I jK^afV CORNELL UNIVERSITY LIBRARY 924 067 741 763 '/I®, 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/cu31924067741763 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. Digital file copyright by Cornell University Library 1993. A TREATISE ON THE LINE COMPLEX Honbon: C. J. CLAY and SONS, CAMBRIDGE UNIVEESITY ^fflSS WAREHOUSE, AVE MARIA ThTnE, ©laBBoin: 60, WELLINGTON STREET. Ijipjig: F. A. BROCKHAUS. pfto gork: THE MAOMILLAN COMPANY. ffiomiaE aiiti ffiakuttfi : MACMILLAN AND CO., Ltd. [^All Ruflitt reserved-l A TREATISE ON THE LINE COMPLEX BY C. M. JESSOP, M.A. FOKMEBLY FELLOW OJ? CLABE COLLEGE, CAMBEIDGE, ASSISTANT PBOFESBOE OF MATHEMATICS IN THE DUBHAM COLLEGE OF SCIENCE, NEWCASILE-ON-TYNE. CAMBRIDGE AT THE UNIVERSITY PRESS 1903 -\ CambriligE : PRINTED BY J. & C. F. OLAY, AT THE UNIVERSITY PRESS. PREFACE. rriHE important character of the extensive investigations into -■- the theory of line -geometry renders it desirable that a treatise should exist for the purpose of presenting these investi- gations in a form easily accessible to the English student of mathematics. With this end in view, the present work on the Line Complex has been written. The subject owes its origin to Plucker, who suggested the idea of taking the straight line as the element of space*. The straight line thus holds to the present subject the relationship in which the point and the plane stand to the older geometry. Types of coordinates of the line were introduced by Cayley and Grassmann ; Plilck'er adopted a coordinate system which is a special form of them. In his work theNeue Geovietrie des Raumes, Plucker intrdduced the conception of a complex of lines, i.e. the oo ' lines which satisfy one given condition, so that one equation exists between the four coordinates of each line of a complex. He investigated in detail the linear and the quadratic complex ; his work contains most of the chief properties of such complexes ; in particular he shows that if any screw motion about a certain axis be given to the lines forming a linear complex, these lines still remain within the complex. He discovered the polar properties, viz. that the lines of a linear complex in any plane pass through a point, the pole of the plane ; that the lines of the complex through anj' point lie in a plane, the polar plane of the point ; and that if a point moves along any given line, its polar plane turns round another line called the polar live of the first line, the relationship between the two lines being reciprocal. The, greater part of the Meue Geometrie is concerned with the quadratic complex, of which it contains many of the leading properties ; in particular, Pliicker shows that while the lines of *"■ System der Gemnetrie des Eaumes, Diisseldorf, (1846). VI PREFACE such a complex through any point form in general a quadric cone, there is a certain surface, the Singular Surface of the complex, for whose points these cones break up into two planes. Likewise the lines of the complex in any plane, which in general touch a conic, in the case of any tangent plane of the singular surface form two pencils. This surface is the one known as Kummer's surface ; it is of the 4th degree and class and possesses 16 nodes and 16 singular tangent planes. The next investigator in this field was Battaglini, who pursued still further the ideas of Pliicker. He adopted as the general quadratic complex one which was afterwards shown to be a special case, viz. the complex formed by the lines for each of which the points of intersection with two given quadrics form a harmonic range ; but many of his results apply also to the general complex. The success of Pliicker's researches was limited by the un- suitable (Cartesian) analysis he employed. The second important step in the development of the subject was' due to Prof. Felix Klein, who, in his celebrated memoir in volume ii. of the Mathe- matische Annalen, introduced the coordinate-system determined by six linear complexes in mutual involution*. By its adoption a simple and elegant analytical mode of treatment of line-geometry is rendered possible. Klein further revealed the existence of a singly infinite series of quadratic complexes which have the same singular surface as any given quadratic complex. In his Dissertation (Bonn, 1868) he pointed to the method of Weierstrass for the canonization of two quadratic forms, as the appropriate instrument for classifying the quadratic complex ; and this classification was carried out by Weiler. Another service rendered by Klein was his discovery of the analogue existing between the lines of three-dimensional space and the points of four-dimensional space, together with the equations embodying this relationship. His enunciation of the fact that line-geometry is point-geometry on a quadric contained * On any line common to two linear complexes a (1, 1) correspondence of points is determined by the planes through the line, viz. by taking the poles of each plane for the two complexes. If a certain condition, connecting the constants of the equations of the two complexes, is satisfied, these pairs of points form an involution. PREFACE Vll in point-space of five dimensions, offers a new point of view of the subject. Other important contributions to the theory are introduced from time to time in the text : of these the most fundamental are contained in the investigations of Lie, in which he showed the connexion of line-geometry with sphere-geometry. He esta- blished a relationship between the lines and spheres of three- dimensional space of such a nature, that to two intersecting lines there correspond two spheres in contact ; and he applied the ideas of both varieties of geometry to the investigation of various types of differential equations. In the present work the analytical method of treatment with Klein coordinates has been generally adopted ; but as it frequently happens that synthetic methods are appropriate, recourse to such has been occasionally made. Since the study of synthetic geometry has been less widely followed in this country than on the Continent, I have not thought it superfluous to insert, by way of Introduction, a short sketch of the simpler portions of that subject which have bearing on the context of the work. The main object of investigation is, as has been stated, the properties of the line complex, and, in connexion with it, the characteristics of the system of oo ^ lines common to any two complexes. To any set of oo^ lines the name congruence is attached ; the study of such systems was extensively pursued at a period considerably before Pliicker's discoveries took place. The chief property of a congruence is that each of its lines is bitangent to a surface, (including as a special case two surfaces, a surface aiid a curve, etc.). Through any point there pass a definite number m of the lines of a given congruence, and in any plane there lie a definite number n of its lines. If the congruence is the complete intersection of two complexes, m = n. Though not necessarily included in the scope of this treatise, nevertheless, on account of its close connexion with the theory of the complex, a discussion has been given in Chapters XIV. — XVI. of the congruence (to, n), and in particular, of the congruences (2, n), so elegantly treated by Kummer. As regards the various authorities on this subject, the student is referred to the work of Prof. Gino Loria II passato ed il presente Vlll PREFACE delleprincipali teorie geometriche, which contains detailed references to the chief memoirs. A useful summary with references is given in Prof. E. Pascal's Repertono di mathematiche superion. The comprehensive treatise of Prof. R. Sturm, Die Oebilde ersten und zweiten Grades der Liniengeometrie, is a storehouse of information ; his method is, however, exclusively synthetic. An introduction to most of the leading ideas is given by Prof. G. Koenigs in his work La gSomStrie regUe et ses applications. An interesting general account of Line Geometry given by Mr J. H. Grace in the Supplement to the Encyclopaedia Bntannica, will be found very serviceable by the student of this subject. I have thought it not desirable to include in this treatise a description of the important investigations of Prof E. Study, on account of their distinctness in aim and method from those of the other writers who have built up this subject. I rather refer the reader to Prof Study's work peometrie der Dynamen. It gives me much pleasure to express my gratitude to several friends for assistance generously given me; and especially to Mr J. H. Grace, M.A., Fellow of Peterhouse, Cambridge, who read the manuscript, and who, by his criticisms and suggestions, has greatly increased the value of the work. My colleague Mr G. W. Gaunt, M.A., late Scholar of St Catharine's College, has read all the proofs ; such accuracy as the book possesses is largely due to his carefulness. I am also under obligations to Mr P. W. Wood, B.A., Scholar of Emmanuel College, who has read the proofs and verified many of the examples. Professor T. J. I' A. Bromwich, Fellow of St John's College, has kindly put at my disposal a collection of examples, most of which were made by him ; they have been incorporated in the Miscellaneous Results and Exercises, and add greatly to the book's usefulness. Finally, I feel it a pleasant duty to express my appreciation of the admirable manner in which the staff of the University Press have carried out the onerous task involved in the printing. C. M. JESSOP. September 1903. CONTENTS. INTRODUCTION. ART. i ii, hi iv, V xi-xiv Double ratio . .... Correspondence of points on the same line Involution Con-espondences on diffei-ent lines CoUineation XV, xvi Involutions on a ciu-ve svii (2, 2) Con-espondences .... PAGE 1 9 3 4 8 12 14 CHAPTER I. SYSTEMS OF COORDINATES. 1 Definition of complex and congruence 15 2-9 Systems of coordinates 16 10-13 Pencil, sheaf and plane system of lines, von Staudt's theorem 21 14 System of sixteen points and planes 23 CHAPTER II. THE LINEAR COMPLEX. 15-16 The linear complex . IV Polar lines 18 Invariant of a linear complex . 19-21 The special complex, coordinates of polai- Lines 22 Diameters and axis of the complex . 23 Standard form of the equation 24 Pair of polar lines of two complexes 25-27 Complexes in Involution . 28 Transformation of cooi-dinates . 29 The fifteen principal tetrahedra 25 27 o^ 28 30 31 32 32 36 37 CONTENTS CHAPTER III. SYNTHESIS OF THE LINEAR COMPLEX. ART. 30-32 Determination of the complex from given conditions 33 Every linear complex contains two lines of any regulus 34-36 Collineation ami Reciprocity 37-38 The Null System . 39 Sylvester's method . 40 Automorphic transformations 41-43 Curves of a linear complex 44^45 Polar surfaces and curves 46 Complex equation of the quadrio 47-50 Simultaneous bilinear equations PAGE 39 41 42 45 47 50 52 53 CHAPTER IV. SYSTEMS OF LINEAR COMPLEXES. 51 ' The linear congruence 52-53 Double ratio of two and of four complexes 54 The special linear congruence . 55-56 Metrical properties .... 57 The cylindroid .... 58 Systems of three terms 59-60 The generators and tangents of a quadric 61 The ten fundamental quadrics 62 Closed system of sixteen points and planes 63 Systems of four and of Ave terms . 64 Invariants of a system of linear complexes 65 Property of the six residuals . 60 61 61 62 64 65 66 68 69 70 71 73 CHAPTER V. RULED CUBIC AND QUARTIC SURFACES. 66 Ruled surfaces . 76 67 Ruled cubics . . . 76 68 Ruled quartics of deficiency unity . 77 69 Ruled quartics of zero deficiency . . 78 70-72 Analytical classification of Voss . 80 CONTENTS CHAPTER VI. THE QUADRATIC COMPLEX. ART. 73 The quadratic complex 74 The tangent linear complex 75 Singulai' points and planes of the complex 76 Singular lines 77-78 Identity of the surfaces *i and *2 79 Polar lines 80 Singulai" lines of the first, second and third orders 81 The equation of the complex in Plucker coordinates 82 The singular surface 83 Double tangents of the surface 84 Determination of a quadi-atic complex . 85 The singular surface is a general Kummer sm'faoe 86 PKicker Surfaces 87 Normal form of the equation of a complex 88-89 Special and hai-monic complexes 90 Symbolic form of the equation of a quadric . 91 Symbolic forms of Pliicker and singular surfaces FAQE 87 88 89 90 91 92 94 97 97 100 101 102 105 109 110 112 113 CHAPTER VII. SPECIAL VARIETIES OF THE QUADRATIC COMPLEX. 92-93 The tetrahedral complex . 94-95 Reguli of the complex .... 96-97 Other methods of formation of the complex 98-101 Complexes derived from projective pencils 102 Reye's complex of axes .... 103 Differential equation of the tetrahedral complex 104-106 Curves of the complex 107-109 The special quadratic complex 110-112 The harmonic complex . 113 Painvin's complex ... 114 116 118 120 125 127 127 130 133 138 CHAPTER VIII. THE COSINGULAR COMPLEXES. 114-115 The cosingular complexes 116 Correspondence between lines of cosingular complexes 117 The complexes R^^, R^'^ 118 The ten systems of reguli of the congruence (2, 2) . 139 141 142 143 Xll CONTENTS AKT. PAOE 119 Focal surface of the congruence (2, 2) .... 145 120 Confocal congruences 146 121 The ruled quartic {C\ A, A') 150 122 Projective formation of the quadratic complex . . . 151 123 Caporali's Theorem 153 124 (1, 1) correspondences leading to oosingular complexes . . 155 125 Equation of C^ referred to special tetrahedron . . , 157 126 Oosingular complexes for this coordinate system . . . 158 127-128 Involution of tangent linear complexes 159 129 Coplanar conies determined by oosingular conijilexes . . 161 130 Elliptic coordinates of a line 162 131 Bitangent linear complexes 164 132 Principal Surfaces .... .... 165 133 Invokitory position of two lines 167 CHAPTER IX. POLAR IJNES, POINTS, AND PLANES. • 134-135 Polar lines 169 136-139 Corresponding loci of polar lines 171 140 Polar Plane 174 141 Polar Point 176 142 The diameters of the complex 177 143 The centre of the complex 177 CHAPTER X. REPRESENTATION OF A COMPLEX BT POINTS OF SPACE. 144 Representation of the lines of a quadratic complex by points of three-dimensional space 179 145 Eeguli of a congruence (2, 2)' 1821 146 Representation of a congruence (2, 2) by points of a plane . 183 147 Representation of a linear complex by isoints of three- dimensional space 186 CHAPTER XI. THE GENERAL EQUATION OF THE SECOND DEGREE. 148-153 Reduction of the equation of a quadratic complex to a canonical form 189 154 Arbitrary constants contained in a canonical form . . 197 155 Complexes composed of linear congruences .... 198 , 156 Double lines 199 CONTENTS Xlll ART. 157-158 159 160 161-214 The cosingular complexes and the correspondence between lines of two cosingulai- complexes . ... 200 The singular surface corresponding to a canonical form . 204 Degree of the complex .... ... 206 Vai'ieties of the quadratic complex . ... 206 CHAPTER XII. CONNEXION OF LINE-GEOMETRY WITH SPHERE-GEOMETRY. 215 Coordinates of a sphere .... . . 233 216 Contact of spheres corresponds to intei-section of lines . . 234 217 Points of A con-espond to minimal lines of S .... 235 218-219 Definition of a surface element. A surface element of A defines a surface element of 2 236 220 Principal tangents of A correspond to principal spheres of 2 . 238 221 Pentaspherioal coordinates. Double tangents of the singular sm-face of C^ correspond to foci of surfaces . . . 238 CHAPTER XIII. CONNEXION OF LINE-GEOMETRY WITH HYPERGEOMETRY. 222 Definition of point, line, h^-perplane of space of four dimensions 244 223 Klein's jinalogy between line-geometry and point-geometry in four dimensions 245 224 Schumacher's correlation 247- 225 Correlatives in S^ of a sheaf, plane system and pencil of A . 249 226-227 Meti-ical geometi-y 251 228 Principal surfaces of A and lines of curvature of S^ . 254 229 Liiue-geometry is point-geometry of an S^^ in an (S5 . . 255 230 Line-geometry in Klein coordinates is identical with point- geometry in S^ with hexaspherical cooi'dinates . . 255 231-232 The congruence {m.,n). . 256 CHAPTER XIV. CONGRUENCES OF LINES. 233 Oixier and class of a congruence ... . . 258 234 Halphen's Theorem ... 259 235 ' Bank of a congruence 260 236 Focal points, planes and surfaces .... 261 237 Degi-ee and class of the focal surface 261 238 Singulai- points of a congruence 262 239-240 Determination of a ray by two coordinates .... 263 241-248 Application of Schumacher's method of projection to detei"- mine the degree, class, and i-ank of the focal surface . . 265 XIV CONTENTS CHAPTER XV. CONGRUENCES OF THE SECOND ORDER WITHOUT SINGULAR CURVES. ART. fAOE 249 The rank of the congruence (2, Ji) is m - 2 . . . • 276' 250 The surfaces (P) '276 251 The .singular points of the congruence are double points of $ 277 252 Double rays 277 253 The class of a congruence (2, n) is not greater than 7 . . 279 254-256 Number and distribution of the singular points . . 279 257 Equation of a surface (P) 282 258 Tetrahedral complexes of the congruence (2, n) . . . 283 259 Non-conjugate singular points : 286 260-262 Reguli of the congruences (2, «) 287 263 Confocal congruences 291 CHAPTER XVI. THE CONGRUENCE OF THE SECOND ORDER AND SECOND CLASS. 264 The congruence (2, 2) is the complete intersection of a linear with a quadratic complex 295, 265 Confocal congruences . . .... 296 266 Distribution of the singular points 297 267 Every congruence (2, 2) is contained in 40 tetrahedral com- plexes 293 268 The Kummer configuration 298 269 The Weber groups .... .... 299 270-271 Reguli of the congruence 302 272 Focal surface of the intersection of any two complexes . . 304 273 Double rays of special congruences (2, 2) . . . . 305 CHAPTER XVII. THE GENERAL COMPLEX. 274 The general complex 307 275-277 The Singular Surface 308 278 The Principal Surfaces 310 279 Number of Constants of the complex . . ' . . . 311 280 The Special ComiDlex 311 281-283 Congruences and their Focal Surfaces 312 284-285 The ruled surface which is the intersection of three com- plexes 316 286 Clifford's Theorem 318 287-288 Symbolic forms of the equations of the complex and its singular surface 321 CONTENTS XV CHAPTER XVIII, DIFFERENTIAL EQUATIONS CONNECTED WITH THE LINE COMPLEX. AKT. PAGE 289 Application of the siu-face element to partial differential equations ... 326 290 The characteristic curves of pai-tial diflerential equations . 327 291 The Monge equation of a line-complex ... . 328 292 The chai-acteristic curves on an Integral surface ai'e principal tangent curves . 329 293-294 Partial Diiferential equation corresponding to a line-complex 330 295 Contact transformations of space 332 296-299 The trajectory circle. The equations Z*;,, Dj,, i)i3 . 333 300 The complex of normals .... ... 337 301-302 Partial differential equations of the second order associated with line- and sphere-complexes 337 303-304 Partial differential equations of the second order on whose Integral surfaces both sets of characteristics ai-e, principal tangent curves or lines of curvature ... . 339 305-306 Equations B^i", -Djj" with one and with two general fii-st integrals 341 307 Application to the quadratic complex . . 345 Miscellaneous Results and Exercises . . 347 Index . . 363 COEBIGENDA. Art. 39 /or 'which meet such a pair' read 'which meet such pairs.' Page 184 for e^Pstlhi ''^"■<^ "iVmVu- INTRODUCTION. Before entering upon the subject proper of the present work, a short preliminai-y discussion of those parts of synthetic geometry which have the closest connexion with line geometry, has been inserted here for convenience of reference. i. Double Ratio. For four points ABGD lying on a straight AB AD line the number -57^ ^ -=r/, is called the Double Sectional Eatio or Double Ratio of the points, the sense of the segments AB &c. being taken into consideration ; the terms Anharmonic Ratio and Cross Ratio are also used to designate this quantity, which is usually denoted by {ABCB). The orders in which the points may be taken are 24 in number, but there are only six different Double Ratios of the four points, for we find that any two orders which difiPer by a double inter- change of two points have theii- double ratios equal, e.g. so that {ABCD) = {BADG) = (CDAB) = (DCBA). Secondlii, if two non-consecutive members of a double ratio be interchanged the double ratio is inverted, e.g. DG ' BG (ABCD) " Thirdly, the sum of the double ratios for two orders which differ in their second and third members is itniti/. e.g. ,,J,r.n^ ^B.DG ,,^prl^ -^^^^^ CA.D B ^^^^^^-BGTAD' ^^^^^^''GBTAD^BGTaD' uid since BC+CA+AB = 0, AD = BD + AB, AD = GD-CA, J. 1 2 INTRODUCTION therefore BG.AD+ CA (BD + AB) + AB (GB - CA) = 0, or BG.AD + GA.BB + AB.GD=0, hence (ABGD) + (A GBB) = 1 . So that denoting {ABGD) by X, we have {ADGB) = \, (AGBD)=l-\, {ABBG) = y^, A, X — A. (ABDG) = 1 - ^= ^4l - iAGBB) = '^ . All the other double ratios have one of these six values. If the value of the double ratio is — 1 the points are said to be Harmonic ; in this case since AB AD_ BC "^ DC ' hence AB{AG - AD)+ AD{AG - AB) = 0, _2___1_ J_ °'' AG'AB'^AD' whence it is easily found that if Ois the mid-point of AG, OG'=OB.OD. [It should be noticed that in this case the points ABGD are arranged consecutively.] ii. Correspondence. If between the points of a straight line a connexion is established such that to each point of the line corresponds one and only one point of the line, there is said to exist a " one-one " correspondence, or correlation, between its points. If X is the distance of any point P of the line from a fixed point of the line, this correlatio n is define d by an equation of the form ^^ Axx' + £« -fCa;' -<- D = 0, where x is the distance from of the point P' which corresponds to P. It follows from this equation that Gx' aD ''^~ Ax'+B and hence if P, Q, R, S are four points and P', Q', R, 8' their corresponding points, {PQRS) = {P'Q'R'S'); for PQ=x^-x^ = (»; - «,') {AD- BG)l{Ax^ + B) {Ax.,' + B), SR = x,-x, = {x; - xi) {AD - BG)/{Ax,' + B) {Ax; + B), Sue, INTRODUCTION 3 hence ' {x^- X3) (a;i - x^) (a'a - «;, ) (x^ - x, ) It follows that if three pairs of points of the line be associated the correspondence is determined, for' if P and P', Q an^- Q', R and R' be made to correspond, then the point S' which cor- responds to any fourth point S is determined by the equality {PQRS) = {P'Q'R'S'). iii. United Points. The coincidence of a point and its corresponding point will occur twice, for putting x' = x we have Ax' + {B+G)x + D = 0, thus in every (1, 1) correspondence there are two " united" points (real or imaginary). If the point midway between the united points (E, E') be the point from which the distances are measured, we must have B + G=(i, and the equation defining the correspondence is of the form Axx -{- B{x — x') + D = 0, while the distance a of either united point from is given by the equation Ao? + I) = 0; com- bining these two equations and writing k for ^, the equation of correspondence becomes xx + K {x — x') — a? = 0, which may be written in the form (x + a) (x' — a.) = (a + K) {x — x), hence Jg, ^ (^ - -) (" ^«) ^ PP' ■ E'E ^ ^^°°® a + K {x'-a){x + a) FE.PE' ^^^^^^ thus the double ratio of a point, its corresponding point, and the united points is constant. The correspondence is therefore de- termined if its united points and one pair of corresponding points are given. iv. Involution. If B = Cth% relation between x and x' is symmetrical, and hence if P' corresponds to any point P then will P correspond to P", and the pointy of the line form " closed sjrstems " of two points. The correspondence is in this case called an Involution. The equation which connects corresponding points being now Axx ■{-B{x + x') + B = 0, it may be written -("§)(-- 1) =^^. 1—2 4 INTRODUCTION or if y and y' are the respective distances of P and P' from the point whose distance from is — -j , , B^-AD yy = — -j^-- If B'' > AD there are two real points each of which coincides with its corresponding point, viz. those given by the equation 'JB' - AD so that if these points be E and E' and M their middle point (the origin for the y's) and P, P' any pair of corresponding points MP.MP' = ME\ This shows that the two " double " points E and E' of the involution form with any pair of corresponding points a harmonic range. V. Harmonic Involutions. If in the two involutions on the same line determined respectively by xx + A{x-\-x') + B = 0, yy' + (f{y + y')^D = 0, the double points of one form a pair in the other, i.e. are harmonic conjugates to the double points of the other, it is clear that B + D-^AC=Q (a). The Involutions are then said to be " harmonic " to each other. In this case, if to two points P and P' which are conjugate in the first Involution the conjugate points in the second Involution are Q and Q respectively, Q and Q' are themselves conjugate in the first Involution ; for by hypothesis OQ G.OP + D QQ,_ G.OP' + D ^^ OP + G ' ^^ OP' + C • hence {OP + G) {OP' + C) {OQ .OQ + A.OQ + OQ' + B) = {G.0P + D){G.0P' + D)-A(C.OP + D.OP' + G + G.0P' + D.0P + G) + B{0P + G){0P' + G) = {G' - D) {OP .OP' + A. OP + OP' + B), from (a), = 0, which proves the result stated. vi. Correspondences on different lines. We shall now consider correspondences between the points of two different lines and the ruled surfaces (or plane curves) obtained as loci or INTRODUCTION envelopes of lines joining corresponding points. In what follows use will be made of the obvious fact that when a correspondence is established between the points of a line and these points are joined to any external point by a plane pencil of lines, a similar correspondence is thereby established between the lines of the pencil ; similarly a correspondence established on a line gives rise to a correspondence between the planes of any pencil of planes {i.e. planes having a common line of intersection). Wheii a (1, 1) correspondence exists between the points of two lines* in the same plane, the joins of pairs of corresponding points envelope a curve of the second class ; for joining any point P of the plane to the points of the two lines a (1, 1) correspondence is established between the lines of the pencil centre P ; since there are two united lines in this correspondence, through P will pass two and only two lines which connect a pair of corresponding points. To this envelope the two given lines are themselves tangents. A special case arises when the point of intersection of the two given lines corresponds to itself, i.e. regarded as a point of the first line has itself as corresponding point in the second line ; in this case, of the two lines through P which join corresponding points one coincides with PO and therefore passes through the fixed point ; the envelope of lines joining corresponding points breaks up into two points, and one other point G, the two rows of points are said to be in pei'spective, and the point G through which pass all lines joining corresponding points is called the " centre of perspective.'' The corresponding theorem afforded by the Principle of Duality is, if between the lines of two plane pencils a (1, 1) correspondence exists, the locus of intersection of corresponding lines is a curve of the second order passing throvgh the centres of the pencils ; for on any line the two pencils determine a (1, 1) correspondence of points, the two double points of which are the points of intersection of the line with the required locus. A special case arises when the line joining the centres of the pencils corresponds to itself j in this case, of the two double points on any line, one lies on the line joining the centres of the pencils, i.e. the locus of intersection of corresponding lines of the two pencils breaks up into the line joining the centres of the pencils and one other line c, the two pencils are said to be in perspective, and the line c which contains * The X of Art. ii here refers to a point P of one line, and .t' to its oorresponding point P' on the other line. 6 INTRODUCTION the intersections of corresponding lines is called the axis of perspective. vii. A (1, 1) correspondence between the lines of two pencils (or the points of two lines) in one plane is established when to three elements of one are assigned as correspondents three elements of the other; for if three lines of one pencil SA, SB, SC meet any given line p in A, B, and C, and three corresponding lines S'A', S'B', S'C of the other pencil meet the same line in A', B', C, then (Art. ii) the three pairs of corresponding points J. ^1', 55', CO' deter- mine a correlation on p, hence if any other line of the first pencil meets p in P the corresponding line S'F' of the other pencil is determined. viii. The joins of corresponding points on two non-intersecting lines form one set of generators of a quadric, that is, a Regidus* ; for the points of the two lines u and v establish on the pencil of planes whose axis is any line I a (1, 1) correspondence, viz., if P and P' are corresponding points on u and v, to the plane (P, I) corresponds the plane (P', I), each of the two double planes of this correspondence will meet the lines u, v in & pair of corresponding points, hence tivo and only two lines joining corresponding points on M and v will meet I, and any line I will meet the locus of lines which join corresponding points on u and v in two points. The two given lines belong to the other system of generators of the quadric determined by the Regulus. It is to be noticed that the lines of a Regulus determine on any two lines of tlie other system two rows of points having a (1, 1) correspondence ; and four given generators determine on a variable generator of the opposite system four points having a constant Double Ratio. The Principle of Duality gives the theorem, the locus of inter- section of corresponding planes of two pencils of planes connected by a (I, 1) correspondence is a Regidus ; for the two pencils of planes determine on any line I a (1, 1) correspondence of points, hence corresponding planes will only meet on I at the double points of this correspondence. The following properties of a quadric should be observed : First, if A, B, C, D are any four points on a generator and a, b, c, d the tangent planes thereat, {ABCD) = {abcd). * In the seqnel the word Begulus is restricted to mean ' one set of generators of a quadric surface,' the other set of generators is called, in reference to it, thd ' complementary ' regulus. The word ' demi-quadrique ' is used by Koenigs in this sense. INTRODUCTION 7 This follows from taking any other generator of the same system which meets a, h, c, d in A', B, C, jy respectively, then AA', BB', CC, DD' are generators and therefore from what precedes (ABCD) = {A'B'C'D'), while (,A'BC'I)')=={abcd), hence {abcd)={ABCI)) ; Second, iiABCD ai-e any given points on the quadric and x any generator, the double ratio of the four planes xA, xB, xG, xD is coTistmit ; for if the generators through A, B, O, D of the opposite system to x meet x in PQRS respectively, xA, xB, xC, xD ai-e the tangent planes at P, Q, R, S respectively, therefore {PQRS) = ixA, xB, xC, .rZ>), but (PQRS) being the double ratio of points of section by a generator of the fom- given generatora through A, B, G, D is constant. ix. Correspondence betvreen the points of a conic and the lines of a plane pencil. If a (1, 1) correspondence exists between the points of a conic and the lines of a plane pencil centre S, there are ihreepoirds on the conic, of which one at least is real, through which pass their corresponding lines ; for take any point S' on the conic f^ and join it to the points of f^, then between the lines of the pencils centres S and y, so that for a definite assignment of the suffixes {X^X,X,X^={Y^Y,Y,Y,). CHAPTER I. SYSTEMS OF COORDINATES. 