the periods themselves. It is indeed readily inferred
from the equation of definition of X (u), that X (u) is a one-valued function of u,
having the character of a rational fraction throughout the whole plane ; and
this character is incompatible with the existence of a pair of periods having
a real incommensurable ratio ; but is not incompatible with the existence
of two periods having a commensurable ratio, and in fact reducible to a
single period. Thus, in order to show that for all values of k 2 , other than
0, 1, oo, X (it) is a doubly periodic function of u, it has to be shown that the ratio
of the two periods is not commensurable. And this may perhaps be most simply
done by showing, as has here been shown, that this ratio is always imaginary.
Theorem II. ' The triangles formed by the vectors
K, -K+iK', -iK', ......... (a)
K, -K-iK', iK', ......... (t)
are both right-angled when k 2 is real ; when k 2 is not real, the triangle (a) is
acute-angled, and the triangle (6) obtuse-angled, or vice versd, according as the
amplitude of k 2 is positive or negative.'
Demonstration (1). If k 2 is real, positive, and less than unity, the angle
between the vectors K and iK' is a right angle. If k 2 is real, positive, and
greater than unity, the integral (24) is real, and the angle between the vectors
K iK' is a right angle ; if A; 2 is real and negative, the integral (24) is a pure
imaginary, and the angle between K iK' and K is a right angle.
(2) Let the amplitude of k 2 be positive, i.e. greater than and less than
180 ; in this case the amplitudes of T and -rp are negative, the former amplitude
K t/C
being absolutely less than the latter, and each being absolutely less than 90.
But the vector K iK' lies between the two vectors T and -^-, , because
* sin 2 v) always lies between k and ik f . Since K lies between + 1 and
p, the angle from K iK' to K IB acute and positive ; and since iK' lies
between T and i, the angle from iK' to K+ iK' is also acute and positive.
Art. 8.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 363
(3) Let the amplitude of & be negative. The vector K+iK' may be
1 i
shown precisely as in the former case to lie between the vectors j- and ^,, each
. K K
n
of which has a positive amplitude less than 90, that of Y> being the greater.
1
Since K lies between p and + 1, the angle from K to K+ iK' is acute and
A/ .
n
positive, and since iK' lies between i and T > the angle from K+ iK' to iK is also
acute and positive.
That the angle from K to iK' in case (2), and from iK to K in case (3), is
acute and positive has been shown in the demonstration of Theorem I.
Cor. The angle opposite to K in the acute-angled triangle is always ab-
solutely greater than the amplitude of k', and the angle opposite to iK' is
absolutely greater than the amplitude of k.
Theorem III. ' When k 2 is real, the right-angled triangles (a) and (b) are
reduced triangles of the parallelogrammic system [K, iK'~\ ; when k 2 is not real,
the acute-angled triangle is a reduced triangle of the system.'
Theorem IV. ' The pairs of periods
[2K, 2iK^, [-2K2iK',+2K], [ + 2iK', 2K-2iK'],
[2iK',-2K], [ + 2K, 2K+2iK'], [2K-2iK', 2iK'],
are respectively the least pairs of primary periods of the functions
\*(u,
or, which is the same thing, they are respectively the least pairs of periods of
X 2 (u, k 2 ) which satisfy the equations
S2 representing the quotient obtained by dividing the second period of any pair
by the first ; the six quotients H are all primarily reduced, and one of them is
absolutely reduced.'
The pairs of periods, and the linear transformations of \ 2 (u), which appear
in the enunciation of the theorem IV, correspond to the elementary matrices
3 A 2
364 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
of Art. 3, which serve to pair in every admissible manner the sides of the
reduced triangle.
It will be observed, that the parallelogrammic system of X 2 (w) is primarily
ambiguous, only and always when k- is real ; and that the angle between
K and iK' is a right angle, only and always when k 2 is real, positive, and less
than unity.
9. Theorem I. ' Let Q be any primarily reduced complex quantity of which
the amplitude is positive ; let also, as in Art. 4,
-,
the rectilinear integrals K and K' denned by the equations (23) are then ex-
pressed in terms of li by means of the equations
K=h(Q), iK' = h'(Q).'
For 2h(Q) and 2A'(Q) are a pair of primary periods of X 2 (u, k 2 ) ; so also are
27Tand 1iK ; we have therefore (Art. 4)
., iK'
or, if w= -JJT,
-/3 + aQ
then = ; p;
the unit matrix
being primary.
By the theorem IV of Art. 8, w is primarily reduced ; Q is so by hypothesis ;
and by the theorem of Art. 3 two primarily reduced quantities cannot be equi-
valent without being identical ; we have therefore w = fl, a = S=l, ft = y 0; or
Theorem II. ' Let Q be any complex quantity whatever of which the
amplitude is positive ; and let
be the primary reducing substitution of
the quadratic form to which Q is associated, so that the vector
Art. 9.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 365
is primarily reduced ; then, the same notations being retained,
For, as in the last demonstration, both [2h(Q), 2A'(Q)] and [2K, ZilF] are
primary pairs of periods of X 2 () ; we have therefore
*'(D)-jtr+ixr
iK -B + AQ ' ..... ' ' ' (26
the matrix
K
A B
'
n being a primary unit matrix. Thus & is equivalent to each
C, Lf .
of two primarily reduced quantities -j^- and Q' ; these two reduced quantities,
and the two reducing substitutions, are therefore identical ; i.e. the equations
(26) coincide with the equations which it is required to prove.
The theorems I and II of this article may be demonstrated in a different
manner.*
When k 2 is real, positive, and less than unity, so that the quotient Q = -^-
is a pure imaginary, and q = e" n is a real positive quantity less than unity, the
equations K=h(Q), iK' = Qxh(Q), (27)
as has been shown by Jacobi, certainly subsist. But these equations cannot
become untrue so long as the extremity of the vector Q lies within the primarily
reduced space, because within that space the rectilinear integrals K and K' are
continuous and one-valued functions of k 2 , which is itself a continuous and one-
valued function of Q. When k 2 is real and negative, there is, as we have seen,
an ambiguity in the determination of the rectilineal integral K', and its differen-
tial coefficients with regard to k 2 become infinite ; the same things are true for
the rectilineal integral K, when k 2 is positive and greater than unity. But k 2
cannot become real and negative at any point of the reduced space unless Q
falls on the boundary X = 1 ; and k 2 cannot become positive and greater than
unity, unless Q falls on the boundary X 2 + Y 2 X = ; because, as we have
already seen, if k 2 is real and negative, K is real, and K iK' is a pure imaginary
* See a note ' Sur les Int^grales Elliptiques completes ' printed in the ' Transuuti della R.
Accademia de Lincei,' vol. i. (3rd series) [vol. ii. p. 221].
366 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note II.
of negative amplitude ; and, if k 2 is positive and greater than unity,
is a pure imaginary of positive amplitude, and KifC is real and positive.
Thus the equations (27) must continue to subsist at every point in the interior
of the reduced space ; and they do not become untrue at the boundaries of
that space ; for the conventions by which we have removed the ambiguities in
the determination of K and iK' are so arranged as to correspond with the con-
vention by which the boundaries X+1 = Q, X 2 + Y 2 + X=Q are excluded from,
and the boundaries X 1 = 0, X 2 + Y 2 X = are included in, the reduced space ;
viz. the values assigned to the integrals (23) in the cases of ambiguity are
such as to make the equations (27) continue true at the included boundaries,
while becoming untrue at the excluded boundaries. A complete demonstration
is thus obtained of the theorem I of this article ; the theorem II, and the
theorems of Art. 8, may be regarded as corollaries from it. This mode of
demonstrating the propositions of Art. 8 has some advantage in respect of
simplicity ; but the method adopted in this note is perhaps more direct and
natural, as it depends only on the elementary properties of the rectilinear
integrals themselves.
T7-'
The theorem that the quotient -^ is always a primarily reduced complex
quantity is of some importance in the theory of the connexion of the elliptic
functions with the Theta functions.* If k 2 is given, the elliptic function \(u, k-)
is given ; but if we wish to express \(u, k 2 ) as a quotient of Theta functions by
means of the formula
we have, first of all, to express the element Q of the Theta functions in terms of
k 2 . The problem, as is well known, and as we have already seen in Art. 4, is
indeterminate ; viz. if Q is any admissible value of the element, & = ^ ^7
6 yM
the matrix being primary, is also an admissible value ; or, which is the same
thing, we may take for 12 the quotient obtained by dividing the second period
* See a note by Professor Weierstrass in the first volume of Jacobi's Collected Works (Berlin, 1881),
p. 545.
Art. 9.] NOTES ON THE THEOBY OF ELLIPTIC TRANSFORMATION. 367
by the first in any primary pair of periods of X 2 (). Since \%K, liK'"] is a primary
TT'f
pair, the quotient ^ is an admissible value of il ; but it is also the simplest of
all the admissible values, for it renders the analytical modulus of q = e' Vn the
least possible, and the Theta series the most rapidly convergent.
XLIL
NOTES ON THE THEORY OF ELLIPTIC
TRANSFORMATION.*
[Messenger of Mathematics, Ser. II. vol. xiii, pp. 1-54 (May-August, 1883).]
III. On the Functions Q (&>) and Q' (o>).
JUET Q (a), Q' (o>) be two functions of denned by the equations
* dK
d
(1)
which imply the equality
because K'(u) = K( -- j ; and which may be replaced by either of the following
pairs:
Q-k'K-J, Q--k'K'-J- ........ (4)
k 2
of which (3) was obtained by writing in (1) for du> its value ,. ,,. ^-, and
'
(4) by substituting in (3) the values of , , 2 and . , . taken from the equations
(i) and (v) of Art. 12.f
[* These Notes contain the fragments, relating to the continuation of the preceding Notes, No. XLI,
which were found among Professor Smith's papers after his death. It is to be understood that
they are only unfinished work which he would have greatly altered and extended. The Notes have
been placed in what appears to be the most convenient order, and headings have been supplied.]
[t These references are to the Memoir on the ' Theta and Omega Functions,' No. XLIII. of the
present Volume, p. 415.]
Note III.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 369
The values of -j-j^ and -y-^ taken from the same equations give
and eliminating between these equations and the equation (3), we have [writing
dtf" = ix(l-x)' ( 6 )
the complete solution being CQ + C'Q'.
The equations (1) and (5) respectively give
Q__dK^ dQ' _ K'
Q = ~~ dK' dQ~~~~ K
of which either is a consequence of the other, because
We have also
1 dO 1
K dta K' du> iir
_ = _ ^
Q d<a Q' dw iir
Q'dQ-QdQ' d.k*
dta du>
Lastly, combining the equations (4) with the equations (vi) of Art. 10,
we have K
Q = I k 2 cos* &m xdx,
Jo ....... (9)
. ~K+iK' |
Q' = i I k* cos 2 am xdx;
JK
thus the functions Q and Q' do not differ essentially from the functions
J=k*K-Q, J' = k*K'-Q-
of M. Weierstrass, nor from the functions
of Legendre ; their introduction serves to simplify the formula relating to the
transformation of the complete functions of the second species.
. x O is any transformation whatever of determinant n, that is,
VOL. II. 3 B
370 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [.Vote III.
Differentiating with regard to k 2 , multiplying by -2k-k'*M, and observing
-t "\ 2 \ '2
rJ X2- v // 2
I* ,/V - i ^, XN ^, _ , /- U/ A/ .
2 22
we find -.
i/3e(0)]= Q (.) -
In these formulae, which serve to express the transformed functions Q (Q),
Q'(&) in terms of the given functions Q (o>), <?'(), K(ia), K'(<a), the differential
coefficient -=j -j-rj- is to be found from the equation of the multiplier.
jyj. (t.K
3
For example, ifo>= '
__
Jf * if if
of the multiplier, we find without difficulty
' ixn = ^O, then, from the equation
U, 1 i
! 6 8(1-2F)
and the formulae then become
O 7,27/2 I2
K (.).
07.27.'2 Jlf2
3 y ( .) - 3 WH + (J(f 1* * ^ y f (.).
where, if w be a pure unaginary, If is the negative root of the foregoing
equation*.
* Cayley, 'On the Transformation of Elliptic Functions,' Phil. Trans., vol. clxiv. p. 421.
Note III.] NOTES ON THE THEORY OF ELLIPTIC TEANSFOBMATION. 371
Some considerations of interest present themselves in connexion with the
last-mentioned general formulae.
(1) The two equations are not independent : viz. multiplying the first by
iK'(a>), the second by K(ta), subtracting and attending to the relations
inMK' (0) = - 7 JT() + ia K (),
we obtain an identical result.
(2) In the case of Jacobi's first transformation
the equations become
n,
0, 1 "~ n
of which the first coincides with a relation considered by Prof. Cayley (' Elliptic
Functions,' Art. 305), and expressed by him in the form
A Jf rfx- Z'
where r A
A = ^(Q), E = f A 2 am (u, k*) du, G=f A 2 am (u, X 2 ) du,
* o / o
so that E = k'*K(u} + Q (), ^ = X /2 ^T (Q) + Q (12),
and the equation of Professor Cayley becomes
If J'
which is the same as the first of the foregoing equations, because
.
K(Q)~ ' nM~*d\~ ' dk
If instead of the first transformation we consider any reduced transformation
3 B 2
372
of the type
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Xote III.
g,
, the first equation becomes
which, on eliminating g by the equation # =
and substituting for
7 -v -v /2 J ' V /
' 2 -7-,^- its equivalent : 1*7-7;: , coincides with equation (ii), i.e. with equation
(i). Thus the equation of Prof. Cayley holds for all reduced transformations,
i.e. for all transformations of which the second element of the matrix of trans-
formation is = 0. This remark is of some importance, as it enables us to show
that the equation
dx
Q
which is satisfied by 2 = 3 o MK~ia\ > when = , i.e. when the transforma-
tion is the first transformation of Jacobi, is also satisfied by the same function
when the transformation considered is any reduced transformation whatever.
By combining the foregoing results we can express the elements of the
matrix of transformation in terms of the multiplier and of the integrals K, K',
Q, Q' ; we thus find Q j
1 iK' (.), nMiQ' (.) - 2n* (1 - ^ 2 ) tX' (.)
x
-t^T(Q),
S (iii)
and the equation of Prof. Cayley is then found by putting f3 = in the formula.
If 7 = 0, we have
When j8 and y are each = 0, i.e. in the real transformation =
a,
0,
a combination of the foregoing equations gives K(<a) K'a> = nM*K(Q)K'(S), which
is true because vt \ -vt \
H- W r>-//->\ ^ W
M
Note IV.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
373
IV. Employment of the Transformation of the Function of the second species
to obtain expressions rational in k 2 and \ 2 for the coefficients in the
formula of Transformation of the n ih order.
The analytical theory of transformation supplies a direct proof that the
coefficients a, b in the developments
.
j;J
and 1 + M 2 + &20 4 ... + fc^-i)*"- 1 = n,- [1 - k 2 x 2 sin 2 am 4/7?]
of the numerator and denominator functions in the formula of transformation of
the n th order are rational in k 2 , \ 2 , M, and -jj- ; that is in k 2 , \ 2 and M, for ~j^-
is always rational in k 2 , \ 2 and M: this follows from the equation of Jacobi
combined with the theorem that at a multiple point on the modular curve the
branches are all linear, and the tangents all different ; so that even in the
case where the modular equation has equal roots the coefficients contain no
irrationalities other than those involved in the expression of M.
To obtain the proof it is convenient to consider the function of the second
species denned by the equation
Z(u) = / k 2 sin 2 am u du,
Jo
and expressed by Jacobi in the form
iK'
q = e <wa , and regarding j/k = ^ (w), K=K(w), J=J(w) as one- valued functions
of <a defined by the equations
r K
where J=Z(K)= k 2 sin 2 am udu. Employing the usual notation =
we consider the transformation
where
a, p
O ^"^ '
r,*
= ^+7
a, (8
*V o
1,0
0,1
, mod 2, and
">74 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IV.
Defining the multiplier M by the equations
i jr^-a^xaj+tfjna).)
....... (2)
I/
we write in the equation of Jacobi ^ for u and i2 for . Dividing by JW we
obtain
u
/o\
where X 2 = ^> 8 (O). With this equation we combine a known formula for the
transformation of the Theta functions, viz.
5 r
H
xe ~' ^^xn^^tl-^sin^am^s^amtt], . ... (4)
in which (7=3 [O]-r^o[a>], and 17 = - , v being the greatest common
measure of a and /8, and /u being defined by the congruence n = yr + Ss, mod- ,
in which r and s are any two integers satisfying the equation ar + /3s = v.
Instead of i] we may take any multiple of 57 by a number prime to n.
Taking the logarithm and differentiating, we obtain from (4) the equation
d
& 2 sin 2 am 77 sin 2 am u\ , (5)
VtfVf
or finally the equation
= 2i 2 sin am M cos am W A am u x y^(-i)
1 k' 2 sin- am 4^'j? sin 2 am
where 2H= n
" "
-
JST() " M*
K ( a ) " M 2 A: (Q) 2 JOT () A' (Q)
Note IV.] NOTES ON THE THEOKY OF ELLIPTIC TRANSFORMATION, 375
The formulae (5) and (6) are given in fact by Jacobi for the transformation of the
elliptic function of the second species.
For our present purpose we have to show however that the constant H is
a rational function of k 2 , A 2 , M, , , ; and that the integrals /(>), J(&), K (to),
cL./c
K(Q) enter into the expression (6) only apparently.
To verify the assertion we write Q (to) + k 2 K (w) for J(u>), and conse-
quently - Q (O) + X 2 K (il) for J (O). Similarly for the function J (u) denned by
the equation K( ^ J' (,) - K' (u) J () = $*, ....... (7)
we write - Q' () + k 2 K' (), so that K' () Q () - K(w) q () = \ v . . . . (8)
Then we have
--
which are the general formulae for the transformation of the complete integrals
of the second kind represented by
Q ()=/ k 2 cos 2 am xdx,
Jo
fK+iK'
= / k 2 cos 2 am x dx.
Jo
Introducing the functions Q(o>), (/(<o) into the equation (6), we have
-2^4 _LiB. ^
M * K () M * K(ti) 2MK () K (Q) '
But from (9)
_
M* ' d.k*
Substituting this value in (10), we obtain
X2
~]^"
or finally, >z l-nl-il] T?i\ rJM
-*- >
--- ~ "~
376 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [.Vote IV.
the terms in the square brackets vanishing by virtue of the equations
If we now employ the developments
sin 2 am = -
(of which, however, the general terms are not known, and which are convergent
only for values of u not surpassing a given certain limit), we can by equating
the coefficients of like powers of u in the two members of equation (5) obtain
successively the values of the sums
S;:^- 1 ^ 2 ' sin 28 am 4/77,
expressed rationally in terms of X 2 , k 2 and H. Thus, for example,
2 & 2 sin 2 am Ijrj = H,
But from the theory of transformation
1 + M 2 + b 2 x*... + &.(_!) x n ~ l = IT; (1 - k*x 2 sin 2 am 4/ij),
so that the coefficients b l , & 2 , . . . are rational because the quantities 2 2 sin 2 ' am 4/
are so. In particular
,__#(!-*) dM ,X 2
"" " +2 2
1 \ 2 (3\ 2 -2) t X 2 {(37t
3 24 ^4 -12 jf, 2 Jlf
t n -/ \ n n
5 - M* \d.k') 24
The last coefficient is known, viz. we have
the expression being rational in k 2 and X 2 .
Lastly, the coefficients a are rational because
whence it follows that a
Note V.]
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
377
V. Relations in the formula of Jacobi between the elements n and the
transformation represented by the formula = TTQ-
The general formulas of Jacobi for any transformation of an uneven order n are
sin am
( u ^t\
( jry, X 2 ) =
W /
r7=4 (i-i)!"-. sn
P ' 1 - -^-
L sm^
sin'amw
, fr .
MD
cos am M ._., ,_r. sin 2 amw
/u \ cos am M .,_,., if ,,_ n r., sn 2 amw "I
cos am ( T>, X 2 ) = - - 7r - IP: ^l-- __,-
\M / D L 8m*ooam49J
A am Q , X 2 ) = ' ~^- n^}*"- 1 ' [1 - Jt 2 sin 2 am u sin 2 coam4y ,],
'
X'2 _ Kin v TTJ=1(''-D
- - x
~
(ii)
representing an element of the form
pK+iqfC
(iii)
where p and q are integral numbers having no common divisor with n.
It is also known that the modules k z and X 2 are connected by a relation
of the form 7 , > , n .
K 2 = 4> (eo), X 2 = < 8 ("), \
(iv)
to
a,/3
xO,
where
is a primary and primitive matrix of integral numbers, having n
for its determinant ; and (p (w) is the function already defined in Note I.
The elliptic transformation is completely determined when either (1) the
integral numbers (p, q), or (2) the integral numbers , /3, 7, S are given. Thus
the two converse problems present themselves, (1) ' given any pair (p, q~), to find
I
all the corresponding matrices |
find all the corresponding pairs (p, q).'
OT. IT
I
; and (2) 'given any matrix
9
, to
VOL. II.
378 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note V.
The solution of these problems is implicitly contained in the congruences
pa + qy=Q, p(3 + qd=(), mod n, ...... (v)
which result from the theory of the transformation of the Theta functions ;
but for the sake of clearness it may be worth while to give the details of the
solution, although the arithmetical considerations involved are quite elementary.
(a) If p l =p 2 , <Zi = <?2> m d n, the pairs (p 1} qj, (p 2 , q 2 ), and the elements
jj,, i/ 2 , may be said to be congruous ; if lip = p. 2 , hq l = q 2 , mod n, where h is an
integral number prime to n, the pairs (p 1} q^, (p z , q 2 ), and the elements 17,, >7 2 ,
may be said to be equivalent. Congruous elements are always equivalent;
and equivalent elements, when employed in the formulas of Jacobi, give identical
results.
(6) To find, for any given uneven number n, the number N of non-
equivalent elements 17, let x ( s ) denote the number of numbers prime to any
given number s, and not surpassing s, and let S, & be two relatively prime
divisors of n. The number of incongruous elements 17, having S and S' for the
greatest common divisors of n with p, and of n with q respectively, is
->); hence the number of non-equivalent elements, having their
greatest common divisors, is
and the whole number N of non-equivalent elements is given by the formula
the summation extending to every pair of relatively prime divisors of n, and the
pairs, S, & and ^', S being regarded as different, except when S' = S = 1. It is
easy to simplify the expression thus obtained for N ; viz. if n = A x B, A and B
being two numbers relatively prime, we find
also if n = 6?, 6 being a prime,
the sign of multiplication extending to all the primes 6 dividing n.
Note V.]
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
379
(c) Two matrices
and
2> ft
are said to be equivalent (or, more
properly, equivalent by primary post-multiplication), when
1 ft
a 2> ft
7z> 4
|| e || being a primary unit matrix.
Equivalent matrices, employed in the formulae (iv), give identical values
for X 2 .
(d) The number of primitive and primary matrices
non-equivalent
1
by primary post-multiplication, is also wIlM + ); so that there are as many
non-equivalent matrices as there are non-equivalent elements r\ ; this we know,
a priori, must be the case ; because the number of different transformations
must be equal to the number of non-equivalent matrices, and also to the number
of non-equivalent elements.
(e) We may take as representatives of the N non-equivalent elements
n the elements of the reduced system
y -2lK+igK'
n
where g is any divisor of n, and 21 is any term of a system of residues, even
n
and prime to g, taken with respect to the modulus - . No two of the elements
can be equivalent; for if 2^ = 2/^4, modw; hg l = hg 2 , modn; we must have
in the first place <7i = <7 2 , because h is prime to n ; and in the second place,
71
writing g for g l or g 2 , h=l, mod-; Ii = l 2 , mod-; i.e. I l = l 2 , and the two
/ c/
elements , which were supposed equivalent, are identical. Again, any given
element - is equivalent to one of the elements ; for, if g be the
. n
greatest common divisor of q and n, let 21 be the residue of -, which satisfies
(1 71
the congruence 2l-=p, mod-, and which may be supposed prime to g, because
J J
p is prime to g ; then the simultaneous congruences
hp= -21
, mod n,
hq= g
are resoluble, because p, q, and n are respectively prime, and because the
302
380
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
[Note V.
determinant pg + 2lq is divisible by n ; further, the value of h supplied by these
congruences is prime to n, because 2 1 and g are relatively prime ; thus the given
element is equivalent to the reduced element - - --
Example. Let 17 = - be the given element. Here n = 4725,
~
~T I ^O
= 25, - = 189, = 160, -22x160 = 581, mod 189; whence -21 = 182, mod 189.
The congruences 581^= 182, 4000^= 25, mod 4725, give h= 1147, mod 4725 ; so
-i co JT \ O 5 y fT'
that the reduced element is congruous to 1147 x /, and con-
sequently equivalent to 57.
(f) As representatives of the non-equivalent matrices of determinant n, we
may take the matrices
HtH
g and 21 having the same significations as in (e).
(g) The reduced pair ( 21, g), and the reduced matrix
0,0
n
, satisfy the
congruences (v) ; and the reduced pairs and matrices cannot be combined in any
other way so as to satisfy those congruences.
(h) The congruences (v), which may also be written in the symbolic form
(p, q) x
a, 13
= 0, mod n,
n) primitive and incongruous
admit, for any given primitive matrix
solutions (p, q) ; which, however, are all equivalent : and if we replace the
matrix by any equivalent matrix, these solutions remain unchanged.
From these considerations it is evident that, to solve the two converse
problems, we have only to determine the reduced pair equivalent to the given
pair, or the reduced matrix equivalent to the given matrix, as the case may be ;
either of them being known, the other is known also ; and the proposed problem
is completely solved.
Example. If n = 4725, and (581, 4000) is the given pair, the equivalent
reduced pair is, as we have already seen, (182, 25) ; the corresponding reduced
25,
matrix is therefore
-182, 189
, and all the corresponding matrices are in-
Note V.]
eluded in the formula
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
25, 01
38!
X e
where || e || is a primary unit matrix.
75, 100
Again, if
168, -161
equivalent reduced matrix is
-182, 189 1
itrix.
is a given matrix of the determinant 4725, the
25,
; hence (182, 25) is the corresponding
- 182, 189
reduced pair, and all the corresponding pairs are included in the formula
(182 h + 4725 a, 25 /t + 4725&),
where a, b, h are any integers, positive and negative, of which h is prime
to 4725.
If we introduce the one-valued functions
we may write the first two of the equations (ii) in the form
am 4; i
(vii)
A* am
Similarly the last of the equations (ii) may be written
J_
M~ ( '
(viii)
In the theory of the transformation of the Theta functions it is convenient
a,B
satisfies the congruences (8=0, 7=0, mod 8,
to suppose that the matrix
= 1, mod 4. In this case the equations (vii) and (viii) present themselves in
the form cos 2 am
A 2 am
A J am
sin 2 am 4
sin-coam
382 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [.Vote V.
in which they occur in the ' Fundamenta Nova.' From these special formulae,
in which ^/X, ^/X', ^/k, ,Jk' represent the one-valued functions <* (ft), \p (ft),
< 2 (tt), ^ 2 (<o), the general formula (vii) and (viii) may be immediately inferred
by linear transformation.
If = yy', where 7 is the greatest divisor of -- which is prime to g, the
number of the residues of which are prime to g (i. e. the number of different
numbers 2Z in (e), p. 379) is 7X (7')- Hence, the whole number of non-
equivalent pairs is 2 7x (y), the summation extending to every divisor g of n.
We have, consequently,
these results are easily verified independently (see a Memoir, on the ' Singu-
larities of the Modular Equations and Curves/ ' Proceedings of the London
Mathematical Society,' vol. ix., Art. 9) *.
It may deserve notice that the equation (vi) is the first of a series of
elementary formulae of the same general type.
The next in order is
= * ( M M) x
x ....
.V A v<v*vj 3
the general formula being
X(n) x II X (-^) x n x(^yy^) x = II x(^) x IT X
The s divisors S l , S 2 , ..., S,, are not necessarily relatively prime ; but, if p be
any prime dividing a- of them, and having a for its exponent in n, JJid b for the
sum of its exponents in S l} S 2 ,..., we must have either b<a, or else b = a % a-<s.
Ifb = a,<r = s, the factor ~ * - must be applied to the left or right-hand
P P
member, according as s is even or uneven.
* [Vol. ii. p. 252.]
Note VI.] NOTES ON THE THEOBY OF ELLIPTIC TRANSFORMATION.
383
VI. Further Theory of the Functions Q (to), Q' (w)*.
The following is a Table of the linear transformations of Q (&>) and
Q' (to). For brevity the argument w is omitted, and the functions J=
J'=Q' + k-K' are introduced :
dfl
w =
a + bl
a, b
c, d
iwi*<rw
iww+*
1
I |1
Q
^V
(_!)*<-
II |*|
Q
v
(-i)**- 1 ^
III |cr
J
t/'
(-l)i ( "-"F
IV |r
J-K
i(J'-K'}
(-l)i*-i>*
V |p
J-K
i(J'-K'}
(-l)i ( + 1) ifc
VI \p 2
J
iJ'
<-lp-'>tf
* [In explanation of the notation, it is to be observed that every matrix of uneven determinant is,
with regard to the modulus 2, of one of the six types
1,0
0, 1
0, 1
-1,0
which are represented by the symbols
1,0
1,1
1, '
1,1
0,1
T, p,
1,1
-1,0
0, 1
-1, -1
and considering these as unit matrices, i.e. supposing
the preceding values, they are represented by
a, b
c, d
to be equal, instead of only congruent, to
.
Similarly the primitive matrices of any even determinant, considered with regard to the modulus 2,
are of one or other of the following nine types :
1,1
1,0
0, 1
1,1
>
1,0
)
0, 1
0,0
0,0
0,0
1, 1
p
1,0
9
0,1
1, 1
1,0
0, 1
0,0
,
0,0
i
0,0
"f 11 "v n "if 1
r f 1 r
"MI ^2'Z> ^3,2
which are symbolized thus :
See Arts. 21 and 23 of No. xliii.
The functions Q (at) and Q' (o>) have opposite signs to the Q (ui) and Q' (to) defined in Note III
(p. 368). The quantity t] denotes *".]
384
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
[Note VI.
