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A Treatise on the Geometry of Surfaces

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A Treatise on the Geometry of Surfaces BOSIDNCOUEGESCiENCEUBR/iri? £rf f2. Digitized by the Internet Archive in 2010 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/treatiseongeometOObass A TREATISE ON THE GEOMETRY OF SURFACES i,^.^ ^^U^i^^ ON THE GEOMETRY OF SUEFACES BY A. B. BASSET MA. F.R.S. TRINITY COLLEGE CAMBRIDGE MATH. DEPT. BOSTON COLLEGE LIBRARY CHESTNUT HILL, MASS, CAMBRIDGE DEIGHTON BELL AND CO. LONDON GEORGE BELL AND SONS I910 [AH Rights reserved'] : (Cambritige PRINTED BY JOHN CLAY, M.A. AT THE UNIVEKSITY PRESS. 1937DE PEEFACE THE last edition of Salmon's Analytic Geometry of Three Dimensions, which was published in 1884, has been out of print for some years ; and although there are several excellent works on Quadric Surfaces and other special branches of the subject, such as those of Mr Blythe on Cubic Surfaces and of the late Mr Hudson on Kummers Quartic Surface, yet there is no British treatise exclusively devoted to the theory of surfaces of higher degree than the second. I have therefore endeavoured to supply this want in the present work. The Theory of Surfaces is an extensive one, and a thoroughly comprehensive treatise would necessarily be voluminous. I have therefore decided to limit this work to the more elementary />^ a i^^ portions of the subject, and have abstained fronTrTntroducing ^>^^/ inves¥igations^"wliich require a knowledge of the Theory of Cm^^ Functions and of the higher branches of Modern Algebra. The|^^j^»</ ordinary methods of Analytical Geometry are quite sufficient to <S**'*^' enable the properties of cubic and quartic surfaces and twisted /Y^JC^^/ curves, and also the point and plane singularities of surfaces, to Mfm^i be discussed with tolerable completeness, and to demonstrate a ^^^vXi number of interesting and important theorems connected with ^rJiWW/i them ; but for the purpose of confining this treatise within a ^S^^*^ moderate compass, I have abstained from any general discussion of ^<«-^**' surfaces of higher degree than the fourth. The properties of a point-singularity may usually be examined I by means of a surface ofHow degree just as well as by one of the 1 nth degree ; but if the degree is less than a certain limit, which ) depends on the character of the singularity, the latter appears in an incomplete form on the surface. Thus the properties of a triple line cannot be fully investigated without employing a surface of the seventh degree, and this fact has rendered it necessary to partially discuss surfaces of higher degree than a quartic. VI PREFACE The resolution of a multiple point into its constituents has been discussed by Professor Segre of Turin, and other Italian mathematicians, in various papers jpublished in the Annali di important Matematica ; and these researches have shown that an analogy exists between the theories of plane curves and of surfaces. The class of an anautotomic plane curve of degree n, and also the reduction of class produced by a multiple point of order n, the — tangents at which are distinct, are both equal to n {n 1) ; whilst the constituents of the multiple point are \n{n — \) nodes. The class of an anautotomic surface of degree n, and also the reduction of class produced by a multiple point of order n, the tangent cone at which is anautotomic, are both equal to n{n — Vf; and from analogy I concluded that the constituents of the multiple point were ^n{n — lf conic nodes. In 1908 I succeeded in obtaining a formal proof of the last theorem, which enables a large number of singular points to be resolved into their constituent conic nodes and binodes. In the present treatise I have incorporated a variety of results, originally due to Italian and German mathematicians, many of which have been published since the last edition of Salmon's work ; and I have endeavoured to modernize the analysis and the terminology by discarding antiquated methods and inappropriate symbols and phrases. I have also to express my obligations to the late Professor Cayley's papers, references to which are denoted by the letters C. M. P. ; as well as to the Repertorio di Matematiche Superiori by Professor E. Pascal, which contains a valuable epitome of the subject, together with an exhaustive collection of references to the original papers of British and foreign mathematicians, who have studied this subject. Fledborough Hall, HoLYPORT, Berks. March, 1910. CONTENTS CHAPTER I. THEORY OF SURFACES. AKT. PAGE 1. Equation of a surface .... 1 2. Four distinct species of surfaces 2 3-4. Quadriplanar coordinates 2 5. Section by a tangent plane 3 6. Conic nodes 5 7. Binodes .... 5 8. Unodes .... 