1. In the analytical treatment by Des Cartes of the Geometry of Space, the point is the space-element ; the researches of Poncelet and other geometers, and in particular the Principle of Duality, lead naturally to the plane as a space-element ; finally to Plucker* is due the conception of a geometry in which the line serves as the element of space. Just as the point and plane are defined by coordinates and the investigation of loci of points and envelopes of planes is conducted by algebraical methods in the older geometry, so in that with which this work is concerned, line-coordinates are employed, and loci of lines ai'e by their means discussed. The object of the present treatise is then, mainly, the investigation of the properties of the assemblage of lines which satisfy one or more given conditions, i.e. of lines whose coordinates satisfy one or more equations of given form. If only one condition is imposed the lines which fulfil it are said to form a Complex, and since a line has four coordinates, it is clear that the lines of a complex are triply infinite, or oo ', in number. If a double condition or two conditions are imposed we have a Congruencef , this is seen to consist of oo " lines. If three conditions are specified we have oo ^ lines forming a ruled surface. A fact which has most important bearings upon our subject is that since a straight line may be regarded either as the locus of its points or as the envelope of its planes, it is found that the pro- positions of line geometry stand in the same relationship to points as to planes, or the subject is dual. The present chapter is mainly given to the description of various kinds of coordinates of the straight line. * Neue Geometrie des Raumes. t The term ' system of lines ' is also sometimes employed. 16 SYSTEMS OF COORDINATES [CH. I 2. The line of intersection of the planes x = rz + p, y = sz + a-, is known, if the values of r, s, p, Pu are proportional to the coordinates of the point of intersection of the line and the coordinate plane Hi, and so on. Conversely six quantities pne connected by an equation P = 0, and such that pik = —pki, are the coordinates of a line. For, in this case, since any one of the equations (ii) ia deducible from two of the others, the four points they respectively represent lie in one line, and taking two particular values of the u, i.e. two particular planes through this line, we derive the equations (v) showing that the coordinates of this line are the pi/c. 5. Intersection of two lines. Two lines p and p' will intersect if PiiP'u + P'uPm + Pi3P'42 + P'laPii + Pup'^ + p'uPii = 0. For a and /S being any two points of p, and a', j3' two points of p', these four points are coplanar, hence «! Kj «3 04 /3i ^2 A A a'l a'2 a'3 a\ 0\ /S; ^'3 /3'4 which gives the condition just stated. Observe that this may be dP = 0, written ^^'^i^='- 4-8] SYSTEMS OF COORDINATES 19 6. Coordinates of Plucker and Lie. Passing from homo- geneous to Cartesian coordinates, or writing x , y', z , 1 respec- tively for «!, 13(2, ftj, tti, and x", y" , 2^', 1 for /3i, ySj, /Ss, /Sj, we obtain Pi2 = soy" - x'y', p^ = y'z" - y"z', ps, = z'x" - z"x', p,t = x'- x", p^ = y' - y", p^ = z'- z". These are the homogeneous coordinates adopted by Pliicker*. If now x" = x'-\-dx', i.e. if the points are consecutive, omitting accents, Lie's coordinates are obtained, viz. j0i2 = xdy — ydx, p^i = ydz — zdy, psi = zdx — xdz. Pu = -dx , p2i = -dy , psi = -dz. If the tetrahedron of reference be formed by three mutually perpendicular planes and the plane at infinity, it is clear that the coordinates p^^ are proportional to the components along the axes, and the moments about the axes, of a force whose line of action is the given line. 7. Transformation of Coordinates. If a new tetrahedron of reference be chosen, the coordinates of the Hne pne will become p'ik, where If the equations of transformation are fl . X i ^^ diiXi -{- di^X^ + Ct{8^3 T" di^X^j p'iic is obviously a linear function of the pa, so that P'ik = ^ik,i2Pii + ■^ik,i3Pi3+ The six coefficients of jp^ on the right are proportional to the coordinates of the edge A-^A^ of the old tetrahedron of reference with regard to the new one, as is seen by putting Pis = Pu = ^23 = Pat ==P42 = on the right-hand side of the equations, and so for the other coefficients. 8. Generalized Coordinates. In place of the puc we may use any six given linear functions of them as coordinates. Denoting for convenience the p's by pip^-.-pe and the new coordinates by Oi^j ... ^6. the identity P = will be replaced by a new quadratic function of the q's equated to zero. Let the latter be co {q)f = 0, then P(p) = co (q), hence dP _.^d(o dq^ dpi ~ i dqt: dpi ' * See Neue Geometric des Baumes, Bd i. S. 2. t The notation u, fi appears to be due to M. Koenigs, see La Geometrie regUe, p. 9. 2—2 20 SYSTEMS OF COORDINATES [CH. I now let p and p' be two lines denoted by q and q' in the second system of coordinates, then ^ , dP _ / , ^ 30) dqk\ ^XC^^P'M k \dqk i opj ^ dca , Thus the condition for the intersection of two lines is 1 oqjc 9. Coordinates of Klein. The simplest case included in the last transformation is the coordinate-system of Klein*. This is obtained by writing , (vi). where t = V — 1. These quantities x are adopted as the coordinates of the line. The equation P = becomes 1 The condition of intersection of two lines x, y assumes the simple form 6 S Xiyi = 0. 1 The last equation will usually be denoted in future by (xy) = 0, and ixf = hy ix^) = 0. Conversely six quantities x which satisfy Ixi' = (ar") = 0, may be taken as the coordinates of a line. For if the Xi are given, by equations (vi) we can find six quantities pik which by virtue of the equation (x^) = satisfy P = and are therefore the coordinates of a line. Intersection of consecutive lines. The condition of intersection of two lines x and y in Klein coordinates is in general {xy) = ; when applied however to consecutive lines this condition is satisfied identically for terms of the first order, and the condition of inter-* * See Math. Ann. Bd. ii., Zur Theorie der Liniencomplexe des ersten wnd zweiten Grades. 8-12] SYSTEMS OF COORDINATES 21 section of x and x + dx is {dot?) = 0. For let x be determined by its two points a and yS, then x + dx is determined by the points CL + da, /3 + dj3, thus if Xi is the (bilinear) function of a and y8 Fi{a, yS) we see that dxi = Fi (a, d^) + Fi (da, /3) + Fi (da, d^) where S Ff (a, d/3) = 0, 2 i?';^ (da, /3) = 0, 2 i?'i'= (da, d/3) = 0; also XFiia, ^)Fi{a, d^) = 0, since the lines have the common point a, and 2i?'i(a, /3)J^i(d=t, y3) = 0, since the lines have the common point /3, so that (xdx) = 0, for terms of the first order. If X and x + dx have a common point, let it be a, then da = 0, and dxi = Fi (a, dfi), hence {dx^) = 0. 10. Plane Pencil of Lines. Referring to the coordinates Pijc it is observed that they are linear in the coordinates of each of the two points a and /3, thus the new coordinates x are so also. It follows that if x and y are the coordinates of two lines which join the point a to the points ^ and 7 respectively, then denoting by z the line joining a to the point /9 -I- Xy, we have Zi = Xi + Xyi. By giving all values to \ we obtain the lines of the plane pencil determined by the two intersecting lines x and y. Conversely if x and y are two intersecting lines we have («^) = o, (xy) = o, (yo = o, hence x + \y is a, line, and thu.s is one of the pencil determined by X and y. 11. Double Ratio of four lines of a pencil. The four lines X + \y, X + X^y, * + f^y, « + ^.4^ will pass respectively through the points a + \i/3, a + X2/8, a + Xa/S, a + X^/S, where a and ^ are points on x and y respectively ; hence the. double ratio of the four lines is (Xi — Xg) (X4 — X3) (Xj — X3) (Xi — X4) 12. Von Staudt's Theorem. If a, /3 are the points in which a line meets the coordinate planes respectively opposite to the vertices At,, A^ of the tetrahedron of reference, then a + XyS, a + fjL^ are the points in which the line meets the coordinate planes opposite to A^, A^ if a2 + X/32 = 0, ai + fi0t = O. 22 SYSTEMS OF COORDINATES [CH. I The Double Ratio of these four points on the line is - = — ^r j but it is easily seen that for the given line hence the D.R. of the four points in which it meets the coordinate planes is -?^^^^. By a precisely similar process it is obvious that the D.R. of the four planes through the line and the respective vertices of the tetrahedron is equal to — — - = —-^-^-^^^ ^ or, the Double Ratio Tl4 ■ TTss P23 -Pu of the points in which a line meets any tetrahedron is equal to the D.R. of the planes through the line and the vertices, of the tetrahedron. 13. Sheaf and plane system of lines. If /8, 7, S are respective points on the three lines x, y, z which meet in the point a, then by the reasoning just employed the coordinates of the line joining a to the point \^ + fj/y + vS are 'f^i + fJ-l/i+f^i, i.e. any line through the intersection of three concurrent lines x, y, z is Xsc + /ly + vz. All such lines are said to form a " sheaf"*. If the lines x, y, and z, on the other hand, lie in the same plane t, then if u, v, w are planes through x, y, z, respectively, the line of intersection of t and any plane \u + ijlv-\-vw through the point (m, V, w), has for its coordinates again \xi + /xyi+vzi. Thus if X, y, z are concurrent, "hx ■\- fiy -V vz includes all the lines of the sheaf through their point of intersection ; if x, y, z are coplanar \x-\- fiy + VZ includes all the lines which lie in their common plane, or the " plane system." Taking the case where x, y and z are concurrent, we see that if P is any point on 'Kx + fiy + vz, the coordinates of this line being proportional to linear functions of the coordinates of P and a ; \, /i and v are each linear functions of the coordinates of P. Now if X = it was seen that the line, and hence the point P, will lie in the plane of y and z, or, \ = is the equation of the plane (y, z) ; M = „ „ „ „ {z, x); v = „ „ „ „ {x, y). * See Introduction, Art. xi. 12-14] SYSTEMS OF COORDINATES 23 14. Closed system of 16 points and 16 planes^. From Art. 4 we see that the line x may be defined either by the equations or by the equations cr . «! = ttij + -nSi, — T .iXi = TTii — TT^, &c. ; (where o- = ir^ilpsi)- A comparison of these forms shows us that we may regard the first sets of equations as the condition either that the line (aJi, x^, x^, Xi, Xs, Xn) should pass through the point («!, Oa, as, a4) or that the line (xi, — x^, x,, —x^, x^, — x^) should lie in the plane whose coordinates are («!, as, «a, cti); thus one condition involves the other. If the squares of the coordinates « of a line have given values, we obtain a set of 32 lines having important connexions ; namely the different lines obtained by taking x^ positively and x^... x^ with either sign. It will now be shown that these 32 lines together with 16 points and 16 planes form a closed system. For if we substitute the x coordinates of the line for the tthc in (iv), (Art. 3), we obtain the conditions that the line x should contain the point a, and from them may be derived the following, (which can easily be directly verified), «i (aitta - «3a4) + »'i»2 (— «i«2 - a3«4) + x^ (ojas + a2a4) \ + iXi{a^ai — ttiOs) = 0, Xi (aittj + a3a4) + ix^ (agUi — a-^a^ + x^ (aitti - Oatts) + ix^ (— ci^o-i — a^as) = 0, , together with two other equations derivable from these. If in these equations the signs of any two of the a's are changed the coefficients of the Xi are thereby either unaltered or altered only in sign. The same is true if any two pairs of the a's are interchanged (e.g. a^ and a^, a^ and aj ; and thus by a combination of these two methods of change the coefficients are altered at most in sign, so that by suitably changing the signs of the x's we return to equations (vii). The arrangement of signs for the six quantities x is easily found to be different in the different cases, we thus have 16 points, viz. "D "2) "31 "i' "ai °1> "41 "3 J °4> "si "21 "l J °3l ''4> °H °2 "1) "21 ""3) ""4! "21 "l; ~°4i ""O3; "41 "3) ~'^H ""l! "31 °4> ~"l) ~ "2 "ll ~<'2> ""3) "4 > °2) ""!) ~"41 °3> "4) ~°3) ~''2l "l > "si ""4) ~"l> "2 "D ~''2t °3> —"4 1 "2> ~"l) °4> ""si °4) -"31 °2l — "l > °3> " °4l "iJ ~ °i through each of which one of the 32 lines x passes. * See Klein, Math. Ann. Bd. n. (vii) 24 SYSTEMS OF COORDINATES [OH. 1 And, by what was shown above, in each of the 16 planes having these coordinates will lie one of the 16 remaining lines ; and having given one point or one plane the other points and planes are determined, and the same system of 16 points and planes is arrived at, with whichever of the above points or planes we start. Any point or plane of space determines such a system ; so that by aid of the table just given, all the points of space are divided into such sets of 1 6 ; similarly for the planes of space. An important fact connected with the above system is the following: — since the point (oi, a«, a^, a,) clearly lies in the six planes (Oa, — «!, 04, — Hs), (Oa, — tti, — O^, Hs), (Ks, — "t, — «], Ha). («s. «4. — «i. — — «s> ("2. — «i), and the plane («!, Oa, as, ««) passes through the six points having these coordinates, therefore, from the nature of the system, if any other point of the system e.g; (oj, Hj, a^, a^) be taken, we should obtain a similar result, whence it follows that — through each point of the system there pass six planes of the system, and in each plane of the system there lie six points of the system. CHAPTER II. THE LINEAR COMPLEX. 15. A COMPLEX of lines has been defined, (Art. 1), as the assemblage of lines which satisfy one condition. Thus if q^ ... q^ are the general coordinates of the last chapter, the lines whose coordinates satisfy the homogeneous equation of degree n fiqi...qe) = 0, form such a complex. We have seen that if a and /3 are any two points of a line the quantities pq^ . . . pq^ are each homogeneous linear functions of the coordinates of a and of the coordinates of /3: thus/=0 is homogeneous of the wth degree in the coordinates of both a and j8. Taking a to be any given point, this equation therefore gives the cone formed by the lines of the complex through the point a ; and this cone is seen to be of degree n. Similarly the q's of a line being proportional to homogeneous linear functions of the coordinates of any two planes u and v through the line, by taking m in/= as any given plane, we have the curve enveloped by the complex lines which lie in the plane u ; and this curve is seen to be of the nth class. 16. The Linear Complex. If n is unity the complex is of the first degree, and we see that in a complex of the first degree all the complex lines through any point lie in a plane, the "polar plane " of the point, all the complex lines in a plane pass through a point, the "pole" of the plane. If we employ the "p" coordinates the equation of the complex will be of the form a^Pn + (hsPis + ^liPu + a-^Pii + Clii Pa + a^p^= 0. Similarly if the coordinates of Klein are used, the equation "^aiXi^Q, or {ax) = Q, represents a linear complex. The equation pik = represents the complex of lines which intersect the edge of the tetrahedron of reference opposite to 26 THE LINEAR COMPLEX [CH. II AiA^\ for if Oi/Sj; ^ Ht/Sj = 0, where a, /3 are two points on a line of the complex pa; = 0, let the point a be that in which the line meets the face opposite Ai of the tetrahedron of reference, then 0^ = 0, it follows that either at = or /3i = and in either case the line intersects the edge opposite AiA^^ of the tetrahedron of reference. The equation a;i = 0, i.e. ^12 + ^34 = 0, represents a complex of the utmost importance in this subject, it will be termed a fundamental linear complex ; there are six fundamental linear complexes, viz. «, = 0, x^ = 0, x^ = 0, Xi — 0, aij = 0, ajg = 0. Now taking ^1, ^j, fs, ^4 to be the coordinates of any point on a line of SojiPij; = through the point a ; or writing in this equation ^i* = «(?*— «i?i! ^'Od arranging the terms, we observe that the pc4ar plane of a is, (if we write for convenience a*; = — Oi*), Ii(«2iff2 + asiHs + a«a4) + ^2(ai20j+as2a3+a42a4)+^3(ajsai+c'.23a2+a43a4) + ?4(ai4«i + a24a2 + a34a3) = (i), so that if M is the polar plane of a, we have the equations O-.M4 = CtiiOi + a24«2+a34°'3+ • I = %ai + M^aa + Wsbts + 1*404 _ / If two complex lines intersect, their point of intersection is the pole of their plane. It is clear from the foregoing that to each point of space a unique polar plane is attached ; this may be also seen directly, for if the polar planes of the points m and n coincide, it is necessary that (mj — jOWs) ttji + (ms — pTij) ttsi + {m,i — pn^ ttji = (iii), together with three other similar equations, and m, n being supposed to be different points, it follows that a^ 031 a^ .(ii). a,4 -'31 0132 034 *41 ^42 ^43 = 0; the determinant is skew-symmetrical because a,A; = — a^i, and has the value (auOsi + ai3a42 + 014023)^. The quantity in brackets is in general different from zero, (see Art. IC), and hence the equations 16-18] THE LINEAR COMPLEX 27 (iii) cannot coexist, or, the polar planes of the points of space are all different. 17. Polar Lines. The equations (ii) connecting a point a and its polar plane u, show that if oii = mi + Xui, i.e. if we suppose a to describe the line joining the fixed points m and n, then a- .Ui = Mi + \Ni where Mi = 'Zau'm]c, Ni = 1aunh, {aii = 0), !c k i.e. u turns round a fixed line (viz. the line of intersection of the planes M and N); conversely, if aUi = Mi + XNi, it follows from (ii) that ai = mi + Xni. Two lines thus connected are said to be polar to each other. Polar lines do not intersect unless they coincide, for m and n being any two points and p the intersection of their polar planes, the polar plane of m is the plane through m and p, the polar plane of n is the plane through n and p, so that if p met the line mn these two planes would coincide, which we have seen to be impossible. If the line mn belongs to the complex it lies in the polar planes of both m and n, and hence coincides with its polar line p. The proposition just established shows that the polar planes of the points of any given line p' pass through the same line p, from which it follows that any line meeting both p and p! belongs to the complex ; hence the polar planes of the points of p will all pass through p'. ' . The polar lines of the lines through any point P lie in one plane, the polar plane of P. Any complex line which meets p must also meet p', for let a! complex line *• meet p in P, then since all the complex lines through P belong to the plane determined by P and p', X must lie in this plane, and therefore meet p'. 18. The Invariant of a linear complex*. Taking the equation of the complex in the form 6 2atg'i= 0, 1 * See Elein, Math. Ann. v., Differentialgleichungen in der Liniengeometrie. 28 THE LINEAR COMPLEX [CH. II and the identical relation (Art. 8) as o>{q) = 0, the complex has an Invariant. For writing a)(q) = ttaqi'+ ... + 2a„g'rg', + . . . , since the a; are contragredient to the qi we know that ttic Kji 066 0,s "bi Oi de is invariable for linear transformations effected on the qi. The numerator of this fraction is an Invariant of the complex. It will be denoted by CI (a). 19. The Special Complex. If the invariant fl (a) is zero we then have the system of coexistent equations ^-^ = 2pa,, (i=l, 2....6), {ab) = 0, and hence also a){b) = p (ah) ~ : or the complex may be written where b is. a line. Thus each line of the complex cuts the line b, (Art. 8), which is called the directrix of the complex. The complex is here said to be special. The coordinates of b are proportional to da for since — fl (a) = A^^Oi' + . . . + 2Argaras + ... where .4„ is the coeflScient of o^j in the discriminant A of w, _isn 2dai' and from the equations ^ = 2pai = Aiiai+ .. we see that or 86.- A.bi = p{Ai,ai+ ...) -p dn bi = 2A'8ai" 18-21] THE LmEAR COMPLEX 29 When the Pliicker coordinates are used, since we see that Xi (a) = 2 (oijOai + 013042 + 014023), or, il (a) and to (a) have the same form. When Klein coordinates are used a>{x) = l Xi\ - n (o) = 2 Oi''. 1 1 Generally, in any system of coordinates in which each coordinate appears in 0) {q) only once, the first minors of A are — and thus A Qjit ■ 20. Coordinates of polar lines. If, when Klein coordinates are used, z and z' are polar lines with respect to a linear complex Soj^i = 0, it will now be proved that p.Zi' = Zi + Xoj ; for the coordinates of any line of (ax) = which meets z, satisfy the equations (ax) = 0, (zx) = and hence the equation % (Zi + \Oi) Xi = 0, for all values of \. The last complex is special if 2 (zi + Xaif = 0, which gives two values for X, viz. zero and — 2 ^-^ ; hence, with this value of \, z + Xa is a line which meets every line x of the given complex which meets z, i.e. z + 7m is the line z' (Art. 17). An important case arises when (oa;) = is a fundamental complex, e.g. x-^ = 0, in which case Oi =1, 02 = Oa = 04 = Oe = Oj = 0. The polar of any line z has the coordinates z^ + X, z^, z^, Zi, z^, z^ which requires that \ = — 2^1, i.e. if z and / are polar for a fundamental complex Xi = 0, then Z]c = Z]^, except for k=i when Zk + ^h = 0. The 32 lines of Art. 14 may be obtained by starting from one of them and taking the successive polars for the 6 fundamental complexes. 21. Relations between the functions <•> and fi. The following identities, which may easily be verified, are useful. Denoting - g— by ^j where Zj...^ are any quantities, \ Za{Z) _ 2'W~~ *' 30 THE LINEAR COMPLEX [CH. II O(^=j2Z,?|;=-A.o)(0), ,(!g)-.-..<.>. From these equations it follows that if a complex line x meets the line z it will also meet another line z'. For if {ax)=^ 0, (^ x\ = 0, X belongs to each complex of the system (.(X.+|)) = 0; of these complexes two are special, viz. those corresponding to the two values of X given by Q (Xa + 2.^) = or \^a{a) + ^XS.(Zi^+ia.{Z) = 0, and since 2 is a line, m(z)=0, i.e., Q(^ = 0. Thus the values of X are and ^ ^r'£taj 4A(ag) a (a) a, {a) The directrix of a special complex is (Art. 19) i.e. taking X 3Q(Xffl + 2.g) do. (a) da (2) 4A (az) Q (a) , 4A {az) da (a) . . , we have „ , . -^^- — 4A0i = pz/. a (a) dai • f • Multiplying by at and adding we obtain 4A {az)—p(az'). Thus finally iL , ■Sj' ^ 1 da (a) (az) {az') a (a) da( is the equation connecting the coordinates of polar lines in general coordinates. 22. Diameters. If a line lie in the plane at infinity its polar line is called a diameter and passes through the pole of the plane at infinity for the complex. Taking a series of parallel planes, their poles lie on a diameter, viz. that corresponding to their common line at infinity, and it follows that all diameters are parallel. There is one diameter which is perpendicular to the planes through whose poles it passes, viz. that which joins the poles of the planes which are perpendicular to the diameters. This diameter is called the Axis of the complex, and is perpendicular to the complex lines which it meets. 21-23] THE LINEAR COMPLEX 31 23. Reduction of the complex to its simplest form. If we take as two opposite edges of the tetrahedron of reference two polar lines, e.g. A-^A^ and A^Ai, the other edges will belong to the complex; thus for instance the edge A-^A^ whose coordinates are given by Pi2=^i4=i323=/'34=P42 = belongs to it and hence ftis = 0. Similarly a^ = a^^ = a^^ = and the equation of the complex reduces to fliaPia + ^34^34 = 0. To refer the complex to Cartesian coordinates take the plane at infinity as one face of the tetrahedron of reference, and then the p coordinates will assume the form given in Art. 6. Let the axis of the complex be chosen for axis of z and as the edge AiA,, the edge A^A^, the polar of J.3J.4 will then be at infinity; the edges A^A-^^, AiA^, being complex lines, are each perpendicular to A^Ai, we take them as being also at right angles to each other ; the complex is now referred to rectangular axes, of origin J.