We add a list of useful particular cases ; the formulae are immediately
deducible from the Table. The functions P(&>) and P'(ft') represent J(a>) k' 2 K(<a)
and J'(<a) k'*K'(a>) respectively. For convenience, the transformations of K
and K' are included in the list, and, as before, the argument to is omitted.
II. .-
= K';
iK'(Q)=
iJ' (Q)=-iJ+iK;
III. co =
iQ' (Q) = - iQ.
1,0
+1,1
xQ=
_!_
1_
/
IV. =
1, +1
0, 1
V. 0> =
+ 1, +1
-1,
P(Q) = _^Z^L;
'- A")].
Note VI.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
0, 1
385
VI. =
-i, +1
= [K'-J' + i(K- /)] ; iJ' (Q) = - J
The nine typical transformations of the second order (see Art. 34) give the
following formulae for Q and Q'. They are derived by differentiation from the
corresponding formulae for K and K' in accordance with the equations (1),
Note III, p. 368.
-L
(a)
1,1
-1.1
x Q =
Q-l
IT (a) =
-i,
1 + Q
; - ' " <
1-12
II.
1,0
1,2
Q (Q) = 2(k' + ik)[Q + ikk'K],
Q' (Q) = (k' + ik) [<? + ikk'K' + i(Q + ikk'K)].
VOL. II.
I8
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. L Note VL
III.
0,1
-2,1
Q-2
Q (Q)= -(fc'
- (Q) = - 2 (F - ik) [Q + 1**' K\.
IV.
2,0
-1,1
x =
0-1
K
= Co. g X O =
'2, 2
V.
0,2
-1,0
VI.
2,0
0,1
x Q = i a
J(Q).
K
J'-K'
Note VI.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
387
i = (7j, 3 X ii =
VII.
-1,1
-2,0
x a =
i-o
vm.
1,0
0,2
x (i = 2a
(0 =
IX.
0,1
-2,0
Let A (w) = i j^rl ; the formulas of the Table show that if
a, b
c, d
be a unit
matrix of either of the types (1) or (^), the equation w =
equation
a, b
c, d
3 ^
a, b
c, d
x il implies the
x h (Q) ;
388 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VII,
or, which is the same thing,
1 1 J_ 1
ai ~---'
the quotients being even and the continued fraction subtractive ; then also
^ = 2^- _ :>! ___. ...... (2)
The property of the function h (o>) expressed by the equations (1) and (2),
and the characteristic property of the elliptic function of the second species, may
be regarded as having a relation to one another comparable to that existing
between the double periodicity of the elliptic functions of the first species and
the quasi-periodicity of the modular functions. The plane on which the ar-
gument of the elliptic functions is represented is divided into elementary
parallelograms, of which the sides are any two simultaneous elliptic periods ;
and the elliptic function of the first species has the same value at corresponding
points in two different parallelograms, while the values of the function of the
second species differ by a quantity of the same type as the difference of the
vectors of the two points ; e. g.
On the other hand, answering to the linear substitutions &> = -- r-- , we have a
division of the semi-plane on which w is represented into lunular spaces equivalent
to the reduced space (B) of fig. 2, referred to in Art. 38. At corresponding
points in two such spaces the modular functions 4> (<o) and ^fr (to) have the
same value, while the values of h (<o) are related to one another as the vectors
of the two points.
VII. Formulce relating to the Elliptic Functions of the Second Species.*
The Formula of Addition.
Let x lt x 2 , a; 3 be three arguments of which the sum is zero ; we have, from
the equation of Jacobi [Art. 10, (iii)],
Z(x 1 ) + Z(x. 2 ) + Z(x s )=-K:l^ c log$ (l%) ..... (i)
In the equation (i) of Art. 6, let
Mi = M 2 = r* 3 = M4 = M, V 1 = v^ = v s = v 4 = v, X l + Xi + X 3 = ;
* [This Note is to some extent equivalent to IV, but the two Notes could not be united together con-
veniently.]
Note VII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
389
differentiate with regard to x t , and write a- t = in the result. Representing for
brevity 5( (0) ^(x,) ^ (x 2 ) 3 X (a%,) by a* and
3. (o) [5; (*0 3, (,) 3, (x,) + 3. (,) 5; (z 2 ) 3, fc) + 5 S K> 3 8 (x 2 ) 3;(x 3 )]
by a,, (s = 0, 2, 3), we attribute successively to p., v the values 0, ; 0, 1 ; 1, ;
we thus obtain the equations
= 0,
- a s - = 0,
or, which is the same thing,
a x = a 2 = c*s = ct .
Transforming the right-hand member of (i) by means of the equality
a = a v we arrive at the equation of addition in its usual form ; viz.
Z(Xj) + Z (x. 2 ) + Z (#3) = k 2 sin am a^ sin am x. z sin am x z ; ... (ii)
#! + 2 + a; s = 0.
The Formula of Transformation.
oc
In the equation of Jacobi [Art. 10, (iii)] we write -^> for x, and O for w ;
we thus obtain
1 y[ x \~\ - ^( n ) x d
M ^IM' J ~ X0^ " do;
o^
' V
2J
where w =
a, 6
c,
x ii, X = < 4 (ii), and M is the multiplier corresponding to the
transformation. For brevity, we suppose that the matrix
a, 6
c, d
is of the
uneven determinant A, and satisfies the congruential conditions 6=c = 0, a = 1,
mod 8, assumed to exist in Art. 33. Putting h = 2K in equation (xii) of that
article, and designating by C' a quantity not containing x, we find
, 0]
390 NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. [Note VII.
or taking the logarithm, and differentiating,
whence finally
, . A v sm-am 4i<f
= 2 A; 2 sin am x cos am x A am x 2.- = ; A ?*. , ,
' 1 sin 2 am4^^sm 2 amcc
the value of the constant C being
(v)
Dividing equation (iv) by x, putting x = 0, and observing that lim *- = 0,
(vi)
we also have
(X \
jrf, X V Art. 33, equation
(xvii), is of the form
x(
11 A \l'-ii2,
X(*
1
XT
x
^ 2 sin 2 am4y ^ sin 2 am 03].
Employing this notation, we have for the right-hand member of (i) the
equivalent expression
1
so that C=-B l .
Note VII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
391
The constant C is a rational function of k' 2 and X 2 ; the integrals K(a>),
), J((t>}, e7(ii) entering into the expressions (iii) only apparently : the value
is in fact
(vii)
It may be observed that the coefficients B 2 , B 3 , . . . like B lt are all rational
functions of k 2 and X 2 ; for these coefficients are linear functions of the quantities
2& 28 sin 28 am 4/, which are rational in k 2 and X 2 , as may be seen by expand-
ing the elliptic functions in equation (ii) in series proceeding by powers of x,
and equating coefficients of like powers of x. The same thing is true for the
coefficients A, which are linear in the quantities 2 sm~ 28 am4 j /, and which are
also connected with the coefficients B by the relation
where the coefficient of .5 is rational in k 2 and X 2 , because II sin 2 am 4/ is so.
The right-hand member of equation (i) may be transformed by means of
the formula of addition ; viz. this formula gives
Z(x + y) + Z(x -y)- 2Z(x) = Z(x + y) + Z(x -y)- Z(2x) - [2Z(x) - Z(2x)],
whence
2k 2 sin am x cos am x Aam a; sin 2 am y
1 k 2 sin 2 ama;sin 2 am^/
(viii)
the summation extending to every value of / from - |(A - 1) to \ (A - 1). The
formula (viii) may also be obtained by integrating the sum of the squares of
the roots of equation (xxv), Art. 33.
To complete the preceding theory we add the following list of the trans-
formations of Z(x) by unit matrices.
I.
For any unit matrix
a,
c,
b
d
i)^
'[(-
1,
o,
-1)
1
'" ~
mod 2,
i) / TA 7. IT
tASy IV 1 ~~ 4.J 1 "*- J
k).
* [C is the same as the II of Note IV (pp. 373-376).]
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VII.
II.
0,1
-1,0
*7 iw lf\ Ylw V\ l<n_
t^y i"**" 9 "* / " V j / "' I
in.
1,0 xa
A'Z (*'ar, + ^) = Z(a?, k) -K\x-
V A/ /
sm am x A am an
|.
cos am x J
sm am a; cos am x n
Aamx J
IV.
O) =
1, 1
o, 1
ia 3
, = Z[x, k].
1,1
V.
x(i= -
-1,0
ikZfikx, +) = Z[x, k] -tf(x-
N flj X \
VI.
snam x cos am
o, i
-1, 1
ik'Z(ik'x, -^) = Z[a;, *]-(-
Aam x
1+Q
sin am a; A am a;
cos am
These formulae may be verified by transforming the function sin am a; in
the equation Z(x)= f k 2 sin 2 am x dx. For example,
'lj \\tCi K ) ~~ j \X. K) ~t~ X
\ ' / \ ' /
f> ix rx
il F 2 sin 2 am (a;, k')dx I ^ 2 sin 2 am(x, k)dx + x
.'o -'o
= - j [F 2 sin 2 am (ix, k') + k 2 sin 2 am (a;, k) 1] dx
f x r , sin 2 am (a;, k) , 7 ~\ ,
/ **- / 7 -( + A 2 am(x, k) \dx
.'o L cos 2 am (x, A)
sin am a; Aam a;
Note VII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 393
The formula IV (which is an immediate consequence of the corresponding
formula for sin am x) changes II into VI, III into V, and vice versd. The
formulae may also be deduced from the equation (iii) supra, but the elimination
of the complete integrals requires the use of some additional formulae.
Taking again as an example the formula II, we have
d
X
d . TTX \ d
d
- o cos am
But
_
~
and the coefficient of x becomes
i , xir ,, ,
, -- , that is, 1,
in accordance with the formula II.
Lastly, for the three typical quadratic transformations, writing for brevity
s, c, d to denote sin am (x, k), cos am (x, Jc), A am (x, k) respectively, we have the
formulae
<!> =
1,0
1,2
xQ, .......... (1)
(l + k)Z[(l+k)x, ^]-2Z(x,k) = 2k[x-^]-, . . . (3)
which may be verified by either of the methods exemplified in the case of the
linear transformation w = --- Observing that
u
2Z(x, k) -Z(2x, k)= - k 2 sin 2 am x sin am 2x.
VOL. II. 3 E
394 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII.
we may write the equation (3) in the form
(1 + k)Z\(l + k) x, |^] - Z(2x, k) = * (2x - sin am 2x).
Employing the notation of Legendre,
>- j
and writing
am (2x, k), ^ = am
-,
so that Z (2o;, ) = /?(, jfc) - E (p, k),
we find
l = - . . .(4)
(p) = k sin<f>,
a celebrated formula, due to Landen, which serves to express an elliptic integral
of the first species having a real modulus less than unity by means of two elliptic
integrals of the second species having moduli of the same character.
A similar, and indeed equivalent, formula is obtained from the equation (2) ;
viz. writing
1 7 '
<t> = am (x, k), ^ = am (1 + k') x, j
we find
/ T !\* /' Ti ^ ^ /g\
tan (\J<- <) = tan $.
VIII. 7%e Functions Al (x) q/ TFeie?'s<rows.
The Abelian functions of M. Weierstrass are defined by the equations
'0, 1, 2, 3],
T
Q (T'X
where, if s = l, '
Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 395
and, if 5 = 0,2,3, a 8 =3 8 (0),
and where T C K , . , 7
J= I k-svtf&mxdx.
.'o
These functions which differ from the corresponding Theta functions of -^-,
,
only by the presence of the exponential factor e ZK , and of a factor not con-
taining x, possess, as is well known, the remarkable property that they can be
developed in series proceeding by powers of x, of which the coefficients are
integral functions of k 2 with integral coefficients, and which are convergent
for all values of x. For our present purpose, there is a slight convenience in
considering, instead of the functions Al (x), multiples of these functions by the
exponential e^**. These multiples we shall call, in what follows, Abelian
functions, and we shall denote them by the symbols A (x), A^x), A 2 (x), A & (x),
so that e.g.
Q ^o(sV)
A u (x) = e-*x* x V2A/ .......... (ii)
3tfO)
These functions respectively satisfy the partial differential equations
= <), .... (iii)
where g = -&, g l = l-2k 2 , g 2 =l-k 9 -, g, = 0;
or, which is the same thing, the four functions A Vk', A l V(kk'), A^/k, A 3 all
satisfy one and the same equation
-^(l-^)a:M = ....... (iv)
The equations (iii) are a little simpler than the corresponding equations
satisfied by the functions of M. Weierstrass, viz.
e M = 0, . . . (v)
where g = 0, g^l-k*, g a = l, g 3 = k*;
but the gain in simplicity is apparent rather than real, as the determination
of the coefficients is not more easily effected in the functions A, than in the
3 E 2
396 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII.
functions A1 4 . The equations (iii) or (iv), as well as (v), may be derived directly
from the partial differential equation of the Theta functions
dx* v doa
which gives
da> K dx da i-n- dx 2
and in which we are to put
and to eliminate du> by means of the equation
da ITT
the differentiations with regard to k 2 being effected by means of the formulae
Q
x
d.k* ~ 2^' 2 ' d.k 2 "*
The Abelian functions are connected by a relation which may be inferred
from the corresponding relation between the Theta functions ; viz., using for a
moment a notation with double suffixes,
"o, i = "(,, A lt x = AH A lt o = A. 2 , A^ o = Ay,
and supposing A m< = A^ , if m = n, n = v, mod 2, then from the formula
we obtain the following
A m , n (x +
where
r_
General formulae for the transformation of the Abelian functions are im-
mediately deducible from the formulae for the transformation of the Theta
functions.
Thus, since in general, whatever be the transformation = - , , we have
Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 397
where T is a homogeneous function of order n of two of the Theta functions
3/-T- , w) ; then, attending to the equation
1 Q(fl) % Q(*) i*l3(
M*
!_ dM
'M d.k*'
we find
where 7 1 , is a homogeneous function of the order n of two of the functions
A (x, k 2 ).
In particular, if n is uneven and the transformation is primary, we have
1 j tf
*
where Z7, may be expressed in terms of the Abelian functions ; e. g.
Z7 3 = A am a; II [1 & 2 sin 2 coam4/)j sin 2 am a;]
For 'the linear transformations of the Abelian functions we find
I. If
a, 6
c, cZ
1,0
0,1
, mod 2,
II. If
a, &
c, c?
-A
0,1
= 1,0
, mod 2,
^^tx, 1-k^^iA^x, k 2 ),
A 2 (ix,l-k*)= A<,(x,k*),
A 3 (ix, l-k*)= A s (x, A"),
A(ix, l-/; 2 )= 4, (a, A 2 ).
These equations signify that the coefficients of x 4n + 1 and x* n in ^ and ^4 3
respectively are functions of 2 (1 - k 2 ), and the coefficients of x 4 " +3 and a; 4 " + 2 are
functions of k 2 (1 - k~) multiplied by 1 - 2 2 ; while corresponding coefficients in
398
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [.Vote VIII.
A.; are of the form p(l 2k*)q or j) + (1 - 2Jfc 2 ) q, according as the
exponent of x is evenly or unevenly even.
III. If
IV. If
V. If
a, b
c, d
1,0
1,1
, mod 2, X 2 =
k*
! (*,*>,
a, 6
c, d
1,1
0,1
, mod 2, X 2 = -=- ,
A s (kx, 1)= e-i
kx, 1)= e-i^
a, b
c, d
1
1,1
-1,0
, mod 2
c? log If
\2
1
, A -
1-fr
A, (ikx, ,-ip^.ae-
Note VI II.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION.
Y7-T T / W* *-r v * ...._ J-
399
a, b
c, d
"*77
o,
-1,
1
- 1
, mod 2,
^ d log M
M
A, (ik'x, 1 - 1) = lyfc'e-
A, (ilex, 1 - 1) = e-
^ 3 (ik'x, 1 - 1) = e-
^ iFx, 1 - 1 = e
(x,
(x,
(x,
. _ * o* v
The functions 2 = -j^d a ' l A(-^, \ 2 J satisfy a partial differential equation
/ 1* \
with respect to x and k 2 . We have from that equation, if A = A(jf, \ 2 \
where
r dA-\ _
~
d (l\
dA M dA \M/
~
~dx
we
Observing that
find
d
I/ 4
.
and dividing by
i. / H/T -J / y * 7
-'/ ' ' , /i f ' ,i
or, substituting for J its value >/.lf.<
5^ 2
But the coefficient of - x 2 2 in this equation is equal to n 2 k 2 (1 - & 2 ). Hence,
we have finally
400 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [.Vote VIII.
which is satisfied by the four functions
ir xt > Vf<^. (* )
The determinant of the transformation n may have any positive integral
value, and the constituents of the matrix of transformation may have any
integral values.
SC \/ Tit
If we replace x by - - and =-= by , we see that the functions
M n
satisfy an equation of the same form as that satisfied by A, (x, k 2 ),
We next proceed to form the equation satisfied by
Substituting for 2 and dividing by A n , and observing that
_ . n . n _^
dx dx dx (r '
we find
d 2 <r I dA d<r da-
-j + 2n -r -5 -j- + 4wA; 2 (l-A 2 ) -T^-
dx* A dx dx ' d.k-
or, after all reductions,
Note VIII.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 401
Lastly, if in this equation we introduce s = sinamx and k* as the inde-
pendent variables instead of x and k 2 ; we have
d(7 da ds ,d<r
_ . _ . _ f\fi _
dx ds dx ds'
dx*~ ds* ' ds
dcr ~\ da- da- ds da- da- 1 ,, 2 ,y-.
+ == ~~ + ^- Z J;
where , is the differential coefficient of a- taken on the supposition that a-
is a function of the independent variables k- and x, and , , 2 is the differential
coefficient taken on the supposition that or is a function of k 2 and s, and that k-
does not vaiy in s, and we obtain finally
(1 - *') (1 - k V) + [(2n - !)* - 1 - 2 (n - l)fc V]
-n(n-l) A; 2 (l-s 2 )(r + 4ni 2 (l-^)-j =0, . (a)
in which equation the coefficient of -j- may be written in either of the two forms
(2n - 1) k*c 2 - d- and 2 (n - 1) & 2 c 2 - A;' 2 .
The equation is satisfied by
A' / XX' / \ . / 1
w^ - Vm 7 "^ 1 ' V MF n 2> V m-" 3-
For our immediate purpose we add the term n 2 & 2 <r to this equation in order
to obtain the equation satisfied by
There is some difficulty in applying the equation
to the actual determination of the functions U , U l} U& U 3 . The following
method serves to exhibit these functions in what may be termed their
' canonical form.'
Let the operating symbol
VOL. II. 3 F
402 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note VIII.
r
of this equation be denoted by V ; write also for brevity
4 *< 1 - l 'hnF-*
V /
__,
/ v lif ' / v "af * n ^ ne th ree cases s = 0, 2, 3 respectively. We find, by
expansion,
so that we have in the three cases alike
Let fa =
~4! D !
be a series satisfying the equation V fa = 0, so that the coefficients a 2 , a 3 , ... are
rational and integral functions of k 2 with integral coefficients ; similarly, let
a; 2 , a; 6 , x*
be a function determined by the equation
a: 4 a; 8 a; 10
be determined by the equation
and so on continually. It will be found that these determinations are always
possible ; and that, as indicated in the case of the functions < , <f> v fa, the first
s coefficients in the function fa are zero, and the values of the two following
coefficients are unity and zero. All the coefficients are rational and integral
a: 2 * + 28 '
in Jc 2 , and the highest power of k 2 in the coefficient of ^ ^~Yf ^ n $' ' ls s ''
then have the theorem
Note VHI.] NOTES ON THE THEOEY OP ELLIPTIC TRANSFORMATION. 403
For if $ represent the series on the right-hand side, we have, evidently,
.e.
= ; also <& = v-n$v.
...
1.
so that the series < satisfies the same partial differential equation as the function
(/y ^
-=-=, X 2 ). while the first two terms of the series coincide with the first two
M '
terms of the expansion of the function ; and this establishes the theorem, because,
given the first two terms of the expansion, it can be continued by means of
the partial differential equation in one way only.
We next denote the operating factor
^ 2 + 2nZ (x) ^ + Ink 2 (1 - Jc 2 ) -^ -n(n- 1) k 2 en 2 x + n 2 k 2
by D ; and we observe that we have not only
but also, separately,
viz. each of these equations is derived from the corresponding equation
precisely in the same way in which the equation Da = is derived from the
equation V2 = 0.
We now expand the functions in series proceeding by powers of s ; we
"0
observe that if *!/> (s) be this expansion, it follows from the properties of the
S 2r
functions <}> r that the first term in ^ r (s) is ^ j , and that
where p m is a rational and integral function of Jc 2 with integral coefficients of
an order not exceeding m in Jc 2 ; the coefficient p is however not zero, but
(2r + l)(2r + 2) 2r (2r + l) (2r + 2)
1.2 1.2.3 v '
404 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IX.
We thus obtain for U the expression
But U is a rational and integral function of order n 1. Hence we may
omit in ^ 2 (s) all powers of 5 above s n ~ l , and consequently all the functions
$*v. ^ 2 (s) after Si (n - l) v . >f-j( n _D(s) ; the higher powers of s disappearing of them-
selves. And if we denote by / the operating factor in the left-hand member
of the equation (a), p. 401, increased by n 2 k*, viz.
we have the equations
by which the functions ^(s), 4i ( s ), m &y be successively calculated ; viz. if
fa + 2
we have
- [4<
- 4n<7 - n) 2 ] C,
+ 2(7 (2<7 - 1) (2(7 -
- 1) (2<r - n - 2)
a;
''
2 "
if jB ff be the coefficient of '-'. in ^ r _ s (s).
.
12 <T !
IX. On Modular Curves *.
Let [a, /3, 7] or w be a given quadratic point, and let it be required to assign
all the modular curves of any species passing through the point
Let
A,B, C
A', B-, C'
be a pair of relatively prime solutions of the equation
so that
I %
-2/8, | =
A,B, C
A', &, C'
9
* [On the back of the last page of the manuscript of this Note Professor Smith has written :
' These papers relate to the problem, &c., " Given a quadratic point, to find all the modular curves
passing through it," &c., &c. It ought to be worked into a memoir " On the ordinary multiple points
of a modular curve ; and on the intersections of two modular curves." ']
Note IX.] KOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION". 405
and so that all the solutions of this equation are comprised in the formula
\'A - \A', \'B - \B', \'C- \C'.
The determinants B Z AC=D, B' 2 A'C' = D' are positive, because
ft 2 ay= A is negative, and two pairs of conjugate imaginary points cannot
be harmonic. Hence the symbols [A, B, C], [A', B', C'] represent real semi-
circles passing through the point w. The determinants of all semicircles passing
through to are included in the formula
where 2 / is the invariant 1BR-ACT -A'C, and where J*-DI/= -A. The
form (U, J, D) is the duplicate of the form (a, /3, 7), and is transformed into
the product of that form by itself by means of the bipartite linear substitution
A,B,B,C
A',H,R,a '
We thus obtain the theorem :
' The modular curves of order D pass through the point [a, ft, 7] as often
as there are primitive representations of D by the duplicate of the form (a, ft,y).'
To determine (by means of the corresponding semicircles) the ovals which
pass through a given point [a, ft, 7] we have the following rule :
' Let (P, Q, E) be the duplicate of (a, ft, 7), and let
Pi, q\, <ii,
Pa, ?2, ?2> 7
be the substitution transforming (P, Q, R) into (a, ft, <y) 2 ; let also
(P,Q,E)x(^,^Y = D
be the given representation of D ; the semicircle
of determinant D passes through [a, ft, 7].'
To determine which of the -modular curves of order D passes through
[a, ft, 7] we should have to distinguish the cases in which a, ft, 7, ft v /u 2 have
different congruential values "for the modulus 2. This discussion, for brevity,
we omit.
If we apply to the form (I/, J, D) the reducing substitution of Lagrange,
we obtain the orders of the two lowest modular curves, which can pass through
the point \ + <& (a>), or P. If on the tangents to these two curves at P we
measure lengths PT, PT equal to the square roots of the corresponding de-
terminants, and if, completing the parallelogram PTPT, we construct the
40G
NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note IX.
parallelism of which PTPT is an elementary parallelogram, the tangents to
the modular curves passing through P are the lines joining P to the nodes of
this parallelism ; and, if N be one of these nodes such that PN contains no
other node besides P and N, PN is the square root of the determinant of the
modular curve touching PN at P.
Gauss has shown that every class of the principal genus of any determinant
is the duplicate of as many classes as there are ambiguous classes ; i.e. classes
producing the principal class by duplication. Thus no point of determinant A
can lie on a modular curve of order D unless D is represented by some class of
the principal genus of determinant A ; and, if there be such representations,
there are, corresponding to each of them, as many points of determinant A
lying on the modular curves of order D, as there are sub-classes of determinant
A producing the principal class by duplication.
The necessary and sufficient condition that a given point [a, /3, 7] should lie
simultaneously on a modular curve of order D, and also on a modular curve
of order D', is that D and D' should both be capable of primitive representation
by the duplicate of [a, /3, 7]. Let
D = (P, Q, R) x ( Ml , /x 2 ) 2 , D' = (P, Q, 72) x (,, ^
be the two representations ; let
"2
= v, and let (P, Q, R) be transformed by
into (D, J, D') of determinant At> 2 . The form (D, J, D') is then
into v 2 x (P, Q, R) ; that is, it is transformed by the
transformed by
7 -/*,
bipartite transformation
"2 ~ "l
Pv
into v z x (a, |8, 7)2. The semicircles
" < -
which are of determinants D', D, and which have / for their harmonic invariant
and v x [a, /3, -y] for their covariant, are the two semicircles corresponding to the
given representation and intersecting at the point [a, ft, y].
The equation DD' = J' 2 + v 2 A always subsists (as the preceding analysis
implies) whenever any point of determinant A lies on two modular curves of the
orders D and D'. But this condition, though necessary, is not sufficient; viz.,
confining ourselves to the case in which D and H are relatively prime, J and
Note IX.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 407
v must be relatively prime, and the numbers D, D' must have each the characters
of the principal genus of determinant A ; when these additional conditions are
satisfied, D and D' are necessarily represented by the same form of the principal
genus ; and the two sets of curves have as many points of determinant A in
common as there are ambiguous classes of determinant A.
Let [a lf fa, 7j] and [a 2 , /3 2 , -yj be two quadratic points w^ a> 2 , of which the
determinants are to one another as two squares ; let A x = A'0'i, A 2 = A'0|, A'
being the greatest common divisor of A x and A 2 ; and let it be required to assign
all the modular curves with regard to which l + ^fai) an d i + ^C^) &re
inverse. Let (P, Q, R) be a form of determinant A' compounded of (a^ (3 V 7^
and (a 2> /? 2 , 7 2 ). If^ + ^(t l ) and - + <fr(< 2 ) are inverse with regard to any
modular curve of determinant D, D is divisible by 6 1 6 Z , and the quotient is
capable of primitive representation by (P, Q, R). Conversely, if these con-
ditions are satisfied, a modular curve of determinant D exists with regard to
which the two points are inverse. Let (P, Q, R) x (i, /w 2 ) 2 = A~Z~ > an( ^ ^
""
PI,
Pz>
be the substitution transforming (P, Q, R) into
(i, Pv 7i) x (2, &, 72) J
the points *> l = x l + iy l , 2 = o: 2 + iy 2 will satisfy the equation
x +iv
^ + y
the determinant of the matrix being D ; i.e. \ + $(]) and \ + $ (w 2 ) are
inverse with regard to a modular curve of determinant D. If D and D' are two
uneven numbers relatively prime, the necessary and sufficient condition that
two given points should be inverse with regard to modular curves appertaining
to each of those determinants is that the two points should have the same
determinant A, and that D and D' should be primitively represented by the
same class of determinant A ; or, which is the same thing, that DxD' should
be represented by the principal class of determinant A. If this condition be
satisfied, every point of determinant A is inverse to another point of that
determinant with regard to a modular curve (Z>) and also with regard to a
modular curve (D'). If the class by which D and D' are represented is not a
class of the principal genus, all the points of intersection thus obtained are
408 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X.
imaginary, no two inverse points coinciding. But if D and D' are represented
by a class F of the principal genus, the classes which by their duplication produce
the class F give real points of intersection.
X. On the Quarter Periods E, \iK.
The fundamental pair of quarter periods (K, ^iK) is not in general the
absolutely least pair of quarter periods appertaining to the doubly periodic
function ; for it does not follow that the parallelism (K, \ iK] is absolutely
reduced because the parallelism (K, iK) is primarily reduced.
Problem. To determine, for any given value of k 2 , the absolutely reduced
parallelism equivalent to (K, iK].
From the preceding discussion it appears that the reduced triangle is formed
by the vectors
K, iK, -K-iK,
or K iK, iK, K,
according as the angle from K to iK is obtuse or acute. It may be proved by
the considerations already employed in the demonstration of the theorem (...)*,
or by a method presently to be explained, that this angle is obtuse or acute
according as the coefficient of i in It 2 is negative or positive.
We have now to show how, for any given value of k 2 , the vectors of the
reduced triangle can be arranged in order of magnitude. For this pur-
pose we employ the principles contained in a Memoir ' Sur les Equations
modulaires,' which will be found in the Transactions of the ' Accademia dei
Lincei,' vol. i. Ser. iii. (1877)f; it will suffice to consider only one of the propo-
sitions to be demonstrated. We understand by [z] the absolute value, i.e. the
analytical modulus, of any complex quantity z.
' The inequalities [K]<[KiK1
subsist simultaneously, if [& 2 ] < 1 : if [A; 2 ] > 1, we have
or [K+iK]<[K]<[K-iK],
according as the coefficient of i in the imaginary part of k 2 is positive or
negative.'
* [Blanks in the manuscript are denoted by dots enclosed in parentheses, thus (...).]
t [Vol. ii. p. 224].
