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An Elementary Treatise on Cubic and Quartic Curves Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementarytreatiOObassuoft 75-^/ 10^ AN ELEMENTARY TREATISE ON CUBIC AND QUARTIC CURVES AN ELEMENTARY TREATISE ON CUBIC AND QUARTIC CURVES BY i. ..^ A\ B? BASSET, M.A., F.R.S. TRINITY COLLEGE CAMBRIDGE t'-i^l.s CAMBRIDGE DEIGHTON BELL AND CO. LONDON GEORGE BELL AND SONS 1901 [All Rights reserved] PRINTED BY J. AND C. F. CLAY AT THE UNIVERSITY PRESS m 56 b PREFACE nnHE present work originated in certain notes, made about twenty- five years ago, upon the properties of some of the best-known higher plane curves; but upon attempting to revise them for the press, it appeared to me impossible to discuss the subject adequately without investigating the theory of the singularities of algebraic curves. I have accordingly included Plucker's equations, which determine the number and the species of the simple singularities of any algebraic curve ; and have also considered all the compound singularities which a quartic curve can possess. This treatise is intended to be an elementary one on the subject. I have therefore avoided investigations which would require a knowledge of Modern Algebra, such as the theory of the invariants, covariants and other concomitants of a ternary quantic ; and have assumed scarcely any further knowledge of analysis on the part of the reader, than is to be found in most of the ordinary text-books on the Differential Calculus and on Analytical Geometry. I have also endeavoured to give special prominence to geometrical methods, since the experience of many years has convinced me that a judicious combination of geometry and analysis is frequently capable of shortening and simplifying, what would otherwise be a tedious and lengthy investigation. ; VI PREFACE. The introductory Chapter contains a few algebraic definitions and propositions which are required in subsequent portions of the work. The second one deals with the elementary theory of the singularities of algebraic curves and the theory of polar curves. The third Chapter commences with an explanation of tangential coordinates and their uses, and then proceeds to discuss a variety of miscellaneous propositions connected with reciprocal polars, the circular points at infinity and the foci of curves. Chapter IV is devoted to Plucker's equations ; whilst Chapter V contains an account of the general theory of cubic curves, including the formal proof of the principal properties which are common to all curves of this degree. In this Chapter I have almost exclusively employed trilinear coordinates, since the introduction of a tnangle of reference, whose elements can be chosen at pleasure, constitutes a vast improvement on the antiquated methods of homogeneous coordinates and abridged notation. Chapter VI is devoted to the consideration of a variety of special cubics, including the particular class of circular cubics which are the inverses of conies with respect to their vertices and in this Chapter the method of Cartesian coordinates is the most appropriate. A short Chapter then follows on curves of the third class, after which the discussion of quartic curves commences. To adequately consider such an extensive subject as quartic curves would require a separate treatise. I have therefore confined the discussion to the simple and compound singulari- ties of curves of this degree, together with a few miscellaneous propositions; and in Chapter IX, I proceed to investigate the theory of bicircular quartics and cartesians, concluding with the general theory of circular cubics, which is better treated as a particular case of bicircular quartics than as a special case of cubic curves. Chapter X is devoted to the consideration of various well known quartic curves, most of which are bicircular PREFACE. vii or are cartesians ; whilst Chapter XI deals with cycloidal curves, together with a few miscellaneous curves which fre- quently occur in mathematical investigations. The theory of projection, which forms the subject of the last Chapter, is explained in most treatises on Conies ; but except in the case of conies, due weight has not always been given to the important fact that the projective properties of any special class of curves can be deduced from those of the simplest curve of the species. Thus all the projective properties of tricuspidal quartics can be obtained from those of the three-cusped hypocycloid or the cardioid ; those of quartics with a node and a pair of cusps from the lima^on ; those of quartics with three biflecnodes from the lemniscate of Bernoulli or the reciprocal polar of the four- cusped hypocycloid ; whilst the properties of binodal and bi- cuspidal quartics can be obtained from those of bicircular