^; then, (Art. 6), p^^^xy' -x'y, p3i = z — z', and since when A^A^ and A^A^ axe polar, the complex is ,. (6a;) = 0is (— ota, a,, /cMj, — kKs); and if ^ is any other point on the line, the polar plane of /3 in {ax) = is (- /Sj, jSi, - «^4, /e^Ss), ., ., „ (6«)=0is (-/Sj, ySi, a:;84, -K/Sg). From this it is clear that if the polar plane of ^ in {ax) = is the polar plane of a in {hx) = 0, then the polar plane of /S in (bx) = is the polar plane of a in {ax) = 0. Hence if p be a line common to two complexes {ax) = and {bx) = for which Sflj^r- = 0, the correlation of planes through p, obtained by taking the polar planes of the points of p for each complex, is an Involution. The complexes are said to be themselves in Involution. A similar method of proof shows that the correlation of points on a line p common to the complexes, obtained by taking the poles in the two complexes of the planes through p, is an Involution. We saw, (Art. 24), that each line common to the two complexes meets the lines z and z' which are polar for each complex. Let N and N' be the points where a common line x meets z and / respectively; it is important to observe that N and i\r' are the double points of the involution determined on x; for the polar plane of N in each complex. is the plane {N, z'), thus N corresponds to itself in the involution, similarly for N'. If one of the complexes, say {ax) = 0, is special, its directrix belongs to (bx) = 0, for the condition of involution gives (b ^ j — ; this is sometimes written O (a 1 6) = 0. 34 THE LINEAR COMPLEX [CH. II If both complexes are special, the directrix of each belongs to the other, i.e. the two directrices intersect. The following properties of complexes in involution should be noticed. If (ax) = 0, (bx) = 0, are any two linear complexes in involution and X is any line of the first complex, the polar of x with regard to the second complex is x', where p.Xi =Xi + Xbi, (Art. 20), Klein coordinates being used ; and since (aai) = 0, (ab) = 0, it follows that (ax) = 0, or x belongs to the first complex. If two lines z and / are polar with regard to a given complex (ax) = 0, p .Zi =Zi + \ai; and if (bx) = is avy linear complex which contains z and z', since (bz) = 0, (bz) = it follows that {ab) = 0, i.e. any linear complex through a pair of polar lines for a given complex is in involution with the given complex. Let the equations of two complexes referred to their respective axes, (Art. 22), be pu — K1P34 = 0, p^^ — K^p^ = ; then if d is the shortest distance of the axes and 5- — ^ the angle between them, the equation of the second complex referred to the same system, of coordinates as the first, is obtained by writing in its equation respectively for x, y, and z, the values x — d,y sin i^ — z cos , y cos (^ + ^ sin 0. The equation of the second complex then takes the form ^12 sin , >, ('^^) = and (bx) = ; and let Oi, 0^ and O3 be the points corresponding to in the involutions determined on these lines (Fig. 1). Then OiO^ is the polar plane of in (ex) = 0, hence, (Art. 25), 00^0^ „ „ „ Oiin(6a!) = 0, similarly OOfiz „ „ „ 0^ in (bx) = ; hence both Ofii and O2O3 belong to (6.r) = 0, and therefore 0^ is the pole of the plane Ofifi^ in (bx) = 0; simi- larly, Oi is the pole of 0^0 JD3 in (ax)= 0, and O3 in (ex) = ; hence the three planes through Oi are the polars of Oj in the three complexes, and so for O2 and O3. Thus each vertex of the tetrahedron is the pole of the faces through it in the three complexes. The following table shows the poles of each face of the tetrahedron in the complexes A, B and C. Oi 0, 03 A 00,0, 0,0,0, 00,0, 00,03 B 00,03 00,0, 0,0,0, 00,0, C 00^0, 00,0, 00,0, 0,0,0, 27. Six complexes mutually in Involution*. It has been observed that each of the Pliicker coordinates of a line equated to zero gives a special complex; for the lines which satisfy ^jt = are all the lines which meet the edge opposite to AiA^^ : and in the * See Klein, Math. Ann. 11. 3—2 36 THE LINEAR COMPLEX [CH. II coordinates of Klein, a;t = represents a fundamental complex, but it is not special, for here H (a) = - Soi" = — 1. Thus in this system the line is referred to six coordinate complexes. These com- plexes are all in involution with each other, for here the equation 2 Oi ,55- = takes the form laibi = 0, which is clearly satisfied for obi any pair of the coordinate complexes. The six poles of any plane for the fundamental complexes lie on a conic. For, in any given plane, denote by aa the line joining the pole of aji = to the pole of x^ = 0, then if Ojs and Ois have respectively coordinates a; and bi, Oa and Ojs „ „ „ oj' and bi, the coordinates of Oij are Oj + Xftf, where X. is determined by expressing that a^ belongs to Xi=0, hence X = — r-; similarly Oje is ttf + fibi, where /jl = — j^. Oe The double ratio of the pencil formed by a.13, ajj, Ou, 0,6, i.e. by a,b, a + '\i, a + fib, is -, (Art. 11), or -^ ; similarly the double ratio fi a^Oi of Ojs, Ojs, a^, a^e, is -^rr7 > ^^'^ these double ratios are equal, for since a^bt Oik belongs to Xi = and x^ = 0, we see that Oi = 6, = a, = 65 = O2' = 62' = as' = 65' = 0, and since a,e and a^ intersect 6404' + beae = 0, and since Ois and a^ intersect q-ibi + agbj = ; therefore — t^=~tt^/> aeOi tte O4 hence, since (fltw 014015016) = (023*24 0^25 c^ae). the six poles lie on a conic. 28. Transformation of toordinates. Two instaDces of trausformation of coordinates have been met with, viz., the change from one tetrahedron of reference to another, and the change from the coordinates of Plucker to those of Klein. We now consider such transformations in more detail. It has been shown that if for six complexes .3;j = ... 575 = we have 2^j^=0, the complexes are in involution. Now let ^1 = "12P12 + «34^34 + •^2=*I2Pl2+ ■'^6=/l2^l2''" 27-29] THE LINEAR COMPLEX 37 then if the complexes Xj^=0 ...x!g=0 are mutually in involution, while a {a)=...=a(f)=k ; it will follow that Sa;i^=0; for, from the six equations O^ia "34 + '^34 %2 + • • • = ^> a,, 634 + 0,4 6,, + . ..=0, we obtain 034 A =^12- A, where A is the determinant of the equations of trans- formation and ^12 is the minor of a^^ in A ; similarly ai2^ = A^. k, etc. Hence solving the equations tor p^^ ■■■P2S in terms oi Xj^,..a!g, we obtain A . Pi2 = ^34^1 + 634^2 +.. . +/34 A'e> ^ • ^34 ~ ^12^1 "^* 6l2*'^2 + • • • ~Hj'l2^6J thus, since Pi2^34+iOi3?'42+iOi4ft3=0, we have 2a;2=0. If the equations of transformation are where 7,x'^ = 'S.v'=0, the transformation is "■orthogonal," the equations connecting the coefficients are from these are derivable 2 a^^=l, from the last equations we learn that the complexes x' are mutually in involution*. If the equations of transformation are Pvl = "■ilPl2 + «3t?'34 + O'laPu + «42A2 + "'liPu + «23?'23. i'34' = 6l2Pl2+ the coefficients in the same vertical line are the coordinates of the edges of the old tetrahedron of reference with regard to the new tetrahedron of reference. And since the lines for which p^^' is zero form a special complex of which the edge A^'A^' of the new tetrahedron of reference is directrix, it follows that the coefficients of the first row are the coordinates of the edge Ag'A^ with regard to the old tetrahedron of reference ; and so for the other rows. We find that in this case, if A is the determinant formed by the coefficients, ftjj A = .^34, . . . etc. 29. The fifteen principal tetrahedra. We have seen that if asi-.-Xg are linear functions of the Pliicker coordinates of a line such that l.a;'= 0, the complexes a;i=0, . . . a-6=0, are in involution in pairs. Also if we write ia;,=Pu-p3i,--- we obtain the equations of Art. 9. * We easily derive that Xi=Sa,^Xi/. 38 THE LINEAR COMPLEX [CH. II These last equations express that the line x has coordi- nates pi2... with regard to a certain tetrahedron ^1^.243^4 whose position in reference to the given complexes a;, = 0... will now be determined. The coordinates of the edges A^A^, A^At of this tetrahedron in reference to themselves are got by putting respec- tively all the p's zero except jo^ and p^t ; thus the x coordinates of these edges are (1, - i, 0, 0, 0, 0), (1, + i, 0, 0, 0, 0). But these are the common pair of polar lines for the complexes Xi = Q and x^ = 0, as will now be shown. For let a;, = . . . aJe = be six complexes mutually in involution : they maybe arranged in pairs in 15 ways, take e.g., the pairs x-^, x^; x^, x^ : x^, «„ ; then all lines which belong to both Xi=Q and 373 = belong also to the special complexes x-i + ix^ = 0, a;, — MJa = 0, i.e. the directrices of these two complexes are polar lines in both aji and x^, and their coordinates are (1, i, 0, 0, 0, 0), (1, - i, 0, 0, 0, 0); i.e. they are the edges J.3.44 ax^dA^A^. In the same manner the common polar lines of x^ and Xt are (0, 0, 1, i, 0, 0), (0, 0, 1, - i, 0, 0), i.e. the edges A^A^, A,Af, while those of x^ and x^ are (0, 0, 0, 0, 1, i), (0, 0, 0, 0, 1, - i), i.e. the edges A^A^, A^A^. There are 15 tetrahedra which bear this relationship to the coordinate complexes : and starting with one of these tetrahedra we thereby determine six complexes in mutual involution, and thus the 14 other tetrahedra. For the six fundamental complexes Pn+P!>i = 0,pu-pai=0, pu+pts=0,pi!,-pii = 0,pii+p^ =0, pu-pfa=0, the polar planes of a point (aj, a^, aj, 0(4) are easily seen to be, (Art. 16), (fltj, - «!, Ui, - a,), (hj, - tti, - 0(4, as), (tts, - a4, - a^ a^), («s, ai.-ai.-Oii), (a4, as.-Kj, -Oi), K. - Ms. "a.-Mi), i.e. six of the set of the 16 planes of the closed system determined by the point (a„ a^, a,, a,), (Art. 14), all of which pass through (Oj, 02, «3, a4), and from the properties of this system, it follows therefore that the six polar planes of any point of the system with reference to the six complexes are six planes of the system. Thus the 16 points and planes of Art. 14 are such tfiat each point is the pole for the six complexes Pv2+Pm = 0, {A + \B), therefore i.e. the lines q' are either concurrent or coplanar. If to one sheaf of lines q corresponds one sheaf of lines q', then to every sheaf of lines q corresponds a sheaf of lines <^. For let the centres of the given corresponding sheaves be and 0', and let P and P' be centres of any two corresponding pencils ; the planes of the latter will not in general go through and 0', hence to two lines of the pencil of centre P and the line PO will correspond two lines of the pencil of centre P' and the line P'O', so that to the sheaf of centre P will correspond the sheaf of centre P'. In like manner if to one sheaf of lines q there corresponds a plane system of lines q', then to each sheaf q corresponds a plane system q. Thus the relationship established between' the lines q and q' must arise either from a collineation or from a reciprocity of space. Consider for instance the equations Pn = <^12Pu + P3i=\2Pu+ > P^=fnPn + +fiaPis- The quantities which are vertically underneath each other are the coordinates of the correlatives of the edges of the tetrahedron of reference regarded as belonging to 2 {e.g. make all the ^s zero except p-^^: the quantities in the same row are the coordinates of the correlatives of the edges of the tetrahedron of reference regarded as belonging to 2*, e.g. for the side A^An, Pi2' = is a special complex formed of the lines which meet A^A^•, thus the cs'a being the coordinates of the directrix of this complex are those of the correlative in 2 of ^3.44 regarded as belonging to 2'. 41. Ruled surfaces and curves of a linear complex. A ruled surface of a linear complex is one whose generators belong to the complex. On each generator a correlation of points is established by means of the planes through it, for each such plane determines a point of contact, and also a pole in the complex. This 48 SYNTHESIS OF THE LINEAR COMPLEX [CH. Ill correlation has two united points and at each of them the tangent plane of the surface coincides with the polar plane in the complex. These points may be determined analytically as follows : if a line 3/ is a tangent to the surface, it meets the generator x and a consecutive generator x + j^dd, where 6 is the single parameter of which the coordinates x are functions, thus dOj Now take the lines y which also cut any given line a and belong to the given complex (ax) = 0, we then have the additional equations (2/a) = 0, {ay) = 0. These live equations give two lines y, which, with x, determine the two points. A curve all of whose tangents belong to a linear complex is called a curve of the complex. At any point of such a curve the osculating plane is the polar plane of the point in the complex, since two (consecutive) tangents through the point belong to the complex. The locus of points just discussed- is such a complex curve, and in it since the tangent plane of the surface is the osculating plane of the curve, it follows that the curve is a principal tangent curve of the surface. If through any point P a plane be drawn which osculates this curve at Q, PQ is then a complex line, hence the points of contact of the osculating planes which pass through a given point P .lie on the polar plane of P. The degree of the curve is equal to the class of the plane section of the surface, for if any plane ir meets the curve in a point A, the polar plane of A (the tangent plane to the surface at 4) passes through F, the pole of tt in the complex, and every such line FA is a tangent to the section of the surface by tt*. Curves of a linear complex. If the coordinates of Lie, (Art. 6), be used in the simplest form of the equation of a linear complex (Art. 23), it becomes xdy — ydx = kdz. The complex curves through any point {x, y, z) are those the direction ratios dx:dy: dz of whose tangents satisfy this equation, and we have seen that they all have the same osculating plane at the * See Pioard, Ann. de I'ecole normale super. Sir. 2, T. vi. ; also Lie and Soheffer's BeriihrungstransformatioTien, S. 235. 41-42] SYNTHESIS OF THE LINEAR COMPLEX 49 point, viz. the polar plane of the point for the complex. Moreover, they all have the same torsion, for ds _ X'^+Y^ + Z^ where X = dyd^z - dzd'y, dr Y=dzd''x-dxd''z, Z = dxd''y-dyd'a;*. dx dy dz d-'x d'y d^z d^x d^y d^z By aid of the equation of the complex this denominator is easily seen to reduce to T (dxd^y — dyd'xf, while X = I (dxdhj - dyd'x), 3^= - f (dxd^y - dyd^x), hence ds x^ + y^ + 1 x^+y^ + ¥ dT~ 1 k > k a result depending only upon the coordinates of the point, and therefore the same for each complex curve through the point f. 42. It has been seen, (Art. 33), that a linear complex contains two lines of every regulus ; of any ruled surface whose degree is n it contains n generators ; for the equations of any generator of such a surface are, in Cartesian coordinates, x = zf{m)+f{m), y = zf {m) +f (m) ; where m is a parameter. Hence the coordinates puc of such a generator are given by equations of the form Po: =fk (m). Since the ruled surface is, by hypothesis, of degree n, the equation ai2/s4(»i) + a34/i2(m)+...=0 (i), gives n values of m, provided that the am are the coordinates of a line, i.e. ai2a34 + ai3a42 + "lAs = 0. ' But this restriction as to the a^ will not, in general, affect the number of solutions of equation (i) ; hence that equation gives n * See Salmon, Geometry of Three Dimensions, 3rd edition, p. 345. t This theorem is due to Lie; see Proceedings of the Society of Sciences at Christiania (1883). J. 4 50 SYNTHESIS OF THE LINEAR COMPLEX [CH. Ill values of m when the ««; are any quantities whatever. That is to say, any given linear complex will contain, in general, n generators of a ruled surface of the nth degree. So that, for instance, any hnear complex contains, in general, three lines of a given ruled cubic. 43. In any rational curve whose points are given by the equations «i=/i(0. «2=/2(0. 1^3= fit), a:t=f{t), where the functions / are of degree n in t, the coordinates of its tangents oc^dx^ — x^dx-^, &c. are seen to be of degree 2n — 2 in t, hence 2n — 2 of them will intersect any given line. The number of its tangents which intersect any line is known as the rank of the curve. It is clearly equal to the degree of the developable of which the curve is the cuspidal edge. Now any twisted cubic can be so expressed, n being equal to 3, and hence the rank to 4 ; also a linear complex is determined by any five tangents of the curve, this complex therefore contains five tangents of the developable, and must therefore contain it altogether (since a linear complex which does not contain a ruled surface of degree n can have only n lines in common with it), hence the tangents of any twisted cubic belong entirely to some linear complex. Every rational quartic curve with two stationary tangents* has the rank 6, hence a linear complex through the two stationary tangents and three other tangents contains seven generators of the developable, i.e. contains it altogether. Every rational quintic with four stationary tangents is seen similarly to belong to a linear complex. 44. The polar surface. If P be any point and tt any plane through it, the pole of tt for a given linear complex A is some point P' in tt, while the polar plane of P is some plane ir' through FP' If now P and tt are so related that while P describes a surface S the plane -rr touches S at P, it follows, neglecting small quantities of the second order, that all positions of P consecutive to P lie in tt, hence all planes tt' consecutive to tt' pass through P', i.e. the plane tt' touches the surface S' the locus of P', at P'. The surface S' thus obtained may be called the polar surface of S for the given linear complex. The tangents of 8' are the polar lines for A of the tangents of S. * See Salmon-Fiedler, Geometrie des Baumes, Bd. n. S. 140. 42-45] SYNTHESIS OF THE LINEAR COMPLEX 51 If the coordinates of P are Wi and of tt are Mj, «/ and ui denoting corresponding quantities for P' and ir', we have p.Ui = aia^a 4- a^^Xs + a^Xi, &c. (Art. 16), hence it follows that the class of S', i.e. its degree in plane coordinates, is equal to the degree of S, and vice versa. If p and p' are a pair of corresponding tangents of S and S' respectively, we have p . pj = 2A.a3i-D. ia)p^i, &c., (Art. 20), where A =1 is the result of substituting the quantities Ofj; for ink in "¥. It follows that with reference to the tetrahedron of which four edges are the generators of F which belong to A, the equations of F and A can be brought to the form 2Z,Z4 + 2X^X3 = 0, X,pu + \pi3 = 0, where Xj, Xj are roots of A (X) = 0. In the reciprocity determined by the equation ^+X/=0, the locus of points .v which are united to their corresponding planes is clearly Soitajfa;^ = 0, or F; for the planes u which are united to their corresponding points the following five equations hold, Mi = S (oik + ^■aift) ^k, k "^UiXi = ; hence eliminating the Xi we obtain as the envelope of such planes a quadric A whose equation is SAiiMiWi, = 0, in which the Ajt are minors of A (X). The left side of the last equation is clearly a contravariant of A+\f, hence B' . 2 ^ikUiUi: = 2X (X= - X/) U, U, + 2X (X= - Xi^) U^ U,. In point-coordinates A will therefore have as its equation X^ {2X,X, + 2X^X,) - 2\,'X^X, - 2\^X^X, = 0. Thus A is a member of a " pencil " of quadrics of which one is F, while another is 2X/Z,A'3 + 2X1^X1X4 = 0. This latter quadric is the locus of the polars of the generators of F with regard to the linear complex A ; for one system of generators of F being Xj = /iXj, /j-X^ + X3 = 0, the polar plane for -4 of a point Xf on the line yu. is \ (X, Y, - X4 FO + X, (X, Y, - Xs Y,) = 0, or iM (XiX, 74 + X,X4 Y,) - X1X4 7i + X,X, 7, = ; 47-48] SYNTHESIS OF THE LINEAR COMPLEX 55 hence the polar of the line ' /i ' has the equations and this line clearly lies on the quadric VF,F, + vy.F3 = 0. Again there is a singly infinite number of linear complexes which have in common four generators of F, one of them being A, another, B, is the locus of the polar lines of A with regard to F; the system of linear complexes through the four given lines is then A + /xB = 0. If p is a line of A its polar p' with regard to F is determined by the equations 'tthc = - — , moreover the equation of A being dA' STTiifc = — = 0, the equation of B will be dpiic doik or, which is the same thing, 2^.^ = 0, dctik dpik where P =Pi2P3i+Pi3P42 + PiiP2s, and is obtained by substituting Oifc for TTa in ■*F. The vanishing of the invariant is thus seen to be the condition that A and its polar complex B with regard to F should be in involution. 48. Linear transformations which leave a quadric unaltered in form. By means of a linear complex a general class of linear transformations is obtained which leave the form of a quadric surface unaltered. For let the given quadric be F='^ai!cXiX!i:=0, and the required equations of transformation ^i-XcikOSk (i). h Now the equations Xi = kti + XTi, ^i = kti-XTi (ii), lead to F{x) = F{^) provided that t and t are the coordinates of two points which are conjugate with regard to the quadric F=0. If therefore ti and tj can be linearly connected, so that from the equations (ii) the equations (i) can be deduced, a transformation of the required kind will be obtained. •(v), 56 SYNTHESIS OF THE LINEAR COMPLEX [CH. Ill Let Ui be the polar plane of ti for F, then ti = 'S.AaUk (iii), k where the quantities Ai^ are the first minors of the discriminant of F, and the fact that tj lies in the plane Ui, i.e. that ti and tj are conjugate points, may then be established by linear equations of the form Ti = l,ait:Uk (iv), provided that oa = — a^ (and hence that an = 0). Substituting in equations (ii) from (iii) and (iv) we obtain Xi = K'ZAikUi: + XSOiifcltii;' k k ^i = Kl,AikUt:-\1aikUk k k the elimination of the u^ from these last equations will give the equations (i) which have been sought. The solution of the first set of equations (v) leads to A{K,X).Ui=tAuWk, k where A («, \) is the determinant of the quantities kAhc + Xoik and Afci a first minor of A («, \). Hence A(K,\)l,AiiUi = '2,'ZAkiAiiXk, i k i also from equations (v) ici + ^i = 2Kl,AiiUi, i so that A {k, \)^i = 2k12 AuAuXt - A (k, X) xi or the required equations of transformation are ^i = 'ZcmXk, Ik'^AuAu where cik = /. ., , , when k^l, A(/£,\) 2K'2.AiiAu-A.{K,\) and cii = — ^ , . ,. . A («, X) Equations equivalent to these may be obtained from the solutions of the second set of equations (v) by changing the sign of X. The above solution of the problem contains the indeterminate quantity -, hence with any given linear complex are associated 00 '■ linear transformations which leave the form of any given quadric surface unaltered. 48-49] SYNTHESIS OF THE LINEAR COMPLEX 57 When F is referred to a self-conjugate tetrahedron its equation may be taken to be F = x^^-\- xi + x^ + x^ = 0, and the equations of transformation are then Xi = KUi + XSojifcWifc, When the four lines which F and the linear complex have (in general) in common are taken as four edges of the tetrahedron of reference, F = 2ZiZ4 + 2X2Z3, A = X,p,, + \p^. The equations of transformation are here, (putting X = 1), Xi = (k + Xi) Ui, Xi = {k + Xs) Us, Xs = (k — X3) U^, Xi = (K-\) Ui\ Hi = (« - Xi) f/4, a = (« - X3) U^, Ss = (« + X3) U^, H4 = (« + Xi) TJ^. Hence 1 = ^ ™1> -^2 = ^ «2) -43= TT —3; -^4— „ . -v "4- K — Xi K — X3 • K + A-a /f + A.1 It is to be observed that this transformation changes not only X1Z4 + X^Xs into H1S4 + S2S3, but also X^X^ + i^X^Xs into 'Bi'ai + fiSii'Siz, i.e. the pencil of quadrics through the inter- section of F and A is unaltered in form by this transformation. 49. CoUineations which leave a linear complex un- altered in form. In the most general coUineation four points coincide with their corresponding points (Art. 34), hence the edges of the tetrahedron thus determined correspond to themselves. If the coUineation is such that it transforms a given linear complex into the same complex, and if one edge of the preceding tetrahedron does not belong to the complex, its polar line for the given complex must remain unaltered by the coUineation, i.e. must be the opposite edge of the tetrahedron ; the other four edges of the tetrahedron must belong to the given complex. Hence when a linear complex is transformed into itself by a coUineation, in general four of its lines remain unaltered in position. The general coUineation is represented by the equations Y, = a^X^, Y^ = a,X„ 73 = 03X3, Yi = a,X„ (Art. 34), and the given complex is by hypothesis, with regard to this tetrahedron of reference, of the form apu -\- bp^ = ; in order that the coUineation should not alter the form of this latter equation 58 SYNTHESIS OF THE LINEAR COMPLEX [CH. Ill we must have Oi«4 = a2a3; and if this condition is satisfied every complex Pit + fip^ = is unaltered by the collineation ; also every quadric of the pencil XiXi + \X^Xi = (i is then transformed into itself Hence the general linear transformation of a linear complex into itself is given by the formulae of Art. 48 by which the siirface 'ZaikXiXt = is transformed into itself; but in the present case the anc are given quantities and the anc undetermined parameters. It is easily seen that every quadric so transformed must be a member of the pencil XiXi + \X^X3 = 0. 50. Reciprocal transformations. It has just been seen that the general collineation as a rule neither leaves a quadric nor a linear complex unaltered in form, but if by it one complex is thus unaltered so are oo ^ complexes ; it will now be shown that the general reciprocity leaves two (and not more than two) linear complexes unaltered in form and also two quadrics. The general reciprocity is given by the equations in which ^it =f ^u ; the surface F which is the locus of points united to their cor- responding planes is and to this surface the reciprocity is similarly related. Taking Xi = x(, (Art. 35), and solving for Xi from each equation we obtain Bxi = 2 BuUk = S BikU^ ; k k hence the planes it^ and zf/ which are united with their corre- sponding points envelope the quadric A = l.'ZBi]eUiUk = 0. Again writing ySfj. + yS^ = 'iayc = 2au, ^ik — ^u = 20^1; = — 2aki, and taking A as the linear complex "ZXa^jcXiyi: = 0, Vi as the polar plane of x for F (or Satta;,^ = 0), Wi as the polar plane of « for ^, Vi + 'Wi = la^Xk + 2 cLikXk = Ui, Vi- Wi = %aiicXk—'Z oncXk = Ui'. 49-50] SYNTHESIS OF THE LINEAR COMPLEX 59 Hence, the most general reciprocity is determined by a quadric and a linear complex. The eqxiations determining the general reciprocity may be, by use of a suitable tetrahedron of reference, stated in the form (Art. 35) C/i = miX4, C/j = 7712X3, U^^m^X^, C/4 = mjXi; hence denoting by pik the coordinates of the line joining two points Xi, Yi and hy pik the coordinates of the intersection of the corresponding planes Ui, Vi we obtain Pn = Zi Fj - X^ Fi , ^,2' =UsVi- Ui V3 = msm^pa , similarly p^^' = ■mimj^js, &c. Hence the coordinates of corresponding lines p and p' are connected by the equations P12 = insiniPa, pj = mimr,pi,, Pis' = ^2^4^513. P42' = Wl] ^3^42, pj = mimspai, p^s = m^miPii. It follows that the linear complex p^ + fip^^ = is transformed into m2msp.2s + fj,viim4Pii = 0, hence for the two values of fi given by — ^ — - = f', tbis complex is unaltered in. form by the trans- formation. It is clear that no linear complex which is not of the form ap-n + 6^23 = can be unaltered by the general reciprocity. Two quadrics of the pencil X^X^ + fiX^X^ = are unaltered by the reciprocity, for the transformation applied to the last quadric gives /.«vn4 and this in point coordinates has the equation X.X. + f^^^^^X,Xi = 0, which is the same as the original quadric provided that fj.- = . CHAPTER IV. SYSTEMS OF LINEAR COMPLEXES. 