Note X.] NOTES ON THE THEORY OF ELLIPTIC TRANSFOEMATION. 409
Let -^r = X+iY; the Inequalities \K]^[K+iK r ] are equivalent to the
inequalities ^ 2 X+ X 2 + Y 2 . But the quadrant of the circle X 2 + Y 2 + 2 X = 0,
which lies within the reduced space defined by the inequalities in question, and
which runs from the cusp at to the point 1 + i, is represented in the plane
k 2 = ^ + x + iy by a semicircle of radius 1, running below the axis of x from the
point x = + -, to the point x = f ; and the two regions containing the points
1 and + 1 respectively, into which the reduced space is divided by the
quadrant are represented in the plane of xy by the two regions, containing the
points infinitely far off on the axis of y in the negative and positive directions
respectively, into which that plane is divided by the axis of x from + oo to \,
by the semicircle, and by the axis of a; from | to oo. Hence according as
k 2 \ lies in the first or second of these regions or on the boundary between
them, we have
or <+t, or
Similarly, if we divide the plane xy into two regions, one lying above the
axis of x and external to the circle
the other comprising the rest of the plane, it will be found that according as
k* | lies in the first or second of these regions, or on the boundary between
them, we have
[K]>[K-iK1 or [K]<[K-iK'l or [K] = [K- iK'].
These two results taken together are equivalent to the proposition which
we have enunciated.
To complete the solution of the problem we have to discuss the inequalities
\K+iK r ] ; their theory depends on the representation of the lines
+1 0, or, rather, of those portions of them which lie within the reduced
space, by the semicircles (x-|) 2 + i/ 2 = 1.
The final result is perhaps most simply expressed in the following form :
The plane xy is divided by the axes and by the circles (x g) 8 + y 2 = 1 into
twelve regions. The regions within both circles are designated by A, those
within one circle only by B, those outside both by C ; the four regions A are
numbered 1, 2, 3, 4 according to the quadrants in which they lie ; the regions
B and C are similarly distinguished (see fig. 1). The Table indicates the ar-
rangement of the four periods K> iK', K+iK', KiK', in ascending order of
VOL. ii. 3 o
410 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X.
absolute magnitude, according to the region in which the vector point x + iy = k 2
is situated.
A.
(1) iK', K, K-iK', K+iK';
(2) K t iK', K-iK', K+iK';
(3) K, iK', K+iK', K-iK';
(4) iK', K, K+iK', K-iK'.
B.
(1) iK', K-iK', K, K+iK',
(2) K, K-iK', iK', K+iK';
(3) K, K+iK', iK', K-iK';
(4) iK', K+iK', K, K-iK'.
C.
(1) K-iK', iK', K, K+iK';
(2) K-iK', K, iK', K+iK';
(3) K+iK', K, iK', K-iK';
(4) K+iK', iK', K, K-iK'.
It will be seen that along the axis of x,
along the axis of y, [{"'] = [K] ;
along the upper and lower semicircles of (y + %
[K] = [K-iK'], [K] = [K+iK f ] respectively;
along the upper and lower semicircles of (x |) 2 + y 2 = I,
[iK'] = [K- iK'], [iK'] = [K+iK'] respectively ;
and that when we traverse any one of these six boundaries the quantities,
which on it are equal, change places with one another.
Cor. 1. The zeros of the Theta functions
Note X.] NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. 411
being respectively 2rK+2siK', (2r+l)K+2siK e ,
it follows that the least zeros of ^(H^) are always K, and the least
zeros of 3 (^) are always iK'. But the least zeros of ^s
(K+iK') or (K-iK'), according as the coefficient of i in the imaginary
part of k 2 is negative or positive; and the least zeros of -^if^rr-) are %K,
2iK', 2(K-iK'), 2(K+iK'), according as k*-\ lies (1) inside the
circle (x +^) 2 + y z = 1 to the left of the axis of y, (2) inside the circle
(y i) 2 + y* = 1 to the right of the axis of y, (3) outside the two circles and
above the axis of x, (4) outside the two circles and below the axis of x.
These determinations assign in all cases the circles of convergence of
the developments of gm am u
, cos am u, A am u,
u
and their reciprocals, in series proceeding by powers of u.
Cor. 2. The Table also assigns in every case the absolutely least pair of
conjugate periods of the functions sin 2 am u, cos 2 am u, A 2 amw; viz. this pair
consists of the least and least but one of the four quantities 2K, 2iK', 2(KiE').
But to obtain the absolutely least pairs of periods appertaining to the functions
sin am u, cos am u, and A am u themselves, we have to consider the modular
curves of the square determinant 4, instead of the lines and circles of deter-
minant + 1.
Problem. To determine, for any given value of k 2 , the absolutely reduced
parallelisms equivalent to (K, \iE'), (^[K iK'], ^[K+iK']), and (\K, iK).
Since the reduced triangle of the parallelism (K, iK') is acute-angled, one
of the two triangles into which it is divided by the line joining any of its
vertices to the middle point of the opposite side is certainly acute-angled, and
the half side is always less than the bisecting line. Hence the reduced triangle
of the parallelism (K, \E') is one of the four
K+liK', UK', -K- iK', (2)
K-\iE', \iE', -K,. (3)
K- iK', \iK', -E+\iE', (4)
302
412 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X.
in each of which the side \iK' is less than the side K+\iK'. In fact, we have
the case (1), (2), (3), or (4), according as
(1) amplitude of ^>i T ; [K]<[K+iK / ];
(2) amplitude of ^-'>I^; [K]>[K+iK'];
(3) amplitude of ~< ITT; [E]<[E-iE']i
iK'
(4) amplitude of ^<\ ', [E]>[E-iE f ].
These inequalities have been already examined ; in addition To these we
have to consider the inequalities
which determine the order of magnitude of the sides in the four triangles ; and
which are equivalent to the following :
In the plane k z = ^ + x + iy, the circle -X" 2 +1 7 " 2 = 4, and the three pairs
of circles
are represented by the four loops of the modular curve (...), beginning with the
innermost and proceeding in order to the outermost. If we designate the
regions (taken in the same order from within outwards) into which the plane
is divided by the curve by A, B, C, D, E, the Table gives the three least quarter
periods corresponding to a value of k 2 \ lying within any given region :
A.
K,\iK'; K\iK'.
B.
-\iE',E; K\iK r
Note X.] NOTES ON THE THEORY OP ELLIPTIC TRANSFORMATION. 413
a.
iiK'', KiK'.
D.
iK'\ K\iK',
E.
KiK',\iK'; K$iK'.
The upper signs are to be taken in the region below the axis of x, and vice
versd. The region C is supposed to be divided into two, C^ and <7 2 , the first
within, the second without the circle (# + |) 3 + 2/ 2 = !
To obtain the reduced parallelism equivalent to (1-5", iK') we have only
to take the modular curve (...) which is symmetrical to (...) with respect to the
axis of y, and to interchange K and K', dividing at the same time by i. We
thus obtain the following Table :
A.
B.
,iK'; iK'\K.
\K,iK'\K; iK.
C 2 .
\K,iK'\K; iK'K,
D.
\K,iK'K; iK'\K.
E.
The reduction of the parallelism (\[K iK'~\, ^[K + iK'~\) depends on the
remaining modular curve of determinant 4, ... which is a symmetrical with
regard to both axes. The reduced triangle of this parallelism always has the
vectors ^(K+iK') for two of its sides, and either K or iK' for its third side:
in this enunciation, which is obtained by bisecting the side K + iK' of the
reduced triangle of the parallelism (K, iK'), the signs of the vectors are neglected.
414 NOTES ON THE THEORY OF ELLIPTIC TRANSFORMATION. [Note X.
Therefore, besides the inequalities [JT]5|[tir], [E+iE f ]^[EiE r ], which we
have already considered, we have to examine the inequalities
or, which is the same thing, the inequalities
(X I) 2 + 7 2 ^ 4 ; (.Y I) 2 + F 2 ^ 4 (.Y 2 + F 2 ).
The four circles
.Y 2 +F 2 2X-3 = 0; 3(A r2 +F 2 )2X-l = 0;
or rather the arcs of those circles which lie within the reduced space are repre-
sented in the plane k z \ = x + iy by the curve (...), of which the form may be
roughly compared to that of a hyperbola having an internal loop at each vertex.
We designate by A the central infinite region between the two branches of the
curve ; by B and R the infinite regions internal to the two branches of the
curve on the right and left of the axis of y respectively ; by C and (7 the regions
internal to the loops : we then have the following scheme, the upper signs being
taken in the upper part of the plane, and vice versd :
A.
iE'); E,iE.
A'.
iE'); iE'.E.
B.
B.
\(EiE'] > iE / ;
a
E, (E+iE') ; \ (E+iE-), iE'.
(7.
-iE', i(EiE') ; (E+iE') t E.
XLIII.
\
MEMOIR ON THE THETA AND OMEGA FUNCTIONS,
Arts. 1-14. DEFINITIONS AND ELEMENTARY PROPERTIES OF THE THETA,
OMEGA, AND ELLIPTIC FUNCTIONS.
Art. 1.] THE Theta Functions.
An exponential series of the type
is termed a Theta series. The necessary and sufficient condition for the
convergence of the Theta series is that the real part of the coefficient a shall be
different from zero, and negative ; subject to this restriction, the coefficients a,
b, c, may be any quantities whatever, real or complex.
For the purposes of this Memoir it is convenient to employ a notation for
the Theta functions somewhat different from that which has been adopted in the
Tables.* Writing q = e*, we define the function S^ v (x, q) by the equation
m ~ V 21 " + A**
m= oo
m= + oo
2 t
m=- /
and v denoting any positive or negative integral numbers. We thus have
(iii)
* [Thia Memoir was written to accompany the Tables of the Theta Functions calculated by
Mr. J. W. L. Glaisher.]
416 MEMOIK ON THE THETA AND OMEGA FUNCTIONS. [Art. 1.
For brevity, we shall often write
i, o 5 = * x > 2 o > 2 = a?, ? ;
we shall also omit the second argument q, when no ambiguity is likely to arise
from this abbreviation.
From the equation of definition (ii.) we infer immediately
TT) =(-)" -V(z) ......... (vii)
^(x), ........ (viii)
Thus there are only four distinct Theta functions :
(equations iv and v). Of these, 5j (a;) is an uneven function, while the other
three are even (equation vi). The Theta functions are singly periodic, having
TT or 27r for their period according as n is even or uneven (equation vii) ; the
quotient of any one of them divided by any other is doubly periodic, the periods
f -3>. v( x ) "=" $n',v( x ) being (1) TT, 2o>x, (2) TT + WTT, v anr, (3) 2ir, wr, in the three
cases (1) fi ft even, v v uneven, (2) M M uneven, v v uneven, (3) t*. r/ un-
even, v v even (equations vii and viii) ; lastly, any one of the four can be
expressed as a product of any other by an exponential factor (equation ix) ; so
that, in particular,
- . . . (x)
write a =
To obtain the Theta function $n, v (x) from the series (i), we have only to
7 1 /
= ^ITT I /J.<D + V + 2
U 2 -
\
Art. 2.] THE OMEGA AND MODTJLAB FUNCTIONS. 417
Thus, in some sort, the function 3- 3 (x) is the simplest of the Theta functions, the
values of the coefficients a, b, c being, for this function, a = iirca, b = ix, c = 0.
2. The Omega and Modular Functions.
The Theta functions are themselves functions of two arguments x and q ;
but if we give to x the value zero, or any numerical value, or, again, any value
depending on the value of q, we obtain a series of functions containing the single
argument q or a>. In this Memoir we propose to direct our attention chiefly but not
exclusively to these functions of a single argument, which we propose to term the
Omega functions. They are important not only in the theory of elliptic functions,
but also in other parts of analysis ; and they are intimately connected with
several interesting questions of arithmetic, algebra, and geometry. An exhaustive
account of the researches which have been undertaken by geometers on this
subject would exceed our present limits ; we propose therefore to give a brief
outline of the most essential parts of the theory, and to select for a fuller treat-
ment certain recent investigations which appear to have some interest.
We shall usually find it convenient to regard <, and not q, as the indepen-
dent variable, although for brevity we shall often employ q as an abbreviation
for e' v<u . It will be observed that o> may have any complex value of the form
x + iy, in which y is positive. If y is negative, the analytical modulus of q is
greater than unity, and the Theta series are divergent ; the value y = is also
excluded, though y may be as small a positive quantity as we please *.
The following are the expressions, in terms of w, of the Omega functions
which appear at the top of each page of the Tables. We write 3 [to]
for
* The collected edition of Eiemann's works (Leipzig, 1876) contains some fragmentary notes
on the limiting values of the Omega functions, when the analytical modulus of q converges to unity.
To these the editor, Professor Dedekind, has added some interesting researches of his own on the
same subject.
VOL. II. 3 H
418
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 2.
r - Tf \ -
~
E = E(w) = K(w)-J(a>),
E' -E f -***
(iii)
When w is a pure imaginary (i.e. a quantity of the form icr, a- being real
and positive), q is real, and k 2 is real, positive, and less than unity ; the
functions K, K', J, J', E, E' are also real and positive. This, in all practical
applications, is the only important case, and it is only for such values of <*> that
the values of the Omega functions are given in the Tables ; but the theory
of these functions requires us to attend to all complex values of o> for which the
analytical modulus of q is less than unity.
The equations (i), (ii), (iii) are to be regarded as defining the functions
Vk, Vk', /K, K', J, J', and E. To these we add four other equations of
definition,
m = oo
-. _
. . (iv)
and we observe that these four functions, as well as the seven preceding, are
one- valued functions of w. (We understand by ^/2 and v/2 real and positive
roots of 2 ; and by q^ and q& we understand e^ iir<a and e^ iira .) Any rational and
integral function of (f> (<o), -^ (<o), ^ (<) is termed a Modular function.
We also note the equations
(vi)
(viij
which are particular cases of equation (ix), Art. 1, and which serve to express
the Theta functions of the half periods in terms of 5 [<o], 5 2 H> ^sM-
Art. 3.] THE THETA AND OMEGA FUNCTIONS AS INFINITE PRODUCTS. 419
3. The TJieta and Omega Functions as Infinite Products.
The theory of the Omega functions is so closely dependent on the theory
of the Theta functions, of which indeed it forms a subordinate part, that we
shall find it convenient to give a brief outline of the principal properties of the
Theta functions, and to deduce from these, as we proceed, the characteristic
properties of the Omega functions. We shall, in the first instance, confine
ourselves to those properties which are independent of the theory of the
transformation of the Theta and Omega functions.
The identity
1 + q (v 2 + v ~ 2 ) + q* (v 4 + v~ 4 ) + q 9 (v 6 + v ~ 6 ) + . . .
- 1 v 2 x
m=l m=l m=l
which has been demonstrated by Jacobi and by Cauchy, expresses a fundamental
property of the Theta functions. If we replace v by q?v, and multiply by q*v,
this identity assumes the form
~ l ) + (ft (v 3 + v~ Si ) + q ? r(v 5 + v- s ') + ... 1
m = oo t = oo " * '
v ) x l
m=l m=l m=l
Writing successively v = e", and v = ie ix , in the equations (i) and (ii), we
obtain
- 2 ]; (iii)
i i
1 - 2 q cos 2x + 2 g*cos 4 x - 2 g 9 cos 6 x + . . .
- 2 ]; (iv)
; (v)
= 2g*II (L-q 2m ) since II[1 -2g 2m cos2x + g 4m ]. (vi)
i i
The zeros of the Theta functions (i.e. the values of x for which these
functions vanish) are consequently as follows :
3 H 2
420
MEMOIB ON THE THETA AND OMEGA FUNCTIONS.
[Art. 4.
$ (x) = 0, if x = r7
^(a;) = 0, if X = TTT + SCOTT ;
$ a (x) = 0, if a; = i(2r + l)*- + SUTT
$,() = 0, if x = i(2
i. e. the zeros of ^ (a) are
. . (vii)
Putting a = 0, we obtain from the equations (iii), (iv), (v) :
=l + 2q + 2q< + 2q+...=
(0)= /^^=i_
V ff
^j (viii)
-g ')'; (ix)
). . . (x)
Since 3j (a;) is an uneven function, we have
^(0) = 0;
but we find
+ ..... (xi)
If v denote any uneven number, we have, by an identity due to Euler,
i/ = oo m oo v = <x>
)...=l. . (xii)
Multiplying together the values of 3 (0), 5 3 (0), 5 2 (0), and attending to
this identity, we obtain
(1-^)3 = ^(0). . . (xiii)
3 (0)5 2 (0)3 3 (0) =
Dividing (x) and (ix) respectively by (viii), we find
< 2 (o>) = Vk, xp(o>) = Jk'
We also have from (xii), and from Art. 2, (iv),
For brevity, we shall often write
(xiv)
). (xv)
4. Expressions for the Modular Functions <(>), ^("O, x( w ) ^ Infinite
Products.
The following different expressions for <(w), for -^(w), and for the Eulerian
Art. 4.] THE MODULAR FUNCTIONS AS INFINITE PRODUCTS. 421
product II (1 + q m ), or, which comes to the same thing, for x ( w + 1) have been
given by Jacobi (Crelle's Journal, vol. xxxvii. pp. 67-77) :
I.
m= +00
(a) <!>()
(5) ^
w*w- /5 q *n
^
II.
l\ ^M- ~" >
- n
'
AM
III.
(d\
422
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 4.
(l-g 2m - 1 )( 1 -g 4m )
- g 8m )
We have also for the square and cube of x( ft) + 1) the analogous formulae :
IV.
V.
fa \
/(0. g)
I -g
These expressions may be verified by comparing their type factors with
the type factors in the equations defining <p (<*>), ^(<a), and x( w +l); the trans-
formation of the products into sums is effected by employing appropriate
particularisations of the formula (i) of Art. 3 : among these, besides the formulae
(vii)-(x), of that article, we may mention the following :
VI.
(a)
(a')
(b)
(60
U(l-q m )
Art. 5.] EXPANSIONS OF THE OMEGA FUNCTIONS. 423
We give three examples of these verifications. In the formula (I. c) we have
n(l-2 2m - 1 )(l- g 4m ) = 2(-) m 5 2 ' n2 + m by (Via),
and H(l - g *- 2 ) 2 (1 - q*<) = 2( - ) m g 2m2 by Art.. 3 (ix) ;
also
n (l-g 2 1 )(l-g 4 *) = n I-? 2 " 1 - 1 _ n _ 1 1 + g 2 '"
' (l_ g *-*)(l_ g *) (l-q* m - z Y (1 -q 2m
since by Art. 3, equation (xii),
Again, to verify the equation (III. g) we have
n(l+q m - 3 )(l-q l2m ) = '2qe m * + 3m by (VI. a') ;
and, writing q 3 for q, and g 2 for v 2 , in the formula (i) of Art. 3, we obtain
also
6m - 3 - 12 " > =
_ TJ _ _ _ TT _ _ _ TT
--
Lastly, to establish (VI. 6'), we write e*** g $ for g , and e* >7r 9^ for v 2 ; we
thus find
5. Expansions of the Omega Functions in series proceeding by powers of q.
Let ^L f (q) ,
VZ.ql
Jacobi has given the formula (Fundamenta Nova, Art. 40, equation 27)
where A (h) is [1 + ( ) X 2] times the sum of the uneven divisors of h. Let
s being any quantity whatever ; since
424 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 6.
we have, if h = 0, [s, 0] = 1, and, if h > 0,
r, ;n-, A W -, s * z A (0 A ( & ) , * 3 2 *() *(&)*(*)
~~ ' 2 ' 2 "~~ " 3 ~ ~~
where 2,. extends to every set of i numbers (equal or unequal) which satisfy the
equation a + b + c + ... =h.
But, when s is a positive or negative integer, the coefficients [s, h] are more
easily calculated by a recurring formula, similar to that given by Euler for the
development of the product H (1 + q m ), and deduced by him from the equation
III. (a) of the last article. Several such formulae are supplied by the equations
of Jacobi. For example, from equation I. (c), we have
4- 2 - l m * m \
whence 2 m ( l)'"x[l, h 2m 2 ] = ( l) ft or 0, according as A is or is not of the
form 2m 2 + m. Similarly, since f' +1 (q) =f'(q) x /(<?)> we nn d, from equation
I- (6), 2 m [s + l, -2m 2 -m] = Z m (>, A-4m 2 -2m],
by which the coefficients in the successive powers off(q) may be calculated.
Formulae, similar to these in their general character, exist for the other
modular functions.
6. The Formula for the Multiplication of Four Theta Functions.
We next give a formula which enables us to obtain the algebraical relations
connecting the four Theta functions, and the differential equations satisfied by
them.
Let 2s = x 1 + x 2 + x a + x t , 2o- = Mi + /* 2 + M 3 + / u 4, 2v' = v 1 + v 2 + v s + v t ,
the integral numbers /*,, /u 2 , /* 3 , /u* and v 1} v 2 , v s , v t being subject to the restriction
that er and </ are to be integral. Multiplying together the four Theta functions
\,r>r), r= 1,2, 3,4,
and transforming, in the general term of the product, the indices of e <ira> , of e ivx ,
and of e ri by the identities
where
we find
Art. 7.]
THE ELLIPTIC FUNCTIONS OF THE FIRST SPECIES.
425
Giving in this formula to the symbols
Ml, M2> t*3> ^4
V-i, "2> "3. "4 [
first, the values
secondly, the values
0, 0, 0, 0,
0, 0, 0, 0,
X lt X z , X z , X t
1, 1, 1, 1,
0, 0, 0, 0,
and adding the results, we obtain the equation
This is the fundamental formula of Jacobi's Lectures (see Rosenhain,
' Mdmoires des Savants Strangers,' vol. xi. p. 361). Owing to the relations
(Art. 1, ix) which connect the Theta functions of different indices, it is no less
general than the equation (i) from which it is derived ; it is, however, less
easily manipulated.
Of the conclusions which may be derived from the formula (i), we shall in
this place mention only those which serve to establish the elementary properties
of the elliptic functions.
7. The Elliptic Functions of the First Species.
Attributing successively to the symbols of the scheme (ii) Art. 6, .the values
we find
o,
o,
o,
o,
o,
o,
o,
o,
o,
1,
1,
o,
1,
1,
o,
X,
X,
o,
o,
o,
o,
1,
1,
o,
X,
X,
o,
(i)
(ii)
(iii)
VOL. II.
31
426
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 7.
or, using the notations of Art. 2,
')-
Again, attributing to the same symbols the values
o,
i,
o,
1
1,
i,
o,
-y,
/y 1 /)/
it/ ^r */ *
o,
o,
i,
o,
1
1,
o,
1,
-y,
z+y,
o,
o,
o,
o,
i,
o,
o,
1
x -y>
x + y,
o,
we obtain the three equations,
(iv)
(v)
(vi)
We divide each of these equations by y, and then cause y to
without limit ; attending to equation (xiii) Art. 3, and observing that
), (vii)
), (viii)
). (ix)
decrease
we deduce the three differential equations
d_ A (x)
/ v
If we write
IT a;
x
for x, and introduce the elliptic functions sin am x,
cosamx, Aama; (i.e. the sine, the cosine, and the Delta of the amplitude of x),
Art. 7.] THE ELLIPTIC FUNCTIONS OF THE FIRST SPECIES. 427
which are respectively defined by the equations
Q / TTX\
sinama;= -7=.- = -^, (xiii)
cos am x
(xv)
the differential equations (x), (xi), (xii) become
c^.sinama;
- -j - = cosamxAama;, ....... ( XV1 )
d. cos am x
- -5 - = smamxAama;, ....... (xvu)
j = 2 cosama;sinamx ....... (xvm)
ax
The equations (v) and (vi) give at the same time
os 2 ama;= 1, >
> (XIX )
A 2 ama; = l. f
The functions sinama;, cosama;, and Aama; are all doubly periodic (Art. 1),
their periods being 4 K, 2iK ; 2(K + iK'), 2(K- 1 JT) ; 2K, tiK. The zeros of the
three functions are respectively (see Art. 3, vii)
x= 2rK+2siK', )
x = (2r + l)K + 2siK', ....... (xx)
x = (2r+l)K+(2s + l)iK'.}
The infinite points are the same for all three, viz.
Lastly, in the formula (i) of Art. 6, we successively attribute to the symbols
(ii) the two sets of value
0, 0, 0,
1, 1, 1, 1
x-y, x + y, 0,
31 2
428 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 7.
1, 1, 1, 1
1, 1, 1, 1
x-y, x + y, 0,
adding the results, we find
$ (x-y)$ (x + y)%(0) = $t(x)$t(! / )-%(x)%(y) (xxi)
Dividing by the terms of this equation the terms of the equations (vii),
(viii), (ix), and introducing the elliptic functions, we obtain the formulae
of addition
. sin am a; cos am T/ A am w + sin am w cos am cc Aamx , ...
sin am ( + ?/) = 79 . : ^- ; -, . (xxii)
1 - k 2 sin 2 am x sm 2 am y
. . cosamiecosam y sin am a; sin am A am a; A am?/ , ....
cosam(as + w) = , . , . (xxm)
1 - k 2 sm 2 am x sin 2 am y
. . A am x A am y k 2 sin am x sin am y cos am x cos am y . . .
A am(z + ?/)=- -2- . -?-. (xxiv)
1 K? sin 2 am x sin^ am y
The quantity K x is termed the complement of x ; the amplitude of the
complement of x is the coamplitude of x, and is written coam x. From Art. 2,
equations (v) we have
sin am K= 1, cos am K= 0, A am K Tt;
and hence . ... cosamcc
sin coam a; = sinam (K x) =
Ic sin am a;
coscoamo; = cosam(A-a;) = -7 ,
Aamx
A coam a; = Aam(^T x) =
A am a;
The theory of the elliptic functions may be treated in two different ways :
(1) We may begin, as we have done in this memoir, with the definition of
the Theta functions. We then define the elliptic functions by the equations
(xiii), (xiv), (xv), and we show, as has been shown here, that these three one-
valued functions are doubly periodic, and that they satisfy the differential
equations (xvi), (xvii), (xviii), the algebraical relations (xix), and the formulae
of addition (xxii), (xxiii), (xxiv).
(2) Or we may begin with the definition of the elliptic integral
du , ,
(xxv)
o\ /- 7 o tA * *
Art. 8.] THE COMPLETE ELLIPTIC INTEGRALS. 429
We then define the function u as the synectic integral which satisfies the
differential equation
g-<l -)(! -JM (xxvi)
and the initial conditions
du , ...
x = 0, u = 0, -j-= 1 (xxvn)
This definition implies the theorem that the equation (xxvi) always admits of
one, and only of one, synectic integral satisfying the initial conditions (xxvii).
For a demonstration of this theorem, we may refer to the work of MM. Briot and
Bouquet*. Assuming the theorem, we evidently have the equation
u = sin am x;
for sinamx is a synectic integral of the equation (xxvi), and satisfies the initial
conditions (xxvii). Hence also
y = cosamo;, w = Aamo;,
if v and w are two functions defined by the equations
v y = l-u 2 , W 2 = l-k 2 ii z ,
coupled with the initial conditions x = 0, v=l, w \.
In this way the identity of the functions obtained by considering the
differential equation (xxvi) with the functions sin am a, cos am x, Aamx, as
defined by the equations (xiii), (xiv), (xv), may be completely established.
That the functions u, v, w are doubly periodic, as well as synectic, can be
inferred directly from the differential equation (xxvi), without employing the
expressions of u, v, iv by means of the Theta functions ; the formulae of addition
can also be established in the same manner.
8. The Complete Elliptic Integrals.
We have already obtained the equations
sinam^T=l, cos am ^=0, Aam^T=^';
we have also (Art. 2, equations vii),
- cos am (K+ iK') = , A am (K+ iK') = 0.
* ' Theorie des Fonctions Elliptiques,' par MM. Briot et Bouquet, ed. 2, Paris, 1875. See livre v.
chap. iii.
430 MEMOIB ON THE THETA AND OMEGA FUNCTIONS. [Art. 9.
Hence, if in the equation
' du
x
-J
Jo
-)(! -**)'
we put successively x = K, x = K+ iK', we find that K is one of the values of
the definite integral
r
Jo
*
and that iK is one of the values of the integral
i
du
'
which, by either of the substitutions
M = (1-*V)-*, u = (l-k'*y% ....... (iii)
is changed into . r 1 du
We thus have the two equations
^ T 1 du , r l du .. .
K=l , K=l . . . . (iv)
-A) /l-w'l-W Jo l-u*l-k'*u*
in which, however, the track of the integration has not been determined. With
this determination we shall occupy ourselves hereafter ; for the present we
observe that, when w is a pure imaginary, k' 2 and k' 2 are real, positive, and less
than unity ; the quantities K and K' are also real and positive. In this case,
therefore, the integrals in the equations (iv) are the rectilineal integrals obtained
by causing u to pass from to 1 through a series of real values, the initial value
of the radical being in each case + 1.
9. The Partial Differential Equation of the Theta Functions.
The Theta functions satisfy the partial differential equation
d^" '^~dx*' ^
which enables us to express their differential coefficients, taken with respect to
<o, by means of their differential coefficients of an even order taken with respect
to x. Thus, from equation (iii) Art. 2, we find
" = ~T1? <\ /rA ' ~V ~ 0. /\' il //YV (U)
Art. 10.] THE ELLIPTIC FUNCTION OF THE SECOND SPECIES. 431
10. The Elliptic Function of the Second Species.
If we differentiate the equation (xxi) of Art. 7 twice with regard to y, and
put y = in the result, we find
TT / y
or, writing ^ for x, and attending to the equations (ii) of Art. 9, (xiii) of
2Jx
Art. 3, and (xiii) of Art. 7,
J TT d
-~ -^-^ -j-
K 2K dx
(ii)
Integrating between the limits and x, we obtain
* J
l k 2 sm 2 3nn.xdx = -^.x - -^ , (iii)
JO -& 4 K Q /TTX\
f^n I f^ r-r
where the right-hand member is Jacobi's expression* for the elliptic function
of the second species defined by the equation
C x
Z(x)= I k* sin 2 am x dx ......... (iv)
/o
The function Z(x) is a one-valued function of x, whatever be the value,
real or complex, which we assign to x, and whatever be the course of the
integration from the lower to the upper limit in the equation (iv). This
may be inferred from the theory of definite integration, since the residue of
sin 2 am x, corresponding to any one of its infinite points, is zero ; or it may
be proved by considering the simultaneous equations
, , r u k-u 2 du r " du
Z(x) = / . x = I (v)
Jo l-u*l-k*u* Jo l-u*l-k*u 2
in each of which the initial value of the radical is + , and the integrations are
to follow any one and the same track.
J C x J
* [ Jacobi'e function Z (x) is equal to -= x I d? sin 2 am x dx, and therefore differs by the term x,
K. Jo ti.
as well as by a change of sign, from Z (x) as defined in the text. The Z (x) in the text, which is the
same as the Z (x) used by M. Hermite, differs only in sign from the Z(x) of "Weierstrass.]