quartics and cartesians. Whenever the medical profession require a new word they usually have recourse to the Greek language, and mathematicians would do well to follow their example; since the choice of a suitable Greek word supplies a concise and pointed mode of expression which saves a great deal of circumlocution and verbosity. The old-fashioned phrase "a non-singular cubic or quartic curve " involves a contradiction of terms, since Plucker has shown that all algebraic curves except conies possess sin- gularities; and I have therefore introduced the words autotomic and anautotomic to designate curves which respectively do and do not possess multiple points. The words perigraphic, endo- dromic and exodromic, which are defined on page 14, are also useful; in fact a word such as aperigraphic is indispensable in order to avoid the verbose phrase "a curve which has branches extending to infinity." At the present day the subject of Analytical Geometry covers so vast a field that it is by no means easy to decide VIU PREFACE. what to insert and what to leave out. I trust, however, that the present work will form a useful introduction to the higher branches of the subject; and will facilitate the study of a variety of curves whose properties, by reason of their beauty and elegance, deserve at least as much attention as the well-worn properties of conies. Fledborouqh Hall, HoLYPORT, Berks. August, 1901. CONTENTS. CHAPTER L INTRODUCTION. ART. PAGE 1. Definition of a quantic 1 2. Discriminants 2 3. Eliminants 2 4. Discriminants of a binary quadric, cubic and quartic . 3 5-6. Theorems relating to the roots of an equation ... 3 7. Conditions that various equalities should exist between the roots of cubic and quartic equations ..... 4 8. Condition that the product of two of the roots of an equation should be equal to —1 5 9. The Hessian of a quantic. Invariants and covariants . 6 CHAPTER II. THEORY OF CURVES. 10. General equation of a curve of the nth degree . 8 11. Two curves of the mth and nth. degrees intersect in mn points . 8 12-13. Multiple points 9 14-16. Contact of two curves 10 17. Definition of tangency 10 18-19. Double points. Nodes and cusps 11 20. Definitions of deficiency, the class of a curve, points of inflexion and undulation, and multiple tangents . 13 21. Definitions of autotomic, perigraphic, endodromic, exodromic and multipartite curves 14 22. Conditions for a double point 14 23-24. Polar curves 16 17 25-28. -Number of tangents that can be drawn to a curve . CONTENTS. i^-^Si Vuwm propertiee of }x>lar curves S8-J8L Tbe HaaUn of a curve and its properties .... 30. The SleiMrian of a curve 40l AqrnplotaB ...•••••••• 41. QemnU aquatioD of a curve in trilinear coordinates 23 48. TIm HeHian paiwfr through every point of inflexion 49-46. Purthar propo8ition« connected with multiple points on a 24 curve, ita Hessian and its polars 25 4i-4a SingularitieB at infinity 27 fla Condition that the line at infinity should be a tangent . 61. IroaginHT}' iiinguUrities 28 CHAPTER III. TANGENTIAL COORDINATES. ftS. The boothian system of tangential coordinates ... 30 69L Taofeotaal equation of a curve whose Cartesian equation is given 30 54. fNirtfiiian equation of a curve whose tangential equation is given 31 6ft. On the curve (r/a)»+(y/fc)»=l 32 66. Eiaminatioo of the different tenus of a tangential equation . 32 67-66. Reciprocal ix>lans inverse and })edal curves ... 33 66. A mde oorraqionda to a double tangent on the I'eciprocal polar . 34 60l a ct»p correeponds to a stationary tangent on the reciprocal polar 35 61. The line at infinity 35 66-63L IfeUKx! of finding the nmltiple tangents to a curve . 36 64-66L Pedal cun'en. The tangential polar equation of a curve 38 67. On the curves r"* a* cos n^ 40 66. OrthopCk and iaoptic loci 40 66>70l The circular |M>inta at infinity 42 71-71. The trilinear system of tangential coordinates ... 43 7X. Taoffsotial equatif>n of a circle 44 74 Tanfntkl equation of the circular points .... 45 76. Pod of ennree 46 7& VMkkm^9 <ie6nition of a focus 47 77. Nunihv of foci when the line at infinity is a multiple tangent . 47 76. da when a curve paesos through the circular points 48 7fc da when the circular {mints are nodes ... 49 •^ <^ <Io. cusps or points of ifi^nitm 49 61. The iateiw of a focus is a focue of the inverse curve 50 66. Bqtaetlon of tangenta drawn from a ix)int (A, ^) to a curve . 51 of the pointe of intersection of tangents at the extremities of a line drswn through a fixed point .... 51 CONTENTS. xi CHAPTER IV. plucker's equations. ART. PAGE 84. Maximum number of double points of a curve of the nth degree 52 85-86. Maximum number of points of inflexion .... 53 87-88. Maximum and minimum values of the class of a curve of the nth degree , 54 89.
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