51. Among the complexes of the system (ax) + X (bx) = 0, where X, has all values, there are two which are special (Art. 19), the corresponding values of X being the roots of a(a) + 2Xil(a|6) + \^a(6) = (i), where n,(a\b) = i'Eai^r=- . obi The directrices d^ and d^ of these special complexes were seen to be polar lines both in {ax) = and in (bx) = 0, (Art. 24), (which will be referred to as the complexes A and B). Every line which belongs both to A and to B belongs also to (ax) + \i (bx) = and to (ax) + X^ (bx) = 0, and hence intersects d^ and d^. Thus the congruence of lines common to two linear complexes consists of all the lines which meet the two lines d^ and d^ thus determined ; di and ds are called the directrices of the congruence (ax) = 0, (bx) = 0. Every line of this congruence belongs to each member of the " system of two terms " (ax) + X (bx) = 0. Moreover, d^ and d^ are polar in each member of the system, for if a line x belongs to (ax) + \(bx) = and also meets d^, i.e. belongs to (ax) + \ (bx) = 0, then it must satisfy both (ax) = and (6a;) = and hence also (aa;) + A,, (6a;) = 0, i.e. it meets d^. The lines of a linear congruence are said to form a system of lines of the first order and first class. For through any point one line of the congruence can be drawn, viz. the line of intersection of the polar planes of the point in the two given complexes ; and in any plane there is one line of the congruence, viz. the join of the poles of the plane in the two complexes. Four lines which do not belong to the same regulus determine a congruence, whose directrices are the (two) common intersectors of the four given lines. There are oo ' linear complexes through 51-54] SYSTEMS OF LINEAR COMPLEXES 61 four such given lines, since the coefficients of the equation of a linear complex through them satisfy four independent equations. Any linear complex contains oo ■" linear congruences ; since a line p may be chosen in oo * ways, its polar line p' for the given complex is then known, and hence a congruence which belongs to the complex. 52. Double ratio of two complexes*- If qj and ft are the coordinates of the directrices d^ and d^ of the congruence determined by (ax) = and (6a') =0, then any two complexes of the system are {ax)-ir\(psc)=0, {aJs)+ii(^x)=0, the coordinates of Klein being here used. Let any plane cut d^ and d^ in two points P, Q and have Pj, Qi for its poles in these two complexes ; P, Q, P^, Q^ are coUinear and their double ratio is X//1 ; for let the plane be determined by two intersecting lines y and z which respectively meet d^ and d^, then {ay) = 0, 02) = 0, (yz)=0; join the point j/z to the poles of the two complexes by the lines y + p^z, y + p2Z, i.e. express that these lines belong respectively to { Rijk are complementary, the quadric to which they belong is yi' + ym' + yn'^o. There are ten such quadrics which have been termed "fu/nda- mental " by Klein. « Two reguli pna, pimn have no common line, ). Pijk' Riji have dmn, dnm 11 common. Thus the quadrics (Pijh, Pmnl), (.Rijl, Rmnk) meet in the quadrilateral formed by ^inni (^nmt ^ijt ^ji* In the quadric {pn^i, Pjmn) dy and dji are polar lines ; for dij 60-62] SYSTEMS OF LINEAR COMPLEXES 69 Pig. 2. and dji meet the four lines in the figure, (which lie on the given quadric since they have in common with it the suffixes kl and mn). Hence d^ and dji are the lines 00^ and O'Oi, and the polar plane of passes through 0' and 0/, also the polar plane of 0^ passes through 0' and 0/, hence 00^ and O'Oi are polar lines. We observe also that "'iji "'jit "'lk> "'kh "'mm ""nm form a tetrahedron. [There are fifteen of these tetrahedra, which are the funda- mental tetrahedra of Art. 29.] From this result we see that the four quadrics \Pikmy Pjnl)> \Pilm> Pjknji KPikni Pjlm)i \Pilnt Pjkm) have the tetrahedron {ij, kl, mn) as a common self-conjugate tetrahedron. Let TT be any plane with poles Oj, Oj, 0^ in the complexes Oi, Gj, and 0^. On OiOj an involution is determined of poles in Ci and Oj, the double points of this involution being where dy and dji meet OiOj, (Art. 25). Hence Oi, Oj and these two points are har- monic; similarly for OiO/c and OjOjc. But dij, dji, di]c, djci, djii, d/cj lie on Pimn, hence the trace of pimn on ir has Oi Oj Oje for a self-conjugate triangle. In the same manner the trace of pijje on ir has OiOmOn for a self- conjugate triangle, bat these traces are the same, hence the six poles being the vertices of triangles self-conjugate with regard to the same conic lie on a conic, and the sides of the triangles OiOjOic, OiOmOn, touch a conic. 62. Closed system of sixteen points and planes'^. Since Ci and Cj are in involution there is a plane tt^- through OiOj which is the polar plane of Oj in Cj and of Oj in d ; there are clearly fifteen such planes as tt^; take three of them TTy, itjic, ttm, and let Oijk be their common point. Now OiOijji belongs to Cj and to Cj, hjk „ Ci „ Cie] * See Art. 26. OjOi, 70 SYSTEMS OF LINEAR COMPLEXES [CH. IV heace Oyk is the pole of inj in (7a, but Oi „ „ „ ITij „ Cj, therefore 0, and Oyk are corresponding points of the involution on OiOijk established by Cj and C^, thus d^j and d^ divide Of and Oij^ harmonically. Similarly dy and dji divide OkOijie harmonically, = (6») + 2 (ac) = (c^) + 2 (bd) = (ad) + (be) = 0. The equation yfr (X, \') = is here (b') + 2 (6c) (X + X') + (c") XX' = 0. There are two singular generators given by the equation (¥) + 4 (6c) X + (c") X' = 0, if 4>(bcyi=(b')(c''). The quantity X may be so chosen that X = 0, X = oo correspond to the two singular generators ; in this case we have (6^) = (c^) = 0, hence (ac) = (bd) = 0. The equation •>/r = reduces to X + X' = 0. From the equations (a") = (b") = (c^) = (d') = (ac) = (bd) = (ab) = (cd) = 0, it appears that the lines Ui, bi, Ci, di are edges of a tetrahedron in which a, d and b, c are opposite. Also since p.Xi = ai + CtX' + X(bi + diX') with the condition (ad) + (be) = 0, it is clear that the lines at + ciK", bi + diX' intersect, therefore x meets the line of intersection 7 of the planes (a, c) and (6, d). The generator x' which meets x is obtained by changing the sign of X (since X + X' = 0), hence X and x' intersect upon the line 7, which is therefore the double directrix of the surface ; the single directrix being the intersection of the planes (a, b) and (c, d). If 4 (60)^^ = (6^) (c") the singular generators coincide. Taking zero as the value of X which gives the singular generator we must have (6') = (6c) = ; and 1^ = becomes XX' = 0, that is, each generator meets the single generator and no other. This gives Cayley's ruled cubic. 72. Ruled quartlcs of zero deficiency. For a ruled quartic of this species we have p.Xi = ai + biX + aX" + diX' 4- eiX\ with the conditions = (a') = (ab) = (e') = (de) = 2 (ac) + (6^) = 2 (ce) + (d') = (ad) + (be) = (6e) + (cd) = 2 (ae) + (c«) + 2 (bd). The quartic belongs in general to one linear complex (7a;) = whose coefficients are determined by the equations (7a) = (76) = (7c) = (yd) = (ye) = 0. 71-72] RULED CUBIC AND QUAETIC SURFACES 83 (ab) (6^) (be) (bd) (be) (ac) (6c) (en (cd) (ce) (ad) (bd) (cd) (dO (de) (ae) (be) (ce) {de) (eO The double curve is of the third degree, its equation is (ac) + (ad) (X + \') + (ae) (X + Xj + (bd) XV + (be) XX' (X + X') + (ce) VX'^ = (II.) There are four singular generators ; if X be so chosen that X = 0, X = 00 give two of them, we have the additional equations (ac) = (ce) = 0, and therefore (b') = (#) = 0. If the linear complex (7*) = is general, we have class III. The complex is special when, (Art. 64), (a') (ab) (ac) (ad) (ae) = 0. By aid of the equations connecting the constants, the last equation is seen to reduce to (bd) {(ad) (be) - (ae) (bd)} {2 (ad) (be) + (c') (ae)] = ; while the equation of the double curve breaks into two factors, if (bd) {(ad) (be) - (ae) (bd)} = 0. There are thus two cases to be considered : (i) (bd) = 0, here X + X' = is a factor of the equation of the double curve, which breaks up into a conic c', and a line 7 meeting c^ 7 is the directrix of the special complex. lix and x' are a pair of generators for which X + X' = 0, we have p.Xi = ai + biX + CjX^ + diX' + e^*, p' . xi = ai— biX + aX" — diX' + eiX* ; hence p .Xi — p . x{ = 2X (6f + X^di) ; therefore a line of the pencil (x, af) is a line of the pencil (6, d), i.e., the plane (x, x') passes through the fixed point (b, d). The figure shows the relative positions of the lines a, b, d, e, 7 and it is clear that X, x' meet on 7 while their plane passes through (b, d). In this case, therefore, the double curve consists of d^ and 7, and the bitan- gent developable consists of 7 and a quadric cone whose vertex is the point (b, d) ; the surface belongs to class V. Fig. 4. 6—2 84 RULED CUBIC AND QUARTIC SURFACES [CH. V (ii) If (ad) (be) — (ae) (bd) = 0, the equation of the double curve breaks up into two equations of the form where A and B are constants. In this case, since p.Xi-p'.Xi' = (\- X') [bi + Ci(\ + \') + di(\ + \' - W) + ei\-\-X'(\ + X'" - 2XV)}, for the lines x, x' which meet upon the curve corresponding to \ + \' = j4,we have pxi — p' . xi = (\ — \') (Oj 4- XX'/3i), where the «; and ySj are constants, and must therefore be the coordinates of two intersecting lines. Similarly, since p.Xi = \* [ci + dj/i + dp?' + bip? + aiiJi% where X.p,= \, it is clear that 2 <7 . Xi — a-' . xl =■ (pi. — p!) [di + Cip, + fji.' + bi(p, + fi' — p,pf) 4- ttip, ■+ fj,' (p. + p,' — 2p,p,')], hence for the lines x, x' which meet upon the curve corresponding to p, + p! = B, we have as before iT.Xi-a'.Xi=(p,- fi') (a/ + p-p,'^i') ; so that, as in the case (bd) = 0, the plane of the two generators through a point of y passes through a fixed point, and we have again the class V. In the next place if 2(ad)(be) + (d')(ae) = 0, the complex (•yx) = is special, but the equation of the double curve does not factorize ; hence we have the two following cases : (i) 7 is a simple directrix of the surface, and therefore does not form part of the double curve which is the general twisted cubic ; and since 7 meets all generators, it meets those which intersect, hence the envelope of the bitangent planes is the line 7 taken triply; this gives class IV. (ii) 7 constitutes the double curve, and the bitangent develop- able is a surface of the third class ; this gives class IX. In (i) there are three generators in each plane through 7,. in (ii) three generators meet in each point of 7. Ifyis itself a generator, let X = represent it, and then since 7' meets each generator, we have (a^) = (ab) = (ac) = (ad) = (ae) = 0, 72] RULED CUBIC AND QUARTIC SURFACES 85 and by reference to the equation (II.) of the double curve, it is clear that two singular generators now coincide with a (or 7). As before we take X = 00 to represent one of the two other singular generators, in which case we have (ce) = 0. The equation of the double curve is here (6c^) + (6e)(X + V) = 0. There are two cases : (i) the generators which meet in a point of the double curve lie in a plane with a, the bitangent develop- able consists of the line a taken triply, and the double curve breaks up into a and a conic &, which gives class VI. ; or, (ii) the generators meet in a point of a which constitutes the double curve and the bitangent developable consists of a together with a quadric cone, which is class XI. It has been seen that the surface belongs in general to one linear complex ; if however, there exist six linear relations of the form Aai + Bhi + Gci + Bdi + Eei = 0, the surface will belong to an infinite number of linear complexes. The double curve consists of the two lines 7, 7', which are given by (f ) = (7a) = (t&) = (yo) = (yd) = 0. In the present case the surface ma)' have a double generator, viz. when two distinct values of X give rise to the same generator ; as before we may take the values \ = and X = 00 as relating to it ; the generator x is then given by p.Xi = ai + bi\ + CjX^ + diX^ + ajX*, and the linear relation is then 04 = ei. From the identity {a?) = (i, we have the system of equations {a?) = {ah) = {ad) = {cd) = {be) = {¥) + 2 {ac) = {c') + 2 {bd) = {d^) + 2{ac) = Q. The equation of the double curve, which factorizes, is {ac) + {bd) XX' + {ac) X?\'^ = 0. This gives class VII. There are two subvarieties in which the lines 7, 7' coincide ; one arises when {bd)^ = 4 {acy, in which case the double curve becomes XX' = 1, 86 RULED CUBIC AND QUARTIC SURFACES [CH. V hence for two generators x, x' which intersect, we have p.Xi = ai + h^\ + aX' + diX' + ttiX*, p . Xi = aiX* + biX^ + CjX,'' + d^X + at ; hence p.Xi — p .xi = {\ — X') (fe< — di), the generators x, x , therefore, intersect upon the line hi — di, and lie in the same plane with it. The double curve consists of the double generator a, and two indefinitely near lines hi — di. This gives class VIII. The second subvariety occurs when (ac) = 0, we then have {a") = {ah) = (ac) = {ad) = {h') = (d^ = {cd) = (6c) = 0. The lines a, b, d all belong to the complex {cx) = 0, while a meets both b and d ; to the same complex belong the lines 7, 7', which must therefore both coincide with a. This is the class which has a double generator in coincidence with two indefinitely near directrices, i.e. class XII. Finally, the surface may belong to an infinite number of linear complexes and not possess a double generator ; of the two lines 7, 7' one is a triple and the other a single directrix; this gives class X. CHAPTER VI. THE QUADRATIC COMPLEX. 73. When the equation of the complex, f(x) = 0, is of the second degree in the coordinates, the complex is called quadratic. By the same reasoning as that employed in Chapter II., it is seen that the lines of the complex through any point form a quadric cone, and that the lines of the complex in any plane envelope a curve of the second class. The general quadratic expression in six variables involves 21 terms, but, by means of the identical relation co (x) = 0, it may be deprived of one of its terms without loss of generality; so that the quadratic complex is seen to involve 19 constants. It will be shown in Chapter XI. that /(x) and w (x) can, in general, be brought by the same linear transformation to the forms and Xi^ + x^'' + x-/ + Xi^ + Xs^ + Xg^, respectively*. This canonical form of the equation of the quadratic complex will be generally used in the present chapter. With this form of equation it is clear that if x belongs to the complex, so also do the 31 lines associated with x, (Art. 14) ; hence the polars of the lines of a quadratic complex (Xx^) — 0, for a fundamental linear complex, belong to (\x') = 0. This is a characteristic property of the fundamental complexes oi f(x); for if L, or {lx) = 0, is a linear complex such that the polar line x', with regard to L, of any line x of the quadratic com- plex /(«)=0, also belongs to f(x) = 0, then, taking o}(x) = (x'), p.x( = Xi-\- ali , where (7 = - -^^ , (Art. 20), * The complexes x^ which appear in its canonical form may be called the funda- mental complexes of / (x) = 0. 88 THE QUADRATIC COMPLEX [CH. VI if f{x') = 0, we have and, since f{x) = 0, y^ 3/(0 2 (^«) f(l\ - If the last equation holds for every line x of /(a;) = 0, it follows that |^ = «J,, (t=L2,...6). These equations determine six values of k, and hence six linear complexes which have the required property. The equations serve, therefore, to determine the fundamental complexes of any quadratic complex f{x) = 0, (x') = 0, where f{x) is general in form. Any given quadratic complex C^, or f{x) = 0, possesses two lines Ml every plane pencil {A, a) of space, viz. the intersection of the plane a with the complex cone of A (or the tangents from A to the complex conic of a). In any regulus A, B, C there are four lines of C^ viz. those determined by the equations (aa;) = 0, (6a;) = 0, (cx) = 0, f{x) = 0, 6)(a;.) = 0. 74. The tangent linear complex. The linear complex lyi(jj. ^ + vxij=0 (i), is called a polar complex of C'^ for the line x* ; if x belongs to C^ it is seen that this complex contains x, and every line x + dx o{ C consecutive to x, since dxj the complex is then called a tangent linear complex of C\ On any line a; of C^ a correlation is established between its points X and its planes u, such that m is the plane whose complex curve has X for point of contact with x, while u is at the same time the tangent plane along x to the complex cone of X. Hence the pencils {X, u) belong to every tangent linear complex of x, so that each of these oo ' complexes determines on x the foregoing correlation. * 0) (x) is here taken as {x^) ; for the general form of la (x) the polar complex is X ^ 1 = 0, {xdx) = 0, yfdx\ = ; 73-75] THE QUADRATIC COMPLEX 89 The special complexes of (i) are given by the equation which, since x belongs to C"^, reduces to "•"(1) = °; hence in general the two directrices coincide with x. 75. Singular points and planes of the complex. It will now be shown that the complex cones of the points of a certain surface ^^i break up into pairs of planes, the complex lines of such a point therefore consisting of two plane pencils; while the complex conies of the tangent planes of a surface 4>2 break up into pairs of points, so that the complex lines in such a plane form two pencils. Let X be any line^ of C^ it will be shown that the complex cones of four- points on x break up into pairs of planes ; for if x + fia is one line of a pencil which belongs wholly to C^ the following conditions must be satisfied, viz. f(x + /tia) = 0, for all values of fi, while So.; ^ = 0. OXi This gives the following four equations : f{x) = 0, %ai^£ = 0, f{a) = If now a be that line of the pencil which meets any given e b, 2i equations f{x) = 0, ta,^£ = 0, f{a) = 0, 2a,||=0. line b, Saj jr^- = 0, and the coordinates of a satisfy the four Hence the lines a are those common to a regulus, and the complex G^, and are therefore ybitr in number; we shall denote them by a\ a^^, a}^"^, a^^. It has therefore been shown that on every line x of C^ there are four points A^, A^, A^, A^ called singular points, whose complex cones break up into a pair of planes ; since this is true in general for each of the oo^ lines of C^ the locus of such points is a surface of the fourth degree which we may denote by ^^j. Again in each of these four planes {x, a*) since there exists one pencil of complex lines there must also be a second, i.e. the complex conic in each plane consists of a pair of pencils ; hence, through any line x of G^ we can draw four planes /8i, ySa, /3s, /S4, 90 THE QUADRATIC COMPLEX [CH. VI called singular planes, for each of which the complex conic breaks up into two pencils; the envelope of these planes is therefore a surface of the fourth class which we may denote by ^3. It will shortly be sho\vn that the surfaces , and Oj are identical. 76. Singular Lines. No two of the four lines a* will in general intersect, since they belong to the same regulus ; if they do intersect, so that for instance ^1 and A^ coincide or /3i and 02 coincide, the regulus must break up into two plane pencils, for which the condition is that the discriminant of "M dto df da> should be zero (Art. 59). If (o (x) = 1,x'', this condition becomes that the discriminant of should vanish or that 2(^Y = 0, since (bx)i^Q*. Fig. 5. In this case the lines a which satisfy the equations (ax) = 0, a ^']=0, (ab) = consist of the two pencils (i) that which has (x,-^) for its plane, and its centre on b; (ii) that which has (x,~\ for its centre, and its plane through b. * For the general form of (o(x) the condition is that I ^ J = 0. 75-77] THE QUADEA.TIC COMPLEX 91 The lines a^, a^^ ; a^^^, aF^ are the complex lines of these respective pencils. Thus two of the singular points A, say A^, A^, come into coincidence and also two of the planes y8, e.g. /3i and ^^ ; so that we have on x the singular points A, A^, Ao and through x the singular planes yS, ^3, 0^. ri /* A line x of G' for which ;r^- is a line, thus touches i at the ox point ^, or Ix, ^) , and is the intersection of the plane-pair of A ; it meets i in two other points A^, A^ which form the point- pair for the tangent plane yS, or (*, ^j of 2. Such a line is called a singular line of C'' Taking 'ZXiXi' = as the equation of C, the singular lines are given by the equations IXiXi^ = 0, IXiW = 0. 77. Singular points and planes of any line. On any line I let its points of intersection with $1 be A^, A^, A^, A^, and let the tangent planes through it to 2 be /81, ^2, J^.,1^4 ; let the point- pair of /3i be Bi, Bi and their join 61, &c. ; let tho plane-pair of ^1 be «!, a/ and their intersection Oi, &c. Then it is clear that the eight singular planes a njeet the four lines b in the points B, e.g. the plane «! meets the plane (l, &i) in a line through A^ which belongs to C^ and hence must pass through B^ or B/, since all lines of G^ in (I, 61) pass through one of these points. Thus through each point Bi there pass four planes a and through Bi the other four planes a; similarly in each plane a there lie four points B ; so that if we take three pairs of planes «i, «i'; «2; «/; «3. "^3 tbey exactly determine the eight points B; let the notation of these points be determined as follows : — 5i = (a/a^as), 5j = (aia/aj), Ba = (a^a^as), B^ = (a^a^a^), B-; = {aia^0L^), 5/ = (a/aja/), £3' = (od'a/a^), 5/ = (a/asW). Then it is easily seen that the only possible remaining arrange- ment by fours of the points B, so that no three of them are in any of these six planes oLi, ... a,', is (BiB^BaBi), (Bi'B^Bs'Bi). Calling these planes a^ and a/, we have the following arrange- ment of the points B and planes a -. 92 THE QUADRATIC COMPLEX [CH. VI «i = (^i'> A. -S3, -S4), a/ = (.Bi, B^, B3', -B/), «2 = {Bi, Bi, B3, Bi), a„' = (JB/, £2, J5s', 5/), Ms = (B^, 7?2, jBs', £4), 0(3' = (5i', JSj', iJa, £4'), 04 = (Bi, ^3, £3, -B4'), «; = (5/, B,', B,', B,). Now consider the tetrahedron whose vertices are BiB^BaB^' ; the planes joining I to its vertices are respectively /Sj, /Sa, /8s, y34; and the plane through B^B^Bi is a/, which meets Z in ^2 ; „ „ BiBs'Bi „ a/, „ „ Ai; „ „ HiJiiBi „ O4, „ „ A.^; „ „ B^B^B^ „ a.3, „ „ A3 ; hence, by von Staudt's theorem, (Art. 12), (A, A, /^a, ^4) = {-^i, ■A.-i, A4, As) — {Ai, A^, A3, Ai). From this result the identity of the surfaces (p^ and 2 follows immediately, for if I touches i, i.e. if two of the points A coincide two of the planes /3 must also coincide, i.e. I touches 2, and conversely; hence 4'i and 2 have the same tangent lines and must therefore be identical. The surface 4>, with which $1 and $2 coincide, is known as the Singular Surface of the complex. 78. The identity of the surfaces *i and *2 ^^^° follows from the fact, that if .r is a singular line of G\ the plane (x, X.r), or -rr, touches at P the locus of the point (x, \x), or P ; for take any point P' consecutive to P, the singular line corresponding 'to P' being x + dx, then since 'S,\iXidXi = 0, it follows that the four lines x, Xx, x + dx, 'K{x-'rdx) form a twisted quadri- lateral, so that if the distance PP' is of the first order of small quantities, the distance of P' from n is of the second order, i.e. n touches the locus of P at P* 79. Polar Lines. If P be any point 011 a line I, v any plane through I, and u the fourth harmonic to I and the two complex lines of the pencil (P, tt), these complex lines must be I + fjLU and I — fiv ; expressing that they belong to 0", whose equation we take in the form (Kos^) = 0, we have {\P) + fjL'{\u^) = 0, {\lu) = 0, {lu.) = 0. * The identity of the surfaces *i and *2 was shown by Paach, Ueber die Brenn- flachen der StrahUnsysteme und die Singularitdtenjlachen der Complexe ; Crelle, Bd. 76, S. 156. 77-79] THE QUADRATIC COMPLEX 93 The first equation gives the values of fi for which l + fiu and I — fiu are complex lines, the second equation states that u belongs to the complex (\lx) = 0, hence since u meets I it must also meet I', the polar of I for the last complex (Art. 24), that is, it meets 7 ^ 7 u 2(X^=) li-K\ili, where k== (-Un^ ' But for different positions of P, all the lines m in tt pass through the pole of I iov the complex conic of ir, so that I' must pass through this pole. Hence we have the following theorem, the locus of the poles of I with, reference to the complex conies of plwnes through I is a straight line I', called the polar of I with respect to G^ The coordinates of I' were seen to be li — xXili, where _2(AP) so that if I belongs to C', k = 0, and I' coincides with I. If both (\l^) and {\H'^) are zero, any line of the pencil (l, Xl) is polar to I. If I is situated in the plane at infinity, the planes through I are parallel to each other and we deduce the result that the locus of the centres of the complex conies in a system of parallel planes is a straight line, which is called a diameter of the complex. Considering again the system of singular points and planes connected with any line I (Art. 77), each of the lines a and b is singular and satisfies the equations (Xai2) = 0, (XV) = 0, (te) = 0, {l\x) = 0; from the last two equations we deduce that liXiili—KXilij^O is an equation satisfied by each of the lines a and b, hence the eight lines a and b meet both I and its polar I' for G\ Referring to the table of Art. 77, it is seen that B^ and B2 lie on both «! and Oj', hence B^'B^ (which meets 61 and h^), meets tti and -tts; similarly B^B^' meets «] and a^, so that the lines «!, a^, bi, 62 being met by I, I', B^'B^ and B^B^', lie on the same regains. Consider the linear complex K determined by Oi, 61; a^, 62 as pairs of polar lines (Art. 31); then in K the polar of as is bg, for Hg' contains the lines B^'B^, B^'B^ which meet a^, 62 ; fti, 61 respec- tively, hence its pole is B3, and as contains B1B3', B^B^, hence its pole is jBj', i.e. a^ and 63 are polar lines; similarly cti and 64 are polar lines in K. 94 THE QUADRATIC COMPLEX [CH. VI It is clear, therefore, that for K the polar plane of A^ is /Sj, of A„ is /Sa, of J.3 is /Ss and of A^ is ^^, whence we again see that There are three other such linear complexes K', K", K'" ; in K' the pairs of lines aj, 63 ; Oj, 6] ; a,^, h^■, a^, 63 are polar ■■■K" ^1 ; «2>*3; »3' *2 The points Ai, A^, A^, A^ have for polar planes in these four complexes the planes i3i, 02> ^3. 04 ; 02. ft, ft. ft ; ft. ft. ft. ft ; ft. ft. ft. ft. respectively, which shows that the four complexes are mutually in involution. 80. The singular lines of the complex of the second and third orders. It has been shown that the lines which satisfy the equations '(^> = '' "(0 = "' ^'' = '' belong to 0" and touch the surface 4", the singular surface of C\ The point at which x touches 4> was seen to be the intersection of the lines x and -^ , the plane of these lines being the tangent plane at the point. Taking for / its canonical form these equations are (\a^) = 0, (XV) = 0. The singular lines thus form a congruence of the fourth class and fourth order, i.e. through any point there pass four lines of the congruence and any plane contains four of its lines, which are touched by the complex conic of the plane. The complex conies in the pencil oj planes through a singular line X touch x in its point of contact with . For a tangent linear complex r of a singular line, being 2^; (^+/jLXi] = 0, is special, and its directrix belongs to the pencil (x, J-) , so that the point fa;, 5^ 1 is the pole in T for any plane through x ; but the pole of such a plane is the intersection of x and x + dx, the latter line being common to T and C", i.e. the point of contact of x with the 79-80] THE QUADRATIC COMPLEX 95 complex curve in the plane, hence all these points of contact coincide with (-1) If in addition to the equations of Arts. 75, 76 which give the singular lines, viz. /(a) = (), (a.) = 0, (a|) = 0, 2(|J = 0, /(.) = 0, we have also / ( ^ ) = 0> i* is clear that one value of a is ^ + fj.x, so that one of the points Ai, A^ must coincide with A, or one of the pencils of complex lines in /8 is {A, /S), and the plane /Ss coincides with /3; so that we have two points J. on a;, viz. A and A^; and two planes /S through x, viz. yS and ^i\ while the lines a^^, aP^ coincide. Thus x meets $ in three consecutive points, i.e. is a principal tangent to , and all the lines {A, yS) are complex lines. A curve on is thus obtained whose tangents are singular lines of C^ and principal tangents of ; at each point of this curve all the tangent lines of $ belong to C". In any plane the complex conic touches the four singular lines of the plane and has apart from them 2.4.3 — 2. 