432 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 10.
From equation (ix) Art. 1, we find
^o (i) = ; 5; (TT + ITTW) = - i$ ( + ) ;
hence, attributing to x in equation (iii) the values K and -K"+ iK' in succession,
we obtain
Z(K) = T k 2 sin 2 am x dx = /,
wj
,,
Z(K+iK') = l
Jo
The equations
..... (vii)
are often taken as the equations defining / and J' ; we have preferred the
definitions which are given in Art. 2, because they exhibit J(w) and J'(u>) more
directly as one- valued functions of <a.
A characteristic property of the elliptic function of the second species is
expressed by the equation
Z(x + 2mK+2niK') = Z(x) + 2mJ+2niJ' > .... (viii)
in which m and n are any positive or negative integral numbers. This equation
may be verified immediately by means of the equation (iii) ; or, again, its truth
may be inferred from the simultaneous equations (v).
Lastly, we may write
. . r l k 2 u*du ft k 2 u 2 du
J(w) = I -==== , iJ (&)) = / . , . (ix)
Jo i-*i-** * i- 1 -^* 2
the course of the integration being the same as in the integrals (i) and (ii)
of Art. 8.
The second of these equations, by the substitutions (iii) of Art. 8, is
changed into
T , T 1 Tfdu -,
J = / =^== = / ^ , du, ...... (x)
'
where the course of the integration is the same as in the second integral (iv)
of Art. 8. It will be noticed that we have also
v T k'*u*du , - .
= K-J=\ == du= I ^ . -du, .... (xi)
which are of the same form as the integrals (x).
Art. 11.] DIFFERENTIAL COEFFICIENTS OF THE OMEGA FUNCTIONS. 433
The symmetrical substitution
u z =
which results from combining the inverse of either of the two substitutions (iii)
of Art. 8 with the other, changes the integrals (x) into one another, and the
second integral (iv) of Art. 8 into itself.
11. The Differential Coefficients of the Omega Functions.
Differentiating twice the equations (ii) and (iii) of Art 7, putting x = in the
result, and substituting for 3i(0) from equation (xiii), Art. 3, we find
whence also, subtracting, and employing equation (i), Art. 7,
Attending to the partial differential equation of Art. 9, we obtain from
these equations the following :
= JL && 3 = _ A ^ JST 8 ; . (iv)
ITT aw iTT
7 * > _7 - . fv .u , j
aw ITT a u> lit a
which may be otherwise "written
d.k* = d.lc'* _ ^jpViKi.
doo dta iv
or again, employing the notation of Arts. 2 and 3,
Since, by equation (ix) of Art. 3,
the equation denning J (Art. 2, iii) may be written in the form
dXK 2
J : ti> J\. U .
aw itr
VOL. II. 3 K
434
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 11.
dk' .
Substituting in this equation for -j- its value from (iv) or (v), we find
(vii)
Again, differentiating twice the equation (i) of Art. 10, and putting x = in
the result, we have
but, by the partial differential equation of Art. 9,
- - ^<<>).= ^3"[4 5^9
hence the equation (viii) becomes
3o
or, finally,
dt
(ix)
In (v), (vii), (ix), we write for simplicity ca = ivr ; we find
, ~-
*). . (x)
These equations enable us to form without difficulty the differential
coefficients of the various Omega functions defined in Art. 3 ; the following
are useful formulae :
. (xi)
dr
^d.logkK
2 d^~
K(J-K)=-KE,
l d.log1f'K
k dr
Art. 12.] DIFFERENTIAL COEFFICIENTS OF THE OMEGA FUNCTIONS. 435
When w is a pure imaginary (in which case T is real and positive), the
Omega functions of Art. 3, as has been already said, are all real and positive.
Hence T is always positive, -'V 1 is always negative ; .so that, as T in-
creases from zero to infinity, ^() continually increases from to 1, and 0()
continually decreases from 1 to 0. On the same hypothesis, K continually
decreases from oo to|- TT, J from GO to 0, E' = J' from ^TT to 1; K' continually
increases from |TT to oo, and E = KJ from 1 to ^ir. The demonstration
of these assertions depends partly on the expressions for K, K', J, J', E, E'
as definite integrals (equations (iv), Art. 8 ; (x), (xi), (xii), Art. 10), partly
on the differential formulae (x) and (xi) of this article.
d n k 2 d n . K 2 d n . KJ
If n > 1, it follows from the formulae (x), that ' , ' n , and ,' n
u/T CkT \JjT
are respectively of the forms
2 n+ Wk'*K n+l V n _ 2 n + ' l K n + 2 ll, and 2 n+1 K n + 1 V n+l ;
where T_ 1} V n , and P n+1 denote integral and homogeneous functions of K
and J, of the orders n 1, n, and n + l, in which the coefficients are rational
and integral functions of k 2 , with integral numerical coefficients.
- , 2 - 1 d n P
More generally, if P = k a k'^K c , -^ x -j-^ is of the form K n x l n , the
numerical coefficients of the powers of & 2 in V n being, in general, fractional ;
444
these coefficients, however, are integral if -, j, - are all integral.
It will be observed that no new transcendent is obtained by differentiating
the Omega functions, since all their differential coefficients of any order can
be expressed in terms of A: 2 , K, and J.
12. The Differential Equations satisfied by the Omega Functions.
Eliminating dr from the last two of the equations (x), Art. 11, by means
of the first, we find
ClJ\ X f -f i n TT-I
whence by subtraction
J dE d(J-K)
_ _ _ ; ' _
~
_
dk dk
3 K 2
436 MEMOIR ON THE THETA AND OMESA FUNCTIONS. [Art. 12.
Eliminating in turn J and K from the equations (i), we have
which are the differential equations satisfied by K and J regarded as functions
ofk.
Observing that by the definitions of Art. 2,
, <a _, w _ % TT
and that by equation (v) of Art. 11,
da>
dk
we find from the equations (i)
dK' 1
Tar
These equations are of the same form as the equations (i), but contain
K', J', instead of K, J. The differential equations (iii) and (iv) are therefore
respectively satisfied by K' and J' ; and the expressions
CK+ C'K', CJ+ C'J'
are the complete solutions of those differential equations.
We have deduced this result from the definitions of K, K', J, J' as
one-valued functions of without making any use of the expressions of these
functions as definite integrals. A different demonstration may be obtained
by employing the definite integrals ; which, however, it must be remembered,
cannot be used to define the one- valued functions. Writing, for brevity, Aw for
k*u'*), let us designate by P and Q any two corresponding values
of the integrals
r l du
(i.e. any two values in which the track of the integration is the same). It
is not difficult to show that the equations (i), and hence also the equations
(iii) and (iv), are verified if we write P for K and Q for / ; for this purpose
Art. 12.] DIFFERENTIAL EQUATIONS. 437
we have only to differentiate the equations (vi) with regard to k, and to make
use of the identities
-
~^""d~u\ Aw
l-k 2 u 2 d /M-
du\ Au
The equations (iii) and (iv) are therefore satisfied by K and J, because these
are values of P and Q respectively; but so also are (2n + I)K+2iniK',
(2n + l)J+2miJ'; i.e. the equations (iii) and (iv) are respectively satisfied
by K' and J' as well as by K and J.
Since the integral expressions for J and J' become K E' and E
respectively, when k 2 is changed into k' 2 , we find that the functions E and
K' E' satisfy the differential equation
or
. cPE l-k 2 dE
and that the complete solution of this equation is CE+ C'(K' E').
If V is any algebraical function of k, K, and J (i. e. a function defined by
an algebraical equation of which the coefficients are rational functions of k, K,
and /), V satisfies a differential equation of the second order containing only
V, k, and the first two differential coefficients of V with regard to k, and a
differential equation of the third order containing only V and its first three
differential coefficients with regard to w or T. The differential equations
satisfied by K and J are instances of the former assertion ; as instances of
the latter we may take the differential equations satisfied by K and k regarded
as functions of r *. These differential equations, which have been given by
Jacobi, are as follows (we write c for -p, and y for k, and we represent by
accents the differential coefficients of c and y with regard to T) :
(l+cV / ); ....... (x)
i. t S
* Jacobi, ' Ueber die Differentialgleichung, welcher die Reihen 1 + 2 q + 2^ 2j 9 + . . . , 2 q* + 2 j* + 2q ' + . . . ,
Geniige leiaten :' Crelle's Journal, vol. xxxvi. p. 97.
438 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 12.
= 0. ....... (xi)
To obtain the equation (x) we deduce from (x) or (xi), Art. 11, the equation
and we eliminate k 2 between this equation and its first derived equation with
regard to T, viz.
C 4 (cc'" + 3c'c") = 16 W 2 (1-2 fc).
/2K
If we put z = 4/ 7r A/ ' anc ^ introduce <a instead of T as the independent
variable, the equation (x) becomes
To verify the equation (xi) we write it in the form
d 2 r, dii~\ r d , dv~\ z /l + i/-\ 2 dy 2
2 7, log-, -j-log-f- + ( , ) -7^ = 0, .... (xin)
ar L arJ Lar clrJ ^y~y ' ( * T
and we substitute in it the values of the first two differential coefficients of
log -7^, deduced from the equations (x) and (xi) of Art. 11, viz.
d_
dr
dr*
V27?
, y=k, are, of course, only particular integrals
TT
of the equations (xii) and (xiii). We proceed to give the general integrals of
these equations, which have been assigned by Jacobi ; the form of the solution
(as we shall see later) is suggested by the theory of the transformation of
the Theta and Omega functions.
Let <= j-^, the quantities d, 6, c, d being supposed real, and adbc = n
being positive, so that the real part of ifi is negative when the real part
of i<o is negative, and vice versd.
Art. 12.] DIFFERENTIAL EQUATIONS. 439
Keplacing to by Q, we find from the equations (v), (vii), (ix) of Art. 11 :
7T
dQ
.... (xiv)
Let
observing that -7 = ( - =-) , we find that the equations (xiv) become
" w v '
= $(Q) x * (Q) x ^1(0),
7T
.... (xv)
Hence any differential equation, derived by differentiation and elimination
from the equations (x) and (xi) of Art. 11, will subsist unchanged if we
write simultaneously
c aa>
a>\
- ) for
ca/
f
for
and
,
x J
aa>
for
When dia is the equicrescent differential, as it is in the equations (xii)
and (xi), this observation enables us to assign the general solution of the
differential equation when a particular solution is known. Thus the particular
solution of the equation (xii), which we have found, is
440 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 13.
If then we put
<* . & C d . y + Sta . Q 1
= = S, -^ - T j8, := = - 7, --= = a, so that Q = *- =-, a<5 -,87 = 1,
the general solution of the equation (xii), containing three arbitrary constants, is
z= y* (l + 2e f ' + 2e* fa + ...) = 4/7x
Va-t-/3o>
Again, the equation (xi) or (xiii) remains unchanged when we take d u> as the
equicrescent differential instead of dr ; hence a particular solution of that
equation being y = ^/^ (w), its general solution is
13. 7%e Abelian Functions.
Integrating the equation (iii) of Art. 10, between the limits and x, and
taking the exponential function of each member, we find
in which the value of the right-hand side is independent of the track of the
integration, because the different values of which / Z (x)dx is susceptible differ
JO
only by multiples of 2 IT. The expression for ^o(^p) a t which we have thus
arrived, is that by which Jacobi originally defined the function 6(0;) =
in the ' Fundamenta Nova ' ; the importance of this expression in the theory
will be seen from the following observations.
The quantity to enters into the right-hand side of the equation (i) only
in k 2 sin 2 am x ; the equation (xxv) of Art. 7 shows that x can be developed
in a series proceeding by powers of u = sin am x, of which the coefficients are
rational and integral functions of k* ; again, this series gives by reversion a
development of sinamx in a series proceeding by powers of x, and having
coefficients which are, in like manner, rational and integral functions
Art. 13.] THE ABELIAN FUNCTIONS. 441
/ x
of k 2 . The same thing is therefore true of Z(x), of / Z(x)dx, and, lastly, of
Jo
e~^ 9 . But whereas the developments of sin am as, sin 2 am x, Z(x), and
/ Z(x) dx cannot be convergent for values of x of which the analytical modulus
'o
surpasses a certain limit, the function
gives rise to a development which is necessarily convergent for all finite values
of x, real and imaginary. For if, in the cosine-development of 5 (
^
(see equation (iv), Art. 3), we expand each cosine in a series proceeding by
powers of x, we obtain an equation which we may write in the form
= 1 -I ' ' ( VX ^ -I- - ^ ' f VX } 4-
^(U) h 1.23 fl (0)V2AV 1.2.3.4 3 (0) V2A/
IT" P T -9 i /TV/ r n ,^,.4,
_-,, ?W^_ y o H x * ,
h 1.2 ^[T] JL 1.2.3.4 T O [T] JT^
if we again put a> = ^TTT, and denote the series
by TQ[T]. The expansion (ii) is convergent for all finite values of x ; so is also
the expansion of e K ; and so, consequently, is the expansion of e ^ ',
which is obtained by multiplying the two together.
We shall now (after M. Weierstrass *) represent by Al (ic) the function
Q f VX \
M f \ -*& v2j:/
Mo(x}=e x
which appears in the left-hand member of the equation (i) ; we shall write
x*
* Weierstrass, ' Theorie der Abel'schen Functionen,' Crelle's Journal, vol. lii.
VOL. II. 3 L
442
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 13.
and we shall show by a direct method that the coefficients A\, A\, A\, ...,are
rational and integral functions of k 2 having integral numerical coefficients.
For this purpose we transform the partial differential equation (i) of Art. 9
into an equation involving A1 (x) and & 2 , instead of 3,, (x) and to. The partial
differential equation gives immediately
d-
dx*
where, on the left-hand side, we have included the partial differential coefficients
in brackets, in order to indicate that K is regarded as a constant in the
differentiation.
'TTX-
1 ft > I I
v, /TTX\ d /I
Substituting for
we find
"(S)
dr
its value
d-
"(3)
dr K
dx dr
dx z
Writing, in this equation, for
the equivalent expression
x e " x A1 (x),
effecting the differentiations, introducing (from the equations (x) and (xi) of
Art. 11) the values of j, -^X -- , -j- (-^), and dividing the resulting
equation by
IWK 4* rt
v ~^ xe x K >
we obtain the partial differential equation required, viz.
d.A\.(x)
ctx ct k>
This equation gives, for the successive calculation of the coefficients
Art. 13.] THE ABELIAN FUNCTIONS. 443
A\, Al, ...... the following equation of mixed differences and differentials
l- l = Q, .... (vi)
whence A\ = Q, A\= -1, A\ = -8(k 2 + k*), Ac.
The equation (vi) shows, (1) that the coefficients of the powers of Jc* in A^
are integral numbers ; (2) that, if n > 0, A" n contains no power of k 2 higher
than A; 2 *"" 1 ) and no term free from k 2 . Hence, if n>l, we may denote the
coefficient of A 2 ^* 1 * hi A lt+v + 2 by a ft where M^O, v^O ; and we then find
from the equation (vi),
M^ = 4 (/*+ 1 )^,,- 1 + 4^ + l)a Al _ 1 ,,-2( / u + v + l)(2M + 2 1 / + l)a /1 _ ]) ,_ 1 . '. . (vii)
This equation of partial differences supplies a formula of reversion by
which the values of the coefficients a^ v may be successively calculated. It
is symmetrical with regard to the two indices n and v ; and, since the value
of Al= 8(k 2 + k*) shows that 0)1 = 8 = a 1>0 , we have generally a^y = <*,,?.
This implies that the coefficients A* are reciprocal functions of k 2 of the
order n1, or, which is the same thing, that
5 > r ) = Alo(x, k).
The function A1 (x), of which we have now obtained the development
-,.... (viii)
"7rx\
is one of the four Abelian functions of M. Weierstrass. The three others are
defined by the equations
-
~*
K.
/
v
2kk'K'
TT
X
+
1.2.3.4.5.6 ' /2K
3 L 2
' /
444 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 13.
SO that
cosamo3
. Al 3 (a-) . .
> Aamx= .. ; : ; . . . . (x)
'
- r r7-r, Tr> ..
Al (o;) > Alo(aj)' Al (x)
and, if we use the notation of the Tables,
-If? 3 *'
A1 ' x - e x
Since, for values of a; which do not surpass a certain limit, the three elliptic
functions sin ama:, cos ama;, A ama; are all developable in series proceeding by
powers of x, of which the coefficients are rational and integral functions of k 2 ,
it is evident efc priori that the same thing must be true for the functions
Al^a;) = Al (o;) x sin am x,
Al 2 (a;) = Alo(o;) x cos ama;,
Al 3 (a;) = Alo(a;) x A ama; ;
while the equations (ix), by which we have defined these functions, show
that their developments must be convergent for all finite values of x. To
obtain these developments, we first deduce from the partial differential equa-
tion of Art. 9, by precisely the same process which we have indicated in
the case of A1 (a;), a partial differential equation for each of the three functions,
containing only k 2 and not . These equations are of the type
'f +0.xA].(s)-0. . . (xii)
where G t = k' 2 + k 2 a; 2 , G 2 = l + k*x-, G 3 = k 2 + k 2 x 2 . Treated in the same manner
as equation (v) they lead to the developments
where
a; 2 *" 1 - 2 "- 1 - 2 , ....
. . . (xin)
liV _ 1) i
Art. 14.] DIFFERENTIAL COEFFICIENTS. 445
so that b^ = &,,,, or the coefficients of the powers of x in Alj (x) are reciprocal
functions of k 2 , as in the case of Al (o;), the coefficients of the powers of x
in Al 2 (aj) and Al 3 (a;) being reciprocal to one another. We thus have the
equations
Alj, (kx, ) = M! (x, Jc) , A1 2 (lex, ) = Alg (x, k), A1 3 (kx, ^ ) = Al 2 (o;, ft).
14. Differential Coefficients of the Omega Functions expressed by means of
the Abelian Coefficients.
The differential coefficients of the four Omega functions
IWK IzWK /2kK /2lf
V~1T' V~^-' V^T' VT'
can be expressed in terms of the coefficients of the powers of x in the expansions
of the four Abelian functions.
We have
(equation (xi) Art. 11) ; also, by a theorem in Art. 11, we may write
where f n (-j^) is a rational and integral function of == and k 2 , with integral
numerical coefficients, and of the order n in -= The equations (ii), (iii), and
(iv) of Art. 13 now give
J\ a? . /J
, /\ x
1 \K) T
whence, on equating coefficients,
/'Gr) = Jf+ A " = W> since ^J = 0;
446 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 14.
and, in general,
(iii)
where (2r- 1) ! is the continued product of the uneven numbers 1, 3, 5, ... 2r- 1,
and C' is the coefficient of of in the expansion of (1 + a;)'.
If in the right-hand side of the formula (iii) we take successively (instead
of Al, Al, ... ) the coefficients A", A?, ... A", A' 2 ", ... in the expansions of Al 2 (z)
and Al 3 (z), we obtain expressions for
K* T 2 [rJ
where T 2 [T] = 5 2 (0, ?)> an d T 3 [T] = 3 3 (0, q).
The expansion of the uneven function Alj (x) leads to a slightly different
formula. Representing x-^. m / \ where TI(T) = x A/- -=^(0,5), by
&J\. J.I\T) TT 7T
we obtain by the same process as before
It will be seen that ( - ) n A n is the term not containing -^ in / (^\ Thus,
without using the partial differential equation (v) of Art. 13, we may obtain
the equation (vi) of that article by differentiating the equation
and putting -^ = in the result. A similar remark applies to the coefficients
A' A" A '"
fl -a> f n > "
Art. 16.] ARITHMETICAL THEOEY OF BINABY MATRICES. 447
Arts. 15-23. ARITHMETICAL THEOR OF BINARY MATRICES.
The theory of the Transformation of the Theta functions and Omega
functions depends in great measure upon the arithmetical theory of binary
matrices, of which the constituents are integral numbers. We shall therefore
give in this place an account of such properties of these matrices as we
require for our present purpose. We omit many of the demonstrations, on
account of the elementary nature of the subject.
15. Composition of Matrices.
If
\A'\ =
the matrix
a', V
c, d'
, and \A"\ =
// T //
a , o
// -jrr
c , d
,A\ =
a'a," + Vc", a'b" + b'd"
ca" + d'c", c'V' + d'd"
is said to be compounded of the matrices | A \ and | A' \ ; and this composition
is expressed by the equation
\A\ = \A'\x\A"\.
The determinant of the compounded matrix is the product of the determinants
of the component matrices.
It will be observed that the order of composition is not indifferent ; or,
which is the same thing, the matrices | A" \ x | A \ and | A \ x \ A" \ are not,
in general, identical. In the expression | A' \ x | A" \ the matrix | A \ is said
to be postmultiplied by the matrix | A' \ , and | A" \ is said to be premultiplied
by \A'\.
The composition of matrices differs from the multiplication of numbers
in the respect just mentioned. But in the composition of matrices the
components (as in the multiplication of numbers the factors) may be grouped
together in any way we please, provided that the order in which they succeed
one another be not changed ; for example, | A \ x \ B x | C \ is either | A \ x | B
postmultiplied by | C \ , or | A \ postmultiplied by B \ x | C \ .
16. Unit Matrices, Primitive Matrices, Reciprocal Matrices.
We shall have occasion to consider only those matrices of which the
determinants are positive numbers different from zero.
A matrix of which the determinant is + 1 is an unit matrix ; a matrix
of determinant n, of which the four constituents have no common divisor,
448
MEMOIR ON THE TIIETA AND OMEGA FUNCTIONS.
[Art. 16.
is a primitive matrix of determinant n. If n has no square divisors, every
matrix of determinant n is necessarily primitive.
The matrix compounded of two primitive matrices, of which the determinants
are relatively prime, is always a primitive matrix.
The matrices
a, b
c, d
and
d, -b
c, a
are said to be reciprocal matrices ; and
\A\x\B\x\B\-*x\A\-*-
the matrix reciprocal to the matrix | A \ is sometimes represented by the
symbol |-4| -1 . Reciprocal matrices have the same determinant. The result
of compounding two reciprocal matrices is the same in whatever order the
composition is effected ; in fact
n,
0, n
n being the common determinant of the two matrices.
The matrix reciprocal to | A \ x \ B \ is | B \~ l x | A \ ~ l ; for we have
ab,
0, ab
if a and 6 are the determinants of | A \ and | B \ .
Theorem. If | A \ and | B \ are given matrices, of which the determinants
are a and 6 respectively, the equations
\A\ = \X\x\B\ (i)
\A\ = \B\x\Y\ (ii)
are irresoluble when b is not a divisor of a, and may be either resoluble or
irresoluble when b is a divisor of a ; if either of them is resoluble, it admits
of only one solution.
For, postmultiplying (i) by (-6)" 1 , and premultiplying (ii) by the same
matrix, we obtain the equations
IJf-| _ z x 1 .4 I x \ JB\~ l
which completely determine the constituents of the matrices | X \ and | Y \ ;
but these constituents are not necessarily integral.
b
Cor. If i A \ =
we have
0,6
, the equations (i) and (ii) are both resoluble, and
Art. 18.] SYSTEMS OP NON-EQUIVALENT MATRICES. 449
17. Equivalence of Matrices.
If | A | and | B \ are any two matrices connected by the relation
|4| = |e|x|B| (i)
where | e | is an unit matrix, | A \ and | B \ are said to be equivalent by pre-
multiplication, or, for brevity, pre-equivalent ; if, again, | A \ and | B \ are
connected by the relation
Ml = ||x|e| (ii)
| A | and | B \ are said to be equivalent by postmultiplication, or, for brevity,
post-equivalent. The relation (i) may also be written
]5| = |e|-xM|; -
and the relation (ii) may be written
|| = MlxH-'-
Equivalent matrices have the same determinant ; and the greatest common
divisor of their constituents is the same. In any matrix the greatest common
divisor of any column is not altered by premultiplication with an unit matrix,
nor the greatest common divisor of any row by postmultiplication with an
unit matrix.
If two matrices are equivalent by premultiplication, their reciprocals are
equivalent by postmultiplication, and vice versa.
If two matrices are equivalent in the same way to a third matrix, they
are equivalent in that way to one another.
18. Systems of non-equivalent Matrices.
Let n be any positive integer, and a- (n) the sum of the divisors of n ;
every matrix of determinant n is equivalent by premultiplication to one, and
only one, of a system of a- (n) matrices.
Consider the system of matrices included in the formula
101- ff '
h,g '
where g,g is any pair of conjugate divisors of n (we may suppose these divisors
taken positively), and h is any one of the g numbers 0, 1, 2, ...,g\. The
number of the matrices | G \ is a (n) ; and it can be shown (1) that no two
of them are pre-equivalent, (2) that every matrix of determinant n is pre-
equivalent to one of them.
VOL. u. 3 M
450
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 18.
Similarly it may be proved that every matrix of determinant n is
equivalent by postmultiplication to one, and only one, of the <r (n) matrices
(71 =
<7',
h, 9
the signification of the letters g, g being the same as before.
We may regard all the matrices pre-equivalent, or post-equivalent, to
a given matrix as forming a class. For any determinant , and for either
kind of equivalence, the number of non-equivalent classes is a- (n) ; and as
representatives of these classes we may take any a- (n) matrices we please,
one out of each class. Thus the matrices | G \ form a representative system
of the classes of determinant n, non-equivalent by premultiplication ; so also,
if | a | and | /3 | are given unit matrices, do the systems of matrices
G\x\fl\,
x||x|/3|.
For each of these systems consists of <r (n) matrices of determinant n ; and
in the same system no two matrices are pre-equivalent. Similarly the system
\G'\, ||xl<?'|, Itf'h
are representative systems of the a- (n) classes of determinant n, non-equivalent
by postmultiplication. Any such system is termed a complete system of
matrices of the determinant n ; the special systems | G \ and | G' \ are said to
be reduced,
a, b
If
c, d
denote a complete system of matrices non-equivalent by pre-
multiplication, the reciprocal system
d, -b
is a complete system non-
equivalent by postmultiplication ; and vice versd.
The <r(n) classes are not all primitive, if n has square divisors. Let
p, p", p'" be the different primes of which the squares divide n ; the
number of primitive classes (in either classification) of matrices of deter-
minant n is
the sign of multiplication II extending to all the primes p which divide n.
Art. 19.] COMPOSITION OF SYSTEMS OF MATRICES. 451
19. Composition of Systems of Matrices.
Let | A | and B \ denote complete systems of matrices, non-equivalent by
premultiplication, of the determinants a and 6, supposed to be relatively prime.
Then the system | A \ x | B \ is a complete system of matrices of the determinant
ab. For (1) the number of matrices in this system is
o-(a) x a- (6) = o-(a6) ;
and (2) no two of the matrices in the system are pre-equivalent. For, if
possible, let
\A \ x \B\ = \a\ x 1.4,1 x {,{,
| a | being an unit matrix. Let
|| = |0|x|(7|, \B 1 \ = \^\x G,\,
| /8 1 and | ft | being unit matrices, | G \ and | G^ \ reduced matrices of deter-
minant 6. Postmultiplying each side of the equation
ftlx l^lxl^l-i.
by | 6 s j" 1 , we find
6,0
0,6
Hence the constituents of the matrix on the right-hand side are divisible
by 6. But the determinant of | a | X 1 -4j | X I /3 1 is a, which is prime to 6 ;
therefore the constituents of | G^\ x | G\~ l are divisible by 6. It is found, on
trial, that this is impossible unless | G \ and | G l \ are identical ; hence, finally,
| B \, \Bi\ and therefore also \A\, \A^\ are identical; which is contrary to
the hypothesis.
We may add that if | A \ and | B \ denote complete systems of primitive
matrices for the determinants a and 6, | A \ x | B \ will denote a complete system
of primitive matrices for the determinant ab.
It follows from the preceding theorem that ifa=PxQxRx ... , where
P, Q, E, ..., are powers of different primes, and if
in \Q\, \R\, ...
denote complete systems of matrices (or complete systems of primitive matrices)
for the determinants P, Q, R, ..., respectively,
I p | v I Q I v I ff I
I * I ^ I \ " \ * * !
(where the matrices are to be compounded in any definite order) will denote
3 M 2
452 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 20.
a complete system of matrices (or a complete system of primitive matrices)
for the determinant a = P x Q x R . . .
When the two determinants a and b are not relatively prune, it will suffice
for our purpose to consider the case in which one of them is the power p*
of a prime and the other is that prime p itself. ' We shall attend only to
primitive matrices.
First, let /* > 1, and let | p* \ and | p \ denote the reduced systems of
primitive matrices of the determinants yP and p, so that \p>* \ consists of
2)> t ~ 1 (l+p) matrices, and \p\ of p + 1. It will be found that the system
I P 1 * I x I P I contains (i) the primitive matrices (each occurring once) of determinant
2)' l+1 ; (ii) the matrices (each occurring p times) which are derived from the
primitive matrices of determinant p 11 ' 1 by multiplying each constituent by p.
In all therefore the system ||>''|x|_p| contains
matrices, as it ought to do.
Secondly, let M = 1 : the system | p \ x | p \ consists (i) of the primitive
matrices | p 2 \ each once repeated ; (ii) of p + 1 matrices of determinant p 2 having
their coefficients divisible by p : in all it contains p(p + l)+p + l = (p + l) 2
matrices.
For brevity we have in this article considered systems of matrices non-
equivalent by premultiplication ; but the theorems which we have enunciated
hold equally for systems non-equivalent by postmultiplication.
20. Reduction of any two Primitive Matrices of the same Determinant to
one another.
If | AI | and | A 2 \ are two primitive matrices of the same determinant n,
we can always find two unit-matrices | a | and | /3 | such that
To establish this it is sufficient to show that we can always satisfy
the equation
\A\-\ H>
with two unit matrices | a | and | j8 | .
Let
a, b
c, d
, where ad bc = n, and a, b, c, d have no common divisor.
