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The Theory of Ruled Surfaces

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The Theory of Ruled Surfaces THE BOOK WAS DRENCHED Jl 66424 Ffi =5 CD OSMANIA UNIVERSITY LIBRARY Call No. ;> tf/2. %'/ 3 7 Accession No. Author This. book s\vyfid be returned on or before the dftte last marked below THE THEORY OF RJJLED SURFACES Cambridge University Press Fetter Lane, London JYVttJ York Bombay, Calcutta, Madras Toronto Macmillan Tokyo Maruzen CompanJ Ltd. All rights reserved THE THEORY OF RULED SURFACES BY W. L. EDGJE KFLLOW OF TRINITY rOLLKCIK CAMlilllDGE CAMBRIDGE AT THE UNIVERSITY PRESS 1931 PRINTED IN GREAT BRITAIN CONTENTS PREFACE pageix CHAPTER I: INTRODUCTORY Section I PRELIMINARIES 1 Birational transformation, 3. Normal curves, 5 Ruled surfaces, 7. Uorrespon- .^ dence formulae, 9. The valency of a correspondence, 10. Brill's formula, 12. Znutheii's formula, 14. Che genus of a simple curve on a ruled surface, 15. The genus of a curve on a quadric, 1 6. The ruled surface generated by a correspondence between two curves, 17. Section II THE REPRESENTATION OF A RULED SURFACE IN THREE DIMENSIONS AS A CURVE ON A QUADRIC il IN FIVE DIMENSIONS ... 22 The six coordinates of a line, 22. Line geometry in [3] considered as the 26. point geometry of a quadric primal in [5], 24. The planes on li, The of 27. quadriu point-cone in [4], 26. The representation a ruled surface, Torsal generators, 28. The double curve and bitangent developable, 28. Triple points and tritangent planes, 30. Cubic ruled surfaces, 32. Section III TllE PROJECTION OF RULED SURFACES FROM HIGHER SPACE . 33 CHAPTER II: QUARTIC RULED SURFACES INTRODUCTORY 45 Section I RATIONAL QUARTIC RULED SURFACES CONSIDERED AS CURVES ON Q 48 The representation of the double curve and bitangent de- velopable ......... 52 Section II RATIONAL QUARTIC RULED SURFACES CONSIDERED AS PROJECTIONS OF NORMAL SURFACES IN HIGHER SPACE 55 The normal surface in [5] with directrix conies ... 55 The normal surface in [5] with a directrix line . 59 vi CONTENTS Section III ELLIPTIC QUARTIC RULED SURFACES page 60 Section IV ALGEBRAICAL RESULTS CONNECTED WITH QUARTIC RULED SURFACES 02 CHAPTER III: QUINTIC RULED SURFACES Section I RATIONAL QUINTIC RULED SURFACES CONSIDERED AS CURVES ON 1 72 The rational quintic curves on a quadric O in [5] . 73 The general surface in [3] . 75 Further consideration of the surfaces whose generators do . 77 not belong to a linear complex . Surfaces whose generators belong to a linear complex which 82 is not special ......... Surfaces with a directrix line which is not a generator . 93 Surfaces with a directrix line which is also a generator . 100 Surfaces whose generators belong to a linear congruence . 102 Section II RATIONAL QUINTIC RULED SURFACES CONSIDERED AS PROJECTIONS OF NORMAL SURFACES IN HIGHER SPACE 104 The general surface in [6] . .104 The surfaces in [3] derived by projection from the general sur- face in [6] 110 The surfaces in [3] derived by projection from the surface in [6] with a directrix line . .123 Section III QUINTIC RULED SURFACES WHICH ARE NOT RATIONAL . .126 Elliptic quintic ruled surfaces considered as curves on Q . 126 Elliptic quintic ruled surfaces in [3] considered as pro- jections of normal surfaces in higher space . .130 The quintic ruled surface of genus 2 135 CONTENTS vii CHAPTER IV: SEXTIC RULED SURFACES Section I RATIONAL SEXTIC RULED SURFACES page 139 The rational sextic curves which lie on quadrics . 139 Surfaces without either a directrix line or a multiple genera- tor 141 Surfaces with a multiple generator but without a directrix line .......... 154 Surfaces with a directrix line which is not a generator . 167 Surfaces with a directrix line which is also a generator . 184 Surfaces whose generators belong to a linear congruence . 196 Section IT ELLIPTIC SEXTIC RULED SURFACE . .199 Elliptic sextic curves which lie on quadrics . 200 Surfaces whose generators do not belong to a linear com- plex 202 Surfaces whose generators belong to a linear complex which is not special ......... 206 Surfaces with a directrix line which is not a generator . 209 Surfaces with a directrix line which is also a generator . 215 " Surfaces whose generators belong to a linear congruence 217 The normal elliptic ruled surfaces of a given order n . 217 The normal surfaces in [5J with directrix cubic curves . 224 The normal surface in [5] with a double line . 240 Section III SEXTIC RULED SURFACES WHICH ARE NEITHER RATIONAL NOR ELLIPTIC 248 The normal sextic curve of genus 2 ..... 248 2 lie . Sextic curves of genus which on quadrics . 249 The sextic ruled surfaces in [3| whose plane sections are curves of genus 2 ........ 250 A sextic ruled surface in [4] whose prime sections are curves of genus 2, and the ruled surfaces in [3] derived from it by projection ......... 