4 = 16 common tangents with the section of 4> by the plane. Each of these 16 lines touches 4> and is not a singular line of C^, hence its point of contact is a point of the above curve; thus any plane meets this curve in 16 points, i.e. the curve is of the order 16. Lines whose coordinates satisfy the equations (Xa;») = 0, {W) = 0, (X^a;^) = 0, are termed singular lines of the second order (Segre). If in addition to the foregoing conditions we have • 9/ •4. • ^ *u ^ NT fdf(sc + py)y ^ where w is tt- , it is easy to. see that z -i—, -^ = 0, or, everv ■^ 8a; "^ f \o{x + py)j line of {A, /3) is singular ; for since df{x + pyy jf{x) df{y) d{x + py) dx '^ dy ' it follows that 2 m^±py)Y ^ ^ m ^ 2,s f.f+ p^x (f d{xi + pyi)] \dxil '^ dxi dyt ^ \dyi =''''y^=&i 96 THE QUADRATIC COMPLEX [CH. VI It is clear that ^ is a line which meets «, and also x, since ■by ^' \x^\ = [y -J-] = 2 (f) = 0. Therefore the lines x, y, -J- are either coplanar or concurrent; in the former case, any line of the complex which lies in the plane being of the form x + py + in fo\ir consecutive points. If z is any line of the pencil (x, y), since ^; is a singular line, it touches $ at its point of intersection with ~ ; hence z touches * at two distinct points, viz. (x, y) and (z, ^) ; this being true for each line of the pencil (x, y), it follows that the plane (x, y) is a singular tangent plane of , i.e. it touches ^ along a conic. In the case for which x, 2/, ^ are concurrent, it is similarly seen that their point of concurrence is a double point of , and all the complex lines through this point lie in the plane (x, y), while the four tangent planes to through x come into coincidence. It will be shown (Art. 82), that possesses 16 singular tangent planes and 16 double points. Taking the equation of the complex as being (Xx'') = 0, the lines given by the equations (\a') = 0, ( W) = 0, (XW) = 0, are the tangents of the principal tangent curve previously deter- mined ; if \x is itself a singular line, we have (W) = ; these four equations determine 32 lines, so that 16 of them are tangents to the (conic) sections of 3> by its singular tangent planes ; if a; be one of these lines in such a plane cr, the lines x, Xx, X^x lie in a, and all the lines of C^ in cr consist of the pencil (x, Xx) ; 16 of these lines are generators of the tangent cones at double points 2) of ; at such a point B the lines x, Xx, \'a; concur, and all the complex lines through D consist of the pencil {x, Xx). Lines of this kind are called singular lines of the third order (Segre). 80-82] THE QUADRATIC COMPLEX 97 81. The complex in Pliicker coordinates. If ia the canonical form of the equation of the complex, we transform to coordinates pt/c by using one of the 15 transformations, (Art. 29), it takes the form Ai2 iPi2 +P3i') + K (pJ +P42) + K (pJ+p^') + ^dpi^Pu + "i-eprsPii. + 2/P14P23 = 0, and a quadratic complex may be brought to this form in 15 ways. A quadratic complex whose equation is + '^'^PnPu + 2« PuPii + 2^14^2.^ = 0, may be brought to the preceding form, for since the coordinates of any point are, as usual, the ratios of the perpendiculars from the point on the faces of the tetrahedron of reference to the perpendiculars from a fixed point E on those faces, if we take a point E' for which X from E' on face a, „ X from ^. Qi " it is easy to see that a line whose Pliicker coordinates are ^a with reference to the former system, are p'ji: with reference to the latter, where ^ji:=»ii»it/)'jj.. The quantities m may now be chosen so that and on substituting for p^j^ the resulting equation is of the required form. 82. The singular surface. It has been shown that <3> is the locus of points of intersection of x and \a;, where a; is a singular line of (7" ; let y be the coordinates of such a point, a the point at infinity on x, and /3 the point at infinity on Xx, we have then 2/1 ^■i 92 - 2/2^-1 a, + yiXiO-i - 2/4X10(3 = p (t/i/Ss - ^5^1 + 2/3/84 - 2/4/33), and five similar equations, p being a factor of proportionality which is the same for the six equations. Also diHi + a2a2 + a3a3 + a4a4 = 0, ai/9i + Oa/Sj + tts^s + tti/Sj = 0, where the quantities a are connected with the tetrahedron of reference. Eliminating the quantities a and /j/S from these equations we obtain as the equation of , J. 7 98 THE QUADRATIC COMPLEX [CH. VI -3/2x1 2/1X1 - 2/4X1 2/sX, -ys yi -y4 ys -y^x. 2/1X2 2/4X, -2/3x2 -2/2 yi y4 -ys -2/3X3 2/4X3 2/1X3 -2/3x3 -2/s y4 yi -y^ - yzK -2/4X4 2/1X4 2/2^4 -y3 -y4 yi y^ -2/4X5 -2/»X6 2/2X5 2/1X5 -y4 -ys ya yi -yt\ ^sXs -yi\ 2/1X5 -y4 ys -y^ yi a. Oj a. a4 tti Oj 03 a. = 0. This determinant is divisible by (oi^i + ci^y^ + a^y, + Utytf, after division by it is effected, the expanded form of the equation is (\2X4Xe - XiXjX,) (ly'f + (x,\,\ - x^xAa) (y.' - y2= - ys' + y4')' + (\1\4\5 - XjXsXe) (y,' - y," + yi - y^f + (X2X3X5 - XiX4X„) (2/1= + 2/2" - yf - y4=)' + 4 (XiXjiXa - X^XjXs) (yi^s - y^y^y + 4 (XiXsX^ - XjXjXj) (2/12/2 - y3y4)'' + 4 (X3X4X6 - XiXjXs) (y,y4 - y^y^f + 4 (XiXjX^ - XjX^Xj) {y^ys + y-^y^" + 4 (X3X5X6 - XjXjXi) (2/22/4 + yiys)' + 4 (XiXsXi - x^x^Xs) (yiyi+ysyif- Finally writing Xi — X2 = ai, Xi + X2=''2i X3 — X4 = Oi , X3 + X4 = 62 , X5 — X5 = Ci, X5-|-Xg = C2, Oi^iCi = A, a,{b,''+c,'-{b,-c,y\ = 2B, bi [ci' + < - (C2 - must have 16 singular tangent planes. The quartic surface which has 16 double points and 16 singular tangent planes is known as Kummer's surface. It should be noticed that the coordinates of the 16 double points are the same as those of the 16 singular tangent planes and form a system already discussed, (Art. 14), so that through each double point there pass six singular tangent planes and in each singular tangent plane there are six double points which lie on the (conic) section of the surface by the plane. The polar line tt'^ of any line pa, with regard to the quadric 2ai§i'=0, is given by the equations n'ik=aia.kPik, (Art. 46). Hence it follows that if ^,i belongs to the complex 0% or a (Pn^ +Pu^) + b (^13^ H-fta') + c {p,,^ +^23^) + Up.^Pu +^ePi3Pi2 + ^fPiiP^ = 0. its polar with regard to any one of the quadrics llH|2HS3' + ^/ = 0, l,^-^2H|3'-^/ = 0, ll' + l2'-|3'-|/ = 0, also belongs to C. The polar jr'ij, of a line p(j„ for the quadric is given by the equations, 'r'l2 = <'Vl2. ^'l3 = <'Spii, ir'i4 = a^p32, ^'si^^'^Psi' ^'i2 = <^^Pl3> ^'2S = <^^Pil '' hence it is clear that if ^^ belongs to C^, its polar line with regard to any one of the quadrics fll2 + f3^4 = 0, Ill3 + l2^4 = 0. ^1^4 + ^2^3 = 0, ^ll2-|3^4 = 0. fl4-|2^4 = 0> ^ll4-?2l3=0, also belongs to C^. These ten quadrics, which occur in the equation of the singular svirface of C, are the fundamental quadrics connected with the fundamental linear complexes, (Art. 61). 83. Double tangents. The line y whose coordinates are XiXi + jMXi where x is a singular line, is a tangent to $ at the point (x, \x) ; and similarly each of the 32 lines ± (XiXi + ^txi) is such a tangent to <1>. If one of these 32 lines y belongs to a fundamental complex so do all the others. If (P, if) is one of these 32 pencils, 15 centres of pencils and 16 planes of pencils belong to the closed system determined by P, 15 of the planes and 16 of the centres belong to the system determined by ir. Join P to the poles Pi.-.Ps of the fundamental complexes in tt. 82-84] THE QUADRATIC COMPLEX 101 these poles belong to the closed system determined by ir ; the line PiP, since it belongs to the complex Xi = Q is common to the pencil centre P and the pencil centre P;, thus PiP touches * at Pi, and therefore is a double tangent to <1>. Hence through any point of O we can draw six double tangents to 4>, each of which belongs to a fundamental complex. This may also be seen analytically ; for, if the pencils {x, Xx), (x' , Xx') have a common line, we have XiXi + fi,Xi = p (XiXi + vxi), whence in general x and x' are the same line, unless fi, = v = — \i, and Xi: = x'k, except for k = i, when Xi= —x{, and the common line of the two pencils belongs to the fundamental complex Gi] this bitangent line therefore belongs to the two complexes 2/, = 0, 2-1^ = 0, {k^i); k ^k — f^i for since 2/i = ; yk = {^k- 'K) ook, therefore S ^ ^\ = S (X^ - X^) x^" = tXixf -\i%xi = 0. k ^k~ M 1 1 There are thus six congruences to which bitangent lines belong, viz. those obtained by giving to i the successive values 1, ... 6. Hence, the double tangents form six congruences of the second order and class, a fact also deducible from consideration of the 28 bitangents of an arbitrary plane section of <1>, which consist of the 16 intersections of this plane with the 16 singular tangent planes, and 12 others which pass in pairs through the poles of the plane with regard to the fundamental complexes. Hence the double tangents of a Kummer's surface form six con- gruences of the second order and class, and each congruence belongs to one fundamental complex. 84. A Kummer's Surface and one singular line determine one C^. In a closed system of 16 points and planes, two of the planes intersect in a line containing two points of the system ; if two such planes are. a and /3 and two such points A and S, we have six planes of the system through A and six planes through B, (including in each case a and j3); this leaves six planes of the system ; hence there are six points on AB through which three planes of the system pass. Now taking any tangent line of * as a singular line of the complex to be determined, we know at the same time 31 other singular lines, the singular point and plane for each singular line being also determined, and we have two closed systems of 16 points and planes. If P be the point of intersection of three planes of one system, we know six complex lines for P, viz. the joins of P to the points A^ and A^, the centres 102 THE QUADRATIC COMPLEX [CH. VI of the pencils in each of the three planes ; hence the complex cone of P is determined. On the intersection of two planes of one of these closed systems there are six such points P, therefore the complex conic of any plane through the line is determined and therefore the complex cone of any point of the line. Lastly, any arbitrary plane meets all the 120 lines of intersection of the 16 planes of either system, and hence we know 240 tangents in this plane, and therefore the complex conic of this arbitrary plane. It follows that a Kummer surface is the singular surface for op i quadratic complexes, viz. those thus determined by the pencil of tangent lines at any point ; hence a Kummer surface contains 18 constants. There are two quadratic complexes which contain a given line touching * at a point P, viz. the complexes determined by the principal tangents at P (Art. 80). If the Kummer surface * and one line x be given, we can construct four complexes which contain the line and have * for singular surface ; for draw through x one of the four tangent planes a to *, and let x meet * in the points A^A^A^A^ ; 09!« of these points is a singular point for a, so that if is the point of contact of this plane with *, the singular line through must pass through one of the points A; thus taking in succession OA-^, OA^, OA^, OA^ as singular lines, we can by the preceding method construct four complexes which contain x. 85. The singular surface is a general Kummer Surface. The general Kummer surface* is the most general quartic surface which possesses 16 nodes, and it will now be shown that the equation of such a surface is reducible to the form (I) of Art. 82. The enveloping cone of a surface with 16 double points whose vertex is a double point, is of the sixth degree; if one of the double points S^ be joined to another jS'/, two tangent planes of the tangent cone to the surface at »Si' pass through S^Si , hence SiSi is a double edge of the enveloping cone whose vertex is Si, so that this enveloping cone has 15 double edges; but an irreducible 5x4 cone of the sixth degree can have only — „ — , i.e. 10, double edges, it hence each of the enveloping cones whose vertices are double points of the surface breaks up into six planes ; each such plane touches the surface along a curve which is necessarily a conic. Therefore through each point S there pass six singular tangent planes and through each pair of points S pass two singular tangent planes. A singular plane o- through a point B is met by the five other planes a through 8 in five lines on each of which a second point 8 lies, therefore in each plane a there are six points 8, * This surface was investigated by Kummer in papers published in the Monats- berichte der Akademie zu Berlin (1864), and the Abhandlungen der Akademie (1866). 84-85] THE QUADRATIC COMPLEX 103 (which lie on a conic). Through each of the 15 joins of the points (S in a plane o- passes one other plane a, hence there are 16 planes cr. A consequence of the above arrangement is that on the inter- section of any two planes a there lie two points 8. Two planes o- contain 10 points S of which two are common to them, and through one or other of two points S pass 10 planes a ; let «! and ajj be two singular planes and X3 one of the six planes which do not pass through the points S on the intersection of x^ and x^; take aii, x^, x.^ as coordinate planes, then on each of the edges AiAi, A^A^, AiA^ lie two pairs of points S and taking in each of these planes one other point of their conic of contact with the surface we obtain nine points through which can be described one quadric "¥ which contains these three conies. It follows that the equation of the surface must have the form ■^-l(iKxiX^XsXi = (I.), and of which, therefore, Xi is a plane cr. Taking for ^V the most general quadric, we may write, '^ = xi' + x^ + x^ + xl + la-i^x^x^ + la^x^Xi + 1(iyi,x-^Xi + la^x^x^ + la-^iX-^x^ + ^a^x^x^ ; in each edge of the tetrahedron of reference there lie a pair of points >S', for instance, in the edge A'iAi there are the points (S34, S'^ whose coordinates are (0, 0, a^, 1), (0, 0, a'34, 1) respectively, where «34, '"'34 are the values of x^\xi determined from the equation m^z + Xi + ^a^x^Xi = ; hence 0(34 . a's4 = 1. Through the line SuS^g there passes a singular plane o- in addition to the plane Ki, and the line of intersection of er and oLg contains two siAgular points, which therefore lie in A-^A^ and A^Ai respectively, say the points S'l^, S^, hence Fig. 6. a'l. 1 = 0, ie., a'li, . a «23 1 «34 1 ai4 1 . ccsi = a,, .(i). 104 THE QUADRATIC COMPLEX [CH. VI The singular plane through S^S'^, (distinct from aO, must pass through (S'la*, therefore a'i2 . a'jj . a„ = a'i4 (ii). It is clear that neither singular plane through SatS^^ can meet the edges A3A3, A^At in singular points, (for the singular plane through /S34/S2S passes through S'l^, &c.); expressing that such a plane through S^S^^ meets A^As, A^A^ in singular points, which may be designated /S',s, St^, we have again, a'ls • «34 ■ «42 = ai2 (iii)- From (i),'(ii), and (iii) it follows that either a^ = a^ , Ois = 042) «i4 = «2s j or, ai2 + 034 = Kia + a^j = a,4 + 033 = 0. The first set of equations require that 'lis ~ '*S4> ^8 ~ '^t *14 ^^ (^asi the second that Oij + a»i = Ois + 021 = ^14 + aas .= 0, but, by changing the sign of x^, it is seen that there is no real difference in form of the equation of the surface in the two cases. The equation of the surface is therefore seen to be {^i" + x^" + xi + x^ + 2ai2 {x^x^ + x^x^ + 2a,3 {x-^x^ + x^x^ + 2cii4 (x^Xi 4- x^x^Y — ^QKx-^x^XsX^ . . .(II.)- By changing to a new tetrahedron of reference, the equation of the surface now arrived at, can be brought to the form (I) of Art. 82 ; for introducing the new coordinate system given by the equations a^i = a.2/1 + "22/2 + as 2/3 + a42/4, x^ = oOi = ajyi + 04^2 + ai2/3 + a52/4. «4 = a4yi + a^y^ + 02^3 + aly^ ; * For if it passed through S^, (and therefore through S\^, we should have, similarly, '*12 • ** 23 * **34 = ** 14 j whence from (i) 034*=!, since u^^ . a',2 = i>23 ■ a'^j^a^. a'j4 = l. But if 034^=1, then 034= ±1, and the points S34, S'34 would coincide; hence expressing that S34, S'jj, S'j^, 5'i4 are coplanar, we have O ,2 . a 23 . €^34 = a 14 . t To determine K we observe that the equations which give the double points are * (»a + "la ^2 + «i3*3 + "14^4) - ili^x^x^Xi = 0, together with three similar equations; and, excluding the twelve double points which lie upon the coordinate planes, the coordinates of the remaining four double points satisfy the equations x^^ + x^(a-^^x^ + a-^^x^ + a,^^x^±•J Kx■^x^x^x^=Q, &c. But these equations become identical with equations (III) of Art. 82, if we replace Xi by y^ ; hence from the result there given it follows that ■K'=V + «is''+ai4''-2ai2'']3"i4-l- 85-86] THE QUADRATIC COMPLEX 105 it is easily seen that each of the expressions x^ + ai^ + cc^ + x^, XiXi + x^Xi, X1X3 + x^Xi, x^Xi + x^x^ ' is linearly expressible in terms of four functions of the yt of this form; hence ^ = 2/1' + y.' + yi + 2// + 2^1 (2/12/2 + 2/3^4) + 25i (2/,2/3 + 2/22/4) + aCi (2/12/4 + 2/22/3). The equation of the surface now assumes the form ^' - \&K' (ai2/i+ 022/2 + 0(32/3 + «42/4) (0(22/1 + • • •) (a32/i + • ■ •) (o(42/i + • • •) = (III.). Lastly take {a^, a^, a,, 0(4) to be the coordinates of a double point of the surface which does not lie on ^ = 0, then writing Z = (ajOj + Ha 04) (HiSa + a^at) (0(104 + a^Us), we obtain {'I' (o)}''— 16^"'. 2o(i''. 8i = 0; and expressing that the four equations to determine the double points of the surface are satisfied by the point fl(j, we have 4^ (a) {a, + A,a, + A a, + G,a,)-16K'. 8i la, + «, ^"'^ 2(aia2+asa4) + "»2(aia3 + «.o(4) ^ "' 2(0,04 + 020(3)} ~ ' with three similar equations ; these four linear equations to deter- K'L mine Ai, B^, G^, and , are seen to be satisfied by j^ ^ tec' jg _ So^ (7=__^?L_ ^ 2(0102+0304)' ^ 2(0103 + 020(4)' ^ 2(0,04+0203)' ^(o) = 4Soi'^ = 32Z'.i. From equation (III) all such terms as 2/i'*2/2j 2/1^2/22/3 disappear, \QK'L since the coeflficient of ^1*2/2 = ^-^1 = ^ ; and the 0,02 + 0304 coefficient of y,^2/22/3 = 4C + 8^,5, - UK' \ ^—- + (0,04 + P4) — \Pi > Pi J P3 1 ■'4 )• Singularities of the surface. The eight singular points of 0^ which lie in the four singular planes through I are double points of the surface ; they are given by the equations {>mF) = 0, (\uv) = 0, (M^) = 0. If a; is the singular line of G^ which joins a corre- sponding pair of these singular points it is a simple line of the surface ; for if u and v are the lines joining any point P of a; to ^4 and B we have , let A be the point of contact of I with 4> and tti the singular line of 0^ which touches at J. ; then if v is any line through B in the plane {I, Oj), since (Xoiw) = 0, it follows that the surface {Xuv) = contains the line Oj ; moreover (Xw'') = 0, representing the complex cone of A, consists of two planes a, a' intersecting in a^; hence the complex surface is of the form which shows that Oj is a double line of the surface. Two of the tangent planes to 4> through I coincide with (I, Oi), hence there are four double points of the surface outside I; denote them by B^, B^ ; B^, B^, where B^B^ and B^B^ meet I; BiA and Bi'A are lines of C, hence the planes (B^, ai), (Bi, a,) are the plane-pair for A ; similarly {B2, aj), (B^', Oi) are also the plane-pair for A, hence the joins of two pairs of points B meet ai; thus BiBi B^B,' form a twisted quadrilateral of which one pair of opposite sides meet I and one pair meet a,. The tangent plane along each side of this quadrilateral is stationary. (iii) If I is a double tangent to by a (Art. 80), and all are singular lines; if A and B are the points where I meets this section, then denoting OA and OB by b and c, X6 meets b in A and Xc meets c in B, hence (X-Z6) = 0, (Xic) = 0; 86-87] THE QUADRATIC COMPLEX 109 and expressing that all lines of the pencil (0, a) are singular lines we have (W) = (Xbc) = {Xc') = (X'b') = (X'bc) = (X'c') = 0. Let d and d' be any two lines through A and B respectively, then {Xhd) = 0, (\cd') = 0, and we have for u and v coordinates given by Ui = pli + Kbi + vdi, n = p'h + «'ci + v'di, where p = is the equation of the plane (bd) &c. (Art. 13), thus v = and v' = are each the equation of the plane a. We obtain, rejecting terms which vanish by aid of the above relations, (Xu^) = p'' {XI') + v" (Xc?=) + Ipv (Xld), {Xv') = p"" (XI'} + v'' (Xd'') + 2p'v' (Xld'), (Xuv) = pp (XP) + vv (Xdd') + Kv (Xbd'} + KV (Xcd) + pv (Xld') + p'v (Xld). Thus from the equation (Xm^) Q<,v') — (Xuv)' = 0, the factor v may be removed, and the resulting cubic surface is seen to have I, b, and c for simple lines. There are no double lines. 87. Normal form of the equation of a quadratic complex. The equation of a quadratic complex may be written in the form F (p) = ai2, 12 pi^' + 2ai2, 34^12^34 + ^ (aj/Sj - 04/83), &c. and the equation of the complex becomes = {(ai^2 - Oa/Si) (x3y^ - x^s) +...}' = «! /3l «i yi «.2 A «!2 2/2 «3 ^3 a^s 2/3 a4 A «4 2/4 or, as it is usually written, (a^xyf = 0. 88. Complex equation of a quadric. The quadratic complex which consists of the tangents of a quadric, or '^ = 0, (Art. 46), is a special case, for which the symbolic form of * A corresponding result holds for a complex of any degree, see Chapter XVII. + "Ueber die Pluckerschen Complexe," Math. Ann. 11. See also Waelsoh, "Zur InTariantentheorie der Liniengeometrie," Math. Ann. xxxvii. 87-89] THE QUADRATIC COMPLEX 111 equation may easily be found directly ; for if (^at^if = 0, or as it is usually written a^^ = 0, is the symbolic form of the equation of the quadric so that aye = Ui-aic; the points in which the line joining any two points X, Y meets the quadric are determined by substituting Xi + XF; for ^^ in the equation a/ = 0, giving ax + ^XaxCiy + ^^fly^ = 0. If these values of X are equal we have if now we employ a second set of symbols a/, so that a/ = a/^, the last equation is seen to be equivalent to ax"^ ■ ay'^ + aY ■ "x'^ ~ 2 {ax ■ ay) (ax' . ay) = 0, or, = {ax- ay -ay . ax'f = {% {aia,^ — aifti') Pijt}" ; which is the required symbolic form of the equation '^ = 0. If U and V are two planes through the line {X, Y), the last equation is, writing ttjs for p^, &c., {aa UVy = 0. 89. Harmonic Complex. The assemblage of lines which meet any two given quadrics a^' = 0, 6/ = in points which form a harmonic range is a complex, usually called the Harmonic complex, whose equation is readily found by the symbolic method ; for the condition that the roots of ax" + 2Xax . aj- + Var' = 0. bx' + 2fibx.br + fi'br' = 0, should be harmonic is known to be ax'.br' + ar'.bx'-2{ax.ar){bx.br) = 0; i.e. (ax.bY-aY.bxf = 0, or, {t{aibt:-akbi)pikY = 0. If the complex equation of a^^ = be ^ = 0, it is at once seen that the harmonic complex has the form for ^ = has been seen to be S (atak - ata/ypit" + 2X {ataff - a^ai) {ahai - aiah')pu:Phi = 0, or, t (an akk - «,-*') pik" + 2l(aih au - a^h aa) Pik Phi = 0, 112 THE QUADRATIC COMPLEX [CH. VI while the equation of the harmonic complex is + 2S (aikhi + auiih - , gives a pair of corre- sponding lines ; hence B=0; similarly A =0, and the equation becomes , Now if a line whose coordinates are puc meets each of these two corresponding lines a>(p\a)* + fi(o(p\b) = 0, w (p \c) + v(o{p\d) = 0, or, o}{p\a)m(p\c) = ko}(p\b)a)(p\d); but in m{p\ a) the only term is p^ since all the coordinates ttik are zero except a^, which is unity; similarly 'o{p\c)=pu, &c., and we have as the equation of the locus of p p^Pu = kpiiPi3, i.e. a tetrahedral complex. This result is shown by Hirst and Sturm f as follows : — let (A, 13) and (B, a) be the two given pencils, G and B being the points given as above ; then if two corresponding lines of the pencils meet CD in Xj and X^, and if a; is a line which meets this pair of lines, ^ (ABCB) = {X,X, CD). In the same manner if y is any other line which meets a pair of corresponding lines of the two pencils, y{ABCD) = {Y^7^CD) but {X^X^CD) = {Y^Y^CD), (Introd. iii), therefore x{ABCD) = y{ABCD). Since any two vertices of the fundamental tetrahedron may be taken as the centres of the projective pencils, the complex may be thus generated in six ways. 94. Reguli of the complex. The complex Ap^^p^i + BpaPii + Cpup^ = contains the regulus whose equations are B + ,x — ( P^ = p-V^--Q~.°'-Pn, C + ^ A+^i ■tS + fl * u (y |o) is a frequently used abbreviation for Sjj^ -^ . t Hirst, " On the complexes generated by two correlative planes," Froc. hond. Math. Soc , vol. x. 93-95] SPECIAL VARIETIES OF THE QUADRATIC COMPLEX 117 for any given values of p, a, r and ^; since, eliminating these quan- tities, we obtain the tetrahedral complex. This regulus is seen to pass through each vertex of the tetrahedron of reference, since the three equations are satisfied by taking Pas = ^54 = ^42 = ; which shows that the regulus passes through the vertex A-^, and so for the other vertices ; hence, a tetrahedral complex contains 00 ' reguli which pass through each vertex of its tetrahedron. Similarly, by writing iri^ in place of p^: in these equations, we obtain oo ' reguli of the complex which touch the coordinate planes. Since the equation of the complex may also, by subtraction of G {pi'iPsi+ PnP4ii+ Pujp^, be written in the form {A - C)p,^p^ + (B- C)p,^p^, = 0, it is clear that the regulus whose equations are Pl2 = PPlS, p(A-G)p^ + (B-C)p,, = 0, Pii = 0^23 + rpis + /J'Pt^ , is contained in the complex, for any given values of p, a; t and fi. These 00 * reguli pass through the vertices A^ and J 4 and touch the coordinate planes a^ and a^. There are five other similar sets of 00 ' reguli, hence, a tetrahedral complex contains six sets of ao* reguli, where every regulus of a set passes through two vertices of the tetrahedron of the complex and touches the opposite faces. 95. If any two lines, p and jo', of the complex, meet the fundamental tetra- hedron in points L, M, N, R; L', M', N', R', then on p and p' are determined by these points two projective rows of points, whose joins, therefore, form a regulus p, (Introd. viii) ; each line of the complementary regulus p meets the lines LL', MM' in points X", M", &c., so that {L"M" N"R')=={LMNR)-=coTista,TA double ratio of the complex ; hence each line of p' belongs to the complex ; there are 00 ^ possible combinations of the lines p and p', but 00 ^ of these belong to the same regulus, so that there are qo * such reguli p' ; since the line LL' of p lies in the plane BCD, this plane contains a line of p'. Similarly we derive 00 * reguli p" which are obtained as the complementary reguli of the loci of intersection of projective pencils of planes with p and p' as axes. The reguli p' are those previously obtained which touch the faces of the tetrahedron, the reguli p" those which pass through its vertices. If three lines of the complex p, p', p'' be taken, and the planes (p, A) {p'. A) {f, A) ; {p, B) (p', B) ip", B) ; {p, C) (p', C) (p", C) be made to corre- spond by threes, the correspondence of the three pencils of planes of which the axes are p, p', p" is determined, (Introd. xii), and the locus of points of intersection of three corresponding planes is a twisted cubic, which passes 118 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII through D, since p, p, p" belong to the tetrahedral complex. If p"' be any chord of this cubic, then since p'" {ABCD)=p{ABCD), (Introd. xii), p'" belongs to the complex. Since there are oo ^ combinations of lines p, p', p", and since three chords of a twisted cubic may be selected in oo ^ ways, there are oo ^ twisted cubics all of whose chords belong to the complex. Similarly by taking the three projective rows of points determined on three lines of the complex we obtain oo ^ developables of the third class, for each of which the intersection of two tangent planes belongs to the complex. Conversely if any twisted cubic is given, and four points A, B, C, D be taken on it, if p is any chord of the cubic, p {ABCD) is constant, and hence all the chords of the cubic belong to a tetrahedral complex whose fundamental tetrahedron is ABCD. Also if x is any generator of a regulus which passes through ABCD we have x{ABCD) constant for this regulus (Introd. viii), hence all the generators of one system belong to one tetrahedral complex, while the generators of the other system belong to a second tetrahedral complex. Since a quadric is determined by nine conditions and two intersecting complex lines may be chosen in oo ^ ways, there are co ^ quadrics which pass through four given points and of which the generators of one system belong to a given tetrahedral complex, and those of the other system to another given tetrahedral complex which has the same fundamental tetrahedron. 