Art. 21.] THE SIX TYPES OF UNEVEN MATRICES. 453
The simultaneous congruences
always admit of solution with relatively prime values of and n (this may
be seen by resolving n into a product of powers of different primes, and
considering the congruences with regard to each factor separately). Let
so that A/UJ \/u. = 1 ; we find
or
1^1 x
\A\ =
n\, \
w,
0,1
X.X,
n,
0,1
21. The six types of Matrices of an uneven Determinant.
Every matrix of uneven determinant is, with regard to the modulus 2,
of one of the six types
1,0
0,1
0,1
-1,0
1,0
1,1
1,1
0,1
1,1
-1,0
0, 1
-1. -1
- (i)
which we shall represent by the symbols
1, ^, <r, T, p, ,o 2 ;
giving rise to the system of congruences (mod 2) :
xp=<r 2 =r 2 =1; p 3 =1;
<rr=T^f =-^f(T=p; TO- =\Jr =ir^r=
^ = oy> 2 = T(0 = p(r = p 2 r = TOT = crra- ;
T = v'p = crp = p\^ = j0 2 cr =
This system is symmetrical with regard to -^r, <r, T and p, p" ; viz. it is
not altered by a cyclic permutation of >//-, <r, T, nor again by a simultaneous
interchange of p, p 2 and of any two of the three \^, a-, r.
The relations between the six anhannonic functions
, 1 -x,
x
X
l-x'
x-1
X '
454
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 21.
are identical in form with the relations expressed by the foregoing congruences ;
for, representing these functions, taken in order, by
x, ^(x), <r(x), r(x), P (x), p*(x),
we have the equations
>p (x) = <r* (x) = r 2 (a;) = x, p a (x) = x,
&c.
&c.
&c.
If we represent the six unit matrices which we have chosen as representatives
of the six types, by the symbols
|i|, m H, |T|, \ P \, |/.|. ....... (ii)
we find
-|i|, M' = -P. \r\* = Q, \ P \ 3 =-\i\,
where we write 1 1 1 , P, and Q for
-1,
0, -1
1,0
2,1
, and
1,2
0,1
We have also
The powers of | T | and | <r \ are all different ; but the matrices +
are square roots of the unit matrix 1 1 1 ; and | p \ and | p 2 1 are cube
roots of the same matrix. Thus each of the equations
admits of as many solutions as it has dimensions. No other equation
included in the formula |JC|"=11| admits of any solution other than
1*1= H.
The following congruences are satisfied by the constituents of unit matrices
a, b
c, d
of the six types respectively; the modulus is 16 throughout.
Art. 22.]
PRIMARY MATRICES.
455
(1) (a-d)(a + db)=0, (a-d)(a + dc)
(b + c)(b ca} = 0, (b + c)(b cd).
(T) (a + d)(a-dc)=0,
. . (iv)
22. Primary Matrices and Primary Equivalence.
A matrix of uneven determinant is said to be primary when it is of type
(1) and satisfies the congruence a = l, mod 4. Thus
matrix of determinant 27 ;
number, the matrices
-3,2
-8, 5
-7, 4
2, -5
is a primary
is a primary unit matrix ; if n is any uneven
G\= (-
(the signification of the symbols g, g', h being the same as in Art. 18) form
a complete system of primary matrices non-equivalent by premultiplication.
Similarly the matrices
|0>
form a complete system of primary matrices non-equivalent by postmultiplication.
If | A | be any matrix of the uneven determinant n, and of the type (1), it
is evident (1) that either \A \ or \A\ is primary, (2) that, if n = l, mod 4,
| A | and \A \~ l are either both primary or else both not primary, but that if
n = 3, mod 4, one of these matrices is primary and the other not.
Theorem, I. " Every unit matrix can be represented in one way, and
in one way only, in the form
r> X | | X |a|,
where 17 = +1, | | is one of the six typical unit matrices, and | a | is a primary
unit matrix ; similarly every unit matrix can be represented in one way,
and in one way only, in the form
Theorem II. " Every primary unit matrix | A \ can be represented in one
way, and in one way only, in the form
(i)
456 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 22.
where X,, X,, ...,X 2 , are positive or negative integers, of which the first and
the last may be zero."
For, in the formula (i), if \ > 0, the two constituents of the lower row of
| A | are respectively greater in absolute magnitude than the corresponding
constituents of the upper row; if X^O, the contrary is the case. Hence
the representation is possible in one way only, because an equation of the form
implies the equations
AI = /*I. A 2 = Mj, ^8 = Ms-
Again, if in | A \ the lower constituents are the greater, we can always
determine an integer X,, such that in P~ x ix \A \, which has the same upper
constituents as | A \, the lower constituents shall be less than the upper ;
we can then determine an integer X 2 such that in Q~ x x P~ x i x | A \, which
has the same lower constituents as P~*ix\A\, the upper constituents shall
be less than the lower ; by proceeding in this way, we shall finally arrive
at an equation of the form
... P-* S x Q-\ x P-*I x | A | - + [ 1 1,
which coincides with the formula (i), and gives the required representation.
Employing the symbols (q^ q 2 , ..., #) and [q 1} q 2 , ..., q n ] to represent
respectively the numerator of the continued fraction
1
2,t +
and the continued fraction itself, we find
... (2X 2 , ..., 2X 2| _!), (2X2, ..., 2X 2| )
" (2X,, ..., 2X 2 ,_ 1 ), (2X 1 ,...,2X S .)
and we may write the equation = | A \ x& (see Art. 24) in the form
a,
If the equations
2X 2 , ..., 2X 28 ,
xB
the unit | e | is primary, the equivalence of the matrices | A \ and | B \ is said
to be primary.
Art. 23.] PRIMARY MATRICES. 457
Theorem III. "If | F | represents a complete system of matrices (or a
complete system of primitive matrices) of determinant n, non-equivalent by
premultiplication, the formula
ixm'xiri .......... (ii)
represents a complete system of matrices (or of primitive matrices), non-
equivalent by primary premultiplication."
For (1) if \Z\ be any given matrix of determinant n, let \Z\ A x | r\ |,
| A | being an unit matrix, and 1 1\ being one of the matrices F. Let
| .4 | 9 X | a | x | | , where | a | is primary ; then
Again, (2) if /i x | x | x 1 1\ | is primarily equivalent to / 2 x | 2 1 x | F 2
we must have, first of all, | Fj | = | F 2 1 , or else the two matrices could not be
equivalent at all ; then the equation
~
in which | /3 | is a primary unit matrix, gives by postmultiplication with |
ii x | 1 | = it x I ft | x | 2 | ,
whence finally | | = 1, n\ = 12, 1 = 2 ; * e - two matrices included in the formula
(ii) cannot be primarily equivalent without being identical.
Similarly the formula
ix|r|xK| f .......... (Hi)
where 1 1" | represents a complete system of (primitive) matrices of determinant
n, non-equivalent by postmultiplication, represents a complete system of
(primitive) matrices of the same determinant non-equivalent by primary
postmultiplication.
If N be the number of matrices (see Art. 18) in the systems F and F',
12 N is the number of matrices in each of the systems (ii) and (iii).
When n is an uneven number these 12 N matrices are equally distributed
between the six types ; i. e. there are 2 N matrices, or rather N pairs of
matrices, the two of each pair differing only in sign, which appertain to each
of the six types. This may be seen by taking for F or F''the primary systems
G or G' of Art. 18, since each matrix of F or F' gives rise to a pair of opposite
matrices of each of the six types.
23. The nine types of Primitive Matrices of an Even Determinant.
The primitive matrices of any even determinant, considered with regard
VOL. IT. 3 N
458
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 23.
to the modulus 2, are of one or other of nine different types, which are
exhibited in the following scheme :
1,1
1,0
0,1
1,1
J
1,0
>
0,1
0,0
0,0
0,
1,1
1
1,0
>
0,1
1,1
1,0
0,1
0,0
)
0,0
,
0,0
and which we shall symbolize thus :
, 2,
1, 3,
^2, 1,
^2, 2,
^2,3)
3, 1
a
3, 2
J 3, 3
No two of these types are primarily equivalent, either by premultiplication
or by postmultiplication ; two types in the same row are absolutely non-
equivalent by premultiplication ; two types in the same column are absolutely
non-equivalent by postmultiplication. This may be seen (Art. 17) by observing
that, in any matrix, the greatest common divisor of a column is not altered by
premultiplication with an unit, nor the greatest common divisor of a row by
postmultiplication with an unit. The types C ltl , C 2t2 , C 3j 3 are their own reci-
procals ; C Tt , is the reciprocal of C i>r . The types (C) are unchanged by pre-
multiplication with the corresponding unit-types in the scheme
and by postmultiplication with the unit-types in the scheme
Again, the type matrices which lie in the same column of (C) are
equivalent by premultiplication with the corresponding unit-types in either
of the schemes
1, T, <T
!' P> P 2
T, 1, ^
,
P*, 1, P
ff , ^, 1
P> P*> 1
NINE TYPES OF EVEN MATBICES.
459
Art. 23.]
and the type matrices which lie in the same row of (C) are equivalent by
postmultiplication with the same unit-types ; thus, for example,
T X C 2) ! = (7 2| 2 = ^ X C 2t 3,
Ci, i = C 2t ! x P = C 3> 1 x p\
If we select any one of the types (C), for example C r> s , and denote by
1; 2 any two of the six unit-types, the expression j^ x C r> , x 2 (which has
36 = 9x4 different values) represents the nine types (C) indifferently, i.e. each
of them four times.
If n be any even number, and N the number of primitive matrices
non-equivalent by premultiplication, the 12 N matrices non-equivalent by
primary premultiplication are equally distributed between the nine types (C).
To establish this, we first consider the case in which n ^ is a power of 2.
The primitive matrices G of determinant 2^ (see Art. 18) divide themselves
into three groups :
(i)
h an uneven remainder of 2? ;
(ii) .
2",
h, I
&->,
h, 2'
< s <
case h
and h an uneven remainder of 2 * >, except when s = n, in which
,....
< 1U >
2",
h, I
h an even remainder of 2* 1 .
Each of these groups contains 2**" 1 matrices; hence the N=3x2' t ~ l
matrices of G are equally distributed between the three types C lt 2 , (7 2 2 , C 3
1,
2> 2 , 3>2
except only that the matrix
0,
Next
is of the type C 2 , s instead of C 2i 2 .
v,
let ?i = 2''xm, m being uneven, and let \M\ represent the general term of
a system of primitive matrices of determinant m, of type (1), and non-equi-
valent by premultiplication. The matrices |Jf|xJ<?| are a complete system
of primitive matrices of determinant 2^xm; these matrices are equally dis-
tributed between the types of the first, second, and third columns of (C),
because |jlf|x|6r| and | G \ appertain to types occupying the same column
of (C). Lastly, to obtain a system of primitive matrices, non-equivalent by
3 N 2
460 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 24.
primary premultiplication, we have to premultiply each of the matrices
|Jf|x|(T| by the twelve unit matrices ?x||; and it will be found, by
means of the properties of the type-system ((7), that four of these units do
not alter the type of |Jf|x|6r , and that this type is changed into each
of the other types which occupy the same column by four of the remaining
eight. Hence the
12 x 3 x 2f- 1 x mil (l +-)
matrices contained in any system of primitive matrices of determinant
71 = 2** x m, non-equivalent by primary premultiplication, consist of nine groups,
each containing
2>'xmxll('l+-)
V p/
pairs of opposite matrices, and appertaining respectively to the nine types ((7).
There is a corresponding theorem for primary equivalence by post-
multiplication.
Arts. 24-33. THE TRANSFORMATION OF THE THETA AND OMEGA FUNCTIONS.
24. Enunciation of the Problem of Transformation.
Let to and Q be two complex quantities connected by the relation
<a =
the coefficients a, b, c, d being integral numbers relatively prime, and the
determinant ad be = A being positive ; so that, as we have already observed
(Art. 12), the real part of til is negative when the real part of ica is negative,
and vice versd. We shall frequently write the equation (i) in the symbolic form
a, b
c, d
or in the equivalent form
xQ, (ii)
a, b
c, d
d, -b
c, a
(iii)
The problem of Transformation is "to express the Theta and Omega
functions containing Q by means of the Theta and Omega functions containing
o> ; " and the general nature of the solution of which this problem is susceptible
is indicated by the theorem :
Art. 25.] THE PROBLEM OF TRANSFORMATION 461
" Any Theta function of the arguments (a + bQ) -j- and Q can be expressed
in the form
where, if A > 1, T is a homogeneous function of the order A of two of the
Theta functions of the arguments -5- and ; and if A = 1, T is a multiple,
A
by a coefficient not containing x, of one of the Theta functions of the argu-
, j r
ments -y- and .
h
The demonstration of this theorem and the determination of the forms
of the functions T may be obtained by a method which is due to M. Hermite,
and which we proceed to explain.
25. General Solution of the Problem of Transformation. Method of
M. Hermite.
Lemma. " Let the values of the complex variable x be represented in
the usual manner by the points of a plane ; and let F(x) be a function of x,
synectic throughout the whole plane, and having the period I, so that
F (x) can always, and in one way only, be expressed by an exponential series
convergent for every finite value of x, of the form
f oo Zmiirx
Z4. r- (i)
00
2 minx
This is, in fact, the theorem of Laurent; for, if z = e l , F(x) is a function
of z, synectic throughout the whole plane upon which z is represented, except
at the point z = ; viz. the values of x corresponding to any given value of z
are all included in the formula x + si, where s is an indeterminate integer. But
F(x + sl) = F(x) ; i.e. F(x) is a one- valued function of z, and is finite and
continuous when z is finite and different from zero. Hence, by the theorem
of Laurent, F (x) is developable, and in one way only, in a series proceeding
by ascending and descending powers of z, convergent for all values of z
which lie between two circles having their common centre at the point z = 0,
the radius of the inner circle being as small and that of the outer circle as
462 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 25.
great as we please ; i. e. F(x) is developable, and in one way only, in a
series of the form (i), convergent for all finite values of x.
Let us now represent by F(x) a function of x, synectic throughout the
whole plane, and satisfying the two equations
........... (ii)
+ ^xF(x-), ...... (iii)
m and n denoting any two integers (which, however, we may suppose, without
loss of generality, to be either or 1), and the symbols h, to, A retaining the
meanings attributed to them in Art. 24. We proceed to show that by the
equations (ii) and (iii) F(x) is determined as a function of x containing A
arbitrary constants, which enter linearly into its expression. From the Lemma
we infer that, to satisfy the condition (ii), we must have
vi \ A
F(x)= 2 A e e * ; ........ (iv)
8= 00
and again that, to satisfy the condition (iii), we must have
-* i " a , ........ (v)
Let j be one of the numbers 0, 1, 2, ..., A 1, and let j' be any number
included in the formula rA+j, where r is an indeterminate integer ; we find
from (v)
~
where aj = e 4A
If then for brevity we write
we obtain, finally, the equation
1 Z A _ 1 (z) ; ..... (vi)
which shows that any function, synectic throughout the whole plane, and
satisfying the equations (ii) and (iii), can be expressed as a linear function
of the A functions
Art. 25.] THE PROBLEM OP TRANSFORMATION. 463
From this result we infer that if F , F lt ..., F^_ l are any A synectic functions
of x, satisfying the equations (ii) and (iii), and independent of one another
(i. e. not satisfying any linear homogeneous relation of the type
C F + C l F l +C t F,+ ... +a_ 1J F A _ 1 = 0,
where (7,, C" 2 , ..., do not contain x), all the functions F, synectic throughout
the whole plane, which satisfy those two equations, may be expressed as linear
and homogeneous functions with constant coefficients of F , F 19 F 2 , ...,F^_ l .
If we suppose that F(x), besides satisfying the equations (ii) and (iii), also
satisfies the equation
(i.e. if we suppose that F(x) is either an even or an uneven function), the
number of arbitrary constants in the general expression (vi) is reduced by about
one half. Substituting in the equation (vii) for F(x) the equivalent expression
(vi), and denoting by j jt j 2 , two of the numbers j (the same or different), we
find that if
ji+ja + m = Q, mod A, ........ (viii)
we have necessarily
a, 1 = (-)"^'a j . 2 , ......... (ix)
where
> 2 +,?***)
We now distinguish between the cases in which A is uneven and in which A
is even, (i) Let A be uneven : the equation (ix) shows that the coefficients
+ ttj are equal in pairs one of them, of which the index j is determined by
the congruence
'2j + m = 0, mod A ,
pairing with itself. If nm + a- is uneven, the equation (ix) shows that this
coefficient is zero, and that there are only |(A 1) independent coefficients;
if nm + a- is even, the odd coefficient is not determined by the equation (ix),
and there are |(A + 1) independent coefficients. We shall presently see that
the case in which nm + a- is uneven does not present itself in the theory of
the transformation of the Theta functions, (ii) Let A be even. And (a) let
m=l be uneven: we then have j\ +j 2 + 1 = A, a-' = n, a J -, = ( l)" + <r aj, ! ; i.e. the
coefficients + a_,- are equal in pairs, and there are A independent coefficients.
(/3) Let ra = 0: of the coefficients &j, A 2 are equal in pairs, each of the
464 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 25.
two coefficients a and ai A pairing with itself; for these two coefficients the
equation (ix) becomes
a = (-) <r a , a iA = (
and thus there are ^A + l, or ^A, or ^A 1 independent coefficients, according
as <r and n + cr are both even, or one even and one uneven, or both uneven :
the last case does not present itself in the theory of transformation.
To apply the preceding theory to the Theta functions, we introduce an
auxiliary function II defined by the equation
r. ...... ( X )
where m and n are integral numbers defined by the equations
(^)
From the elementary properties of the Theta functions (see Art. 1, equa-
tions vi, vii, viii) it follows that the function IT verifies the equations
n m .,.. v v
(xii)
which, if we put <r = M", coincide in form with the equations (ii), (iii), and (vii).
We now distinguish between the cases in which the determinant is uneven,
and in which it is even.
I. The determinant is uneven. In this case the equations (xi) give rise to
the congruences
m-l=a(-l) + b(v-l) )
' .; [, mod 2 ; (xm)
n-l=c(fi-l) + d(v-l) }
from which we infer that if n, v are both uneven, m, n are both uneven, and
vice versd. Hence mn + n v = mn + a- is always even ; and every synectic function
which satisfies the equations (xii) can be linearly expressed by means of any
2 (A + 1) independent functions which satisfy that system. Let p represent any
one of the \ (A + 1) even numbers 0, 2, 4,... A - 1 ; and let 3 ~ , denote one of
Art. 25.]
TRANSFORMATIONS OF AN UNEVEN ORDER
465
the three Theta functions other than 3 m>n (-j- , wj ; the J(A + 1) functions
...(" "XT * [*(" ")]' .......
are independent, and satisfy the equations (xii). We have therefore the equation
a?
Q ftrX \ rr , N
xX, n ' ft ' ' ' ( XV )
where T is a homogeneous function of the order |(A 1) of the squares of $ m>
and 3 1 . This equation establishes the theorem of Art. 24, for the case in which
A is uneven.
The following Table gives, for each of the six types (Art. 21) to which
an uneven matrix may appertain, the values (mod 2) of m and n corresponding
to given values of M and v.
1
*
or
T
p
P 2
m =
M
V
M
H+v + 1
fJL+V+1
V
n =
V
M
H + V+l
V
M
H + v + 1
II. The determinant is even. In this case the theorem of Art. 24 may be
established in the same manner ; and the general character of the result is the
same, though there is a little more variety in its form. The number of functions
linearly independent of one another which satisfy the equations (xii) is always
either ^ A + 1 or \ A, but never \ A 1 ; for, since by hypothesis the matrix
a, 6
c, d
is primitive, we infer from the equations (xi) that we cannot have simul-
taneously m = n = 0, M = v = 1 ; i.e. these equations exclude the case in which
the solution of the system (xii) contains only ^A 1 arbitrary constants.
According as that system (xii) admits of ^ A + 1, or A linearly independent
solutions the expression for 3^ ([a + 6Q] -j-, n) is of the type
x A,
or
7rX
. . . (xvi)
. . . (xvii)
VOL. II.
30
/-
Of THE THETA AXD OMEGA flJECTIAHB. [Ait- 25
function* of the aqoazet of any two of the
-,r and *, and are rwpectnrety of the orders
and |A 1; 3 and 3* ate two Theta functions of which the indices
<* *, r and on the type C> , (see Art. 23) of the even matrix of the
The (oSknriag Tattf afaow how to determine, for any tranafimnation of an
even determinant, the form of the expression fix- 3 A
TABLE L
/A =
a,
fl + V+1
M
/
M
M
TABLE II
r
r
inv
m
71
^ ^
^a //(
5
1
1
3 S-
~^I, -^0, 1
i
/^
i
1
1
3 S-
"'I, 1 > ^O,
ii
1
3 %
~* I, > "^0,
iii
J^
1
1
3 . <x
"ft, 1 ' ~^0, 1
iv
/y
1
3 < ^-
^0, 1 > ^0,
V
yy
1
1
3 ^
"*li 1 ^1,
vi
.1
Art. 25.]
TRANSFORMATIONS OF AN EVEN ORDER.
TABLE III.
467
5.
^
C5u
CM
A
J
B,i
3i
B,ii
B, ii
B, ii
S 2
A
5,i
^4
S 3
B,i
4
J
Vl. 2
Q 2
c, 2
ft
Q
A
A
B,y
3i
B, vi
B, vi
, vi
ft
^4
#, v
^4
S 3
B,v
A
^
c,, 3
c^
c,,
ft
A
A
, iii
ft
B, iv
B, iv
5, iv
ft
%
A
B, iii
^4
ft
B, iii
4
A
Table I. gives the values, for the modulus 2, of the indices ra and n,
corresponding to given values of /x and v for each of the nine types of the matrix
of the transformation.
Table II. gives the form of the expression equivalent to
for each admissible combination of the values (mod 2) of the indices M x v, m,
and n. Thus, if/uxv = 0, m = n = l, mod 2, that expression is of the type (xvii) ;
and the Theta functions designated by 3 a and ,% are Sj, and 5 0> i . This
assertion is proved by observing that the ^ A functions included in the formula
(where p is any one of the \ A uneven numbers 1, 3, ..., A 1) are independent,
302
468
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 26.
and satisfy the equations (xii), if we attribute in those equations to n x v, m,
and n values satisfying the congruences /uxv = 0, m = n = l, mod 2.
Table III. refers throughout to Table II., and is founded on the Tables I.
and II. It gives the form of the expression for
answering to each of the four values of the index s, and to each of the nine
types of the matrix of transformation. Thus, if
i.e. if
a, 6
c, d
1,
1,
a, 6
c, d
be of the type
, mod 2, the expression of
is of the type (B, i) in Table II., so that we have the equation
\
, > x B,
where B is a homogeneous function of the order ^ A 1 of the squares of any
two of the Theta functions
~r>
26. The Multiplier.
Since to = i -^ , & = i W(Q\ *^ e equation (i) of Art. 24 may be written
in the form
or, which is the same thing,
1
T>X
= aK (Q) + b.iK'(Q),
x iK'( u ) = cK(Q) + d.iK'(Q).
a, 6
c, d
Either of these equations may be regarded as defining, for any given matrix
, the quantity M (which is termed the multiplier) as a one- valued function
Art. 27.]
COMPOSITION OF TBANSFOBMATIONS.
469
of to. We observe that if the sign of the matrix be changed, the sign of the
multiplier is also changed. If in the equations xv-xvii of Art. 25 we put
h = 2-fiT(ft>) = 2 K, and write for brevity A = K(0), these equations assume the form
(iii)
where T is a homogeneous function of order A of two of the Theta functions
TTX \
2K' /
We have supposed in Art. 24 that the transformation
a, b
c, d
is primitive ;
but, by introducing the multiplier, we can adapt the formulae to the case of a
non-primitive transformation. Let
= 9*
a, b
c, d
ga, gb
gc, gd
where
a, b
c, d
is primitive and g is any positive or negative integer ; and let
M, M l be the multipliers appertaining to the transformations
a, b
c, d
c lt d 1
From the definition of the multiplier we have MgM l ; and, hence, if we write
gx for x, the equation (iii) becomes
where 7\ is deduced from T by writing gx for x the indices of the Theta
functions which occur in the expression of Ji (equations xv-xvii, Art. 25) being
still determined by the equations (xi) of Art. 25, and not (at least when g is
even) by the equations
n =
27. Composition of Transformations.
Theorem I. "If the primitive matrix
of other matrices, so that
a, b
c, d
is compounded of any number
, b
c, d
t ,
72,
X
470
the transformation o> =
c, d
[Art. 27.
x Q results from the successive application of the
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
a, b
transformations
., P,
%, <*.
0) =
Thus we may first express the Theta functions containing Q by means
of the Theta functions containing ta,_ 1 , these by means of the Theta functions
containing w,_ 2 , and so on continually until we have expressed the Theta
functions containing in terms of those containing a>. Since the equation
a, b
c, d
x Q can be written in the form
*., -P.
x ... x
X w,
we can also by a converse process express the Theta functions of w in terms of
the Theta functions containing Q.
Theorem II. " If any number of transformations be compounded, the
multiplier appertaining to the resulting transformation is the product of the
multipliers appertaining to the components."
, Pi,
If, for example, w =
72,
x w 2 , and M lt M 2 , M are the
multipliers respectively appertaining to these transformations, and to the trans-
formation compounded of them, we have
and
whence, eliminating K(w^) and iK' (wj), we find
= (?!, + ^
and these equations imply that M=M l x.M 3 .
A + Wi
Art. 27.] COMPOSITION OF TRANSFOBMATIONS. 471
In the first theorem of this article we have expressly supposed the resultant
matrix (and consequently the component matrices) to be primitive. In the
second theorem the matrices compounded may be any whatever.
When we compound two primitive matrices, of which the determinants are
not relatively prime, the resultant matrix is not necessarily primitive. Let
1, ft
ga, gb
gc, gd
7z>
(i)
where g is some integral number different from unity ; and let A, A 1; A 2 be the
determinants of
a, 6
c, d
2>
The transformations
and
O) =
ge,gd
i, ft
xfi
2) Pi
72 > 4
(iii)
give different results ; viz. if M be the multiplier appertaining to the trans-
formation (0 =
a, b
c, d
x II, the transformation (ii) expresses ^ (*>M \ > ^) as a
homogeneous function of order A of the Theta functions 5('^ , w J ; but the
compound transformation (iii) expresses ,3Y ,,. , Qj as a homogeneous function
of the order ^ 2 A = Aj A 2 of the Theta functions $(~^, ) If, in particular,
i, Pi
and
72.
are reciprocal, we have g = \ = A 2 , A = 1,
a, b
c, d
1,0
0. 1
w = Q, A = K, M=I: the transformation (ii) becomes an identity; while the
transformation (iii) expresses 5 (jrjr , w) as a homogeneous function of order g 2
of the Theta functions 3 ( ^-^., <a\. We are thus led to the theorem of Jacobi :
" The composition of two reciprocal transformations of determinant g serves to
express the Theta functions of the argument gx by means of the Theta functions
of the argument x."
This theorem is, in fact, a corollary from the Theorem II., which shows
472 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 27.
that the product of the multipliers appertaining to two reciprocal transforma-
tions of determinant q is -
9
Since every matrix can be expressed as a product of matrices of which the de-
terminants are primes (Art. 19), it appears from the Theorem I. that in the theory
of transformation it is sufficient to consider transformations of a prime order.
Again, if we regard the theory of linear transformations (i. e. of trans-
formations of which the matrix is an unit), as known, we need only consider,
for any determinant A, o-'(A) different transformations. For (Art. 18) every
primitive transformation of determinant A is included in the formula
o, = | G' \ x | a | x O,
where | a | is an unit matrix and | G' \ is a system of o-' (A) primitive matrices
of determinant A, non-equivalent by postmultiplication. Or, again, we might
employ the formula
= ] a | X | G | X Q,
where | G \ is a system of o-'(A) primitive matrices non-equivalent by premultipli-
cation.
If
. is an unit matrix of type (1), the equations (xi) of Art. 25 show
that m = n, mod 2, n = v, mod 2; and we shall presently see that3f= +1 ; hence
by a linear transformation of type (1) the Theta functions are unchanged, and
only acquire an exponential factor. It is therefore convenient, instead of the
o-'(A) transformations | G\ or j G'\, to consider the 6o-'(A) pairs of transformations
included in either of the formulae
=| r 'l X | a | XO,
CD = a | x | r | x a,
where |a| is a primary unit, and |r|, |F'| are systems of primary matrices
non-equivalent by primary postmultiplication, and by primary premultiplication
respectively.
Lastly, since every primitive matrix of determinant A can be exhibited
in the form
! w
(Art. 20), it is possible, in the theory of the transformations of order A, to confine
Art. 28.] LINEAR TRANSFORMATION OF THE THETA FUNCTIONS. 473
ourselves to the consideration of the single matrix
A,0
, and to obtain the
0, 1
results relative to the other transformations of determinant A by compounding
this transformation with linear transformations.
28. Linear Transformation of the Theta Functions.
When A = 1, T is a constant, and the formula (xv) of Art. 25 becomes
a
<i f/ . T.r^\ 7rX n~\ /-i -iiri(a + &n)-j Q /TTX \ ...
3* ,[(<*+ &Q)-j-,Oj-Cx A x.V (--,*>) ... (i)
where
The determination of the constant C, which depends on u>, on the indices n
and v, and on the integral numbers a, b, c, d, has been effected by M. Hermite
in the following manner *. We shall at first suppose that b is different from
zero, and positive.
Putting h = l, we have from (i)
r= +00
,'mn y I \nr <
% & \ i e
r= oo
r= +00
r= oo
V
Denoting the right-hand member of this equation by i*" x S, multiplying
by e~ mivx dx, and integrating from x = to x = 1, we find
/.i
_ i>r-* e -}<*f x / e~ miirx Sdx.
JQ
Let
so that
S= 2 e
* Liouville'a Journal, 2nd series, vol. iii. p. 26.
VOL. II. 3 p
474 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 28.
we find immediately
f(x
whence we infer
/.I = &-! / +
/ <,-<**&! = 2 / e { '
8-0 ^ oo
By a formula due to Cauchy,
/ +
\ e i
'-
the coefficient of i in the imaginary part of P, and the real part of the radical
N/ - iP t being each of them positive and different from zero. Evaluating the
integrals
f e'^'tidx
J -00
by this formula, we obtain for C the expression
where
-(a/j 5 2nm + dm? aV)
Jp n fAV fnn \f p 4o
. . .(iii)
ai'c)
= I U<rf<J v gii
= 6_1 _/ i}.xi
e ~^ ( } , ............. (iv)
8 =
the real part of the radical ^/ - ib(a + 6 11) being positive.
