THE

THEORY OF PLANE CURVES

THE

THEORY OF PLANE CURVES

BY

SURENDRAMOHAN GANGULI, D.Se.LECTURPlR IN HIGHER GEOMETRY IN THE UNIVERSITY OF CALCUTTA,

PRElIICHAND ROYCHAND SCHOLAR, MEMBER OF THE

CALCUTTA IUTHEMATICAL SOCIETY, AUTHOR OF

ANALYTICAL GJCO~E'l'R;Y" OF HYPERSPACB:S,

,ETC., ETC, ,

VOLUME I

SECOND EDITION(Tborougb/y rev/sed slid ellisF'led)

PUBLISHED BY THE

UNIVERSITY OF CALCUTTA1925

~ll ,ig1tta reaerveil

,-t~~PAINTED BY BiuPBN»&'LAL BANEBJSE

AT ft. UNI"BSITY P•• BB

In

Memory of

SIR ASUTOSH MOOKERJEE

••

" PREFACE TO THE SECOND EDITION

The first edition of the present volume, published'ome years back under the title " Lectures on the Theoryof Plane Curves," was designed to meet the syllabusprescribed by the University of Calcutta forthe Master'sDegree, and intended as an introductory course suitablefor students of higher geometry, scarcely assuming anyfurther knowledge of higher analysis on the part of thereader than is to be found in most of the ordinary textbooks on Calculus and Plane Analytical Geometry. Sincethen it has been suggested that the book should be sorevised and enlarged as to include materials which wouldnot only be of use to the students for the Master's course,but also encourage independent thinking in students ofhigher studies engaged in research work. In the prepara-tion of the second edition, therefore, special care hasbeen taken to incorporate recent researches as far aspossible and to indicate references to original sources as faras practicable. In fact, almost all the chapters havebeen re-written and the articles re-numbered, while fiveadditional sections-Chapters VII, X, XI, XII, XIII-havebeen inserted and a large number of examples givenillustrating the subject-matter and serving as exercises forstudents. The volume contains an exposition of the generaltheory of plane algebraic curves in its various aspectswith applications to conics, cubics, quarties, etc.

In writing on Higher Geometry, it is always aproblem to determine what matters to exclude, and indealing with a subject, so wide in its scope, whichattracted so many workers and has been so muchdeveloped in recent years, specially by the Italian

viii Pltll:l'AOll:

Geometers, it has not been possible to do full justice to allthe importa.nt topics; in consequence, some have receivedfuller attention, while others of equal or greater importancehave been little noticed or even omitted altogether. Itis hoped, however, that the book will afford somescope for independent thinking and research, when thestudent enters upon a systematic study of plane curv•.

In the preparation of the volume, constant recoursehas been had to the classical works of Salmon, Clebscb,Cayley, Cremona, the works of Basset, Teixeira, Scott,Wieleitner, Loria and the papers of Zeutben, Brilland Nother, Castelnuovo and others, published in the.arious Journals and Periodicals. Prof. Pascal's .R~.tori'Unt der ltolteren Geometrie was of great use in supplyinga number of important references. Since the publicationof the first edition, Prof. Hilton has published his"Plane Algebraic Cnrves," which has also been studiedwith much advantage. My obligations to these authors,greater than I can confess, are gratefully acknowledged,and it is impossible to record in detail my obligation ~ thegreat inspiring work of Salmon-Higher Plane Curves.I had no access to the recent work of Enriques,-Lezionimlta teoria geometrica deUe equazioni e deUe JunzionialgelJriclle-2 YoIs, 80 highly spoken of, and it is likelythat the present volume could have been made much moresuggestive, had I had the opportunity of consultingthis book.

In concluding this preface, I must take the opportunityof recording my indebtedness to Prof. J. L. Coolidgeof Harvard for his very valuable suggestions for theimprovement of the work, and to Mr. A. C. Bose, Controllerof Examinations in the University of Calcutta, for hisextreme kindness in giving valuable hints and suggestions.

puP.•..C. IX

. My best thanks are also due to my former pupil Mr. L.Murthi, M.A., who made a number of important sugges-tions and pointed out several printing and other errors inthe preparation of the edition.

With a keen sense of sorrow and gratitude, I record myindebtedness to the late lamented Sir Asutosh Mookerjeeformer Vice-Chancellor and President of the Post-graduateCouncils in the University of Calcutta, whose untimelydeath has been deeply felt by all Indian workers in thefield of mathematics, pure or applied. It was at hissuggestion that I was induced, difficult as the task was,to revise the original lecture notes for the Press, and itwas he who helped and encouraged me to bring out a.second and revised edition) but it is a matter of profoundregret that he was not spared to see its completion. Asan humble token of gratitude, this volume is dedicatedto his revered memory.

