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Cornell ftntuetgUg Babrarg Strata, SJem 5«>*h BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE 1891 ( Cornell University Library QA 565.H65 Plane algebraic curves, 3 1924 001 544 216 Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001544216 PLANE ALGEBRAIC CURVES OXFOKD UNIVEESITY PEESS LONDON EDINBURGH GLASGOW NEW YORK TOEONTO MELBOURNE CAPE TOWN BOMBAY HUMPHREY MILFORD PUBLISHER TO THE UNIVERSITY PLANE ALGEBRAIC CURVES BY HAROLD HILTON, M.A., D.Sc. PROFESSOR Or MATHEMATICS IN THE UNIVERSITY OF LONDON AND HEAD OF THE MATHEMATICAL DEPARTMENT OF BEDFORD COLLEGE FORMERLY FELLOW OF MAGDALEN COLLEGE, OXFORD, AND ASSISTANT MATHEMATICAL LECTURER AT THE UNIVERSITY COLLEGE, BANGOR OXFORD AT THE CLARENDON PRESS 1920 PEEFACE THOUGH the theory of plane algebraic curves still attracts mathematical students, the English reader has not many suitable books at his disposal. Salmon's classic treatise supplied all that could be desired at the time of its appearance, but the last edition was published some forty years ago, and has been long out of print. It seemed therefore as if a new book on the subject might be useful, if only to bring some more recent developments within reach of the student. In the preparation of this volume I have made frequent use of the books written by Salmon, Basset, Wieleitner, Teixeira, Loria, &c. But most of the con- tents and examples are extracted from a very large number of mathematical periodicals. With the ex- ception of the list at the end of Ch. XX, I have not attempted to give systematic references. In fact, in a field which has attracted so many workers, it would be almost impossible to trace the steps by which particular results have reached their present form. In some cases I cannot eveli remember whether a result is my own or not ; but Chapters IX, XI, XVII, and XVIII contain most of my own contributions to the subject. The solutions are mine for the most part, even in the case of examples derived from other authors. vi PREFACE In a book dealing with so wide a subject I can hardly hope to escape the criticism that I have in- cluded just that material which happens to interest myself, and have excluded other matter of equal or greater importance. I have not seriously dealt with problems of enumeration, such as ' How many conies touch five given conies?' I have treated all curves with the same degree and singularities as forming a single type, and have not attempted to subdivide the type by considering all their possible positions relative to the line at infinity. I have not given the properties of ' special plane curves % unless they are representa- tive of some general type, such as, for example, Cassinian curves, into which any quartic with two unreal biflecnOdes can be projected. I have not in- cluded any discussion of curves of degree n for special values of n other than 2, 8, or 4. A thorough dis- cussion of quintic curves would be very welcome, but at present the difficulties seem insuperable. At any rate very little work has been published on their properties. The reader will doubtless detect other important omissions. But on the whole I have tried to cover the limited ground I have selected with reasonable completeness. No one can really master a branch of mathematics except by working at it "himself. I make no apology, therefore, for the long lists of examples. The reader can select from them few or many, as he pleases. I give hints for solution in most cases. I hope that these will be of real assistance to the student, setting him on the right track if he is in difficulties, enabling :;; PREFACE vii him to check the accuracy of his results, and giving him a guarantee that the examples are not of un- reasonable difficulty. The beginner should not attempt to read the book straight through, but should select for himself those parts which he thinks easiest and likely to interest him most. As a rough guide I recommend the omis- sion of the following portions on a first reading Ch. VI ; Ch. VII, §§ 8 to 10 ; Ch. VIII, $ 4 and 5 Ch. IX, §§ 3 to 12 ; Ch. X, §§ 7 and 8 ; Ch. XI Ch. XII, $ 7 to 10 ; Ch. XVI ; Ch. XVII, $$ 6 to*8 3 ; Ch. XVIII, §§ 9 to 15 ; Ch. XIX, §§ to 7 Ch. XX, #10 and 11; Ch. XXI. My best thanks are due to friends and pupils who have made suggestions and pointed out inaccuracies while the book was being written. I owe a special debt of gratitude to Miss G. D. Sadd, who has given me very valuable lists of corrections required in the MS.