To sum the series (iv) we employ the formulae of Gauss :
2 e " = ( ) i^"- 1 )' ,Jn when n is uneven ;
_n v n / "
2 e " = when n is unevenly even ;
f. _
* =
,_1 ,2tir
(v)
V ' /^\/^ \ y~ /*\ -2 /7^ i-l *
2- e = f )(! + * ) v> = ( - - ) x % x ^/'2 n when n is evenly even.
' The symbol t* m stands for e^ m ' w , and therefore presents no ambiguity of sign. A similar
abbreviation is occasionally used in the sequel.
Art. 28.] LINEAR TRANSFORMATION OF THE THETA FUNCTIONS. 475
In these formulae n is a positive integer, m a positive or negative integer
prime to n ; the symbols ( Y ( Y ( ) are the quadratic symbols of
Legendre, as generalized by Jacobi ; *Jn and ,^2n are the positive square roots
of n and 2 TO.
To apply these formulae to the evaluation of the series (iv), we consider
separately three cases.
(1) Let a = 2 a' be even, so that b is uneven ; we have
s-b 1 2tira'
2 e - t -b- = e -l abi *
8=0
observing that
If a = 0, 6 = 1, the method by which this formula has been obtained is
inapplicable ; but if we attribute to the symbol (-) the value +1, the formula
gives the true value of the series (iv), viz. H=l.
(2) Let a and b be both uneven ; we find
, =6-l ,
H=e-* M 'x 2S (-)'e b =e-* ai ' v x 2 e b =( x
=o =o ^0'
the form of the result being the same as in case (1).
(3) Let a be uneven, b even ; then
= 26-1
2 e
=0
x ^(6-a) x /
-
But
as will be found by applying the laws of quadratic reciprocity ; hence
3 p 2
476 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 29.
This formula is also applicable to the case (2) ; in fact when a and b are both
uneven
= x
The value of C is therefore
TT
where the real part of the radical is positive, and where S^ and H (a, b) =
are eighth roots of unity, of which 8^ v is determined by the equations (iii), and
H (a, b) by the formulae
H(a, b) = (-) i~* a , if a is uneven,
> Cv^
.... (vii)
= ( 7 ) i~l a x 4-4(0-1)*-^ if b is uneven.
If 6 is negative, and different from zero, the value of C is given by the
equation
the real part of the radical having a positive sign.
If 6 is zero, we have a = d= +1, =+c + Q = oc + Q; and the determin-
ation of ^ iK ( -, , HJ is, in this case, obtained immediately from the equations
defining the Theta series ; we have, in fact,
This result is in accordance with the general formulae (vi), (vii), if in them we
attribute to the indeterminate symbol (-) = f~i) tne value +1.
29. Linear Transformation. Determination of the Multiplier.
Putting in the formulae (i), (iii), (vi), (ix), x = 0, M = v = 0, we have
Art. 30.] LINEAR TRANSFORMATION OF (j> (<u) AND -^ (at). 477
/2K ^
V
if 6 is positive, and different from zero ; and if 6 = 0, a = d= + 1, then
/2A~ Q. r T
\f = \c[<]
Hence
^-^(a + &0) = xft,xS(a>&)x^|^, .... (Hi)
if b is not zero ; and
3
if b is zero, or any even number.
30. Linear Transformation of 4>(<o) and
Since (Art. 2, equation (i))
we can employ the general formulae of linear transformation (Art. 28, equations
(i), (iii), (vi), (vii)) to express < 2 (Q) and ^(ft), in terms of $ 2 (o>) and ^P(w).
By extracting the square root we obtain, but with an ambiguous sign, the
corresponding expressions for <(Q) and ^(&) in terms of <(>) and \^(&>) : this
ambiguity can be avoided by employing the equations of Jacobi (Art. 4, I (b),
(d) ; II (c), (d)) :-
....
The resulting formulae are contained in the following Table ; they were first
given in a complete form by M. Hermite. The symbol n is written for e* ir .
478
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 30.
TABLE OF THE LINEAR TRANSFORMATIONS OF <(<) AND
0,6
c, d
X".
1,0
0, 1
or|l|
0, 1
-1,0
or |*|
1,0
1,1
**-
(-)-<
A 2
1,1
0, 1
or|r|
1, 1
-1,0
or|p|
0, 1
-1.-1
or |^|
1 .W
(_)*+-*
I/*
If
F*
Art. 30.]
LINEAR TRANSFORMATION OF (f) (w) AND ^ (w).
479
The matrix of the linear transformation being
a, b
c, d
, so that
(0 =
c + dD,
a+bil'
ad bc =
the Table gives, for each of the six types of the unit matrix, the values of <p (O)
and vf' (Q) in terms of <f> (eo) and -^ () ; and also conversely the values of < (<B)
and -^ (o>) in terms of < (O) and ^ (Q). This second set of determinations is
implicitly included in the first, and is given for convenience only ; for the same
reason, a column is added giving the values of <E> (O) = X 2 , ^ (li) = X' 2 = 1 X 2 , in
terms of $ (<*>) = 2 , and ^ (e) = &' 2 = 1 k 2 . The value of the multiplier, cor-
responding to each transformation, is also given (Art. 29, equations (iii) and (iv)).
To demonstrate the formulae of the Table, we first consider the case in
a, b
which
and
c,d
is of the type 1. We have
c + dfl
<B= =-
Hence, by the formulae of Art. 28,
The combination of these equations gives
Again,
whence
or
(iii)
(iv)
480
Next, let
hence
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
a, b
[Art. 30.
c, d
2 CO
be of the type ^ ; we have
(v)
and also
or
(vi)
Each of the formulae (v) and (vi) involves the other.
The cases in which
a, b
c, d
is of one of the types <r, r, p, p 2 may be treated
similarly, but the formulae relating to them may also be obtained as follows :
Putting
a, b
c, d
0,1
-1,0
= [ \fr | in (v) and (vi), we find
Again, if
a, b
c, d
1,
1, 1
= | a- 1, we have from the equations of definition
of </> (a>) and vf' (a>), Art. 2, equation (iv),
The formulae relating to the unit-matrices | T | , | p \ , \ p 2 \ may be obtained by
combining the formulae (vii) and (viii). If, for example,
a, b
c, d
let
Art. 30.] LINEAR TRANSFORMATION OF (f> (ca) AND
we find successively
481
(ix)
Similarly, since
we have, if
a, b
c, d
(x)
and, if
a, 6
c, d
Combining the special formulae (viii) to (xi) with the general formulae (iii)
to (vi), we obtain the formulae of the Table. A single example of the process
be of the type p 2 , and let us write
will suflSce. Let
c, d
a, 6
c, d
a', 6'
c', d'
O), =
/ T/
c , a
the matrix
' 1 '
a, o
c', cf
being of type 1.
From (iii) and (ix) we have
But
a', 6'
/ 7'
c , a
a, b
c, d
a c, b d
a, b
whence, by Art. 21, equation (iv),
and finally
in accordance with the Table.
VOL. it.
3Q
482 MEMOIR OX THE THETA AND OMEGA FUNCTIONS. [Art. 31.
31. Linear Transformation o
The formulae for the linear transformation of the functions x( a> \ which
were first given by M. Hermite, have been presented by M. Schlaefli in the
following form.
Let f ***, and distinguishing the six cases in which the matrix of trans-
formation
a, b
c, d
is of the type 1, >^, a-, T, p, p 2 respectively, let C be an integral
number, denned for the modulus 48 by the congruences
C=(a + d)(b-acd) ; cases ^, p 2
C = (b- c) (bed a) ; cases 1, a-,
/*\
= (b c) (abc d) ; cases 1, T,
C = (a + d) (abd c) ; cases \J/-, p.
We then have
cases 1,
"X.
cases jo, a,
(ii)
To establish these formulae we have from Art. 3, equation (xv),
and
Hence x ( -- ) -r x ( w ) i g a cu ^e root of unity ; but this cube root is + 1 ; for,
since x (>) and x ( -- ) are one- valued and continuous functions of <a, the cube
\ Q}'
root of unity must be the same for all values of <o, and cannot be imaginary when
the real part of <a vanishes, in which case x (w) and x ( -- ) are both real.
From the two equations
Art. 31.] LINEAR TRANSFORMATION OF x( ft) )-
we find, attending to the relations
483
T = <r
H 2 =
; co = I T I X fi,
'r ; at = I p | X n,
xn.
(iv)
We have also
Let, as in Art. 22,
X()=
. . (v)
c, d
= |<r| 2 >'x|T| 2l 'ix|G-| 2 ' l X ... X|\|,
where | X \ is one of the six unit matrices
I 1 !. 1^1. H H IP!' \p\ 2 >
the formulae (ii) are verified, if in them we attribute to the exponent C the values
C=^fJL Zi/ ; cases 1, v/', \
(7=2^ 2i/ + l ; cases p, o-, I (vi)
(7 = Z/u Zi> 1 ; cases jo 2 , T. J
These values of C coincide with the congruential values given by the
a, 6
formulae (i) when
c, d
is one of the six simple unit matrices ; to establish the
coincidence generally it is sufficient to show that
C" = C-Zv, mod 48,
(vii)
C' and C" denoting the values assumed by the expressions in the right-hand
members of the congruences (i), when | o- 2 >* x
a, b
c, d
and
T "X
a, b
c, d
re-
spectively are substituted for
a, b
c, d
The congruences (vii) are readily
verified in the several cases ; for example, if
3Q 2
a, b
c, d
is of the type \^, so that
484 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 32.
o, d are even, b, c uneven, we have
C=(a + d)(b-acd) = b(a + d), mod 16,
C=(a + d)(abd-c) = -c(a + d), mod 16,
the two values coinciding because
(b + c)(a + d) = 0, mod 16.
Hence (7 = b(a + d + 2b) =(7+2/x, mod 16,
= C-2 v> mod 16.
Again, the four values of C in the congruences (i) are all congruous to one
another for the modulus 3 ; as may be seen by distinguishing the three cases
ad = l, bc = 0, mod 3 ; ad = Q, bc = 1, mod 3 ; ad= bc = 1, mod 3. And
using either of the last two values of C we may immediately verify the con-
gruence (?'=C+2/u, mod 3, while either of the first two values will serve to
establish the congruence C" = C 2 v, mod 3.
It will be noticed that the formulae (vi) are obtained by direct reasoning ;
but that the congruential values (i) (which may be exhibited in many different
forms) are established only by an d, posteriori verification.
32. Linear Transformation of the Elliptic Functions.
For the linear transformation of the elliptic functions the determination
of the constants C of Art. 28 is not necessary ; and it is sufficient to assign
the ratio of any two of them to one another. This may be done by means of
the formula (see Art. 33, equation (x))
= C x e><
^, )
in which /*,, v l are any two integral numbers, and m l} % are determined by the
equations
the other symbols having the same signification as in equation (i) Art. 28.
In the following tables, however, we have given, for convenience, the values
of the transformed Theta functions, as well as those of the transformed elliptic
functions ; the latter set of values being derived from the former by writing
Art. 32.]
LINEAR TRANSFORMATION OF THE ELLIPTIC FUNCTIONS.
485
h = 2K(a>), , - = TWJ?-, \ and forming the quotients in accordance with
the equations of definition (Art. 7, equations (xiii), (xiv), (xv)). We consider
the general transformation of type 1, subject only to the restriction that b is
not to be negative, but only the simplest transformations of the types ^, <r, T, p, p 2 ;
the real parts of the radicals V(a + bn), v fi, V(fl + 1) are supposed to be positive,
and y denotes, as before, e^' v .
I.
,
b
1,
, c + dfl
mnn *' A>
c,
d
0,
1
> *-**" "j *" "L f\
-l(a-l)
p=
^ (n) =
The elliptic functions are unchanged ; viz.
sin am (~, X 2 ) = sin am (( - )5 (a - 1 'cc,
(O^
^-j-, XM = cos amx, A am
II.
= ( - )*<- si
k
, X 2 ) = Aamx.
a, b
c, d
0, 1
-1,0
P =
*'
C '"A 5
^= >
MEMOIR ON THE THETA AND OMEGA FUJfCTIONS.
[Art. 32.
/ 7/o\ .sinama;
sm am (ix. k 2 ) = i ,
cos am a;
cos am
,. 7 , . Aamcc
t, A; 2 ) = - .
cos am a;
III.
a, b
c, d
S
1,
1, 1
7,2 1
X 2 = - V
V M~
sin am ( k'x, p- ) = k' - cos am (k'x
V K'/ Aarnx V
_ cos am x
A uni.'- '
A 7' \
A am IA; x, r ) =
* '
A amx
Art. 32.] LINEAR TRANSFORMATION OF THE ELLIPTIC FUNCTIONS.
IV.
a, b
c, d
1, 1
0, +1
i P
" +1 "'
o(-r( 1 + n )' n ) =
^-( + 1+0), n) = ,-
sn am
f ^x, T) = ^ si
x, = Aarax,
A am
, T) =
cos am a;.
487
a, b
c, d
1, i
-1,0
fcl :
43S
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
1
[Art. 33.
"--IF-
i
JT
sin am
(&' 2
*** * F
., snamx
k'
Aanix'
/ ., & 2 \ cos amx
A am ( IKX, y- ) =
V & 2 /
A amx
VI.
a,
c,
b
d
0,
-1,
v/:
+
T.I
1
1
n
l + ft e *'
' *' /o
' <V
/ .,, 1\ . 7/ smamx /.,, 1\
sm am ( %k x, JT, } = ik - - , cos am ( ik x, JT, ) =
V 2 / cos amx V '#/
A am ( ii'x, TT, ) =
V V cos a
cosamx
cos amx
33. Trausfoi-rnation of any uneven Order. Development of the Solution.
We have now to determine the function Tin the equation (xv) of Art. 25.
The zeros of 3,,, v [(a + bty-^ , ft] are, by Art. 3, equation (vii),
Art. 33.] TRANSFORMATION OF ANY UNEVEN ORDER. 489
..... (i)
where r and s are any two integers whatever. For given values of r and s,
the equations
= -
are always resoluble in integral numbers p and q, in such a manner that to
every different pair of values of r and s there corresponds a different pair of
values of p and q. When we attribute to r and s all pairs of values in
succession, the pairs of values assumed by p and q are, in their totality, the
same as the pairs of values satisfying the simultaneous congruences
pa + qc =i/ l, ]
mod 2, ........ (iii)
pa + qc= 0, )
\ mod A ......... (iv)
pb + qd= 0, j
The solution of the congruences (iii) is (see Art. 25, equation (xiii))
p = n 1, q=ml, mod 2 ; ...... (v)
the congruences (iv) admit of A sets of incongruous solutions
p = 0, 2 = 0; p=pj, q=q jt mod A, .... (vi)
where j=1, 2, ..., * (A 1), and where we may suppose that p j} q^ satisfy the
congruences (v). Thus all the numbers p and q satisfying the simultaneous
congruences (iii) and (iv) are included in the formulae
[(2r' + -l)A, (2/ + m-l)A], [2/A+#,., 2s' A + </;], . (vii)
r and s denoting any two integral numbers whatever. Since a, b, c, d have no
common divisor, a certain number of the pairs \_p it q^\ (viz. as many as there
are numbers less than A and prime to it), have no common divisor with A. If
[p, q] is such a pair, the A 1 pairs may be all represented by the formula
[jp, jq], in which j is any one of the uneven numbers 1, 3, ..., A 2 ; or,
again (when /u = v = 1 , mod 2) any one of the even numbers 2, 4, 6, . . . , A 1. The
zeros of 5 Mi v [(a + bn) -j- , ii] are accordingly
VOL. n. 3 R
490 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 33.
(a) -3 ^[2 7
(b) \h\2\
By Art. 3, equation (vii), the formula (a) represents the zeros of 3,, 1)n ( j-, a>j-
Again, if ?= - 9 ?*" the (A- 1) formulae (b) represent the zeros of the 1(A- 1)
functions
3., / and 3),, ft > denoting any two different Theta functions of the arguments -j-
and &). For the periods of the even, uniform, and doubly periodic function
3^ a '( - - ") -f 3i a- ( T- J being h and ha, and one solution of the equation
' \ h ' Pl p v h /
being x=jhn, all the solutions of that equation are included in the corresponding
formula (b). Thus the equation (xv) of Art. 25 becomes
(ix)
x e
C denoting a constant which can always be determined by putting x = 0, when
the matrix of the transformation and the indices p, v a, a' ; ft, ft' are given.
The constants C, appertaining to the same matrix, but to different values of
these indices, are not independent of one another. To establish the relations
between them, let MI, ^ be any two integral numbers ; let also
Art. 33.] TBANSFORMATION OF ANY UNEVEN ORDER. 491
and let us write in the equation (ix) x - ^(m^ta - n t ) for x, and consequently
Changing the indices of the Theta functions in (ix) by the equation (ix) of Art. 1,
we find, after some reductions,
= Cx
x
x -
M x )
x e
The formulse (ix) and (x) complete the general solution of the problem of
transformation of the Theta functions when A is uneven. The following special
a, b
cases are convenient in the theory of elliptic functions. In the matrix
c, d
let &=c = 0, mod 8, a = l, mod 4 ; this supposition is admissible if we regard the
theory of linear transformations as known (see Art. 27) ; also, in the formulse
(ix) and (x) let
M = ,= l, a = a'=l, = 0, 0'=I,
whence
m = a, n=.d, mod 8 ; p=q = 0, mod 2 ;
so that we may attribute to the index _;' the series of even values 2,4, 6, . . . , A 1.
X*
-mr -i- r\ e -i*b(a + 1>n}- .
Wntmg Q tor e A , we obtain from (ix)
t . . (xi)
and from (x), in which we write successively,
(A) MI = !> "i = 0, m^a,
(B) /! = (), !/!=!, ra! = 0,
(C) A*! = !/! = !, mj=a,
3 R 2
= 0, mod8,
= c, mod8,
= c?, mod 8 ;
492
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 33.
. . (xii)
. . (xiii)
. . (xiv)
Let h 2K; = - =K*i; introducing the elliptic and modular
functions, and attributing to j the values 1, 2, ..., (A 1), we find from (xii),
(xiii), (xiv),
Again, dividing (xi), (xiii), (xiv) respectively by (xii), and attending to
(xv) and (xvi), we have
sm am
/x \
( 1l> ,X) =
vM /
sin am x
sin 2 am a;
x IT
sin 2 am x
cos
/ x \ sin 2 coam 4 ;T
am ( ^TT, X ) = cos am x x II - To .
\M / l-A; 2 sii
^, . .., . --
2 sin 2 am 4^ ^sm 2 am a:
( vv r,\
..... ( XV11 )
/.,, ;-\
.... (xvm)
A am
x ^ \
if> x ) =
M /
am x x
-x
^ . (xx)
The equations (xvii), (xviii), (xix) have been transformed by Jacobi as
Art. 33.] TRANSFORMATION OF ANY UNEVEN ORDER. 493
follows. The equations for the addition of the elliptic functions (equations (xxii)-
(xxiv) of Art. 7), give immediately
/ \ , \ sin 2 am x sin 2 am y
sin am (x + y) x sin am (x y) = - ? r . - ,
1 - & 2 sm 2 amxsm 2 am2/
, , . mw A a amwsin 2 ama; , , .,
cos am (x + y) x cosam(cc y)= ~-r- - , } . . (xxi)
l- 2 si 22
, . A 2 am7/ + & 2 cos 2 am w sin 2 am a;
A am (x + y) x A am (x y) = - ^ r
l-A 2 sm 2 ama;sm 2 am2/
Employing these formulae to transform the type factors in (xvii), (xviii),
(xix), and attending to the values of ^/A, ^/\', -^, we obtain the equations
/x \ yl'i A
sin amf-^, Xj = x II sin am (x + 4/), .... (xxii)
/x \ A'** A ....
cos amf -^., Xj = ,,, x II cos am (x + 4j (j, .... (xxin)
Aam(-^,\)= pj xll A am (x + 4/), .... (xxiv)
in which f is to receive all values from to A 1 inclusive.
A third set of expressions for the transformed elliptic functions has also
been given by Jacobi. Comparing the equations (xvii) and (xxii) we see that
the roots of the equation
zll [z* - sin 2 am 4jT] - A sm am (~ , A) x II [" z 2 - . y . / , .J = 0, (xxv)
^ J M vlf / ^sm 2 am4J
which is of order A and in which j = 1, 2, . . . , |(A 1), are all comprised in the
formula
z = 8inam(a; + 4/), j = 0, 1, 2, . . . , A- 1.
Hence, equating the sum of the roots to the coefficient of z*~ l , we find
sin am (TT>, A ) = -T- 2) sin am (x + ty'Q ', .... (xxvi)
and similarly, from a comparison of (xviii) with (xxiii), and of (xix) with (xxiv),
494
MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34.
X* \l
(1* X * l
jj., XJ = (-1)* (A - 1) 2 cos am (a; + 4/), . . . (xxvii)
), (xxviii)
the sign of summation 2 in the three equations extending to the values
0, l,...,A-
34.] Quadratic Transformations of the Elliptic, Theta, and Modular
Functions.
The matrices of determinant 2 (see Art. 23) are of nine different types ; and
if we regard matrices differing only in sign as identical, two classes of matrices
non-equivalent by primary postmultiplication, and only two, appertain to each
of the nine types. There are thus nine pairs of different quadratic transfor-
mations ; but, so far as the elliptic functions are concerned, the difference
between the two transformations of any one pair is not essential. As repre-
sentatives of the eighteen classes of matrices we take the matrices C^p, F 0i3
exhibited in the annexed schemes ; the first index referring to the columns, the
second to the rows.
1, 1
-1, 1
9
1,
1, 2
y
0, 1
-2, -1
-1, 1
-1, -1
>
1, 2
-1,
|
2, 1
0, 1
0, 2
-1, -1
9
2, 2
-1,
9
2,
0, 1
2,
-1, 1
>
0, 2
-1, -2
J
2, 2
-2, -1
1, -1
0, 2
9
1, 2
-2, -2
>
2, 1
-2,
1, 1
-2,
1,
0, 2
J
0, 1
-2, -2
It will be seen (1) that C a> fl and F^ ^ are of the same type, (2) that C^ B
and r0 >a are reciprocal, (3) that the matrices C, and also the matrices F, are
transformed into one another (irrespectively of sign) by premultiplication and
postmultiplication with the matrices of the scheme
1, P,
P 2 , 1,
P*
P
1
in the manner explained in Art. 23.
Art. 34.] QUADRATIC TRANSFORMATION. 495
In the following Table I. of the eighteen quadratic transformations of the
elliptic functions, M is the multiplier, X is the transformed modulus, s, c, d are
respectively
sin am (x, k 2 ), cos am (x, k 2 ), A am (x, k 1 ) ;
the columns (M X.S), (C), (A) exhibit the numerators of the expressions for
(cr \ / *T* \ / x \
~, A 2 ), cos am (^, X 2 j , Aam(^, X 2 J
respectively ; the common denominator being placed in the column (D). Thus
the first line of the Table supplies the three equations
w f
ikk x,
L
sn am - am x x A am x
sin am
cos am
k(k ik') sin 2 am x
k' + ik ~\~\ 1 k (k + ik') sin 2 am x
I, I. \ clrl a QTV,
A/ tA/ I Dill 1 1 ! 1 1 i
COS i
i 2 am x '
and indicates that they arise from the transformation <a - , or, more
generally, from any transformation of the type
1, 1
(a =
-1,1
X a X
where | a | is a primary unit matrix. It will be observed that if in the Table I.
we substitute for one of the matrices C or T any other matrix equivalent to it
by primary postmultiplication, the signs of M and X may change ; but this does
not affect the elliptic transformation.
When the index /3 is 1 or 2, a- 2 x T a is primarily equivalent by post-
multiplication to C a< ; the same relation holds between r 2 x T a @ and C a , e
when ft = 1 or 3. This observation explains why it is that the formulae for F a e
may be obtained from those for C a< 3 by changing the sign of either k or k' when
/8 = 1, by changing the sign of k' when /3 = 2, and of k when /S = 3.
Again, the matrices of the nine pairs C 3i p, T 3i ; C,, #, r 2 >$ ; C iiP ,T^ $ ; /3=1,
2, 3 are connected by a relation of the type
Iri x| (7| = |r| x ia|
496 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34.
where | a \ is a primary unit matrix. Thus the transformations corresponding to
the two matrices of any one of these pairs may be reduced to one another by
means of the linear transformation IV. of Art. 32.
The two reciprocal transformations (7 3j 2 and F 2 3 are of special importance,
Ixjcause they alone, of all the eighteen, transform real elliptic functions having
moduli less than unity into functions of the same kind. The former, which is
the transformation of Landen, is also termed the transformation " ct, majori ad
minorem" because it diminishes the modulus ; the latter, which increases the
modulus, is the transformation " a minori ad majorem," and is sometimes called
the transformation of Gauss, though it is apparently due to Lagrange.
If we employ the notation of Legendre for elliptic integrals of the first
species, viz.
sn as
the transformations of Landen and Gauss respectively give rise to the equations
, ... - (A)
. . . (B)
which supply expeditious methods, known as the descending and ascending
modular scales of Legendre, for computing the values of elliptic integrals by
successive transformations. In these formulae k and k' are positive and less
than unity, and only the real value of the integral is considered ; (A) is derived
from C 3> 2 in Table I. by writing
</> = am ,,
and (B) from (I\ 3 ) by writing < = am [2x, k], or
,
( 1 + fyx, j^J.
The formulae A and B are not essentially different ; in fact, if in either of them
we call the transformed modulus k, and interchange <p and ^, we obtain the
other. As before >; denotes e* 1
Art. 34.]
QUADKATIC TRANSFORMATION.
497
TABLE I. QUADRATIC TRANSFORMATIONS OF THE ELLIPTIC FUNCTIONS.
Matrix.
1
A
F
MxS
C
A
^>
*,
*Vff
*r + a
8(2
l-Hs(k+^)s 1
c
i-* ( *-0^
r,,,
*i-*V&
*'-a
sd
l-k(k-W)f
c
l-*(* + Oa
<Vi
v-a
27," V^
*
c
l-k(k-ik')s*
l-k(k + iV)*
r,.,
k'+ik
2,,VM'
sci
c
l-k(k + ik')s*
l-kik-iV)?
C M
k-ik-
4+a*
sd
l-k(k-W)t
l-k(k + ik')s*
c
r ,i
k+ik'
*-a*
sd
i-*(*+*V
l-k(k-ik')s*
c
01.1
2i Vk'
(ii+yj
sc
1_(1_A') 8
d
l-(l+/fc')s 2
r,.,
aSu
,-(i-*0
sc
1-(1+^) S '
d
1-(1 -*')
C 2 3
i(\-k f )
2-/*'
sc
d
l-(l+/f)s 2
1_(1_A')^
r s> ,
*(i+*0
2.V/1'
sc
d
l-(l-k')s*
l-(l+^) 2
^3,
!+*
I-/!/
sc
l-(l+k')s*
l-(l-k')s*
d
r 3, S
\-1f
l+/f
se
l-(l-k')<?
l-(l+k') a *
d
C 1 ,,,
2</k
1+4
s
\-k
cd
1+W
r,,,
2iVk
1 *
s
l+k?
cd
1-A S J
<?*
l-k
2.VA
s
cd
1+jtf
1-^
r 2lJ
1+k
2^/fc
s
cd
l-A g 2
1+A 2
r,, s
1(1 -i)
<(!-*)
s
s
1 + U
l-^ s
1+jW
cd
Every matrix of determinant 2 is comprised in
He'll x||||, ||rjjx|i
where || a || is a primary unit matrix ; or, again, in <
1;J x ii- ii, l' } l * ii n, ;
i one of the eighteen formulae
ill (i)
sne of the three formulae
2,0 n n /--N
' X II a II, . . . . (ll)
'i -^
where || a || is any unit matrix whatever.
The quadratic transformations of the Theta functions are of eighteen dif-
ferent types, corresponding to the eighteen formulae (i) ; but the formulae (ii)
VOL. ii. 3 s
498 MKMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 34.
are more convenient for the determination of the transformed Theta functions
corresponding to any given matrix of determinant 2. For, if we were to use
the formulae (i), it would be necessary, whenever the second element is different
from zero in each of the two compounded matrices, to multiply together two
radical quantities and to determine the sign of the product ; viz. if the trans-
formation
C+DQ.
= T+Bil
is regarded as compounded of the quadratic transformation
<o =
a
and the linear transformation
we have to consider the product
V(A + Bty = V (a + &i) x V (a + fin),
where the signs of the radicals /(a + b<a^ and V(a + /3fl) can be determined by
the rule that their real parts are to be positive; but the sign of V(A + Bfl),
which alone appears in the final result, depends on the signs of its two factors,
and cannot be determined by inspection, or (as it would seem) by any simple
rule, without first resolving it into its factors. The inconvenience thus arising
is obviated if we use the formulae (ii) in which the second coefficients of the
matrices of determinant 2 are zero ; and it is accordingly preferable in the
theory of the Theta functions to consider only the three transformations
^2, 1> If, 2) * 2,3-
In the following Table II. e* n is still represented by q, and 3 is written
>r
ana
BLE II. QUADRATIC TRANSFORMATIONS OF THE THETA FUNCTIONS.
The formula.
we call the 1 n " -
other. As beft
Art. 34.]
QUADRATIC TRANSFORMATION.
499
I *T~~ > ** / == -" 1 ^2 ^ ^0,
"
>,[]
2A"
(7 3)2 XQ =
(ii.)
2,0
0,1
= i a
/ T
X A/ 2S ,
, . .i
' '
/ -
> V 2 A"
3S 2
500
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 34.
= r 2i 3 x n =
( \
111.)
1,0
0,2
= 211,
~T2 \~JT ' ) = 3 X ~2
The equations in the preceding Table connecting the given and transformed
modular functions are included in a complete system exhibited in Table III.
TABLE III. QUADRATIC TRANSFORMATIONS OF THE MODULAR FUNCTIONS.
A.
Matrix.
*^fcl> 1 2,1
r* F
^3,1 > x 3,1
r . p
^%l , r 1,2
P F
*^M > - 1 2,2
C. I 1
J,2 , ' 3,2
( 1 F
l/ l,3 *l,i
2,3
C, s ; F
. 9 :
3,8
Equation.
= 72X0.