96. Second method of formation of the complex. The collineation of two spaces S and 2' gives rise to a tetrahedral complex; for, if the united points of S and 2' be taken as vertices of the tetrahedron of reference the equations connecting corresponding points are then, (Art. 34), fixi = UiXi, hence if pne is the line joining x and x', fJ'Pik = /*(««; — Oi) miOOk p^.pu (.(h-(h)(ai-as) . . or, ■* — ^ =y r^ -,= constant. Pu -Pa {at- tti) (as - a.,) If u and «.' are a pair of corresponding planes v.Ui = Oiiti, hence, if TTat is the line of intersection of u and u, "Tu • •"'34 _ (^2 — Ol) (tt4 — aa) . TTij . TTjs (ttj — tti) (tts - Oj) ' that is, the locus of intersection of corresponding planes is the same tetrahedral complex. Finally if jOit is the line joining tvfo points xu yi and pi^f the line joining their corresponding points «/, y/, /J-^-Pik=aiakPik- 95-97] SPECIAL VARIETIES OF THE QUADRATIC COMPLEX 119 Hence, the locus of lines puc which intersect their corresponding lines is (aiftj + agat) pisPst + (aiCts + a,iad P^Pti + (chat -t a^a^) p^ p^ = 0, which is easily seen to be the same tetrahedral complex. Let A BCD be the united points of the spaces S and 2', and P, P' a pair of corresponding points, then in the sheaves (/") and {P') certain corresponding lines intersect, the locus of such points of intersection being a twisted cubic, which passes through A, B, C, D, F and P' (Introd. xii). There are oo^ such cubics obtained by taking for P all positions in 2 ; two such cubics have a regulus of chords in common ; for if Q and §' are any other pair of corresponding points, since PQ, PQ' are corresponding lines we have two projective pencils of planes, with axes PQ, PQ' respectively, the intersections of whose corresponding planes form a regulus, (Introd. viii) ; any generator of this regulus is a chord of the cubic of both P and Q, for if a and a are two corresponding planes of the pencils which meet in p, the pencils (P, a), (P, a') determine two projective rows on p having two united points, hence p is a. chord of the cubic of P, and similarly is seen to be a chord of the cubic of § ; it follows therefore that if x is any chord of any one of these oo •= cubics , ^ n^ ^s ^ ^ a; {ABCD) = constant. Hence, as just shown analytically, the joins of corresponding points P and F of 2 and 2' form a tetrahedral complex. 97. Third, method of formation of the complex. A third method of formation is the following : hxmng given a (1, 1) correspondence between the points P of any plane a, and the lines p of any sheaf {A), the lines of the pencils {P, p) constitute a tetrahedral complea;. For the lines of the sheaf give rise to a second point-system in a, coUinear to the given one ; let BCD be the united points of this correspondence (Introd. xiii), then if to P in the first system corresponds P' in the second, and if to P in the second corresponds P" in the first, to the line PP' in the second system will correspond the line PP" in the first system. Now take any .^ line PT of the pencil (P, p) meeting AP' in T, and any point S in the plane {PT, PP"), then the pencils {S, PT), {A, PP') are projective and determine two pro- jective rows on PT; hence, cor- responding lines of the pencils meet twice on PT. But the sheaves which project the two coUinear plane systems from S and A have, as locus of intersection of corre- sponding lines, a twisted cubic Fig. 7. (Introd. xii) which passes through A, B, C, D, and of which PT is therefore seen to be a chord. Hence PT (ABCD) is the double ratio of the chords of this twisted cubic; by taking difierent positions for S we obtain oo 3 cubics, and since any two 120 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII of them, as will be shown immediately, have a regulus of chords in common, the double ratio of the chords is the same for each of these cubics, or, PT {ABCD) is a constant, which proves the I'esult. That any two of the twisted cubics have a regulus of chords in common may be seen as follows : take any two points S and + >-Pis = 0, Pu + h-Pk = ^ (II); hence eliminating - _ we obtain as the equation of the complex 97-99] SPECIAL VARIETIES OF THE QUADRATIC COMPLEX 121 Secondly, if the plane of one pencil passes through the centre of the other, and if also to the join of the centres corresponds the intersection of the planes of the pencils, then A1A3 corresponds to AsAi, and we may take AiA^ and A^A^ as corresponding lines. In the foregoing relation (I) between \ and /x, since \ = 0, /M=; X = oo, fj- = 0, give corresponding pairs of lines, it follows that B = C = 0. A line puc which meets a pair of corre- sponding lines satisfies the equations Pis + ^?>i2 = 0, pu + fipvi = ; giving as the equation of the complex ApuP^ + Dp,i = Q. Fourthly, if the pencils have the same centre, 0, the complex consists of the lines of the planes which pass through every pair of corresponding lines of the two pencils. The two projective pencils determine upon any plane a which does not pass through 0, two projective rows of points whose joins envelope a conic (Introduction, vi), therefore the planes which contain the lines of the complex meet a in the tangents to a conic, i.e. they touch a quadric cone and the complex consists of the tangents of this cone. Fifthly, if the pencils have the same plane, the intersections of corresponding pairs of lines lie upon a conic and the complex consists of the lines which meet this conic. 99. Complexes determined by two bilinear equations. The preceding complexes may also be arrived at by means of two equations which are bilinear in two sets of coordinates. For in the equations Xi'taieiOOk + x^ta^aXk + Xa'ZaiciXk + x^^.a^X}/ = 0) Xi'ZhiXk + x^h^x^, + ^sSfcjfcsa:*' + x^lbiciXk = ol ' 122 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII regarding the «< as the coordinates of a point P in a space S, and the xl as the coordinates of a point P' in a space 2' ; if P' is given, a line ^ of 2 is determined, and if P is given a line p' of %' is determined. Moreover, regarding the xl as parameters, the equations (i) establish a (1, 1) correspondence between the planes of 2, hence the locus of p is, in general, a tetrahedral complex (Art. 96), similarly for f . This may be seen otherwise, as follows: — such values of xl as make the two planes of 2 identical give a singular plane of the locus of p, i.e. a plane all of whose lines belong to the complex ; these values of a:/ are given by the equations "LakiXk — pZhinXk = ; la^xi,' — pZh^iX^, ta^^Xk' — /326t3%'. By elimination of the x// we find the quartic equation for p «3> ^4 = 0), (021 °4> ^3 = 0) and planes qj, aj respectively, projective to each other. If now the plane a^ passes through the centre of the pencil in oj we have concurrence of the planes (aj, a^, 03) and ^4=0, hence Also writing ^^-^v-^x-^+v^x^ + v^x^ + v^x^, the solutions of the equation which corresponds to (ii) are in this case zero, infinity, and the values of p given by the equations X^+UiX^=pX^Vi u^x^=p{x^+v^x^) , , f Y X^ "T" Wo*4 p Xn Uo 0=p(x^+v^x^)^ One of the two values of p given by these equations is zero, hence (ii) must have a pair of equal roots. The remaining value of p given by (v) is easily seen to be *— ; this is infinite, i.e. (ii) has two pairs of equal roots, if v^ = Q; which, with the con- dition already satisfied, viz. 11^=0, makes the plane of each pencil pass through the centre of the other. If, in the two pencils (iv), the line joining the centres corresponds to the line (oi, oj), then, for some value of the ratio x^'jx^', the first equation reduces to UiXi+u^2—^t while the corresponding line of the second pencil through (oj, 02, 03) ; this requires that x^+Xi'Ug=0, Xf+x^'v^=0, .(V). 124 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII i.e. \-v^Uj=Q; but this is the condition that (iii) may have three roots equal. The centres of the two pencils will coincide if the six planes ^,-1=0, ^2=0, ^3=0, «4=0' ^3=0, l4=0 concur, in which case the eight original planes must concur. 101. It has been seen that there are in general four points P' which make the two planes in equations (i) coincide, giving four different planes : in the following special case there are oo '■ points F which make the planes (i) coincide with the same plane ; for, when the coefficients in (i) are such that two points P' can be found, which identify the two planes (i) with the same plane «!, then any point on the line joining these two points P' will make the two planes (i) identical with a; for take this supposed pair of points P' as the vertices (0, 0, 1, 0), (0, 0, 0, 1) of the tetrahedron of reference for S'.. then the equations "^5 ^1 i" *^\ ^3 ~r ^3 64 ~" ") ^4 '^l ~r *'!'l ^4 ~r ^2 53 ^^ ^» give rise to a connexion of the supposed kind, since it is clear that any point P' on the line x^ = 0, x^ = 0, makes the preceding planes identical with x^ = 0. Every line p meets the conic x^, = 0, 0:3^3 — x^j^i = 0, hence the complex in S consists of the lines which meet this conic c". To find the complex formed by the lines p', we may for convenience write Is = «2. ^4 = «!«! + "^i + a3«3 + a4a'4 ; then the complex is obtained from the equation aJs' + Mi^a' OaiCa' X^ + a^X^ diXi yi-^^yi ^l 2/1' + 032/2' a.^i Xi X2 x-^ yl y^ y; i.e. p\i [p\i + oLip'a + a3i''32 + "-iPu + «-ip\i\ = ; and consists, there- fore, of two linear complexes. Hence, to the points of c* coifespond in %' the lines which meet A^Al ; to each of the other points of 1, corresponds a line of the complex - aiP'i2 + Ka/u - a3Jo'23 + ^iP'«. -P'ls = 0- = 0; 100-102] SPECIAL VARIETIES OF THE QUADRATIC COMPLEX 125 An important special case arises when the planes ^4 = 0, f 4 = are taken to be identical, i.e. when aj^ = a2 = 0^3 = 0, we may then take 04 = — 1 ; in which case c^ has as its equations and the complex in %' reduces to p'42+j''i3 = 0; the relationship between S and S' is then determined by the equations ^3 ^'1 i "'I **3 """ "^2 '*4 ~^ ^» a^i'^i + a;i'a;4 + x^x^ = 0, and is such that to each point of %, except those of «! = 0, x^Xs + Xi = 0, (or c^), there corresponds a line of the complex p'i2+p'i3 = 0; to each point of S' corresponds a line which meets c^. Finally, writing • / / / -J ^ rf/2 ^~ vC ^~ V") "^2 ^~" ~*" J 1 """ 1 — ) aJs = a; + yi, x^ = z', Xi = z, Xi = 2/', the spaces S and 2' are each referred to Cartesian coordinates, and the bilinear equations become x + iy + zx'+ z — *, X (x — iy^ — z — y' = 1^ ; while, with reference to the new coordinates, the S complex is that formed by lines which meet the trace on the plane at infinity of x^-\-y'^-\- z^ — ^, or the sphere-circle; the 2' complex is that formed by the lines of the complex p'u=p'3i- So that to each point of 2' corresponds a line which meets the sphere-circle, i.e. a minimal line ; to each point of 2 corresponds a line of the complex y,2— ^'34 = ; the only exception being that to the points of the sphere-circle of 2 correspond all the lines of a plane parallel to the plane x' = 0. 102. Reye's Complex of Axes. Reyef denotes by an axis of a quadric, a line which is perpendicular to its polar line for the quadric; such a line is an axis of a plane section of the quadric ; for if p and p' are a pair of such lines, and tt' the plane through p which is parallel to p, the pole of the section of the quadric by ir' is the point at infinity on p' ; hence, since the pole * See Lie and Sohefiers, Beriihrungstr. Bd. 1. S. 445. t See Eeye, Geometrie der Lage, 11. 126 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII oi p for this section is at infinity in a direction perpendicular to p, it follows that p is an axis of the section. a? tfl z^ Taking — 1-^+— =1 as the equation of the quadric, if the direction cosines of p are I, m, n, and any point on it is (xyz), then I'm'v! being the direction cosines of p' , we have, IV + mrn^ + ivn! = ; also the polar planes of {ocyz), and of the point at infinity on p, being respectively a c a b c therefore V ■.m':n' = a (yn — zm) : h{zl — xn) : c {xm — yl), hence al (yn — zm) + hm {zl — xn) + en {xm — yl) = 0; while for a tetrahedron of reference of which one face is the plane at infinity we have I = p^^ &c., yn — zm = p^ &c.; hence the axes form the complex apuPa + ^13^42 + cpy^pst = 0. To this complex also belong the normals of the quadrics confocal to the given quadric, and of those similar, similarly situated and concentric to it ; for if the line through the point (xyz) having I, m, n for its direction cosines, is normal at {xyz) a? ifl z^ to the quadric — I- t- H — = 1, we must have ^ a c o_ ft _ c X y z' I m n whence af-^ --) + 6(--?) +cff - ^^ =0. \m n) \n I] \l m/ The quadric being one of a series of similar, similarly situated and concentric quadrics, the oo ' normals of the quadrics form a complex, which, from the last equation, is seen to be tetrahedral. Again ii a = A^ ■\-\,h = B' + \ c= G^ + \, i.e. if the quadric is one of a series of confocal quadrics, the normals again form a complex which is seen to be the same tetrahedral complex* * From this property the complex is sometimes called the Normal Complex. x^ v^ z^ We easily find that any line of this complex meets the quadric — i- i- + _ doc in points at which the normals to this quadric intersect each other. 102-104] SPECIAL VARIETIES OF THE QUADRATIC COMPLEX 127 103. Differential Equation of the Complex. The tetra- hedral complex has been seen to possess an equation of the form aPl4i>23 + bpi3Pii + CP12P34 = 0. Any projective transformation will change the given tetrahedral complex T'^ into another such complex; since, by a projective transformation, the double ratio of four points on a line is equal to the double ratio of the four corresponding points on the corresponding line. A projective transformation which leaves the fundamental tetrahedron unaltered, will interchange the lines of T^, but leave the complex as a whole unaltered ; such a trans- formation is given by equations of the form /la;/ = aiXi, and of such transformations there are oo ', which may also be observed from the fact, that when the united points of the collineation are given, the projection (or collineation) is determined (Introduction, xiv) by connecting a given point P with any point of space Q, and Q may be chosen in oo ' ways. If T^ be projectively transformed into a complex which has the plane at infinity as one face of its fundamental tetrahedron, since we have now p^^^xdy — ydx, ...p^ — dx, ...{Axl. 6), the equation of the new complex is adx {ydz — zdy) + hdy (zdx — xdz) + cdz (xdy — ydx) = 0. The complex represented by this equation, (that of the last Article), is therefore the projection of any T', one of whose double ratios is j- ; its complex curves are parabolas (since they touch the plane at infinity). If P, Q, R be the points in which any complex line meets the coordinate planes of x, y, z PQ respectively, since {PQB ) = constant, we have jy^ = constant. 104. The line element. If with any given point of space a definite direction be associated, we obtain the idea, due to Sophus Lie, of a line element. Connected with any point there are od'^ line elements (corresponding to the different directions through the point), and in space there are altogether oo " line elements. By any differential equation of Monge, of the form f(x, y, z, dw, dy, dz) = 0, homogeneous in dx, dy, dz, oo * line elements are selected from the 128 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII 00 ' line elements of space. It was seen (Art. 6), that an equation of the form f {ydz — edy, zdx — xdz, xdy — ydx, dx, dy, dz) — represents a line complex ; a line element {x,y,z; dx : dy : dz) which satisfies the equation f— 0, will be said to belong to the complex represented by /= 0. In Chapter XVIII. Lie's investi- gations of this differential equation will be considered. 105. Curves of the Tetrahedral Complex. Any point P of a curve and the tangent p at P determine a line element of the curve, the number of such line elements being qo \ A curve of a complex is one whose line elements belong to the complex. Along any given curve of a tetrahedral complex T" the coordinates of its points are functions of a single variable t, and this parameter t may clearly be chosen so that dx dy _ 1 _ 1 * X ' y a + t ' b + t ' and since the equation of T^ may be written ,, .dydz,, .dz dx _ . , . dx dy . (6 - c) ^ . — + (c - a) — . \-(a-b) — . -^ = ; ^ ' y z z X ^ ' X y it is easily seen that for a curve of T'^ we have dx dy dz _ \ 1 1 X ' y ' z a-'t-t'b-\-t'c-\-t' whence the curves of T^ are seen to be the form of F being arbitrary and \, /t, v being arbitrary constants. The following cases are of special interest : (i) F{t) = \, x = X{a + t), y = fj,{b + t), z = vic + t); this gives the complex lines ; (ii) P(0=2, x = X(a + ty, y = ^(b + ty, z = v{c + ty; this gives the (parabolic) complex conies ; (iii) P(,)=-l, .=^^.,= -^^,. = ^-|_; • See Lie, Beriihrungstr. S. 327. An exception occurs if — :— : — =-:-;- X y 2 a /S 7 where a, /3 and y are constants. In that case x"^ : y° : z^ = A : B : C. 104-106] SPECIAL VARIETIES OF THE QUADRATIC COMPLEX 129 this gives the oo' twisted cubics through the vertices of the fundamental tetrahedron the chords of which form T^. 106. Non-Projective Transformations of the Complex. Any transformation of the form x^ = Xa;™, y^ = yinf^, z^ = vz^ applied to a line element of T" transforms it into a line element which also belongs to T^, since here dx dx-, dy dt/i dz dz, X x^ y y-, z z^ and substituting in the differential equation of the complex the form of the equation is not changed. Complex curves are there- fore by this transformation changed into complex curves ; the cases m = 2, m = — 1 are of special importance. For m = 2, any Utw Ax^ + By^ + (7^. + D = 0, A'x^ + B'y^ + C% + B' = 0, becomes, for X = /a = v = 1, the twisted quartic Ax^ + By" +Gz" + I) = 0, A'x" + By + G'z" +D' = 0. Hence the line elements of this twisted quartic belong to the same T^, and by projection it follows that any tangent to the curve of intersection of two quadrics meets their common self- conjugate tetrahedron in four points of constant double ratio. To a line in the space (xyz), e.g., x = a.+ Ir, 2/ = /3 + mr, z='y + nr, there corresponds Xj = '\.(a + Iry, i/i = /i (/3 + mry, Zi = v{y + nrY ; i.e., a complex conic. For m = — \ the transformation is Involutory ; the complex line x = a + lr, y = ^ + mr, z^y+nr becomes the twisted cubic X /J, 2/i - o I ^^ ' ^1 ~ ; ' oi+lr'- '^ P + mr' y + nr' The proposition already established (Art. 95), that the chords of these oo s cubics form T^, is shown by Lie in the following manner : — ^the transforma- tion 3!]^= - , yi = -, h — ~ is such that by suitably choosing X, /i, and v any two points of space may be interchanged, i.e. so that to P and Q of the space (xi/z) there correspond Q and P of the space (a^iyiZj); now take any two points P and Q on one of these twisted cubics, then to the cubic will correspond a complex line which passes through P and Q, i.e. the chord PQ is a complex line. J. 9 130 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII 107. The Special Quadratic Complex. One species of quadratic complex consists of the tangents to a quadric surface. This complex is said to be special. On each line of the complex there is one point for which the complex cone becomes two (coincident) planes through it, viz., its point of contact with the quadric. Hence every line of the complex is singular, so that for every line of the complex F{x) = we must have where a and ^ are constants. If in place of F {x) = 0, the equation f{x)^F{x)-l%{x^)=0, which represents the same complex, be taken, we have = aF {x) + /3Sa;i» -aF{x) + ^ txi Thus the equation of a special quadratic complex being given in the form F{x) = 0, by the addition of a determinate multiple of (a?), this equation may be replaced by f{x) = 0, where The equation of the complex being expressed in the form /(a;) = Oi, «!=+... + laijtXiX): + . . . = 0, the result last obtained requires that i i A being the same for all values of h. It follows also that if the quantities Xi are the coordinates of a line so also are ——, more- over this line is the polar of x with regard to the quadric ; for the polar line of x with regard to the complex is XXi + u,^, whence X.fi = 0, and therefore \ = 0, OXi (since a line coincides with its polar only when it is a complex line), and this polar line, being the locus of poles for x of the 107-108] SPECIAL VAKIETIES OF THE QUADRATIC COMPLEX 131 sections of the quadric through x, is therefore the polar line of x for the quadric. Taking as the equation of the quadric its complex equation is (Art. 46), '^ = a^a^p^^" + asaipai^ + ciiCis^is' + ^402^94/ + a^aiPu" + a^agp^s'' = 0. On substituting for pi^, as usual, ^(xi+ix^), &c., and multi- plying by two, we obtain for /(«) the form f=^[aia2Xi + ix2 + asa^Xi — ix^ + } = 0, which gives S ( ^ j = ioia^asai . («") ; showing that A is equal to the discriminant of the given quadric. The determinant of the coefficients of f has the value — A^ ; for from the equation connecting the coefficients of f we see that \aijc\'^ = ^^, and taking a^, a^, a,, a^ as being each equal to unity J ^ Xy "T ^2 "I '^'b '^S *^4 "^6 ) whence it is seen that in this case | a^ | = — 1, therefore generally |aa| = -A^ 108. System of two special complexes''^. If we have any second quadric, its complex equation may, as has been observed, be brought to the form = 0, where The discriminant x (p) of /— P • ^os ofi^y reeiprocal roots ; for writing Bit=Srbir . a^, (r=l, ..., 6), we have — A^.X=\C'ik\-X—\^~P- ^it I ) ^ appearing only in the principal diagonal, -A'3.x = l^*l-X = l-^«-p-^'l, A' „ „ „ „ ; showing that if )• is a root ot x = 0, so also is /, where r' = - . — ; thus we have three pairs of roots r, r', each pair being connected by the equation rr' = — . If the quadrics have as equations in point-coordinates * The following is an aooonnt of an investigation by Voss, " Die Liniengeometrie in ihrer Anwendung auf die Maohen zweiten Grades," Math. Ann. i. 9—2 132 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII then / and <^ the complex equations of U and V expressed in Plucker coordinates are, If now we make the substitutions* •Ja^ai.p^^ty.p'^, • becomes CG'+y,''T=0. This form of the equation shows that contains two families of 00 1 twisted quartics, viz., G+\yi' = 0, T-\G' = 0; and C + X'yi' = 0, T-X'G = 0. 136 SPECIAL VARIETIES OF THE QUADRATIC COMPLEX [CH. VII Through any point of there passes one curve of each family. Any curve t of one species and any curve t' of the other species lie on a quadric, since There are two harmonic complexes which have a given tetra- hedroid as singular surface ; for when b c c a a b c b' a c' b a are each given there are two sets of values for a : 6 : c ; if one set is a : /8 : 7, the other is 111 so that for the two complexes T is the same, while G and C are interchanged. By reference to the equations of Art. 82, in which E is to be takeu as zero, it is seen that the singular points of the surface lie by fours in the coordinate planes ; the singular tangent planes of the surface pass by fours through the vertices of the tetrahedron of reference ; thus the coordinates of the singular points /3 in the plane a^ are given by the equations A' : /3s' : /S4' = a (c= -¥):b (a' -c''):c {b^ - a% and the coordinates u of the four singular tangent planes through -4i are given by the same proportion. These four singular points are seen to be the intersections of the conies (aj, G), («!, (?'). 112. There are x ^ pairs of quadrics such that for each pair the Harmonic complex is the same as the given complex. In the first place it is easy to see that the Harmonic complex for the quadrics U+aV, C/" - o-F is ^-ff='^' = 0, where ^ and '^' are the complex equations of U and F respectively ; for if U= kyi' + l^yi + ky,^ + l^y^, V= m^y," + m^y^^ + riHyi + m^^, the complex ■^, for JJ-^trY, TJ - = Xop^a + X6p\3 + XcpSi + bcp^M + cap\i + abp''^, ^ {\a — be) (kb - ca) {Xb — ca) (Xc — ab) (Xc - ab) (Xa — be) ' hence 4> - a-yKj, = Xap\ + \bp\3 + Xcp^u + hcp\ + cap\ + 06^^23 „ (Xa — be) (Xb - ca) (Xc — ab) ,- r „ , ^-r , - o-^ ^^ ^ ^ ?) V ^ ('*''' ~ ^^P^i + X6 - capV - (abcf + \a — be p\) = X . H^, if a" = y- , x /^ ? w^: l^ • ^ (Xa — be) (X6 — ca) (Xc — ab) Any pair F—aK, F+aK intersect on a curve t which lies on . If a line of the complex U^ meets t in P it must meet one of these two quadrics in a point consecutive to P, i.e. it must lie in the tangent plane at P to one of the pair ; and the tangent planes to the quadrics at P form the two pencils of complex lines through P. In like manner, by interchanging G and C, we obtain oc ' pairs of quadrics such that for each pair the harmonic complex is H'^ = - {p\,+p\) + \ {p\ +p\) + \ ip\.+p'2s) = 0. a o c The given Harmonic complex is also the locus of lines through which four harmonic tangent planes can be drawn to any one pair of 00^ pairs of quadrics. For, starting with the complex equation a (-jtS, + 7r-^34) + i ('^^3 + vr^O + c {7r\, + 7r'^) = 0, and repeating the previous analysis, using plane instead of point coordinates, we arrive at the result just stated. Let Fi + fiLi, F1 — /J.L1 be a pair of these quadrics, then any complex line which lies on a common tangent plane tt of the quadrics must lie in two consecutive tangent planes of one of * From the values of the coordmates recently obtained for the four singular tangent planes through A-^, it is seen that K touches each of these planes. 138 SPECIAL VAKIETIES OF THE QUADRATIC COMPLEX [CH. VII them, i.e. must pass through one of the points of contact P of tt with the quadrics; hence tt is a singular plane of the complex, the centres of the pencils in it heing the points of contact of tt with the quadrics. Now through P two of the preceding quadrics F+ a-K, F— a-K will pass, and it was seen that -tt was the tangent plane to one of them at P, say to P + aK, hence F+ 13P42 + «Vl4P23 = 0. and is therefore the complex of normals for the given ellipsoid. • See Sturm, Lin. Geom. Bd. m. S. 344. CHAPTER VIII. THE COSINGULAR COMPLEXES. 114. It has been seen that there is a singly infinite set of quadratic complexes which have a given Kummer surface as singular surface (Art. 84). This will now be shown independently, as follows : — using the coordinates of Klein, let XXiXf = be any given quadratic complex G^, the complex :0, where (i has any definite value, has the same singular surface as C\ For, denoting this complex by C/, if yi is a singular line of 0^", then —^ — is a line, (Art. 76), and intersects yt in a point of the singular surface of (7/, the pencil of tangents thereat being Now if we write (- %^> SCi Xi + fj,' it follows that ^\,oof = 0, SVa;/ = 0, hence a; is a singular line of C (Art. 76), and touches the singular surface of G^ at a point for which the pencil of tangents is {'ki + /MXi, asi); but this is identical with the pencil [yi, —^ . hence, the singular surfaces of 0,^^ and C, having the same pencils of tangent lines, must be identical. The complexes (7^^ are said to be cosingular. 115. It appears that if y is the singular line of 0/, at a point P of , at which the singular line in G^is x; then y-i = {\ + m) «i ; 140 THE COSINGULAR COMPLEXES [CH. VIII by varying /i we obtain the pencil of tangents to at P, hence each tangent to «1> at P is a singular line for one of the cosingular complexes G^\ If /x be eliminated between the equations '^ --— = 0, 2 J^=o, Xi + fi- ' {'K + fJ'T we obtain the complex equation of 4', i.e. the complex formed by its tangents. Since it is seen that C is included in the series of complexes C^^, and corresponds to the value /i = oo . The complexes (7/ also include each fundamental complex taken doubly ; as we see by taking successively /u, + \i = 0, ... /n, + Xg = 0. Through any line I there pass four complexes C^, (Art. 84), viz. those determined by the equation. X- — "^ — = 0. If two of these values of fj,, say /x, and fj^, coincide, then since in that case it follows that Z is a singular line of the complex (7^,^, hence, the singular lines of a complex C^^ are its lines of intersection with a consecutive complex of the system.. If a plane be drawn through I to touch $ in 0, the four singular lines corresponding to the above four complexes are the joins of to the points of intersection of I and 4> (Art. 84). It should be noticed that if two complexes 1.kiXf = Q, t-kixf=Q, are cosingular, we have h = 7^.