253 The sextic ruled surfaces of genus 2 which are normal in [3] 256 The sextic ruled surfaces whose plane sections are of genus greater than 2 258 viii CONTENTS CHAPTER V: DEVELOPABLE SURFACES Introduction ........ page 260 The developable surface of the fourth order . 266 The developable surface of the fifth order .... 267 The projections of the developable formed by the tangents of a normal rational quartic curve ..... 270 The sections of the locus formed by the osculating planes of a normal rational quartic curve ..... 274 The developables of the sixth order considered as curves onii .......... 278 The classification of the developable surfaces . .284 * CHAPTER VI: SEXTIC RULED SURFACES (CONTINUED) Further types of rational sextic ruled surfaces . .286 Sextic ruled surfaces with a triple curve .... 295 TABLES SHEWING THE DIFFERENT TYPES OF RUL SURFACES IN [3] UP TO AND INCLUDING ; ^D THr JE OF THE SIXTH ORDER ..... 302 NOTE. The intersections of two curves on a ruled surface . 311 PREFACE In this volume all the ruled surfaces in ordinary space of orders up to and including the sixth are studied and classified. Tables shewing the different types of surfaces are given towards the end of the book, and the tables for the surfaces of the fifth and sixth orders are here obtained for the first time. results so are of but the It seems that the obtained great importance ; incidental purpose which, it is hoped, may be served by the book is perhaps of still greater importance. For there exists at present no work, easily accessible to English readers, which tests the application of the general ideas here employed in anything like the same detail. One might mention especially the use of higher space and the principle of correspondence, and these two ideas are vital and fundamental in all modern algebraic geometry. It is hoped therefore that the book may be of use to a wide circle of readers. I wish to express here my thanks to the staff of the University Press for their unfailing accuracy in the printing and for the ready courtesy with which they have accepted my suggestions. Notwithstanding the large number of surfaces which are herein in- vestigated, the book would be incomplete were I not to make an acknow- ledgment of my obligations to Mr White, of St John's College, and Professor Baker. Even those who have only a slight knowledge of the inultifarious- ness of Mr White's mathematical public services will be surprised to learn that he found time not only to read the proof sheets but also to read through the whole of the manuscript, and I am very grateful to him for his criticisms and suggestions. My gratitude to Professor Baker is something more than that of a student to his teacher. He it was who first suggested that I should under- take this work, and his encouragement has been given unsparingly and effectively in times of difficulty. I have derived great benefit not only from my personal conversations with him but also from attending his courses of lectures. I thank him for many things; but especially for his interest, which has never flagged, and for his trust, which has never wavered. W. L. E. TBINITY COLLEGE CAMBBIDOE October 1930 CHAPTER I INTRODUCTORY SECTION I PRELIMINARIES 1. The system of points on a line is determined by two of them, third of line derivable these two the same line any point the being from ; is equally well determined by any two of its points. Similarly, if three points are taken which are not on the same line they determine a plane, the same plane being equally well determined by any three non-collinear points of it. Proceeding in this way we say that n + 1 independent points determine a linear space of n dimensions, the points being independent when they are such that no one of them belongs to the space of less than n dimensions determined by the others; the same space of n dimensions is equally well determined by any n -h 1 independent points belonging to it. We shall use the symbols [ri\ and Sn to denote a space of n dimensions. In [n] two spaces [m] and [n m] of complementary dimensions have, in general, one point in common and no more. A space [p] and a space [q] have, in general, no common points if p -f- q < n, while if p + q > n they have, in general, a common [p + q n] . If they have in common a space r [r] where > p -\- q n, then they are contained in a space [p + q r] or where s = r n. For : two lines in [n s] , p q -{- example ordinary space do not intersect in general; if they do so they lie in a plane. If we call the intersection of [p] and [q] their meet and the space of lowest dimension which contains them both their join, then the sum of the dimensions of the meet and the join is p -f q.
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