() = ox/2 f () (n) ^ (O),
(n) ^ 2 () = V^ (") </> (),
() i 2 (") = i V 2 ^ () <^> ()
X.
a(c-6)
a (c - 6)
- a(6 + 2 c)
a (2 6 + c)
Ait. 34.]
Matrix.
r,,
,
3>2
QUADRATIC TRANSFORMATION.
B.
Equation.
501
X
2a(b + c)
-2a(b + c)
-lab
2ab
-2ac
2ac
2a(b + c)
-2a(b + c)
-2ab
2ab
-2ac
2ac
2a(b + c)
-2a(64c)
-2ab
2ab
502
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 34.
Matrix.
Equation.
X.
-2ac
2ac
To obtain the corresponding equations between <(!) and <(<*>), proper to
any transformation
=|| Cllxllalixfl,
= lir||xi|a||xn,
where II a II =
a, b
c, d
is a primary unit matrix, we have to multiply the right-hand
side of the equations in the Table by >? .
Table III. contains the complete theory of the quadratic transformations of
the modular functions <p (to) and ^ (>) ; viz. given any equation whatever of the
type
^.LrH)
ad bc = 2,
to =
no rational relations subsist between < (&>), \f/ (a>), and <(Q), ^ (&) which are not
included in the formulae furnished by the Table. The two formulae (A) and (B)
corresponding to any given matrix are not rationally deducible from one another ;
but they coincide with one another when rationalized, so as to contain only $ (o>)
and 4> (Q). To effect this rationalization we raise each member of the equations
in Table A to the power 8 ; in Table B each member of the equations is to be
raised to the power 4, and the two equations (C itJ ) and (r,.^) are to be multiplied
together. The nine equations thus obtained are the nine modular equations
of order 2.
It only remains to show how the formulae of the Tables I., II., and III. are
to be verified.
In Table II. the form of the expression of the Theta functions containing 12,
in terms of the Theta functions containing , is determined by the general
theory of Art. 25 ; the reduction of the six coefficients, indicated by that theory,
to only two, A and B, is effected by substituting x + \h, x + \h(a, x + ^h + \h<a
for x, and employing the formulae (ix) of Art. 1 ; the values of A and B, which
are found by putting x = 0, are conveniently expressed in terms of the modular
Art. 34.] QUADKATIC TBANSFORMATIOX. 503
Vojf
- by means of the equations of Jacobi (Art. 4, I. and II.),
which give immediately
(i) 3,[*- *]
or, by Art. 32, III.,
or, by Art, 32, III.,
5R.-
The equations of Jacobi (Art. 4, I., rf, and II., d) also give by multiplication
whence, writing 2o> for w,
(6) ^> s ()0*(2)
and again, writing <o 1 for <o, and substituting for < (co 1), ^ (co 1 ) their values
in terms of (f> (<*>), ^ (<),
The equations (a), (6), () are included in the Table II. ; the values of
\J/ 2 (|a> 1), of ^) 2 (2<B), and of \p(^a>), which are given in the same Table, for the
cases (i), (ii), and (iii) respectively, are obtained by forming the quotient of the
two equations which contain the coefficient B, and putting x = 0. The values
of TJ. in the same three cases are, by the definition of the multiplier,
35 W ' "HW* "35 M '
504 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 35.
and then, attending to the equations (2), (4), (6) of this article, become
in accordance with Table I.
The formulae in Table I., relating to the transformations C 2)1 , C 3)2 , F 2 3 ,
result immediately from the equations defining the elliptic functions (Art. 7) and
from the formulae of Table II. Lastly, the remaining formulae of Table I. and
the corresponding formulae of Table III. are to be deduced by linear transfor-
mation from the formula? appertaining to the three fundamental transformations
<7 2il , C 3 . 2 , F 2iS . To facilitate these verifications, and the determination of the
transformed Theta functions corresponding to any of the matrices C or F, we
add a scheme of the postmultiplying unit matrices, which serve to reduce the
matrices C and F to the three fundamental matrices (7 2|1 , (7 3i2 , F 23 .
x X C 1 C x X C F x X
$ t 1 ) ^"^8,2 ^""^3,2 i5,2' ^"*| 3 2| 3 8,3'
1, 1
-1,
'
1,
0, 1
>
0, 1
-1, -1
0, 1
-I, -I
I
1, 1
-1,
>
1,
0, 1
1, -1
0, 1
>
1, 2
1 1
j
2, 1
-1,
(ii.)
F - r* v y T - n v v r~r v
* ,1 ^2,1 * J ,! * ,2 ~ ^3,2 * * ,2> * ,3 ~~ L 2,3 *
-1, 1
1, 2
2, 1
0, -1
1
-1, -1
|
-1,
1,
0, 1
1, 1
M-
-1, 1
9
-1, -2
I
-2, -1
1, 1
1,
0, 1
-i, o
'
0, 1
-1, 1
-* ,3
Arts. 35-45.] GEOMETRICAL REPRESENTATION OF BINARY QUADRATIC FORMS.
Art. 35.] Quadratic Forms of a Negative Determinant.
Adopting the usual conventions, we represent the complex quantity <o = x + /.'/
by a vector in a plane ; the extremity of this vector we term the point w. As y
Art. 36.] BINARY QUADRATIC FORMS. 505
is essentially positive, we have only to consider the region which lies above the
axis of x ; by the plane (xy) we shall understand this region only.
Every complex quantity o> is the root of a quadratic equation of which the
coefficients are real and the determinant negative ; the ratios of the coefficients
are given when <o is given ; for, if a> = x + iy satisfy the equation a + 2 b<o + c<*> 2 = 0,
we have
x 2 +y 2 -x
Vice versd every binary quadratic form of a negative determinant is
represented in the plane (xy) by a point ; for we may regard the equation
a + 26ft> + ca> 2 = as corresponding to the quadratic form (a, b, c) of determi-
nant A (viz. we have A = ac 6 2 ), and the point CD = - - as representing
c
the quadratic form. This representation is admissible whatever real values the
quantities a, b, c may have. In the general case the ratios only of the coefficients
are given when the representing point is given ; but, if we suppose the coefficients
to be integral numbers, and the form to be positive and primitive, the three
coefficients themselves (and not merely their ratios) are given when the repre-
senting point is given.
, is an unit matrix, the two points co and fi are
7 , a
a, ft
,
If co = i , where
equivalent ; and the equivalence is primary when the matrix
7,9
is primary.
When two points are equivalent the corresponding quadratic forms are also
equivalent ; viz. if (a, 6, c), (A, B, C) are the quadratic forms corresponding to a>
Q
and Q respectively, (a, b, c) is transformed into (A, B, C) by ' .
36. The Reduced Space.
A positive quadratic form (a, b, c) of negative determinant is said to be
reduced when the absolute values of its coefficients satisfy the conditions
(1) [2 6], c^[26], a^c; j
of which the second is a consequence of the other two. . . (ii)
(2) If a = [26], b>0; ]a = c, b>0.
If co = x + iy is a reduced point, i. e. a point corresponding to a reduced form,
the conditions (ii) give
VOL. II. 3 T
506
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 36.
(1)
(2)
If x*
x<0;
x
<0.t
(iii)
In these formulae the symbols [2U] and [2x] denote the absolute values of
the quantities 26 and 2x.
The geometrical interpretation (see figure 1) of these conditions is that the
reduced point <a lies within the space A A' external to the semicircles x 2 + y 2 = 2x,
and internal to the semicircle cc 2 + y* = 1 ; we include in the space A A' that part
of its contour which lies to the left of the axis of x, and we exclude from it the
other part of its contour.
Since every binary quadratic form of a negative determinant is equivalent
to one, and only one, reduced form, we have
the corresponding theorem, " Every point in
the plane (xy) is equivalent to one and only
one point in the reduced space AA '." Again,
if we regard two substitutions as identical
which differ only in sign, we have the arith-
metical theorem, "Every quadratic form is
equivalent to its reduced form by one substi-
tution only ; except when the form appertains
to the class (1, 0, 1) or (2, 1, 2) ; the number
of reducing substitutions in these excepted
cases being respectively two and three." We
therefore infer that " any given point is equivalent to its reduced point by
one substitution only, except when the reduced point is + i or \ ( 1 + i V3), in
which cases there are respectively two or three reducing substitutions."
For our present purpose we have to modify the preceding enunciations by
limiting ourselves to the consideration of primary equivalence. A form (a, b, c)
is primarily reduced, when it satisfies the conditions
(1) [6]^ a, [6]<c.
(2) If[6] = a, 6>0; if[6] = c, Z0.
Similarly, the point a> = x + iy is primarily reduced when it satisfies the conditions
(1) O^z' + y', []<!.
(2) If[x] = x* + y 2 , x<0; if [x] = l,
It may be proved, by a slight modification in the demonstration of the
Art. 37.]
BINARY QUADRATIC FORMS.
507
theorem of Lagrange, (i) that " any quadratic form of negative determinant is
primarily equivalent to one and only to one primarily reduced form ; " (ii) that
"the primary reducing substitution is also unique." If, therefore (see figure 2),
we designate by 2 the space included between
the parallel lines x= 1, and external to the
semicircles x 2 + y 2 = + x, we obtain the theorem,
" Any given point in the plane xy is primarily
equivalent to one, and only to one, point in
the space 2 ; and the tw6 points are primarily
equivalent by one substitution only " *.
We reckon the line x = 1, and the semi-
circle aj 2 + y 2 = x as appertaining to the space
2, but we exclude the line x = 1, and the semi-
circle x 2 + y z = x. The lines x = 1 and x = + 1
we denote by *S_ t and S +1 respectively; the
semicircles x 2 + y" = x and x 2 + y 2 = x by T_j
and T +1 . The space 2 we shall henceforward term the reduced space. The
cornicular points +1, 0, oo, we denote by p, q, r.
37. The Circular Affinity of Moebius.
If o> and ft are two vectors connected by the relation a> = -^ - where
J a + pft
a,/3
7,9
are any quantities whatever, real or imaginary, but ao* ,87 is not zero,
any two corresponding loci described by the points <o and ft are connected by a
relation known as the circular affinity of Moebius. If /3 = 0, the relation between
the two figures is one of similarity ; in every other case the relation is one of
inversion. For if /? be different from zero, the equation connecting w and Q may
be written in the form
* See a Note on the primary periods of the elliptic functions in the ' Messenger of Mathematics,'
new series, vol. xii. p. 73, Arts. 1-3 [vol. ii. p. 342] ; where, however, the point corresponding to the form
(a, b, c) is taken to be instead of to, or, what is the same thing, the point is taken to be
. , j r 6 + tVA
instead of
e
, T 2
508 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 37.
which shows that the vectors joining to - and fl to -= are inversely pro-
portional. The fixed points and - 3, which, in each of the two figures, answer
to the infinite in the other figure, are termed the centres of inversion. Figures
related to one another by circular affinity possess the following properties :
(1) Circles in either figure which pass through the centre of inversion of
that figure are transformed into straight lines, and vice versd. All other circles
are transformed into circles.
(2) The anharmonic ratio of four points on a circle or straight line in the
one figure is equal to the anharmonic ratio of the corresponding four points in
the other figure.
(3) The two figures are similar in their infinitesimal parts ; except at the
points - and ^ which (as has been said) correspond in the two figures respect-
ively to the infinite of the other. The truth of this theorem may be inferred
from the observation that to an infinitesimal circle (c) in one figure there corre-
sponds an infinitesimal circle (C) in the other figure ; and that to the centre
of (c) there corresponds a point distant from the centre of (<7) by an infinitesimal
of the second order, the radii of the two circles being regarded as infinitesimals
of the first order. But the theorem is only a particular case of the general
proposition of Lagrange and Gauss, that if
X+iY=f(x
where X, Y, x, y are real quantities, the points X+iY&nd x + iy describe figures
which are similar to one another in their infinitesimal parts, the only cases of
exception presenting themselves at points where the derived function of f is zero,
infinite, or indeterminate.
(4) To any simple contour in either figure, not passing through the centre
of inversion, there answers in the other figure a simple contour, not passing
through the centre of inversion ; and the two contours are similarly described ;
i. e. positively or negatively in both figures alike*. It is sufficient, in order to
* By a simple contour we understand a closed and finite contour which does not intersect itself.
Such a contour is said to be described positively or negatively according as the describing point has
the enclosed i-pace on the left or on the right ; or, which is the same thing, according as the element
of the contour and the element of the internal normal are situated with regard to one another as the
vectors 1 and t, or as the vectors 1 and f.
Art. 39.]
BINAEY QUADRATIC FORMS.
509
establish this assertion, to verify it in the three cases <o = a + n, w = an, &> = ,
a being any constant quantity.
38. The Subdivisions of the Reduced Space.
The twelve spaces into which 2 is divided by the lines and circles of figure 2
form, when taken in pairs, six regions which are in circular affinity with one
another in the manner indicated by the equations
A_
\l\
A_
rn
B C D E
F
\(7\
C'
m
D'
\P\ \P*\'
E_ F'
\p'\ li /2 l '
The matrices 1 1 1, | -^ |, | a- , \ r \, \ p \, \ p 2 \ are those of which the values are
given in Art. 21 ; the matrices denoted by the accented symbols are derived from
these by changing the signs of the second and third constituents ; thus, for
example,
\P\-
and we have
1,1
-1,0
1, -1
1,
l^'l = -I'H k'l
i / i i i i n
IP 1= - p\ x kl -,
1-1
T = T
X T
2
The equations (I.) are to be interpreted as follows : The space A is transformed
into the space C by the equation
C = k | x A, or C = 1 + A ;
the space C is transformed into the space D by the equation
| o- |- J x C= I T I- 1 x D, or (7= a- 1 x I T h 1 x D
1,0
1,1
o, i
1,
xZ>=
., &c.
39. Quadratic Forms of a Positive Determinant.
A form (a, 6, c) of positive determinant D ( = I 2 ac) is represented in the
plane (xy) by the semicircle
= (i)
510 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 39.
which we shall designate by the symbol [a, b, c]. The equation of the semicircle
may be written in the form
a + b(x+iy)
/ _L ly - iL _
b + c (x + iy)
b,c
a, b
(ii)
The point x + iy (but not, except when 6 = 0, or x = 0, the point x + iy) is a
point on the semicircle. If c = 0, a supposition which implies that D is a square,
[a, 6, 0] is a straight line ; for convenience, however, we shall regard this straight
line as a semicircle of infinite radius.
Theorem I. " If (a, b, c) is a form of positive determinant, and (a, (3, 7) a
form of negative determinant A (where A = ay /3 2 ) connected with (a, b, c) by
the invariant relation ac 2 6/3 + yet, = 0, the point representing (a, /3, 7) lies on the
semicircle representing (a, b, c)."
For, if x + iy be the point representing (a, /3, 7), we have
y
and the invariant relation becomes
I
a + 2bx + c(x 2 + y 2 ) = 0.
Thus the semicircle [a, b, c] may be regarded as representing the quadratic form
(a, b, c), because the points of the semicircle represent quadratic forms of negative
determinant which are harmonically associated to (a, b, c). If 7 and c are of the
same sign,
ac-2fe/3 + 7a> 0,
according as the point (a, /8, 7) lies outside or inside the semicircle [a, b, c].
By equivalent semicircles we understand semicircles which are related to
one another by a circular affinity, the coefficients of the transformation being
integral numbers, and the determinant a positive unit. We then have the
following theorems :
Theorem II. " Equivalent points lie on equivalent semicircles ; and equi-
valent semicircles correspond to equivalent quadratic forms."
Or, which is the same thing,
" If the form (a, b, c) of positive determinant is transformed into (A, B, C)
by the linear substitution
, the semicircle [a, b, c] is transformed into the
7,*
semicircle [A, B, C] by the circular transformation = ^ - ."
Art. 39.]
BINARY QUADRATIC FORMS.
511
To establish this theorem, let <a = x + iy, l = X + i Y, and let w be a point on
[a, b, c], so that
b, c
x + iy =
a, b
changing the sign of i in the equation
we have also
whence
x (x + iy);
x (X + iY),
x + iy =
b, c
a, b
x ( x + iy) =
a, /3
7,0
b,c
a,b
B,C
A,B
or X + iY lies on the semicircle [A, B, C], and [a, b, c] is transformed into
[A,B,C].
This Theorem is the geometrical expression of the invariantive character
of the equation
Theorem III. " If
or, which is the same thing, if
b, c
a,b
b, c
a, b
x (! + -
the two points x l + iy } and x 2 + iy 2 are inverse points with regard to the circle
[, 6, c]."
For, equating real and imaginary parts, we find
, - y 2 ) + c (y t x 2 - y, xj = ;
= 0,
and of these equations the first expresses that x l + iy l , x. 2 + iy. 2 are conjugate
points with regard to [a, b, c] ; the second expresses that the line joining them
is a diameter.
512
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 40.
40. Automorphics of a Quadratic Form.
The automorphics of the quadratic form (a, b, c), the coefficients of which
we now suppose to be integral numbers, are included in the general formula
1
- x
m,
T-bU, -cU
aU T+bU
(iii)
where T, J7are any integers satisfying the equation T 2 DU 2 = m 2 , and m = 1, or 2,
according as the form (a, 6, c) is properly or improperly primitive. The meaning
of course is that if
a,/3
%*
1
x
m
T-bU, -cU
aU , +bUT
(a, b, c) (x, y) 2 = (a, b, c) (an + $y, yn + Sy) 2 .
then
By these automorphics the rational semicircle [a, b, c] is transformed into
itself, and the transformations are homographic (Art. 37, 1 and 2). Consequently
the chords of the homography (i. e. the chords joining the pairs of corresponding
points) all touch a conic of which the major axis is the diameter of the semicircle.
This conic is an ellipse of which the equation is
T 2 \
i J* 2 | A
and of which the eccentricity e is given by the equation
To establish these assertions we write the equation of transformation,
CO =
m
in the form
and, denoting by Z, z the points
T-bU, -cU
aU , T+bU
T-bU
xii,
(iv)
T-IU T+bU
cU ' ' ~~cTT
.,... . .
(which lie on the axis 01 x,
outside the semicircle, and at a tangential distance - from it), we draw any two
C- i-J
lines ZPQ, zpq, making equal angles with the axis, and meeting the semicircle
Art. 40.]
BINARY QUADRATIC FORMS.
513
in the points PQ, pq respectively (see fig. 3). The equation (iv) shows that the
pairs pP, qQ are pairs of corresponding . points. Hence the tangent at the
extremity of the minor axis of the ellipse passes through the points of contact
of tangents drawn to the semicircle from z and Z. Thus the foci of the ellipse
are the points inverse to Z and z with regard to the semicircle ; from this con-
clusion the equation of the ellipse and the expression for its eccentricity are
immediately derived.
If pP are any two points on the semicircle connected by the automorphic (iii),
.. MJ> M 2 P . l + e ,,,-
the constant anharmonic ratio -=-? r -^ is equal to ~ - . MiM? being the
M ,p M 2 p l+c,
diameter of the semicircle. As all the automorphics of (a, b, c) are powers
(with integral exponents positive or negative) of the fundamental automorphic
(i. e. of that automorphic in which T and U are the least positive integral
numbers satisfying the equation T 2 DU' 2 = m 2 ), so the homographic trans-
formations into itself of the semicircle [a, b, c] may be all obtained by
repeating one of them continually in either direction. Let 2 be the funda-
mental automorphic, E the ellipse touched by the chords of the corresponding
homography; also let PP,, P.P., P 2 P 3 ,..., PP^, P_.P_ 2 , P_ 2 P_ 3> ... be the two
series of chords of the semicircle which, beginning with any given point P
on the semicircle, can be drawn to touch E ; we may suppose that PP l are
connected by 2, PP_ l by 2 -1 ; then the homographic transformations (PP 2 ),
(PP 3 ), . . . , (PP_ 2 ), (PP_ 3 ),... , which arise from the duplication, triplication, &c.
of the fundamental transformation (PP^ or (PP_J, correspond to the auto-
morphics of which the symbols are 2 2 , 2 3 , . . . . 2~ 2 , 2~ 3 , . . . For all semicircles
of the same determinant and of the same order (the semicircle appertains to the
properly or improperly primitive order according as m = 1 or m = '2) the auto-
morphic transformations are similar, since the eccentricities of the regulating
ellipses depend only on T, U, D, and m.
VOL. ii. 3 u
514
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 41.
Every point on the semicircle is equivalent by some one automorphic trans-
formation to a point on the arc of the semicircle cut off by any tangent to the
ellipse E ; and no two points of such an arc can be equivalent by any automor-
phic transformation.
41.] The Subclasses of Primarily Equivalent Quadratic Forms.
The introduction of the conception of primary equivalence renders it neces-
sary to divide each class of equivalent quadratic forms (as denned by Gauss) into
subclasses of primarily equivalent forms. The number of subclasses in each class
is six, three, or two, as will be seen from the following enumeration of the
different cases :
We consider first under the headings I. and II. the case where the deter-
minant is a positive number not a square ; secondly the case where the deter-
minant is a square or a negative number other than 1 or 3 ; and thirdly the
case where the determinant is = 1 or 3.
First : I. Properly Primitive Classes.
Every form (a, b, c) of a properly primitive class belongs to one or other of
the three sets :
(A) a = c = l, mod 2;
(B) a = 0, c = l, mod 2;
(C) a = l, c = 0, mod 2;
(i)
and the result of applying unit transformations of the six types of Art. 21 to
forms of these three sets is indicated in the following scheme :
A
B
C
A =
V*
p\ *
P' r
B =
P, *
1, T
P*, +
Q_
P*, r
P, ^
1, o-
. . . (ii)
so that, for example, (A) = (Z?) x p 2 = (B) x a-, the symbols (B) x p 2 , (B) x <r, repre-
Art. 41.] BINARY QUADRATIC FORMS. 515
senting the sets obtained by applying unit transformations of the type /o 2 , and of
the type a-, to forms of the set ().
We must now distinguish the cases in which the least number t/i, satisfying
the equation T 2 DU 2 = 1, is even and in which it is uneven.
(a) Let U l be even ; then U 2 , U 3 , ... are all even, and all the automorphics
of any given form f are primary. Hence the forms
fxl, /x^, /xo-, /XT, fxp, fxp 2 (iii)
are all primarily non-equivalent ; since if any two of them were primarily equi-
valent, f would have a non-primary automorphic. Thus, in this case, each class
contains six sub-classes of forms primarily non-equivalent.
(b) Let U l be uneven ; then U 3 , U s , ... are uneven, U z , Z7 4 , ... are even ; i. e.
the automorphics of an even order are primary, and the automorphics of an
uneven order are of the type ^, T, or a- according as the form is of the type
(A), (B), or (C). In this case the six forms (iii) are primarily equivalent in pairs ;
for example, if/is of the type (A),f&ndfx \f/- are primarily equivalent, because
/has automorphics of the type ^ ; i.e. f can. be transformed into/x ^, not only
by transformations of the type -vf/-, but also by transformations of the type ^ x ^
or 1. Each class therefore contains three subclasses of forms primarily non-
equivalent.
II. Improperly Primitive Classes.
(1) Let D= 1, mod 8 ; the values of T, U in the equation T 2 -DU* = 1 are
all even, and the automorphics are all primary. The forms are of three types :
(A') a = c = 0, mod 4, \
(R] a = 2, c = 0, mod 4,1 (iv)
(<7) a = 0, c = 2, mod 4,)
which are related to one another as the types (A), (B), (C] in the scheme (ii) ;
and each class contains six subclasses of forms non-equivalent primarily.
(2) Let D = 5, mod 8. The forms are of one or other of the two types
(A") 6= +1, mod 4, |
(B"] b= -I, mod 4, I '
3 u 2
516 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 41.
which are related to one another in the manner indicated in the scheme
B"
A" =
, a; T
(vi)
Here, again, we must distinguish between the two cases in which the
equation T 2 D U' 2 = 4 admits, and in which it does not admit, of uneven solutions.
(a) Let T z D U 2 = 4 admit of no uneven solutions. In this case all the
automorphics are primary ; and each class contains six different subclasses viz.
ify is any given form, f,fx p,fxp 2 are of the same type asy, and/x ^,/x <r,
f x T of the other type.
(b) Let T' 2 DU* = 4: admit of uneven solutions : then the least solution is
uneven, and every third solution is even. There are thus automorphics of each
of the types I , p, p 2 , and each class contains but two subclasses, viz. one of each
of the types (A") and (B").
There is no known criterion for distinguishing CL priori between the two
cases (a) and (b).
When D = l, mod 8, the criteria (v) as well as the criteria (iv) apply to the
improperly primitive forms ; and thus, if (a, 6, c), (a, b', c') are two equivalent
improperly primitive forms of such a determinant, they are or are not primarily
equivalent according as the congruences
a = a, b = b', *c = c', mod 4 (vii)
are or are not satisfied.
When D = 5, mod 8, the conditions (vii) are necessary, but not sufficient,
for the primary equivalence of two equivalent improperly primitive forms.
The conditions
a=a, c = c, mod 4 . . (viii)
are necessary (but, of course, not sufficient) for the primary equivalence of any
two equivalent forms (whether properly or improperly primitive).
Secondly : If the determinant is a square, or a negative number other than
1 or 3, each class contains six subclasses of forms primarily non-equivalent.
Art. 42 ] BINARY QUADRATIC FORMS. 517
Thirdly, for the determinants 1 and 3 : there is but one class of forms of
determinant 1 ; this class contains three subclasses, viz. one belonging to each
of the sets (A), (S), (C) ; and each of these subclasses (besides the identical pair
of automorphics + 1) has a pair of automorphics which are respectively of the
types ^, T, +0-.
The class of improperly primitive forms of determinant 3 contains two
subclasses, viz. one belonging to each of the sets (A") and (B"\ and each sub-
class has three pairs of automorphics, viz. a pair of each of the types +1, +p,
+ p 2 ; the properly primitive class of determinant 3 contains six subclasses.
Returning to quadratic forms of a positive determinant, two quadratic
forms of a positive determinant, such as (a, b, c) and ( a, b, c) correspond to
one and the same semicircle : in certain cases these two forms are equivalent,
but it follows from the conditions (vii) and (viii), that they can never be primarily
equivalent. Hence the number of subclasses of semicircles of a given determinant
primarily non-equivalent to one another is always half the number of the sub-
classes of forms of the same determinant. Let H and H' be the numbers of
subclasses of properly and improperly primitive semicircles ; h and h' the numbers
of classes of properly and improperly primitive forms of the same determinant.
Let v = 1 or 2, according as the equation T 2 - DU 2 = 1 admits or does not admit
of solutions in which U is uneven ; and let v = 1, or = 3, according as the equa-
tion T 2 DU- = admits or does not admit of solutions in which T, U are uneven ;
we then have
H=%xvxh, H' = vxh' (ix)
42.] Reduction of Quadratic Forms of a Positive Determinant.
A semicircle which enters the reduced space (figure 2, p. 507) is a reduced
semicircle ; its reduced arc is the arc lying within the reduced space. The
boundaries of the reduced space, viz. the semicircles x 1 + y 2 x = and x + 1 =
are improperly primitive semicircles of the square determinant + 1 ; of these,
x + x 2 + y- = 0, and 1 + x = are reduced. The semicircle x = is also an im-
properly primitive semicircle of determinant + 1 ; and the three semicircles
x + x 2 + y 2 = Q, 1 + a; = 0, a; = represent the three subclasses of improperly primitive
semicircles of determinant 1, which exist in accordance with the general theory.
The semicircles (a) - 1 + x 2 + y 2 = 0, (/3) + 2x + x 2 + y 2 = 0, (7) + 1 + 2 x = 0, repre-
sent the three subclasses of properly primitive semicircles of determinant 1.
No other semicircle of determinant 1, besides those which we have named, can
enter the reduced space ; for the improperly primitive semicircles this is evident,
518 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 42.
because the whole of each reduced semicircle lies within the reduced space, and
any point on an improperly primitive circle of determinant 1 must be equivalent
to a point on one of them, and cannot therefore be equivalent to a reduced point
not lying on one of them ; in the case of the properly primitive semicircles
the reduced arc of either semicircle of each pair is equivalent to the unreduced
arc of the other semicircle of the same pair; and thus the semicircle 1 + x 2
+ y* = 0, and the reduced arcs of + 2 x + x z + y 2 = 0, and of + 1 + 2 x are the only
properly primitive reduced arcs of determinant 1. We may observe (i) that the
reduced semicircles of determinant 1 form the boundaries of the twelve regions
considered in Art. 36 ; (ii) that the only semicircles which lie wholly within the
reduced space are the two boundaries 1 + x = 0, x + x 2 + y 2 = 0, and the semicircles
x = and 1 + x* + y* = ; (iii) that every rational semicircle which passes
through one of the cornicular points p, q, r is a semicircle of square determinant.
Any given semicircle of determinant D is equivalent to a certain series of
reduced arcs. For if to be any point on the given semicircle, and 2 be the
corresponding reduced point, as <a describes any arc of the given semicircle, &
will describe the corresponding arc of the equivalent reduced semicircle : but
when Q reaches a boundary of the reduced space, the reducing substitution of
changes, and Q reenters the reduced space at the opposite boundary ; viz. if
before the arrival of Q at the boundary, the reducing substitution is w = | a | x Q,
we shall have immediately after the passage of the boundary w | o | x 1 3 1 x Q,
where | jjj | is | a- ~ 2 ,\a- 2 , | T ~ 2 , | r \*, according as S2 arrives from the interior of 2
upon the boundary $_,, S +l , T_ lt or T +1 . If O arrives at one of the cornicular
points, ia arrives at the same time at the extremity of the given semicircle, and
vice versa.
The number of reduced rational semicircles of a given determinant is finite.
For every reduced semicircle must intersect one at least of the lines x = Q,
+ 1 + x ; but the number of rational semicircles [a, b, c] of a given deter-
minant D, which intersect the line x = 0, is evidently finite, because, a and c
being of opposite signs (except when one of them is zero, as may happen in the
case of a square determinant), the Diophantine equation b- ac = D admits of
only a finite number of solutions ; and, if we shift the semicircles obtained by
its solution to the right or to the left through the distance of a unit, we obtain
the semicircles which intersect the lines + 1 + x - 0.
Art. 43.] BINAEY QUADRATIC FORMS. 519
We have now to consider separately the cases in which D is, and in which it
is not, a perfect square.