UNIVERSITY OP CALCUTTA,

Marek, 1925.S, M. G.

'III

CONTENTS

CHAPTER I

INTllODUCTION

Co-ordinatesThe Special Line at InfinityCartesian as a. Special System of Hcmcgeneena

Co-ordinatesTangential or Line Co-ordinatesRela.tion between the Co-ordinates of a, Line and

those of a Point on it ...Tangential EquationThe Circular Points at InfinityThe Co-ordinates of the circular points at infinityProperties of the Circular Points at InfinityProperties of the Line at InfinityTheory of ProjectionAnalytical aspect of ProjectionFigures in PerspectiveAnalytical Treatment of Plane PerspeetiveTheory of InversionReciprocation

CHAPTER II

PLANE ALGEBRAIC CURVES

Section I-General Properties

Notion of algebraie CurvesRepresentation of Functions

...P.lH

12

4r67t

1011111814-151719

2112

1r

xii CONTlINT8

The general equation of a curve of the nthdegree ... ..••.. 2S

Number of points determining a curve oforder n 24

Degenerate and Non-degenerate Curves 25

Intersections of Curves 26Curves through !n(n+8)-1 points !8Chasles' Theorem on the intersections of two

cubics 31:OergQl',lJ)e'sTheorem on the intersections of two

curves 32Cayley's Theorem 84

.!

Section II-Theory if Reliduation" .

.Theory of ResiduationPrinciples of Residuation explainedAddition Theorem on ResiduationSubtraction TheoremMultiplication Theorem ...Brill-Noether's Residual TheoremExtension of the Residual Theorem

86373839404344

CHAPTER III

SINGULAR POINTS ON CUR VIIS

Singular Points on CurvesPoints of InflexionPoint of UndulationMultiple Points

47495051

CONTlINTI xiii

Investigation in Trilinear Co-ordinatesMultiple point of order Ie ••.

·Conditions for a Double PointSpecies of Double Points ...Investigation of the Species of Double PointsIntersection of Curves at Singular PointsLimit to the number of double points ...Deficiency of a curveU nieursal CurvesComplex Singularities·Singular Points at Infinity,Multiple Tangents·Reciprocal Singularities

PAOB52535456576062636570717374

CHAPTER IV

THEORY OJ' POLES .AND POLARS

Theory of Poles and Polars 77Polar Curves defined 78Mixed Polars... 83

·Equation of the tangents drawn from any pointto the curve 84

Geometrical interpretation of the equation of PolarCurves .. 86

Centre of a Curve 88A Theorem of Mac La.urin... 91Polar Curves of the Origin 91Polar line of a point 95Poles of a. right line 98Polar Curve of a.point on the curve 99The class of a curve 101

lt

xiv

P.lGII:

The first~ polar of any point passes through thedouble points 103

Multiple points on polar cunes 104,

CHAPTER V

COVUUNT CUXVU-TBl!: HESSIAN, TaB SUUllmUN

AND THE CA TL~TAN

Covariant Curves 107The Hessian defined 108The Steinerian 11SThe class of the Steinerian 116A Theorem 120The Cayleyan 121The class of the Cayleyan 121The Hessian passes through the double points 123Multiple points on the Hessia.n U7Harmonic Polar of a point of inflexion 128Number of points of inflexion on an n-ic 121Discrimination of a double point from a point of

inflexion ... 180A Theorem on the inflexions of the first polars 181Points of inflexion on a curve with singular

points 188

CHAPTER VI

POUR RECIPROCAL AND OTHER DERIVED CURVES

Polar Reciprocal Curves defined 185Polar Reciprocal in homogeneous co-ordinates 137

•...

CONTBKT8

PA.C.Tugential equation derived from point-equation 138point-equation derived from tangential equation 140polar Reciprocal by the Principle of Dua.Iity 142Singularities on reciprocal curves USDegree of the polar reciprocal curve 144EDvelopes (one parameter) 145Envelopes (two or more parameters) 148Evolutes 153Normal of the evolute 154Tangential equation of the evolute 155Caustics defined 156Equation of katacaustic 157Caustic by reflexion of a circle 158Tangential equation of the Caustic 160Intersections of the Caustic with the reflecting

circle 162Caustic by refraction of a straight line 163Secondary caustics 165Pedal Curves ... 167The Cartesian Equation of the Pedal 168Inverse Curves 170Parallel Curves 171Isoptio Loci ... 173Orthoptic Loci 115Equation of the Orthoptic locus when the polar

equation of a curve is given 179

CHAPTER VII

CHARACTERISTICS OF CUltVES

Plucker's EquationsPliick~rian cbaracteristics defined

un181

XVl CONT:&NT8

P.lGJ:The Bitaogential CurveDeductions from Plucker's FormulaeThe point and line deficienciesCurves with the same deficiencyThe Characteristics of the HessianThe Characteristics of the SteinerianThe Characteristics of the CayleyanThe Characteristics of the Inverse CurveThe Characteristics of the PedalThe Characteristics of the EvoluteThe Characteristics of the Parallel Curves'I'he Characteristics of the orthoptic locusThe Characteristics of an isoptie locusOther Derived Curves