^ + 8' -(7Xi + S) + /3-^ since h = l 7 ^«^g7z_«j 7\i + 7 7 115-116] THE COSINGULAR COMPLEXES 141 116. Correspondence betvveen lines of cosingular com- plexes*. If between the coordinates of two lines x and y, the following six equations exist, 6 it follows that x belongs to the complex 2 'kiwi' = 0, or C, and y 1 y^ n ^r. rj i belongs to the cosingular complex % :-^— — = 0, or Gy. Thus the above equations establish between the lines of C and (7/ a (1, 1) correspondence, by aid of which many important properties of the quadratic complex can easily be demonstrated. A fundamental fact of the correspondence is the following : — if x and X are two lines of C and y, Y their corresponding lines in C/, it is clear that txiYi = 'S,yiXi ; hence, if x intersects Y, then y intersects X. vY- If LxiXi = 0, then 1 /' ' = 0, but y, Y will not, in general, intersect ; if y and Y do intersect, i.e. if Xyi Yi = 0, then since V yi _. 0^ 2 p^^ =0, t ^^ = 0, it follows that 2 ^^' "^ ^ '^ = for all values of p, or y and Y Xj + yU. belong to the same pencil of lines of (7/ ; and since lyi 7i = 2 (Xi + /a) XiXi = %XiXiXi, it follows similarly that x and X belong to the same pencil of lines of C^ ; i.e. if a pair of lines of G^ intersect, and also the corresponding pair of C^, each pair belongs to a pencil of lines of its own complex : to a pencil of either complex corresponds a pencil of the other. Any point P is the vertex of a complex cone of C^ and of C,^ \ the two loci of corresponding lines of G^ and C" respectively will not be cones (or conies), as just seen, and are such that any line of one locus meets all the lines of the other locus ; the loci are therefore the two sets of generators of the same quadric, hence, to any complex cone {or complex conic) of 0° corresponds a regulus of 0/ and conversely. There thus arise oo " reguli of G^, the 'images' of the complex cones and of the complex conies of C^^ ; they will be said to form a ' triplex ' of reguli of G^ and will be denoted by * See a paper by the author thus entitled. Quarterly Journal (1903). 142 THE COSINGULAE COMPLEXES [CH. VIII X^, 2/, where S^ consists of the images of the complex cones of C^" and S/ of the images of the complex conies of C/J'. Take any two non-in.tersecting lines x, X of 0', and take the quantity /i determined from the equation as that which determines a cosingular complex C^' ; then to x and X will correspond two intersecting lines y and P"; these latter lines belong to one complex cone and one complex conic of 0/ ; hence, any two lines x, X of C which do not intersect, determine one cosingular com/plex C^ in which the two corresponding lines y, T intersect ; and there are two reguli of G^ through x, X, viz. the images of the complex cone and complex conic of G,^ determined by y,Y. It has been seen that the complex cones and conies of any cosingular complex C^^ have for images ao ' reguli of G^ forming a triplex, and since there is a singly infinite number of complexes G,^ there is a quadruply infinite number of reguli of G' (Caporali, Geometria). The directrices of the reguli of a given 2^ form oo^ reguli, which are the images of complex cones of G', and therefore reguli of Gf,,'. It is known that the polar line with regard to a funda- mental complex, of any line of a quadratic complex G', belongs to G^, (Art. 73); hence, the polars of the lines of a complex cone of C^^ form a complex conic of 0/, and from the equations of correspondence it is seen at once that the polar line of y corre- sponds to the polar line of x ; hence, if a regulus of S^ is polarized with regard to a fundamental complex, we obtain a regulus of 2/. Through the vertex of a complex cone of G^^ there pass oo^ planes, each of which contains a complex conic of G/^", each conic having two lines in common with the cone ; we have, in correspon- dence, 00 ■■' reguli of G' belonging to a 2/, which have two lines in common with one regulus p of 2^ ; these x ' reguli are said to form the 'field ' of p. 117. The complexes Rj", Ht'^. If a complex 2ajiri = contains a regulus p of G^ which belongs to "Z^, the complex 2 ■ r Xi = contains the cone of G^' which corresponds to p. 116-118] THE C0SIN6ULAR COMPLEXES 143 hence the last complex must be special, i.e. The complexes 'S.aiXi = which satisfy the last condition, /j, having a given value, form a quadruply infinite system, which will be denoted by R^^ ; the system passes through all the reguli of a triplex, and two members of the system are contained in any ' pencil of linear complexes,' i.e. a set of complexes of the form {ax) + p{l3x) = 0, where p is variable. In any one complex (ax) = 0, of a given R^^, are contained two singly infinite systems of reguli of S^, S/, viz. those which correspond to the complex cones and complex conies, whose vertices and planes respectively are united to the line Similarly, if a complex XaiXi = passes through the directrices of a regulus p of 2^ forming a regulus of Cf,,", the complex Xai'J{\ + fj>)xi = contains a cone of 2 and is therefore special, hence laf{\i + fi) = 0. This quadruply infinite system will be denoted by iJV ; if a linear complex belongs to an i2/ and to an R'J' for the same value of fi, both -7- — - — r and ai >J{Xi + p,) are lines ; denoting them by I and I', it follows that 1/ = k (Xf + p-), hence I' is the polar line of I with regard to C, and the lines I, I' belong respectively to the quadratic complexes 118. The congruence [2, 2]. The lines common to C^ and any given linear complex A, form a congruence which is of the second order and second class; for the lines of this congruence through any point P are the two intersections of the polar plane of P for A with the complex cone of P for G^ ; in any plane tt, the lines of the congruence are the tangents from the pole of tt for A to the complex conic of tt. If the complex ^atxt = is given, the equation 2 ~ — = ni + pj 144 THE COSINGULAB COMPLEXES [CH. VIII gives five values for ft,, so that if, for instance, /Xi be one of them, the complex cones of C' whose vertices are on the line -77- — ^- — r correspond to 00 ^ reguli of C which belong to XoiXi = 0, similarly for the complex conies of C^ whose planes pass through this line \ from the five lines we thus derive 10 systems of qo ^ reguli of the congruence (C, A). This method of correspondence thus enables us to investigate the congruence (C, A), by consideration of the simpler congruence consisting of the lines which belong to a quadratic complex and meet a given line. The two systems which correspond to the complex cones and to the complex conies connected with the same line —77^ — - — 7-, may be said to be associated ; since each of these complex cones has two lines in common with each complex conic, it follows that any two reguli belonging respectively to two associated systems have always two generators in common. Conversely, if two reguli of different systems have two generators in common, those systems must be associated, for if the reguli pi and p^ belong to two non- associated systems, say those connected with fj^ and p,^, the transformation asi = ,,^ -. turns p, into a cone, and p„ into a regvlus of (7"^,, which cannot have two generators in common with a cone. At the same time it is seen that, if pi and p^ belong to two different and non -associated systems, they have one generator in common, for p^ is transformed as above into a regulus of (7%^ of which the line -77 — - — r is a directrix, and this regulus will have one generator in common with each cone of C^^^ , whose vertex lies on this line. The congruence {C, A) contains 16 pencils of lines ; for, taking one of the roots u, of the equation 2 r — - — = 0, to each line of C'^ which meets the line , '' there corresponds one line VXi + /^ of (C, A), and vice versd; but the congruence ( (7%_, '' ) \ V \j- -I- p,^/ contains 16 pencils, viz. the eight pencils of G"^, at the four points in which *• meets the singular surface, and the eight vXi-l-yu, 118-119] THE COSINGULAR COMPLEXES 145 pencils in the four tangent planes ^i through ' ; hence, since a pencil of lines y corresponds to a^ "-pencil of lines x (Art. 116), there are 16 pencils in the congruence (C", A). Though, in general, there pass through any point only two lines of a congruence (2, 2), it is now seen that there are 16 exceptional points, through each of which there pass oo ' lines of the congruence ; such a point is called a singular point of the congruence. Hence there are 16 singular points in a congruence (2, 2) ; also there are 16 singular planes, each of which contains 00 1 lines of the congruence which form a pencil. ' a- \ . In the congruence ( Ci^,,", . '' I , if the centres of the pencils V ' VXj H- /x/ in /Sj are Bi, Bi, the pencil (S^, /3i) has a line in common with five other pencils, viz. (5/, ySj) and four pencils whose centres lie on '• , (Art. 77) ; hence the corresponding pencil in {G'\ A) has V A,; + /^i a line in common with each of five other pencils, i.e., passes through the centres of these five pencils, hence, each singular plane contains six singular points; and, similarly, through each singular point pass six singular planes. Each complex cone of Cy^' whose vertex is on VXi + ^i contains one line of each of the eight pencils in the planes /S^; hence, each regulus, of the corresponding system ofreguli in (G", A), passes through eight singular points of the congruence ; and the sixteen singular points are divided into ten sets of eight points by the ten systems of reguli. 119. Focal surface of the congruence. The lines of a congruence (2) 2), or (C^, A), which meet any line p form a ruled quartic of class I, having another directrix p', the polar line of p for A, and upon p and p' the lines of the congruence determine a (2, 2) correspondence of points ; of this correspondence there are four branch points on both p and p (Introd. xvii.), hence it occurs four times that two consecutive lines of the congruence intersect on p. The locus of the oo ^ points of intersection of consecutive lines of the congruence is therefore a surface of the fourth degree ; this surface is called the Focal Surface of the congruence. If p belongs to the congruence (but does not pass through a singular point), the quartic surface formed by lines of the con- j. 10 146 THE COSINGULAR COMPLEXES [CH. VIII gruence which intersect p, is of a different character ; for, any line y, which meets p, meets only one other generator of this surface, since in the plane (p, y) there is only one other line of the congruence, hence ^ is a triple line of the surface, which is of class XII, possessing a simple directrix and a double generator ; so that p is met by tw.o consecutive lines of the congruence in points P, F' respectively ; and there is no other point of the focal surface upon p except P and P', for if Q were such a point, then through Q would pass three lines of the congruence, viz. p and the two (consecutive) lines through Q; hence p touches the focal surface at P and P', thus the lines of the congruence are hitangents of the focal surface. Lastly, taking any line p through a singular point 8 of the congruence, the lines of the congruence which meet p consist of the pencil through S together with a ruled cubic of which p is the double directrix. The two generators through each point of p meet the single directrix of the ruled cubic in points Q, Q', so that there is determined upon the two directrices, a [1, 2] corre- spondence, which is therefore given by an equation of the form xu + v = (i), where u and v are quadratic expressions in y, the coordinate of a point Q, X being the coordinate of a point P upon p. For points P in which p meets the focal surface, the points Q, Q' coincide, i.e. the quadratic equation (i) has equal roots, this gives two such points P upon p\ from which we learn that, exclusive of S, p meets the focal surface in two points only, hence )Si is a double point of the focal surface. The focal surface, being therefore of the fourth degree, and possessing 16 double points, is a Kummer surface. 120. Confocal congruences. We shall now investigate the complex represented by the equation {Xx") + {Xd") {ax)- - 2 {ax) (Xax) = (I). This complex meets the linear complex A, or {ax) = 0, in the congruence (C=, A); it may be brought into its canonical form in the following manner. Consider the system of complexes A, B^, B.^, B^, B^, B^, where A = laiXi, Bi = XhtiXi, Bs=XbaXi; 119-120] THE COSINGULAR COMPLEXES 147 we take these six complexes as being in involution in pairs and . choose their constants so that lai' = l, 26ii^ = l , lb,^ = l. Then, (Art. 28), we have Xi = ttiA + buBi + h^iB^ + 631-83 + 641-84 + fesi-Bg. We shall now suppose that hu = pk c— r — > where fijc is a root "^ + /** of the equation S — ^ — = ; this ensures the involution between each pair of complexes, since S 7- r^r ; = 0. Taking the six complexes A, B^ as coordinate complexes and substituting for the Xi in (I), this equation assumes the form SXi [aiA + IhkiBkY + {\a-')A^- 2AtXiai [aiA + 26^5*} = 0. i h i h Now since IXihiib^ = 2 (X-i + Ml) Siiiaj - fi^thiib^i = P'P'^ X^, " ^'^ (?W + /^:)(X, + /X,) = 0, the term B^^B^ disappears, similarly for all the product terms;' hence the equation (I) becomes B,^lXi he + B.'tXib^^ + B,' l.Xihe + B,^I,XAe + B,^ %\ihe = ; while since SXi6ii= = 2 (Xj + fj-j) 6,/ - fiiXhe = - fh., &c., the final form of the equation is 1 The complex represented by this equation has been seen to intersect A in the congruence (C^, A). Now, (Art. 83), the singular surface of the complex l;(Xi-X,)a;i^ = 0, 2 is the focal surface for the six congruences x, = 0, 2 ''\ =0, (i+1), A/j — A,i x, = 0, 2^-^ = 0, (i + 6). 10—2 148 THE COSINGULAR COMPLEXES [CH. VIII We conclude therefore, in the first place, that the singular surface of the complex 2^=0 (II), 1 Mi is the focal surface for the congruence A=0, ifikSk" = 0, , 1 i.e. for the congruence {€', A). Next, substituting in the equation of the complex (II) from the identical relation - B,^ =A'' + B^ + ^3= + ^/ + B,\ we obtain as an equivalent form of (II) /^2 /^S H-i M6 therefore the sinsrular surface of %~ =0 is the focal surface of the congruence £, = 0. -^^ + 2— ^^^ .£a= = 0, (A;4=l), &c.; the five congruences confocal with (G^, A) are therefore seen to- be B, = 0, -A' + ^—^^.B,'' = 0, (A;+l), B, = 0, -A"- + t-^^^.B,'=0, (A; + 6); f^t — H'ic Bi? their focal surface being the singular surface of S — = 0, where A = taiXi, B]c = '^huXi, bki = pk-^r—, — > ~i = ^7^ — r — \i' provided that the fik are the solutions of the equation 2-^ = 0. Again consider the complex 2^^ + A'2-^-2AS-^i^^ = (III). This complex clearly contains the congruence (B^, Cfj,^); on substitution for the xi in terms of A, Bi ... B^, the equation assumes the form 120] THE COSINGULAH COMPLEXES 149 - 2A2 --^^- (ai^ + . . . + hiB,) = (IV). A^ + ^l Now since we have' V '^i --OS ^ n (^ + /^i)(A-J + M4;) ' (Xi + /ii)(Xi+/i,)" therefore 2 -, ^ — r^ = provid ed that fj,;,;^ nj^ /j^. Hence it is easily seen that all the terms of the equation (IV) disappear except those involving the squares of B^, B3, B^, B^, and the equation takes the form The complex represented by the latter equation, therefore, contains the congruence (£1 , Gf^') ; but since it is seen that Ha — fJ'i Q^ + /J'i)('K + f^if The equation of the last complex therefore becomes 2-^^=0; and the congruence B, = 0. B,' + t-J'^B,* = 0, 2 ^1 — Mi is identical with the congruence (£1, G^^). But 5,^ + 2-^^ B,' = -A' + i-f^ B„\ 2 IM — fik 2 M'l — P'k therefore by the preceding it follows that the five congruences confocal with {A, G') are the congruences {Bi, G^'') ... (B^, G,!,^). It should be observed that since the congruence (A, (7°) is identical with A = Q, G^+AA' = 0, where A' is any linear complex, it follows that any congruence (2, 2) is contained in 00 ^ quadratic complexes. 150 THE COSINGULAR COMPLEXES [CH. VIII The lines of a congruence (2, 2) which meet any given line I form a ruled quartic of class I ; sixteen of these lines meet any given quartic curve c* ; hence the lines of the congruence which meet c' form a ruled surface whose degree is sixteen. If c* is a section of ^, the focal surface of the congruence, by any plane tt, the two generators of this ruled surface through each point of c* coincide, and the surface degenerates into two coincident surfaces of degree eight. Denoting by S this surface of the eighth degree, it is clear that S touches O along c*, the other curve of inter- section of >S and $ being formed by the second focal points on the generators of S, while S and touch also along this latter curve. Since these two curves of contact form the sole intersection of S and ^ it follows tha,t the order of the latter curve must be twelve. The points of intersection of this curve and tt consist partly of the four points of contact with <1> of the lines of the congruence in tt, and partly of points of contact of such lines as meet in four consecutive points. Hence, there is a curve of order 8 on ^ at each of whose points there is a tangent of '^ which has four-point contact with <1>*. Since a Kummer surface is the focal surface for six congruences (2, 2) it follows that on this surface there are six curves of four- point contact. 121. The quartic surface (C, A, A'). The lines common to a quadratic complex C and two linear complexes A and A', where A and A' are {ax) = 0, {a'x) = 0, respectively, will in general form a ruled quartic of class I, whose double directrices are the common polar lines of A and A'. If this quartic surface splits up into two reguli, since they have two common directrices, they must have also two common generators, (Art. 58); i.e., each belongs to the field of the other ; hence, there is some cosingular complex G^ in which these reguli correspond to a complex cone and complex conic having two lines in common. It follows that the lines , must intersect. V\i-t-/i ^/\i + n The conditions required, therefore, in order that the surface (C, A, A') should consist of two reguli, are \i + /i M + fl Xi-f/X * See Sturm, Liniengeom. Bd. ii. S. 42. 120-122] THE COSINGULAR COMPLEXES 151 hence, for all values of p, -^i^lll^^Q i„ the system of two terms, (Art. 51), determined by A and A', belongs entirely to one Ri. The foregoing condition may also be expressed as follows : — the equation S Vl^C-JiZ = is satisfied for the two values of p which make A + pA' a special complex, i.e. by the common polar lines of A and A' ; hence, the common polar lines of A and A' m,ust belong to the same cosingular complex G^ and he directrices of some regulus of '2^. 122. Projective formation of C^. If three linear com- plexes A, A', A" give a regulus p of G^, we have the six equations «'■ =0, 2^-^^^— =0, 2^^^^— = '=0, 2-^^=0, 2-^^ = they state that the three lines •(1); meet in a point or lie in a plane, thus the complex cone of this point or complex conic of this plane, for G/, corresponds to p. If the first two lines pass through a given point P, then in any given plane tt which does not pass through P, there is one line satisfying the conditions, viz. the intersection with tt of the plane through the two lines. Let p be the regulus of (? which corresponds to the complex cone of P for 0^', and p' that which corresponds to the complex conic of tt for C/, our result states that any two complexes A, A' through p, and any third complex A" through p', intersect in a new regulus of G', provided that the preceding equations are satisfied. Let now the equations of ^, A', A" be XttiXi =olL +/3if +7iV" =0 ] 2a/«i=a'Z +/3'lf +r^'N =0 i (2), 2a{'xi = a"L' + 13" M' + y"N' = J where L = 0, M = 0, iV= are three given linear complexes through p, L'==0,M'=O,N'=O p'. 152 THE COSINGULAR COMPLEXES [CH. VlII Then since p and p' are reguli of 2^, 2/ respectively, the first four above conditions (1) are satisfied independently of the values of a, /S, &c. ; and if A and A' are given, the ratios a" : /3" : 7" are determined by the equations hence, p being a given regulus of 2^ and p of S/, any two com- plexes A, A' through p, determine a third complex A" through p', such that the complexes A, A', A" intersect in a regulus of C, (which belongs to the field of p); this is called the projective forma- tion of C. Each of three complexes L, M, N through the given regulus p contains two undetermined constants; similarly for the complexes L', M', N' ; taking L as likxi, &c., we are therefore at liberty to suppose the following equations to exist between the constants ^ l^mi' _ ^ liUi _ ^ mjlj ^ „ miTii' Xi + fi A^ + M \ + fJ- \ + fi nil ^gil^^o (3)_ The two equations „ ttjCti ^ a^a-i ^ now assume the form aa" 2 r^ + y8/8" 2 ^^^ + 77" 2 -^^ = Aj + /A Af + M Af + /i a a 2,—- — +/3/3 2 r— — + 77 2—— — =0 Xi + /i ^ + M' ^ + /i Eliminating the variable quantities o, /S, 7, &c. between (2) and (4), we obtain the equation of G'^ in the form LL' . MM' NN' .(4). 1(1/ „ mf»rei' ^ WiTif' From the foregoing process we observe that the equation of any quadratic complex O^ may be brought to the form just given, if L, M, iV" are three complexes through a regulus p of 0^, and L', M', N' are three complexes through a regulus p' of O, where p and p' belong respectively to a 2^ and the corresponding 2^', provided that the equations (3) are satisfied. The geometrical significance of these latter equations is that the six lines h mi 'JQn + p.)' ^{Xi + p) , &c. 122-123] THE COSINGULAE COMPLEXES 153 form a tetrahedron. Hence, to get the equation of C^ in the form now obtained for it, we take any tetrahedron and multiply the coordinates of its edges by the quantities V(^^ + A^) ; thus each edge gives rise to one of the six linear complexes L, M, iV, L', M', N'. Hence the equation of 0^ may be brought to this form in oo " ways. The 00 = reguli of l,^ and 2/ respectively, are now seen to be given by the equations L=pM'-- aW, \ ( L' = p,M- - C7,N, a -r, , ,,, T.r « M=tN'-^pL', and i M' = T^H - r piL, N'=<7,L--T^M; c I \ c Wlicic • 1 • — -^A. ■•^^l *~*-vi ' a C Ai + yU. A.^ + Z"' A,i + /i by giving all values to p, a; t, pi, a^, r^. 123. Caporali's Theorem. It has been seen that the equation of any general quadratic complex can be formed by aid of a tetrahedron in the manner described. An application of this method will now be given to prove the theorem of Caporali, that any congruence (2, 2) is contained in 40 tetrahedral complexes*. Taking L, or 2te, as any given linear complex, the equation 2 r-^ — = gives iive values of fi, which correspond to the five Xj + ft pairs of associated reguli of the congruence (C^, L). Take one of these values of p-, say /ii, this determines a line ,, ' — r with which are connected the vertices of the complex cones, (and the planes of the complex conies) connected with this pair of systems of reguli of (C^Z). Through the line .. — r there pass four tangent planes V(.A^ + P') /Si, /Ss, ySs, /S4 to the singular surface; let B^ and 5/ be the centres of pencils of 0" in /3i, B^, 5/ in ^2, &c. (Art. 77), and let I' the line joining Bi and B^ be -.y—^ ; , this determines L'; again of the eight pencils of G" through the points Ai, A^, A,, A^ in * This theorem is due to Caporali, Sui complessi e sulle congruenze di 2° grado, Atti dei Lyncei, (1877-1878). 154 THE COSINGULAR COMPLEXES [CH. VIII which ' . meets the singular surface, each such pencil passes through one of the points B^, B^' ; B^, B^, and so on, i.e. two of these pencils whose centres may be denoted by A-^ and A^ pass through both B^ and B^ (see Table, Art. 77). Hence the lines A^B^, B1A2, AJB^, B^A^ form a twisted quadrilateral formed of lines of O, in which any two which intersect belong to the same pencil of C. Taking these lines respectively as VC^t + Mi)' \/(>a + /Ui)' VC^ + yiii)' VC^^i + Zii)' the quantities mi, rii, ml, nl, must be coordinates of their corre- spondirg lines of C^^^, therefore these latter lines must form, a twisted quadrilateral, in which any two lines which meet, form part of a pencil of G^^^ (Art. 116). Hence the form of equation of C^ derived from the quadri- lateral A^B^A^B^ being LL' MM' NN' ■ + — -rrrr + — ttt- = 0, lik' „ mjmi' „ ntni the intersection of G^ with L = gives the tetrahedral complex MM' FN' ■ + — rrrT-=0- ^ mtrnj ^ riiUi Xi + fj^i \ + fii Now the line ,,^ '' — : may be determined in five ways, and ^/(Ki + fi) I' for each such line there are 24 lines -77;-^^ ; , since the centres of V(Xt + /i) the eight pencils {Bi, ^i), (Bj, ySj) may be joined in 24 ways. Hence the equation of G^ may be written in 5 x 24, or 120 ways, in each of which the result of putting Z = gives a tetra- hedral complex. Moreover any form of equation of C', aLL' + hMM' + cNN' = 0, which complies with this condition, must be derived from one of these 120 tetrahedra; for if mi, ni, m/, n/ are two pairs of opposite edges of the tetrahedron of such a tetrahedral complex, 123-124] THE COSINGULAR COMPLEXES 155 M'- aM + N=0 (i) ^ tM + N'=0 (iii) -<7^N'=0 (ii) ■ and M'-T~N=0 (iv) i = J z = o give two systems of reguli of ((7^ L), and since (i), (ii), (iii), and (iv) taken simultaneously are equivalent to only three equations, giving a regulus p, it follows that any regulus of one system has two lines in common with any regulus of the other, viz. the inter- section of p and L. Hence these two systems are associated, (Art. 118), and correspond to the same value of fi, say /Aj. The complex o-Jlf +iV=0 is special, having for directrix a line of the pencil (m^, Wj); the lines of this pencil, therefore, being directrices of reguli of (C, L) belong to G^^^ and meet '' ; V X, + /ii similarly for the pencils {m(, n{), (mi, n-), (m/, r?i). Hence, as before, the lines V(Xi + /Ai) ' <^(Xi + /ii) ' ^(Xi + /Ai) ' V(^i + fn) form a twisted quadrilateral in which any two lines which inter- sect belong to a pencil of G^, while these four lines all meet the line whose coordinates are k VXj + /Ai ' Again, if aLL' + hMM' -h cNN' = is an equation of the required form, and K, K' are the special complexes whose directrices are the remaining pair of opposite edges of the tetrahedron formed by the lines ?/ij, w^, mi , n/, there exists an identical relation of the form aKK' + fiMM' + -fNN' = ; so that by eliminating in turn MM' and NN' between the last two equations, we derive two other forms of the equation of 0^ of the required type, thus each tetrahedral complex which contains (C, L) gives rise to three of the 120 stated forms of the equation of G^, i.e. there are 40 tetrahedral complexes which contain (C", L). 124. Condition for (1, 1) correspondence in any coordi- nate system. We will now consider the analytical conditions to be satisfied, in order that the equations yi = 1,aikXk may give k a (1, 1) correspondence between cosingular complexes, for any coordinate system. 156 THE COSINGULAR COMPLEXES [CH. VIII The equations 2/, = 2aa:«i (1), where Xi and yi are both coordinates of lines, give rise to two quadratic complexes X"^, Y"^ when we substitute from (1) in to (y) = for the yi in terms of the xi, and similarly for the Xi in ' i.e. if, whenever x meets y', y meets x', the complexes X^, Y" are cosingular. For if P is any point on the singular surface of X^, to the two pencils of X^ through P will correspond two pencils of Y' having a common line, while to the complex cone of ]P through P will correspond a regulus of X'^ such that each line of it meets the above two pencils of Y^; hence this regulus must break up into a pair of pencils, and therefore the complex cone of Y^ through P will consist of two pencils, that is, P is a point in the singular surface of both X^ and Y". If ft) (x) = 'S.OiiiXiXic, by equating the coeflScients of XiXk on each side of (2), we obtain as the necessary conditions r r for all values of i and k. Denoting these expressions by Aik, Au, we have, Aik = Au, and on multiplying the equations (l) by an,..., oje, we obtain 1 9<» _ V ^ Oyi r and the equations (1) are equivalent to the following K- = ^^> ^^^Aij^XiX),, eyi oxi Hence any quadratic expression in the six variables Xi, 124-125] THE COSINGULAR COMPLEXES 157 gives rise to a (1, 1) correspondence between the lines of two cosingular quadratic complexes*. 