- I must also express my gratitude to the Delegates of the University Press for so kindly undertaking the publication of the book. H. H. June 1919. CONTENTS CHAPTER I INTEODUCTOBY PAGE § 1. Coordinates 1 2. Projection 3 3. Plane perspective . 4 4. Asymptotes . 7 5. -Circular points and lines 6. Higher plane curves 7. Intersections of two curves 10 8. Pencil of curves 12 9. Tangents 13 10. Inversion 14 11. Theory of equations 16 CHAPTER II SINGULAR POINTS 1. Inflexions, &c. 18 2. Double points 21 3. Multiple points 23 4. Conditions for a double point 25 5. Points at infinity . 29 6. Relations between coefficients 33 CHAPTER III CUBVE-TRACING § 1. The object of curve-tracing 37 2. The method of curve-tracing 37 3. Newton's diagram . 38 4-10. Examples of curve-tracing 39 CONTENTS CHAPTER IV TANGENTIAL EQUATION AND POLAR RECIPROCATION PAGE 57 § 1. Tangential equation .... 58 2. Class of a curve ..... 59 3. Tangential equation of any curve . 4. Tangential equation of the circular points 61 5. Polar reciprocation .... 62 6. Equation of polar reciprocal . 62 7, 8. Singularities of a curve and its reciprocal 63 CHAPTER V FOCI 69 § 1. Definition of foci ..... 2. Singular fpci ...... 71 3. Method of obtaining the foci . 73 4. Inverse and reciprocal of foci . 74 CHAPTER VI SUFERLINEAR BRANCHES § 1. Expansion of y in terms of x near the origin 76 2. Intersections of two curves 81 3. Tangential equation near a point . 83 4. Common tangents of two curves 85 5. Polar reciprocal of superlinear branch 86 CHAPTER VII POLAR CURVES § 1. Polar curves ...... 88 2. Equation of polar curves 89 3. Polar curves of a point on the curve 92 4. Harmonic polar of an inflexion on a cubic 95 5. First polar curve ..... 96 6. Equation of Hessian .... 98 7. Intersections of a curve with its Hessian 99 8. The Steinerian , 104 9. The Cayleyan 107 10. The Jacobian of three curves . 108 CONTENTS xi CHAPTER VIII plOcker's numbers PAGE § 1. Plueker's numbers ....... 112 2. Deficiency IIS 3. Multiple points with distinct tangents . .115 4. Multiple points with superlinear branches . .116 5. Higher singularities 118 CHAPTER IX QUADRATIC TRANSFORMATION § 1. Definition of quadratic transformation . 120 2. Inversion * 128 3. The number of intersections of two curves at a given point ......... 124 4. Class of a curve ....... 125 5. Tangents from a singular point to a curve . 126 6. Latent singularities . .127 7. Deficiency 127 8. Deficiency unaltered by quadratic transformation . 128 9. Deficiency for higher singularities .... 129 10. Intersections of a curve with adjoined curves . 130 11. Another transformation ...... 131 12. Applications of this transformation . 133 CHAPTER X THE PARAMETER § 1. Point-coordinates in terms of a parameter . 137 2. Line-coordinates in terms of a parameter . 139 3. Deficiency not negative ...... 144 4. Unicursal curves . • .146 5, 6. Coordinates of a point in terms of trigonometric or hyperbolic functions ...... 153 7, 8. Curves with unit deficiency 156 Xll CONTENTS CHAPTER XI DERIVED CURVES PAGE § 1. Derived curves 161 2. Evolutes. 161 164 3. Inverse curve . 166 4. Pedal curve . 5. Orthoptic locus 169 6. Plticker's numbers of orthopti locus 171 7. Isoptic locus . 174 8. Cissoid . 175 9. Conchoid 177 10. Parallel Curves 179 11. Other derived curves 182 CHAPTER XII INTERSECTIONS OF CURVES § 1". Conditions determining a curve 185 2, 3, 4. Cubics through eight points . 186 5, 6. Intersections of two n-ica 191 7. Intersections of any two curves 193 8, 9, 10. Theory of residuals 194 CHAPTER XIII UN1CURSAL CUBICS 1. Types of cubic 201 2. Geometrical methods applied to unicursal cubics . 201 3. Cuspidal cubics ....... 204 4. Nodal cubics ........ 207 CONTENTS xm CHAPTER XIV NON-SINGULAR CUBICS PAGE 1. Cubics § with unit deficiency . 214 2. Circular cubics ..... 217 3. Foci of circular cubics .... 220 4. Pencil of tangents from a point on a cubic 226 5. The cubic {x -r y + zf + Qltxyz = . 227 •6. Symmetry of cubics .... 229 3 3 7,8. The cubic x? + y + + 6mxyg = . 233 9. Syzygetic cubics . ... 237 CHAPTER XV CUBICS AS JACOBIANS § 1. Jacobian of three conies .... 241 2. Any cubic as a Jacobian of three conies . 244 3. Ruler-construction for cubics . 245 4. Cayleyan of cubic ..... 248 5. Tangential equation of Cayleyan of cubic 249 CHAPTER XVI USE OF PARAMETER FOR NON-SINGULAR CUBICS § 1. A standard equation of the cubic with two circuits . 252 2. Coordinates of any point on a cubic with two circuits 252 3. Parameters of points on a conic or cubic . 254 4. A standard equation of the cubic with one circuit 255 5. Unicursal cubic ...... 256 6. Applications of the parameter 257 7. Another standard equation of the cubic .
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