43.] Case when the Determinant is not a square,
No reduced arc can pass through one of the points p, q, r thus the reduced
arcs are of twelve different types, distinguished from one another by the
boundaries which they traverse,
[-SU, S +1 ], [S +l , SLA [7-1, T +l ], [T +l , 71,], )
[SLt.l-J, [T-L-SLJ, [fi-i, ZVJ, [T +l ,S_A V . . . (i)
[S +1 , 71,], [71,, S +1 ], [S +l , T +l ], [T +1 , S +1 ~\ ; j
so that, for example, [S_ lt 7" +1 ] is an arc entering the reduced space at S_ l , and
quitting it at T +1 . Two types such as [<S_ 1} S +] ] and [S +l , $_,], or again such
as \S_ l , T +1 ], \T + i, S_A are not essentially different; but it is convenient to
distinguish between them in order to indicate the direction in which the reduced
arc is described. If A l is a reduced arc having S ( , or T t) for the second con-
stituent of its type-symbol, and if A 2 is the reduced arc immediately succeeding
A l (i. e. the reduced arc primarily equivalent to the continuation of A-, immediately
beyond the boundary in which A 1 terminates), the continuation of A 1 is trans-
formed into A 2 by | a- | 2e , or | T | 2 ; A z has for the first constituent of its type-
symbol S_ ( , or T_ t , and enters the reduced space at the point on S_ t , or T_ t ,
symmetrically situated (with regard to the axis of y) to the point on S f , or 7J,
at which A l quits the reduced space ; lastly, the corresponding directions on
A l and A 2 make equal angles with the corresponding directions on the boundaries
S t , S_ ( , or T t , T_ t .
The series of reduced arcs is periodic ; for, if a point describe a given
semicircle, the reduced point Q will describe a series of reduced arcs. But we
have just seen that a reduced arc quitting the reduced space at a point on S_ 1}
S +l , 71,, T +1 , is necessarily succeeded by an arc entering the reduced space
at a point on S +1 , $_,, T +1 , T_ l} symmetrically situated to the former point
with regard to the axis of y. Thus the series of reduced arcs can never terminate,
and as there is only a finite number of different reduced arcs, the same arcs
must recur ; further, each reduced arc determines the reduced arcs which
immediately precede and follow it ; hence the reduced arcs must arrange them-
selves in a periodic series in which they succeed one another in a definite order.
All the reduced arcs must be included in this period, because as <a describes the
given semicircle, the reduced point Q either describes a reduced arc continuously,
or passes discontinuously from the end of one reduced arc to the beginning of
520 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 43.
another perfectly determinate arc ; i. e. can never leave the period of reduced
arcs, and this point must therefore comprise every reduced arc equivalent to the
given semicircle.
To determine the law of succession of the arcs of the period, we shall term
arcs of any of the eight types (ST) or (TS) transitive arcs ; and arcs of any of
the four remaining types (SS) or (TT) intransitive arcs. A succession of intran-
sitive arcs cannot form a period ; and, as the transitive arcs must be alternately
of the types (ST) and (TS), there must be an even number of them in any period.
Let us take as the first arc of the period a transitive arc A lf which we may
suppose to be of the type [T_ fo , S (l ], e , e z representing positive or negative units ;
let A l be followed by MI 1 intransitive arcs of the type (S_ (l , S tl ) ; let these be
followed by a transitive arc (S_ ei , T ( ^), and this in its turn by M 2 1 intransitive
arcs of the type (T_^, T (3 ); let the next arc be a transitive arc of the type
(S_ (2 , S (3 ), and let the series be continued until we arrive again at the transitive
arc A-L with which we began ; the last transitive arc (that immediately preceding
A } ) will be of the type [SL^.j, T J, the suffix 2s 1 being uneven because arcs
of the types [ST] occur in even places in the series of transitive arcs. Writing
2g for f , we find that the matrix
\A\ = \cr\ 2e i^X\T\ 2 ^^X\(r\ 2t 3^X ... X | T | 2t s ** ...... (ii)
is an automorphic of the transitive semicircle AI. For the transformations which
change A l into the successive semicircles of the period are respectively
so that (ii) is a transformation which changes A l into the arc immediately suc-
ceeding the last arc of the period, i. e. into itself.
Let A 1 = [a , /3 , oj, and let 6 represent either root of the corresponding
equation
^ = 0; ......... (iii)
the formula (ii) gives rise to the equation
2 M2 f 2 , ..., 2M 2 / 28) 0]; . . . . (iv)
and hence if 6 2 be the greater of the two roots of the equation (iii), (one of the
roots of that equation is, in absolute magnitude, greater and the other less than
unity, since [o , /3 , a,] is a transitive semicircle) the quotients 2/Ui*i, 2/x 2 e 2 ... may
Art. 44.] BINARY QUADRATIC FORMS. 521
be obtained by developing 6 1 in a continued fraction in which the integral
quotients are the even integers, positive or negative, which approach most
nearly to the complete quotients. The roots of the equations corresponding to
the successive reduced semicircles are
-2e 3 , 3 -4e 3 ,..., 6 3 -
where
^n ^2> ^3, , being all greater in absolute magnitude than unity, and the roots of
the equations which correspond to the transitive semicircles being
*1, -, 3, -,.-..
Thus the succession of the reduced arcs in the period is regulated by the
development in a continued fraction with even quotients of either root of the
quadratic equation corresponding to any transitive semicircles of the period.
According as we develope the greater or the lesser root of the quadratic equation,
the transitive semicircle is of the type [TS] or [ST] ; we may thus describe the
period either backwards or forwards, obtaining two continued fractions which
consist of the same quotients in reverse order.
44.] Comparison of the Geometrical and Arithmetical Reduction.
(1) Given any reduced arc, we can by geometrical construction only, and
without calculation, trace the complete period of reduced arcs ; for the point
of entrance of the arc succeeding any given arc is known, and also the tangent
at that point. Again, if in the diagram exhibiting the complete period of reduced
arcs we begin with the transitive arc A, or [a , /3 , aj, we can, as we have
already seen, obtain the equation (iv), Art. 43, by counting the reduced arcs, and
VOL. II. 3 X
522 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 44.
attending to the order in which arcs of different types succeed each other.
Lastly, if we write that equation in the form
A .. /3
= C, D * 6 '
the integral numbers, a , /3 , c^ are determined by the equalities
C ^D-A -B_
a ~ 2/3 a,
Thus when a single reduced arc is given, we can not only complete the period
geometrically, but can also ascertain the equations of the semicircles of which it
is composed.
(2) On any given semicircle (see figure 3, p. 513) let PP l be an arc cut
off by any tangent of the ellipse corresponding to the fundamental primary auto-
morphic. While the point describes the arc PP 1} the point Q describes the
complete period of reduced arcs, and describes that period once only. For as
soon as a>, after travelling from P to P^, passes beyond P 1} it begins describing
an arc primarily equivalent to the arc PP l ; therefore, as soon as <a passes P lt
tt must begin the period over again ; and, until &> has passed P, , this repetition
cannot begin, or we should obtain an automorphic of the given semicircle trans-
forming points of the arc PP 1 into points of the same arc, which is impossible if
PPi is an arc of the fundamental homography.
But, while <a describes the arc PP l , ii may occupy the same position more
than once, because points on the arc PP l may be primarily equivalent, though
not by an automorphic of the semicircle. Whenever this happens, the chain of
reduced arcs intersects itself ; and a certain number of such intersections does in
general occur in every period.
(3) We have here assumed that all the primary automorphics of a quadratic
form are powers of one of them. But this theorem may itself be established
geometrically ; for this purpose it is sufficient to consider reduced forms only.
Let F be the reduced arc of the reduced semicircle A ; and let there be an
automorphic V of A by which F is transformed into y, y being an arc of A which
lies wholly outside the reduced space. If a point a> setting out from any point
F, of F travel along the circumference of A until it arrives at the equivalent
point y 1 of y, the equivalent reduced point Q will in the mean time describe the
period of reduced arcs a certain number of times M in the positive or negative
Art. 44.] BINAKY QUADRATIC FOBMS. 523
direction, returning at last to the point I\. Thus the equation connecting y l
and I\ will be
ri = |^|"xr i;
i. e. the automorphic V is a power of the fundamental automorphic (A).
(4) If we fold the reduced space over itself so as to bring the boundary ^
on *S_! and T^ on Jl u we may regard the chain of reduced arcs as forming an
unbroken curve. We may, in fact, regard the reduced space as forming a quasi-
cylindrical surface, having the singular point p at an infinite distance, and
closed at its lower end by the semicircular curve qr joining the two singular
points q and r ; we may imagine the surface inflated so that qr is not a singular
line upon it.
(5) To obtain the period of semicircles primarily equivalent to a semicircle
of which the arithmetical symbol [e^, 6 , c^] is given, we may employ an algorithm
resembling that of Gauss for the reduction of a quadratic form of positive deter-
minant. We compute the series of equivalent (but not primarily equivalent)
forms,
(a , 6 , a^, ((%, b n a 2 ), (a 2 , Z> 2 , a 3 ), (a 3 , & 3 , a 4 ), ----
where a, +1 is determined by the equation
and & which lies as near as it can to ^/A by the congruence
6 + &,_, = 0, mod 2 a,.
From this series we derive the series of primarily equivalent semicircles
[a a , & , aj, [a 2 , - b l} aj, [a 2 , & 2 , a 3 ], [a 4 , - 6 3 , aj, .....
which eventually becomes a periodic series of transitive semicircles, the first
period beginning with the first transitive semicircle. If [ , /3 , a x ] = [a z A , 6 2 h , a 2 h + ,]
is a transitive semicircle occupying an uneven place in the series, we obtain the
formulae of the last article by writing
l_ / \j
~ \ ) e j^-
In the period of transitive semicircles the coefficients ft are alternately
positive and negative ; and two consecutive semicircles are of the types (TS),
3x2
524 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 45.
(ST), or of the types (ST), (TS), according as they have the same last coefficient,
or the same first coefficient.
If we use ^/D instead of + ^/D throughout the process we obtain the
same period of transitive semicircles, but in a reverse order.
If the semicircle which we have to reduce is given, not by its arithmetical
symbol, but geometrically, the preceding algorithm indicates the following series
of operations : Let Ci be the given semicircle, C, either of its terminal points.
(i) We substitute for C, an equal equivalent semicircle C 2 , such that the point c 2
equivalent to c t shall lie between 1 and + 1. (ii) We substitute for C z an
equivalent (but not a primarily equivalent) semicircle (7 3 , inverse to C 2 with
regard to x 2 + y z = l, so that the point c 3 , equivalent to c z , lies outside of the
segment ( 1, +1). The alternate repetition of these operations gives eventually
the period of transitive semicircles ; the alternate semicircles of the period not
appearing themselves, but being represented by their inverses with regard to
To the algorithm for the reduction of a quadratic form of negative deter-
minant there corresponds (in a similar manner) a geometrical process for the
reduction of a point ; and the substitution which reduces a form < of negative
determinant is also a reducing substitution for every quadratic form f of a
positive determinant harmonically related to < ; and, vice versd, every reducing
substitution of /also reduces an infinite number of forms ^> of negative deter-
minant harmonically related toy!
45.] Case when the Determinant is a Square.
Let D = A 2 ; we may suppose A positive ; and, as we have already discussed
the reduced semicircles of determinant 1, we may also suppose A > 1.
An extreme semicircle is a reduced semicircle passing through one of the
points +p, q, r ; two of these points cannot lie on the same semicircle, since A > 1 ;
we may therefore regard the extreme semicircles as being of three different
types, (p), (q), and (r).
Theorem. " The reduced semicircles, equivalent to a given semicircle F of a
square determinant, comprise two, and only two, extreme semicircles ; the
remaining reduced semicircles form a chain of reduced arcs connecting the
reduced arcs of the extreme semicircles."
" If F is properly primitive, the extreme semicircles are of the types (p,p),
(2. ?). ( r > r )> according as T is of the type (A), (11), (C)."
Art. 45.]
BINARY QUADRATIC FORMS.
525
" If F is improperly primitive, the extreme semicircles are of the types
(q, r), (r, p), (p, q), according as F is of the type (^4'), (B'), (C")."
Let F = [a, b, c] ; and let us consider the quadratic form/ corresponding to F,
/ = (a, b, c) (x, y) 2 = m (px +p'y) (qx + q'y),
where m = 1, or 2, according as F is properly or improperly primitive ; p, p', and
2A
again q, q', are relatively prime ; and pq f qp' = - -
To demonstrate the theorem, we observe
(1) That if f is transformed into
F=(A,B,C)(x,yy = m(Px + Fy) (Qx + Q'y)
,/3
by the matrix
we have the equation
p,p
7,
Q,Q'
and hence, if ' ^ be primary, the congruence
P'P
, mod 2.
(2) That the form corresponding to an extreme semicircle is of one or
other of the types
2 A
m (rjX + n'y) (\X + \'y), r)\' - q
m
rj'-\'r) =
m
where 17 and y' do not surpass unity in absolute magnitude, and X and X' are
2A.
different from zero, and do not surpass in absolute magnitude.
(3) That if F is properly primitive,
1,1
1,1
0,1
0,1
P>P
q> c i
1,0
1,0
is of the type
, mod 2,
526
MEMOIR ON THE THETA AND OMEGA FUNCTIONS.
[Art. 45.
according as F is of the type (A), (B), (C) ; and that if F is improperly primitive,
P>P
is of the type
1 or vj' ; p or <r ; p 2 or T ;
according as F is of the type (A'), (R), (C'}.
Hence the extreme semicircles, if they exist at all, are of the types
assigned in the theorem. It only remains to show (1) that yean be transformed
by a primary matrix into one, and only one, form of each of the two types (i) ;
(2) that the two forms thus obtained give essentially different extreme semicircles.
Consider the equation
P'P
X
n, i
X,X'
in which
is a primary unit matrix, and n, q', X, X' are subject to the in
equalities stated above. We regard p, p', q, q' as given numbers ; a, /8, 7, S, TJ, rf,
X, X' as numbers to be determined, and we have to show that the equation
admits of one solution, and one only. For brevity, we attend only to the case
in which 17 and rf are both units. We first determine a and 7 by the conditions
1/7 = 7, a=l, mod 4, 7 = 0, mod 2,
* m '
which are always satisfied by one, and only one, set of values for a, 7, and i\
the equations
then give
We have also
X' =
The last of these equations determines the unit 17', for i?'X must be negative
2 A
in order that \' may be less in absolute magnitude than .
Thus the form
<j> = m (nx + tj'y) (X x + X'y),
equivalent to/, is completely determined ; similarly a form
<t>i "= m (i/jX + ijjy) (\x + \iy) t
Art. 45.]
BINARY QUADRATIC FORMS.
527
equivalent to/, may be found from the equations
p,p
In
And these two forms correspond to essentially different semicircles ; for the
equation <p l = < is impossible, and the equation fa = <p, or
i, n
\,\ r
= +
2A
implies the equation = 2;orA = m = l, contrary to the hypothesis that A > 1.
We may obtain the reduced arcs equivalent to a given semicircle F of a
square determinant by the following geometrical construction : Determine a
reduced semicircle equivalent to F (Art. 44); and continue its reduced arc in
both directions (or in one only if it is an extreme semicircle) by a chain of reduced
arcs. The same considerations which we have already employed in the case of
determinants which are not squares show that every reduced arc must be
included in the chain. But when we arrive at the extreme semicircles (or at
the extreme semicircle, if the semicircle from which we set out be itself extreme)
the chain must terminate ; for an extreme semicircle cannot be continued (in the
direction of its extreme point) beyond the boundary of the reduced space; in
fact, the extreme points of the two extreme semicircles answer to the extreme
points of F ; and while ca describes the circumference of F from one of its extreme
points to the other, the reduced point Q describes the complete chain of reduced
arcs running from the extreme point of one of the two extreme semicircles to
the extreme point of the other.
When F is given arithmetically by its symbol, we may obtain the symbols
of the chain of reduced semicircles in the following manner : We first determine,
by the arithmetical process given above, one of the extreme semicircles primarily
equivalent to F ; and then by the same process we obtain the substitution which
transforms this extreme semicircle into the other. Let this substitution be
where the first and last exponents may be zero ; the substitutions
528
MKMOII! OX TIIK TIIKTA AND OMEGA FUNCTIONS.
[Art. 45.
applied successively to the first extreme semicircle, give the complete chain of
reduced arcs taken in order.
a,
When the chain of reduced arcs is given graphically, the matrix
7,3
IS
found by counting the reduced arcs of the different types ; the equations of the
extreme semicircles are then also known ; viz. one of the factors of the form
(a, b, c) answering to the first extreme semicircle is x + y, y, or x, according as
that semicircle is of the type + p, q, or r ; and the other factor is
(yS)x-(aP)y, yx-ay,or$x-py }
according as the second extreme semicircle is of the type +p, q, or r.
Example.
Let the proposed semicircle be [8, 9, 7], D = 81 - 56, = 25. We have
(8,9,7) (x,yY = (2
and we find successively
2,1
4,7
0,1
-10, 7
1,0
-2,1
- 7, -2
- 10, - 3
0,1
-10,7
-10, -3
0, -1
- 7, -2
-10, -3
= |<r| 2 x|T|- 2 x|cr| 4 .
Hence the chain of reduced circles equivalent to [8, 9, 7] is
[0, -5, 7], [8, 9, 7], [8, -7, 3], [-8, -1, 3], [0, 5, 3].
Again, if we suppose the circle [8, 9, 7] to be given geometrically, but its
equation (or its symbol) not to be given, we have first to construct, as in Art. 44,
the chain of semicircles ; the aspect of the chain gives us the matrix
x|r|
-2
X I (T | =
- 7, -2
-10, -3
which transforms the first of the extreme semicircles into the second ; whence,
finally, observing that the extreme semicircles are each of the type (q), we find
that the symbol of the first of these semicircles is [0, - 5, 7].
Art. 46.] THE MODULAR FUNCTIONS. 529
*
Arts. 46-51. GEOMETRICAL EEPRESENTATION OF THE MODULAR FUNCTIONS
<(>) AND ^(co) [ = < 8 (>) AND ^ 8 (o>)].
46. Discussion of the Equation <&() = A.
Let A be a given complex quantity having any value whatever except
0, +l,oo. The following theorems (of which the third results from the com-
bination of the first and second) are of fundamental importance in the theory of
the modular functions.
(A) ' A complex quantity (having the coefficient of i different from zero
and positive) can always be assigned satisfying the equation (f> s (Q) = A.'
(B) ' If < 8 (Q) = < 8 (w), the complex quantities and to (in each of which the
coefficient of i is different from zero and positive) are primarily equivalent.'
(C) ' The equation <"(<) = A always admits of one, and only of one, reduced
solution.'
These propositions are immediate consequences from a general theorem of
Biemann (' Inaugural Dissertation,' Art. 21), relating to the representation of one
plane surface upon another ; but they may also be established by means of known
properties of elliptic functions.
(A) The differential equation
^), ......... (i)
taken with the initial conditions
x = 0, u = 0, - = 1, ........ (ii)
defines u = \(x) as a uniform doubly periodic function of x, perfectly determined
at every point of the plane upon which the complex variable x is represented.
It is always possible (and, indeed, in an infinite number of different ways) to
take as conjugate periods of \(x) a pair of primary periods, i. e. a pair of periods,
4Z, and 2iL', verifying the equations
(a) A() = +l, (1) \(L + iL')=-
' T '
and also satisfying the condition (c) that the coefficient of i in the quotient -j-
shall be positive ; this coefficient cannot be zero, because the quotient of any
two conjugate periods of a uniform doubly periodic function is always imaginary.
VOL. ii. 3 Y
530 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 46.
For if we represent by Z t and tZ/ the rectilinear integrals
du . T , r~ V2 du
===== , tZj = /
Au' 2 ) 'i
the initial sign of the radical in the former integral being positive, and the sign
of the radical in the second integral being so taken as to satisfy the condition
(c), while the sign of- - is the same as in the equation (b), it will be found that
JA
, and ZiLf are conjugate periods of \(x), and that the equations (a) and (b)
are satisfied, i.e. that [4.L lt liLf] is a pair of elliptic periods of \(x)*. There is
an infinite number of pairs of primary periods, for if ' be any primary unit
matrix, the equations (a), (&), and the condition (c) are satisfied by the quantities
iL' =
(the sign of >^/A in the equation (b) changing when y is unevenly even). If
* For the method by which this result is obtained, see MM. Briot et Bouquet, 'Theorie des
Fonctions Elliptiques ' (Paris, 1875), pp. 351-368.
By a pair of conjugate periods we understand a pair of periods forming an elementary parallelo-
gram of the doubly periodic function (ibid. pp. 231-234). The primary periods defined in the text
differ from the elliptic periods of MM. Briot and Bouquet, only because we have left the sign of the
radical i/A undetermined in the equation (b). This enables us to enunciate the theorem (D) :
' If [4-t 1; 2 ?'-/] is any given pair of primary periods, all pairs of such periods are included in the
formula [4.L, 2iL'] where
a ./3 ./r -r >\
(L, iL')=
and
a,/3
is a primary matrix.'
If we adhere to the definition of MM. Briot and Bouquet, who suppose the sign of vA to be
fixed, we must add the condition that y is to be evenly even. The theorem (D) is equivalent to the
theorem (B) of the text; for both [4A"(<o), 2tA'(o>)] and [4A"(ii), 2t'A'(il)] are pairs of elliptic
periods of one and the same function u = \(x).
In the Report on the Theory of Numbers (Reports of the British Association for 1865), Art. 125,
pp. 330 et seq. [vol. i. p. 295], the theorems (A) and (B) were enunciated; and the principle of the
demonstration here given was indicated. M. Hermite has recently called attention to the importance
of the theorem (B) in a note on a Memoir by M. Fuchs (' Borchardt's Journal,' vol. Ixxxiii. p. 29) ; and
a demonstration of this theorem, depending on the theory of elliptic functions, has been given by
M. Dedekind (ibid. pp. 266-269), who has also observed (p. 274) that the properties expressed by the
theorems (A) and (B) follow from the principle of Hiemauu. See also the Note on the primary periods
of the elliptic functions, cited in the footnote, p. 507.
Art. 46.] THE MODULAR FUNCTIONS. 531
[4i, ZiU] is any pair of primary periods, the zero points of X (x) are
and its infinite points are
2 mL +(2n+l)iL',
m and n denoting any positive or negative integral numbers ; and the zeros and
" 7"'
infinities being all simple. Let -j- = >, and let K and K' be determined by the
/"' Tf
equations K=K(a>), K' = -K(w); since y- = -??, we may write
1 JL/ ji.
X = , L' = ;
and we shall find that the doubly periodic function
x - - = sm am
has the same periods 4L and 2iL' as X (x) ; it has also the same zero points and
the same infinite points ; hence, by the principles of the Theory of Functions,
X (x) = C x sin am (v-x),
C denoting some constant multiplier ; but X (L) = 1 ; and
sin am (nL) = sin am (K) = 1 ;
i.e. C=l, and \(x) = sin am (/u x). Lastly,
j.I/= =
but
sin am (/mL + inL') = sin am (K+ iK') = , . ;
whence we infer
We may add (though this is not required to complete the preceding demon-
stration) : (1) that M = + 1 ; for, when x = 0,
,. sin am (ux)
am -- s - = 1.
ftX
but also
,. sin am (u cc) ,. X()
lun -- '- = lim - 1 - ' = 1.
x x
3 Y 2
532 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 46.
(2) that the equation (b) is always satisfied when the equation (a) is satisfied ;
viz. if, of two conjugate periods 4Z and 2iL', one satisfies the relation \ (L) = 1,
the equation X (L + iL') = + is always satisfied ; for any two conjugate
periods may be expressed by the formula
+ 2/3 1'4', 2iL f =
where aS (3y = + 1. But if A (L) = 1, we must have a = 1, mod 4, and /3 = 0, mod 4.
Hence is uneven, and
_ I \ /__
JA
(B) If < 8 (<) = < 8 (Q) = A, we form the two elliptic functions
C\ / TiC
Each of these functions satisfies the differential equation (i) and the initial
conditions (ii). Hence the two functions are identical, and their zero points and
their infinite points are the same. Considering the zero point 2K(a>), and the
infinite point iK'(<a), of the former function, we must have
K(u) = aK(0) + ipK f (Q), iK'(<o) = ' Y K(Q) + iSK'(Q); . , . (iii)
where y is even and uneven. Similarly, we should find
JST(Q) = a 1 ^'(a>) + t j S 1 ^'(< B ), iK f (Q) = <y l K(<*) + i3 l K'(*),. . . (iv)
where y l is even and S 1 uneven. But, if a ^y m, we may obtain from (iii)
K(Q) = - KM - i ^ K'(), tA"(Q) = - - ^() + 1 - K'(>) . (v)
m ^ ' m ' m m ^ '
* E 7 "'/ \
The systems (iv) and (v) must be identical ; for otherwise the quotient -. . = w
would be real. Hence m divides a,(3,y,$; i.e., m=+l. The negative sign
, $o
must be rejected, because the sign of i in Q and in a>, = - ^- , is the same ;
therefore m = aS fiy = + 1. Since a = S ly a is uneven : and since
sin am [aK(Q) + i/3K'(Q)] = + 1,
Art. 47.] THE MODULAR FUNCTIONS. 533
/8 must be even, and a = 1 , mod 4. Thus the matrix
is primary, and w,
are primarily equivalent.
47. Limiting Values of < (a>).
We shall now suppose that the point o> is confined within the reduced
space (see figure 2, p. 507), and we shall examine the values assumed by the
function < 8 () in the vicinity of the cornicular points p, q, r. Let T be a
constant quantity included between the limits 1 and + 1, so that 1 < T < + 1 ;
and let a- be a positive quantity increasing without limit.
(i.) Let o> = T + icr, so that a> moves along the straight line X = T, and
approximates to the cornicular point oo, or r. It is evident from the equation
of definition of (p () that
i. e. 8 (w) approximates without limit to zero, travelling in the direction indicated
by the vector e ivT .
(ii.) Let w = --- , so that as or increases without limit, o moves along the
T "7" % (T
semicircle T (x 2 + y 2 ) + x = toward the cornicular point 0, or q. Then
= 1 1 6 e ~ w<r (cos TTT + i sin TTT) ultimately ;
i.e. < 8 (eo) converges to +1 in the direction indicated by the vector e" T ; that
direction depending on the curvature of the circular path pursued by co.
(iii.) Let T be positive, and let a> = 1 -- r- , so that a> converges to the
T ~\~ i/ &
cornicular point + 1 or p, moving along the semicircle
T [(x-l) 2 + y*] + x-l = 0.
Here ^ )8 ( a) ) = l~^7 -- ; = ^~ ^""(COSTT ismvr) ultimately, omitting quan-
<p yT "T ?- 0"j
titles of the order e"""; so that $ (<o) becomes infinite, and travels, as <r increases,
in the direction of the vector e~ rir . If T is negative, we write w = 1
so that co converges to the cornicular point 1, or p, moving along the semicircle
T [(x+ I) 2 + y 2 ] +x + 1 = 0, and we obtain the sameresult as before.
634 MEMOIR ON THE THETA AND OMEGA FUNCTIONS. [Art. 48.
The function $ 8 (a>) cannot attain the values 0, 1, oo at any point of the
reduced space other than the cornicular points oo, 0, +1, respectively. For
if <r be finite and different from zero, the infinite products II (l + q' M ~ l ) and
II (1 + 5 2m ) are finite and different from zero ; again, if <r converge to zero, so that
w approximates to the cornicular point or +1, <"(&>) converges (as we have
just seen) either to 1 or to oo ; hence < 8 (w) cannot be zero, unless <r be infinite,
i. e. unless to approximate to the cornicular point oo ; and by using the trans-
formations w = TT , <o = + 1 o we mav P rove the corresponding assertions for
i2 &6
the other cornicular points.
48. Transformation of the Reduced Space by the Modular Functions $(&>)
and ^(w).
Denoting by X and Y real quantities, we shall now write
and taking a pair of rectangular axes X, Y, in a new plane (X Y), we shall regard
the point X + iY as answering to the point o> in the reduced space. We are
thus in fact mapping the reduced space upon the plane XY ; so that by a well-
known theorem to which we have already referred, the transformation is homceo-
meric, i. e. the infinitesimal parts of the two figures are similar except at points
at which the first derived function of the mapping function is infinite or zero.
The derived function of <(<) is (Art. 9)
4i
TT
which is always finite and different from zero, so long as the real part of ia> is
finite and different from zero ; i. e. so long as o> does not approximate to one of
the cornicular points. If (o = r + i<r approximate to the cornicular point oo,
we find
Lim <&'(a>) = i7rLim<I ) (co) = IQive'"" (cosirr+l'sinTrr) ;
i. e. $'(*>) converges to zero. Writing Q = T + i<r, and employing the substitutions
I i_i
which give
Art. 49.] THE MODULAR FUNCTIONS. 535
we cause to approximate to the cornicular point GO, and we infer that if
approximate to the cornicular point 0,
Lim <' (co) = - 16 i-7!-(a- + ir) z e~ m (coa TTT + i sin XT) = ;
and that if a> approximate to the cornicular point + 1,
Lim <>' (co) = ygi'Tre' (cos ITT i sin TTT) (T + i<r) 2 = GO.
Thus the infinitesimal similarity of the two figures holds at all points of the
reduced space except at the cornicular points GO, 0, + 1, to which there correspond
in the plane (XY) the points ( f, 0) or A_ l} ( + ^, 0) or A + l , and the point at
an infinite distance ; spaces in the vicinity of the cornicular points GO and
being infinitely contracted in the plane (XY) and spaces in the vicinity of the
cornicular points + 1 being infinitely expanded.
The correspondence between the reduced space and the plane (XY) is a
correspondence one to one. For to every point in the reduced space there
answers one, and only one, point in the plane (XY); and vice versd to every
point in the plane (XY) there answers one, and only one, point in the reduced
space ; this results either from the general theorem of Iliemann, or from the
proposition (C) of Art. 46, and from our examination of the values of $(&>) in the
vicinity of the cornicular points.
49. Lines answering to the Semicircles of Determinant + 1.
We shall now determine the lines which in the plane (XY) answer to the
reduced arcs bounding the twelve spaces into which the reduced space is divided
(Art. 38). See figure 2 (p. 507) and fig 