183185187188191192193193195197201202207207

CHAPTER VIII'

FOCI OJ' CURVES

Plucker's conception of fociFoci of curves with siogularitiesThe co-ordinates of the fociFoci in tangential equationEquation of confocal curvesDetermination of singular fociA new theory of fociFoci of Inverse CurvesReciprocal with respect to a focusFoci.of Circular Curves

209211214216216217220223224!25

CONTENTS. xvii

CHAPTER IX

TRACING OF CURVES

Section I-Approzimate Forms of Curves

PAGEAna.lytical Triangle 227Practical Method 229Properties of the analytical triangle 229Use of the analytical triangle in three variables 232Newton's method of approximation 234Application of Newton's method in three variables 236Infinite branches 236Branches with higher singularities 237Determination of the asymptotes 239Special Methods 240Asymptotic Curves 242Parabolic branches 243Circular Asymptotes 244

Section II-Tracing of Curves

Curve Tracing in Cartesian Co-ordinates 245Curve Tracing in homogeneous Co-ordinates 245

CHAPTER X

RATIONAL TRANSFORMATIONS

Rational and birational TransformationsLinear TransformationCollineationCollineation treated geometricallyDualistic TraDsformation '"

2~9250251252253

xviii

Pole and Polar conicsQuadric InversionAnalytical treatmentQuadric Inversion as rational transformationThe Inverse of special pointsThe Inverse of a straight lineProper ,InverseInverse of the line at infinityInversion of special points on a curveEffeets of inversion on singularitiesEJIeets of inversion on a curveApplication of Quadric InversionCircular InversionSpecial Quadric TransformationsNether's TransformationCremona conditionsCremona transformation reduced to a series

Quadric InversionsDeficiency unaltered by Cremona TransformationRiemann Transformation ...Reduction of the order of transformed curveReduction of a curve with multiple pointsAdjoint CurvesIntersections of a curve with its AdjointIntersections with a pencil of Adjoints ...Transformation by Adjointa

CHAPTER XI

UNICURSAl CURVES

Parametric representationClebseh Method

PAGE

25625725825926126126~263263!65266.2£72682..69271272

of2742752762792.81283284284285

281288

COlfTBNTS XIX

PAGEThe order of the unieursal curve 289The class of the unieursal curve 290Parametric representation in line co-ordinates 290Singular Points 292Inflexions 294Bitangents of unieursal curves 295Special class of Rational Curves ~98The circuit of a. unicursal curve 298Unipartite Curves not necessarily unicursal 299Curves with unit deficiency 299Co-ordinates in terms of elliptic functions 300Simplification by Weierstrass's notation 302The Converse Theorem 304

CHAPTER XII

THEORY OF CORRESPONDENCE

Correspondence of points on a curveAnalytical discussionUnited Points ...Chasles' Correspondence TheoremCorrespondence IndexCommon elements of two correspondencesCayley-Brill's Correspondence FormnIaApplications of the Formula

307309310312313314317319

CHAPTER XIII

HIGHER SINGULARITIES ON CURVES

HietoricalSpecies of cusps

821822

xx CONTENTS

P~GEDouble Cusps ... 323Classification of triple points 324Equivalent singularities 325Analysis of higher singularities 326Successive Transformations 328Practical Applications 329Linear and Superlinear Branches 331Application of the method of expansions 334Practical Method 335Expansion of a Function '" 336Discriminantal Index 337Inter.sections of two curves at a singular point 339Expansion in line co-ordinates 343Polar reciprocal of a superlinear branch 347Cuspidal Index 348Extension of Plucker's Formulae 350Curves of closest contact: Osculating curves 352Conics with four-pointie contact 354Transon's Theory of Aberrancy 355Angle of aberrancy 356Aberrancy Curve 358

CHAPTER XIV

SYSTEMS OF CURVES

A pencil of n-U8 361Points having the same polar line with respect to

two curves 362Curves which touch a given curve 363Particular cases 364Tact-Invariant of two curves 364

;'.

CONTENTS :0;1

Generation of a curve 365The Jacobian of three curves a66Net of curves 368The Jacobia.n (or Hessia.n) of a net of curves 368Net of first Polars 369Invaria.nts and Covariants of two ternary forms 370Characteristics of a system of curves 372Relation between the characteristics 373The characteristics of conditions 375

Index 379