6 9. Tangential equation of a surface 7 10. The spinodal, flecnodal and bitangential curves 7 11. The six singular tangent planes 8 12. Boothian coordinates . .9 13. Polar surfaces 10 14-5. Class of a surface . 13 16. Tangent cones . 14 17-8. Nodal and cuspidal generators of tangent cone 15 19. Class of a surface which possesses C conic nodes and B binodes 16 20. Number of tangent planes which can be drawn through certain singular points . 17 21. Double and stationary tangent planes to tangent cone 17 22. Points of contact of stationary planes to tangent cone lie on the spinodal curve 18 23. Maximum number of double points 18 24. Segre's theorem. Multiple points 20 25-7. Different kinds of binodes 20 28. Three primary species of unodes 21 29. Tropes 23 30. Bitropes and unitropes 24 31. Singular lines, the tangent planes along which are fixed 25 32-3. A line lying in a surface is torsal or singular 25 34. Surfaces of a higher degree than the third cannot in general have lines lying in them 26 Vlll CONTENTS ABT. 35. Every tangent plane at a point on a torsal line is a multiple tangent plane .... 36. Three distinct species of nodal lines 37-9. Pinch points, and their properties 40. Tangent plane along a nodal line 41. Nodal lines of the third kind . 42. Number of points of intersection of three surfaces which possess a common multiple line .... 43. Reduction of class produced by a nodal line 44-5. Other properties of lines lying in a surface 46-8. On the intersections of surfaces 49. The Hessian 50. The Hessian intersects the surface in the spinodal curve 51. A conic node on a surface produces a conic node on the Hessian having the same nodal cone 52. The spinodal ciu-ve has a sextuple point at a conic node 53. A binode produces a cubic node on the Hessian and an octuple point on the spinodal curve .... 54. A unode produces a quartic node on the Hessian 55. Lines lying in a surface touch the spinodal curve 56. The curve of contact of a trope forms part of the spinodal curve ........... 57. AVhen a fixed plane touches a surface along a straight line, the latter twice repeated forms part of the spinodal curve 58-9. Cayley's theorems with respect to the degrees of the spinodal and flecnodal curves 60. The Hessian possesses conic nodes 61. The Steinerian .... CHAPTER II. CUBIC SURFACES. 62. Twenty-three different species of cubic surfaces 63. Equation of a cubic surface 64. Intersection of a quadric and a cubic .... 65. An anautotomic cubic surface possesses 27 lines lying in it 66. Such a surface possesses 45 triple tangent planes . 67. It has 54 singular planes, whose point of contact is a tacnode on the section 68. Double-sixes. ......... 69. The Hessian of a cubic surface. Conjugate poles . 70-3. Properties of conjugate poles 74-5. The polar quadric of the cubic with respect to a conic node on the Hessian, consists of two planes. 76. Sylvester's canonical form of a quaternary cubic 77. Sylvester's pentahedron ....... 78. Singularities of cubic surfaces CONTENTS IX ART. CONTENTS AKT. 120. Its mode of generation . 121. Reciprocal relations . 122. Unicursal curves usually possess cusps . 123. Other singularities of such curves 124. The singularities a, or, t, i and S do not in general exist 125. Deficiency of a twisted curve 126. The Pliicker-Cayley equations for certain twisted curves 127. Ditto, when the deficiency is zero 128- 9. Twisted cubic curves . 130. Equations of ditto . 131. Developable of which a twisted cubic is the edge of regression 132. A property of a twisted cubic 133. Twisted cubics are anautotomic curves . 134. Characteristics of a twisted cubic .... 135. Connection between plane and twisted cubics 136. Twisted quartic curves 137. Table of their characteristics ..... 138- 9. Quartics of the first species ..... 140. Connection between twisted quartics and plane cubics 141- 2. Connection between twisted quartics of the first species and plane binodal quartics . 143. Quartics of the first species may be the partial intersection of a quadric and a cubic surface 144. Quartics of the second species .... 145. Such a curve cannot possess actual double points 146. All autotomic quartics and quartics of the second species are included amongst the edges of regression of a certain sextic developable ..... 147- 8. Properties of those developables . 149- 50. Nodal quartics and their reciprocals . 151- 4. Cuspidal quartics and their reciprocals 155. Anautotomic quartics of the first species constitute a class of curves sioi generis .... 156. Quintic curves. Four primary species 157- 8. Quintic curves of the first species 159. „ „ „ second „ 160. „ „ „ third „ 161. fourth „ CHAPTER IV. COMPOUND SINGULARITIES OF PLANE CURVES. 162. Object of investigation 115 163. Point, line and mixed singularities 115 164. Determination of the point and line constituents of a singu- larity 115 CONTENTS XI ART. PAGE 165. Constituents of a multiple point 116 166.
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