125. Equation of the complex referred to a special tetrahedron. In any singular tangent plane tt of the singular surface of C" let the poles of the fundamental complexes 0„, C^, Gyhe A, B, respectively. Then through BC, GA, AB there pass singular tangent planes tt^ , ttb, ttc whose intersection D is also a point of the closed system determined by the three complexes Ca, C|3, Gy, (Arts. 26, 61). This point D is therefore a double point of the surface (Art. 82), and the tetrahedron ABGD is such that each vertex is a double point and each face a singular tangent plane of the singular surface. Taking this tetrahedron as the one of reference, the equation of the complex assumes the form A^ + 2Lp,,p^ + 2Mp,,p,, + INpuP^ = 0. For, since all the lines of the complex in a singular plane of the Kummer surface form one pencil, (Art. 80), and similarly all those through a double point of the surface, hence, if the equation of the complex be fiPn, Pis, Pu, P23, P24, P4i) = 0, these eight conditions reduce / to a perfect square, save as to terms J^PiiPsi + Mp^s^pi^ + Np^ip^ ; for, let the centres of the pencils in the coordinate planes be Pi, P2, P3 and P^, respectively, and the planes of the pencils through the vertices ttj, tt^, ttj and tt^, then, since the points A^, Pj, P3, P4 lie in ttji, the line A^Pi meets P3P4; hence the four lines A^A^, AgA^, P1P2, P3P4 are intersected by A^P^, similarly they are intersected by A1P2, A^P^, A^P^ ; hence they belong to the same regulus. There is therefore one linear complex A in which AiA^, P3P4 and -4-3.44, P^P^ are two pairs of polar lines; each of the foregoing eight pencils of /belongs to A. By identifying the polar plane for A of each vertex Ai with the two (coincident) planes which form the complex cone of Ai for /, and proceeding similarly for the coordinate planes, it easily follows that /= J^ + 2Lpi2Pu + ^MpisPii + 2NpuP2s- * It is easily seen that no (1, 1) correspondence which is not thus formed can lead to two cosingular complexes. . 158 THE COSINGULAR COMPLEXES [CH. VIII 126. The complex A is of the form (h. (Pi2 - 2jo«) + ttj (p,3 - /3p42) + «6 ipu - yp^a) = 0, but we may take such multiples of the point-coordinates as will make a = jS = 7=l, (Art. 81), this will not affect the form of the invariable relation ^12^34 + ^13^42 +^14^23 = 0. The equation of the complex now assumes the form f{x) = — {a^, + a^Xi + UeSCeY + L {x,' + X,') + M {x," + x^) + N {x,' + ^/) = . . .(I.), where, as usual, a;i=pi2+p34, ix2=Pn~p3i, &c., with reference to the coordinate system now adopted. The equations which connect a line x of this complex with a line y of a cosingular complex are the following : — 3/2 = ax^ +fx^ + eXi , 2/1 = «i V(-^ +p)] yt=fx, + bxi + dxe, i/s = x,^/(M+p) I (1), ys= ex^ + dxi+cx^, y^ = x^^J{N + p) ] where L + p-a^^ = a?+p + e^, - a^a^ = af+fh + ed, ] M + p-a.'^p + b^ + d'', -a,a, = ef+bd + cd, i ...(2). N + p — as^ = e^ + d^ + c', — a^a^ = ae +fd + ce. j For this transformation (1) gives, as has been seen, (Art. 124), two cosingular complexes ; also 2 {yf) =f{x) + p2 {x{'), while iix^)^iK.y,+K.y,+K,y,Y+y^+yp-yi+y^...^ii,y, provided tbat values can be found for K^, Ki, K^ which make coexistent the equations A^ fe + j^) = A' + F^ + E\ A'K,K, = AF+FB + ED, A' (k," + 2gir-) = F' + B' + D^ A'K,K, = EF+BD + CD, A' (k,' + -^^ = ^^ + i)^ + G\ A'K,K, =AE + FD + GE, , (3); a f e where A is fid e d c Now it is easily found from (2) that EF + BD + CD and A, B, &c., its first minors. L + p = -' ef+ bd + cd 126-127] THE COSINGULAR COMPLEXES 159 and from the last equations, that L + p~ EF + BD + GD' hence the equations (3) aie coexistent. To determine the manner in which p enters into the coeffi- cients -STa, Ki, Kg, we notice that (II) becomes, for six values of p, the square of a linear function of the yi, since the six fundamental complexes taken doubly form part of a cosingular system ; this can only happen if ^2) i^i> -^s become infinite together, or if K^ is infinite for p + X = 0, K^ for p + if = 0, Z'e for p + iV= ; also p = oc gives the given complex (I). All these conditions are satisfied by the form {p-p,)(p-p,){p-p,) j yi' + y; y/ + .y/ , y^' + y^' \^n '^(p + L){p + M)(p + N) [p + L ^ P + M '^ P + N] • where , {L + p,){L + p,)iL + p,) ,_ {M+p,){M+p,)(M+p,) "^ " {M-L){N-L) ' ' {L-M){N-M) ,jI[+_pMI±PsM+pA {L-N){M-N) This is the form of the equation of the cosingular complexes referred to the stated tetrahedron. 127. Involution of tangent linear complexes. The tangent linear complexes of any line I, with regard to the four cosingular complexes through it, are mutually in involution; e.g. for the complexes corresponding to /Xj and fx^ the tangent linear complexes are and since we have by subtraction (Xi + fj,-^) {Xi + p,^ hence the two complexes are in involution. Moreover the involution determined on I by one pair of these complexes, e.g. G^^ and C^^, is the same as that determined by the other pair, G^^ and G^^\ for the double points of the first »J60 THE COSINGULAK COMPLEXES [CH. VIII involution are the points in which the two lines - — 1- k \i + fjh. \ + /J^i meet i, while the double points of the second involution are the intersections of the two lines — 1- k' - — - — with I ; and, since \i + ylts Xt + /44 each of the first two lines meets each of the second, it follows that the double points of each involution are the vertices of this twisted quadrilateral which lie on I. A tangent plane through I to 4> will contain a pencil, to which I belongs, of each of these four cosingular complexes; denoting them by Ci", C^, C^, and G^ and the tangent linear complexes of I for them by T^, T^, T^, T^ respectively, the pencil which belongs to C^ also belongs to T, , and so on ; if the four points in which I meets are A-^, A^, A^, A^, and the four tangent planes through it to $ are /Sj, /Sj, /Ss and ySi, the notation of the points A may be so arranged that (^1, /3i) belongs to T, and C^', (A, A) T.andCA (^,/80 T,andC73^ (^4, ySi) TtavdC,". Now let ^2 he that plane whose pole for T^ is A^, then will A^ be its pole in T^ (since T^ and T^ are in involution), also At is its pole for T3 and ^3 for 2!,; similarly for /Sg and ySi, and we have the following scheme of pencils which belong respectively to the four complexes Of^, G-' {Au A) {A,, A) (^3, /8s)' (^4, /S4), Gi {A„ A) (A, /SO {A,, A) (^3, /S^), C,' {A„ /3i) {A„ /3.) {Au /83) {A,, /3,), c/ (^.,A) (^3, /30 {A„ A) {A,, /3.). 128. The lines common to three of the cosingular complexes which pass through I, form a ruled surface R^^ of the 16th degree which passes through I; for the lines of this surface which meet any line h are given by the equations 2^:^ = 0, 2^-^ = 0, 2-^ = 0, (^) = 0, {hx) = 0, and are therefore 16 in number. 127-129] THE COSINGULAE COMPLEXES 161 Corresponding tangent linear complexes are 2 -^*-^=0, £-^^^ = 0, 2-^i^ = o (i); "•i + /"■! ^i + 1^-2 A,i + fl^ if these equations are taken simultaneously, they determine a regulus which contains I and the line consecutive to I in R^,^, i.e. the regulus touches R-i^ along I; now the tangent planes of this regulus along I are determined by / itself and the directrices of the special complexes of the ' system of three terms ' determined loc- by (i); but the complex S '' '' =<> is in involution with this Ki + fLi system, hence these directrices belong to the latter complex, (Art. 58), tlierefore, the surface of intersection of three cosingular complexes G-^, C^, G," which contain a given line I is touched along I by the complex cones of the points of I for 0/. 129. Conies determined in a plane by cosingular complexes. The cosingular complexes determine in any given plane a sj'stem of conies; four of these conies can be drawn to touch any line in the plane, since four complexes pass through the line. If in the equation S - — - — = 0, we write Xi = a; + kL, we obtain \i + fl X; + /A Xj + /4 If, now, (a, b) is a point on the complex conic corresponding to fi, the roots of this equation in k must be equal, i.e. 2 -^ 2 -^ - fs -^y = 0. Xi + M Ai + /it V Xi + /i/ Let Pik^io-ibk — akbiYi tben we easily see that 1Pi!c = 0, since k Xakbk = 0, and the equation to determine /j, may be written Pik = 0. In this equation the coefficient of fi^ is 22Piii; which is zero, and the coefficient of fi^ is 22PiA; jSX-(Xi + Xi)}, which- is also i k zero ; thus the equation for fi reduces to a quadratic ; hence, through any point of the given plane there pass two conies of the system. J. 11 162 THE COSINGULAR COMPLEXES [CH. VIII 130. Elliptic coordinates of a line. The parameters jj, of the four cosingular complexes of G^ which pass through a line y may be taken as the coordinates of this line*, and if we have for since /'(-M /'(-^O /'(-^) /'(-^) = ^rPx.) = «" ^^' it is easily verified that y belongs to the complexes corresponding to fj^, fii, 1X3, and /l^, where the latter quantities are the roots of A^t + M we shall denote these complexes by G^-, G^, G^, Gi. The quantities , are coordinates of the singular lines of C^ which v'/'(-X,) satisfy the equations (X'.r2)=0 ; (\*.v'')=0, (Art. 80), i.e. the singular lines of the third order. Taking one of the four complexes as G^, which corresponds to jjL = 00 , (Art. 115), the lines x of G^ are given by the equations and the singular lines of G' by the equations p../=^^y^)^y^'> ("^)' for the lines determined thereby satisfy the equations (\ar^) = 0, (Va;") = 0, for all values of fi^ and fj^. The lines p.x^ = jri — \'^ satisfy also the equation (X.'a;'') = 0, and hence are the tangents to the principal tangent curve deter- mined by C on $, (Art. 80), i.e. the singular lines of the second order. If in equations (i), (x^ and fi^ have givm. values, and fi^ = ii,i = li, we obtain the 32 pencils of lines ..,,= ±(X. + ^)/:5^i|p) (iv); * This method is due to Klein, see Math. Ann. n. 130] THE COSINGULAK COMPLEXES 163 these pencils are the tangent hnes of 4> at the points of contact of the 32 singular lines v/ i.e. a tangent of is (\i + fj.)xi, where Xi is a singular, line of 0^ The quantities /j.^ and /ij may be regarded as determining a point on $. If in (iv) ^1 be taken a.? constant and ^nj vary, the pencils of lines y belong to Cj^, and touch ^, but are not singular lines of Ci^, hence they are the tangents to at the points of the principal tangent curve whose tangents are singular lines of Gi^ of the second order ; these latter tangents are obtained by putting yu. = /Xj ; hence, the tangents to the principal tangent curve of related to the complex C^^ are obtained by putting /Ji,3 = fi4 = /Xi and varying fx^: taking /j.i = constant, and (1^ = fis = fJ'4, gives the other principal tangents of ^ at the points of this curve. If X and x' are the singular lines of 0" at the points of contact P and P' of a bitangent line of 4>, Xk = x'tc except for one value of k, say i, for which Xi = — xl, (Art. 83) ; hence X]^ = x'j^, and the values of ytti and /x^ in (iii) are the same for each singular line ; but the tangents yi at P being given by the equations /) . 2/^ = (Xi + t^J (A-i + y^ (Xi + A'2)//' (- A.i). we see that P lies on the principal tangent curve of Gy and also on that of G^, and the same holds for P', so that these points P and P' lie on the same two principal tangent curves. When two pairs of values of /x in (i) are equal, e.g. yi^ = yi^, lii = fi^yX is a singular line of both the complexes C^ and G^, and would therefore seem to be a bitangent line of ^, but in this case (Xj + Ml) Oh + M2) . whence x meets the three lines 1 A.^ A-i V/'(-x,)' V/'(-x,)' V/'(-x7) which themselves belong either to a sheaf or to a plane pencil; hence x is any line in a singular tangent plane or through a double point of $. 11—2 164 THE COSINGULAE COMPLEXES [CH. VIII The equation (dy^) = 0, when expressed in terms of the coordinates /t; of the line y, becomes ^ Too ^^^ / W ^ ^ o (M3 - Ml) (/^ - Mi) (/^3 - /^O _^ 7 ., (M4-Mi) (M4-M2) (M4 - Ms) ^ Q /(Ms) * /(M4) If /lig and fjii are constant, we have as the differential equation of the curves of the congruence {G^, (7/) V /(Ml) V /(/ia) if fj^=fii, the differential equation of the curves whose tangents are singular lines of C,", is seen to be t^Mi (Mi - Ma) ^ ^Ma (m^ " Ms) v/cmT) v/oir) 131. Bitangent linear complexes. Of the 00^ tangent linear complexes l,(\i + fi)xiyi = 0, of C'^ which have in common with 0" not only x but all lines of C^ consecutive to x, there are six which are bitangent, viz. those obtained by writing successively Xi + M = 0, Xs + /i = 0. For the complex \-\ "kVi + >^3 - ^-1 ^3 2/3 + X4 - \i Wtyt + \-\xsys + \e-\ oc-ey^ = " touches " C^ both in x and in the line all of whose coordinates are the same as x except x^, i.e. the polar of x for the complex x^ = 0: denote these six complexes by T^, T^, T3, T^, T^, Tg. If A be any linear complex, (ax) = 0, it will have a pair of polar lines z in common with 2\, of which the coordinates are given by the equations cr.Zi = imi, a .Z2 = (Xj — Xi) jTj + fia^, a-.Zi = (\a — \)X3 + fMi, cr .Zi = (X4 — Xi) Xi + fjMi, a-.Ze = (\e — Xj) Xg + fm^ where fi has either of the values obtained by expressing that {z"^) = 0. The locus of the lines ^ is a quadratic complex S-^^ ; for, eliminating the Xi from these equations we find Xg Xj X3 Xl X4 Xq (i); 130-132] THE COSINGULAE COMPLEXES 165 We obtain in this way six quadratic complexes Si", which we shall prove to be cosingular*. having the focal surface of (C", A) as singular surface. For, from the equations (i) we deduce '^'^"aa^"" '' X-\ ^^'^" (* = 2, 3, ... 6), hence ^o- _± = _ o-idi . -^ — —^ = - a^XaiXi = ai^Xi, if "^aiXi = 0. So that if X helongs to A, ~- is a line, viz. x, and z is a singular line of /Si". Hence a; is a tangent to the singular surface of (S/ at the point of contact of the singular line z. The same thing applies to the other line, z', associated with x by the equations (i). Therefore, any line x of the congruence (G", A) is a bitangent of the singular surface of /Si". This result holds, similarly, for the other five complexes O2 , ... Os . Hence these complexes have as their common singular surface the focal surface of the congruence {C^, A). 132. Principal Surfaces f. The tangent linear complexes of a line x which belongs to a complex F= of degree n, being (2//) + /.(2/a;) = 0, (Art. 74) (i), riJP where fi= ;^- . the tangent complex corresponding to a consecutive CXi line x + dx is .{y(df+fidx + xdfi,)} + {y(f+fix)}=:0 (ii), where the dxi are connected by the equations ixdx) = 0, {fdx) = 0, (kdx) = 0, in which (kx) = 1, (the quantities ki being constant), is an equa- tion arbitrarily assumed between the coordinates Xi of any line|. If the complexes (i) and (ii) are the same, we must have that dfi + fjidxi + Xidfi = dt ifi + fjLXi) (iii). * This theorem was communicated to the author by Mr J. H. Grace. t These surfaces are of interest from their analogy with lines of curvature in four dimensions ; see Art. 228. J This method is due to Voss, see Math. Ann. ix. "Ueber Gomplexe und Congruenzen." 166 THE COSINGULAR COMPLEXES [CH. VIII This gives nine equations between dos^, ,dxg, d/ji, dt; but since {adf) = 0^ they are equivalent to eight equations. Eliminating the differentials we obtain /u + M /l2 /l3 /l4 /l5 /l6 «! /l+/^l /ei /62 /63 y64 /65 /eo + /* *6 /e + /^6 /t/j /t2 fC^ tC^ A/g n/g U V Now multipl)ring the last column hy n—1 and subtracting from it the first six columns multiplied respectively hy x^, ...Xg and the seventh hy /i(n — 2), we find /u + /* fw /is Ju JlC /l6 /ei /62 /es /64 y6B yee + M ^6 "'1 '*'2 "'3 "'4 '*'6 As — {hx) «;« = 0; or, finally, /ii + A' /la /is /i4 /i6 /i6 X, fes + fJ' Xq = 0. This is an equation to determine /t which is of the third degree, since the coefficient of /u.* is equal to n{n—\)F, which is zero ; hence, there are in general three finite and distinct values of ft, say /ii, /ij, Aij ; each of them gives one set of values of dxi, hence there are three lines of F consecutive to x for each of which one tangent linear complex is the same as one of x. Thus starting from X we may proceed to the one of the three consecutive lines which corresponds to /i^ and then from that line to the one which corresponds to fj^ + d/yL.^, and so on ; thus we have a singly infinite set of lines forming a ruled surface ; such a surface is called a Principal Surface of the Complex : in each line x oi F three Principal Surfaces intersect. In the case of the quadratic complex C, or (\ar'): determinant for fi is = 0, the 132-133] THE COSINGULAR COMPLEXES 167 \ + fi X^ + fi ... = 0, i.e. 2 \ + fi = 0. ^6 + M Xg a.'6 Hence the values of fi are those which give the three complexes through x cosingular to C' ; so that if dx/ are the increments of the Xi corresponding to /Xj, dx" corresponding to fi^, dxi" corresponding to /A3, the equations (iii) become in this case dx^+P^ = dt.Xi (iv), \ + IJ-i. hence A-i + /Aj (?^i + Ml)(>.l+/t2) = dtX \i + fii' therefore = 0, similarly S t: — ; — = 0, i.e. the line Xi + dxl belongs to C^^^ and G^j,^ ; hence the Principal Surfaces for C^ are the ruled surfaces Ri^, R23, -Rg,; i.e. the inter- sections of C' with two of the three quadratic complexes through x ' cosingular with C\ •133. Involutory position of two lines. It is to be noticed that from equations (iv) it follows that {dx'dx")=Q, {dx"dx"') = 0, {da;"'dx') = 0: we shall show that this holds for any complex. In equations (iii) let the line x be taken as the edge A^Ai of the tetrahedron of reference, and let the equation {kdx)=Q be dxg=0, then since A^A^ has the coordinates (0, 0, 0, 0, 1, —i), the equation {xdx) = becomes da!^-ida;ii=0, hence dx^^O. In the present case, therefore, the equations (iii) take the form dfi+ij.dx^=dt.fi, df2+tidx2=dt.f2, dfg+fidx^=dt.f^, df^+|ldx^ = dt.f^, or, (/ji 4- /t) c^i +/12 dx^ +/13 <^3 +fu d^i =dt-fi,\ fu dx-^ +/24 dx^ 4-/34 dx^ + (/44 + v) dXi = dt.fi, together with /i dx^ -f/j dx^ +fs dx^ 4-/4 dx^ = 0. These equations show that --=j , -jj , -r- dx^ ~dt are proportional to the coordinates of a point in which a quadric of the pencil touches the plane /lll+/2&+A^3+/4l4 = 0- 168 THE COSINGULAR COMPLEXES [CH. VIII It is easily seen that there are three such quadrics* and that any two points of contact are conjugate with regard to any quadric of the pencil, and hence with regard to ti^+i^^ + ^s^+^t^^O; therefore (cfo'(fo;")=0, {(ix"dx"') = Q, (dx"'dx')=0. To determine the property expressed by these equations we notice that any plane tt through A^A^, having coordinates (0, n^, ttj, 0), meets the line joining the points qj and ^j in the point a, + X/3i, where 7r2fl2 + 'r3a3 _ and meets the line joining oj' to ft' in the point aj'+juft', where _ _ ^rgQg +7r3a3 then, eliminating n-j and ttj we obtain a relation between X and fi which is symmetrical if fi^a^' - ^,02' + asft' - a2ft' = 0. The last equation may be written 02 + a3)02' + a3')-(/S2-O3)(/32'-a3') + 03-°2)(/33'-a2')-03 + O2)O3' + «2')=0. Now if the points a^ and 0/ are consecutive to A^ and the points ft and ft' to A^, then taking ^ii=aift— a^ft, Xi=Pii+P3t! ^^- i it is clear that <^12 = ^2. '^?'34 = ''3> ^Pl3 = P3' <^?'42= " "21 &"., and the last equation becomes dx^ dxj" + dx^ dx^ 4- dx^ dx^ + dx^ dx^ = ; this equation is therefore the condition that the planes through the line x should meet the two lines x + dx, x+dx' in pairs of points ultimately forming an involution. The two lines consecutive to x are then said to have an Involutory position with regard to each other. ' Since the sectiona of the quadrics f/+XF=0 by any plane form a pencil of conies through the four points in which the curve (17, V) meets the plane; and three of these oonics consist of a pair of Uoes. Moreover the diagonals of the complete quadrilateral formed by the four points form a self-conjugate triangle with reference to any conic through the four points. CHAPTER IX. POLAR LINES, POINTS, AND PLANES. 134. Polar lines. The polar line V of a line I with reference to a complex C,^, cosingular to G^, has coordinates li — k - — ^- — . Xj + /[i ■ (Art. 79), where 7 2 2S- .7.. ^» + M I' coincides with I for values of /i which make ^ = 0, i.e. for the four values given by ^ '" =0. Regarding fi as variable, the lines V generate a ruled surface, on which I is therefore a fourfold line ; to find the degree of this surface, we determine the number of lines V which meet a given line a, i.e. for which {aU) = 0, or, for which if I and a intersect this equation reduces to kl -^L = 0. Hence there are eight values of /j,, viz. the four obtained from the equation k = 0, and the four from Sr—^* =0; the degree of the surface is therefore eight. An exceptional case is that of a line which is common to four fundamental complexes, e.g. for either of the lines 170 POLAR LINES, POINTS, AND PLANES [CH. IX the polar of one of these lines, with regard to any complex included in the series {kx^}=0, is the other ; these are the only lines, in the case of the general complex, for which the polar relationship is reciprocal. There are nine lines I for which I' is the polar with regard to C^ ; for since pk' =li{l — kXt), where k = \. , the values of k are those given by the equation 7' .2 and are nine in number. These lines form a closed system ; for any one of them being given, I' is determined, and hence the other eight lines I. 135. Between the ten lines consisting of I' and the nine lines I for which /' is the polar for a quadratic complex the following remarkable relation exists*. In any quadratic complex (?„ ^ or S r — - — = 0, the lines x for A^ + /*! which a given line F is the polar for (7^,°, are afforded by the equations k or, i{ fjL = fii — k, p ■ k^ (\ + fh) = «i C^t + m)- The equation to determine u. is S ,^ — ^^r- = 0. which is of the tenth degree, and of which one root is /i, ; let the others be denoted by fi^, fi^, ... /[tjo, then between l^ and the nine lines l^ ... l^, of which l^ is the polar for C^,", we have the equations p . k^ (\i + /*:) = k^ {\i + fl2) = y" {\i + fH)=...=li^OH + Atio). Again starting with l^^, we find as the nine lines of which l^^ is the polar for C^, those given by the equations where fi is one of the roots of the equation ^- 0^ + p.y -^' an equation of the tenth degree of which one root is /ij. Inserting in this equation for Z/^ (Xi + fj^), its value we obtain, to determine /i, the same equation as before. * This theorem is due to W. Stahl ; Crelle's Journal (1883). ^•^»' = ^»(i-^T^)' 134-137] POLAR LINES, POINTS, AND PLANES 171 fw Hence we conclude that there are 10 complexes G^^^, ... 0, cosingular with C^ such that for G^^ the line l^ is the polar of Z", i™, ... l^ ; ... G^: /" l\ ;m ... F; and so on ; and having given any line I and any quadratic complex, nine other lines, and nine other cosingular complexes are determined having these relations. 136. Corresponding loci of polar lines. From the con- nexion between the coordinates of a line x and its polar x for (7^ it is clear that when x' describes a complex of degree n, x describes a complex of degree Svi, since «/ is proportional to a cubic expression in the coordinates xi. If x' belongs to the linear complex (ay) = 0, the locus of x is {ax) — k (Kax) = 0, i.e. (ax) (Va!^) - 2 (\x^) (\ax) = 0, a cubic complex, to which the singular lines of G^ belong. 137. If X describes a plane pencil, its polar x' describes a ruled cubic. For let Xi = ai + fjhi, where a and b are the lines of G' in the given pencil, then pxi' = a; + /tt^i — Jc\i (at + /xbi), hence writing (X'a') = A, (X'b') = B, (Va6) = 0, (\ab) = B, we have Vi) -k{A + 2fiG + fi'B) = 0, therefore , at the points of contact of x and y respectively. The lines Xx + fix, Xy + jiy only meet for /i = 00 , or ^ = k ; they determine on PO^ an involution, of which P and Oj are the double points. The lines Xx + fix, Xy + fi'y will meet on POi provided that {fi + fi) (Xxy) + (X^xy) = 0, or fi+fi= 2k. If such a point of intersection lies on , and v is the singular line associated with the point, we have TVi = p {XiX{ + fiXi) + a- {Xiyi + H''yi) + %> where u is the line POi; together with the condition that Ui, XiXi + fiXi and Xiyi + f^'yi belong to the linear complex Xv ; (which involves that this linear complex should be special and therefore (XV) = 0) ; therefore we have for the determination of the lines v, i.e. the singular lines associated with the respective points in which POi meets <&, the equations fi+ fi'= 2k, p (X'xu) + a- (X^yu) + (Xw^) = 0, p (XV) + a- {(X'xy) +(fi + fi') {X'^xy) + fifi (Xxy)} + (X^xu) = 0, p {(X'xy) + (fi + fi') (X^xy) + fifi (Xxy)} + o" (xy) + (X'yu) = 0. Since these equations are symmetrical with regard to fi and /i, it follows that if Xx + fix, Xy + fi'y meet in a point Ay of <&, then 176 POLAR LINES, POINTS, AND PLANES [CH. IX will \x + fix, \y + fj,y meet in a point A^ of ^; the lines v for these points will have the same pair of values for p and a-, and differ only through the interchange of fi and //. Now the lines \x + fjjD, \x + kx, \x-\-fjfx, x form a harmonic pencil, since fi + fj! = 2k, therefore 2 1,1 P0,~ PA^'^ PA,' in a precisely similar manner we obtain 2 1 1 PO, ~ PA, "*■ PA, ' therefore A. = ^ + X_^ + ^ + J^^ , i.e., the point Oj lies on the third polar (polar plane) of P with regard to $; and this being true for each point 0, we see that the polar plane of P with regard to (P is the third polar of P with regard to 4>. 141. The four singular lines which lie in any plane tt are connected by an equation similar in form to (i), and in that case the lines a, /3, 7 are the diagonals of the complete quadrilateral formed by them. The equations (ii), (Art. 140), again hold; the first three of them assert that a, /8,'7 form a self-conjugate triangle with regard to the complex conic of their plane, the latter three show that the polar lines for C of a, /S, 7 are concurrent. The point in which these polar lines meet is called the pole of tt with regard to C ; by duality, this point is seen to be the third polar, (polar point), of the plane tt with regard to ^, considered as a surface of the fourth class. The point P for which ir is the polar plane is 7iot the pole of tt ; cor- responding to any plane ir there are 11 points for which n is the polar plane, viz. the 27 intersections of the first polars with regard to * of any three points of n diminished by the number of double points of *, i.e. 27 — 16=11. In the present case from consideration of the equations (ua) = 0, {\au) = 0, (u") = 0, {u^) = 0, (X/3m) = 0, (a/8) = 0; if the vertices of the triangle formed by a, /3, 7 be denoted~hy A, B and C, the polar line of ir with regard to the complex cone of A is the polar line of a for O, and similarly for the complex cones of B and G. 140-143] POLAR LINES, POINTS, AND PLANES 177 142. The diameters of the complex. The polars with regard to C of lines in the plane at infinity are called diameters, (Art. 79). If P is any point on a line I, the polar plane tt oi I with regard to the complex cone of P passes through I', the polar of I for C" ; hence if I lie in any plane e, the polar line a for e with regard to the complex cone of P lies in v, i.e. meets I'. If 6 is the plane at infinity « is the axis of a complex cylinder, and we have, since l' becomes a diameter d, every diameter d is met by the axes of all complex cylinders which meet the line 8 to which d is polar, {i.e. which are parallel to the direction determined by h). 143. The Centre of the comiplex. It has been seen, (Art. 141), that in any plane ir there is one triangle, self-^conjugate for the complex conic of C^ in tt, and such that the polar lines for & of its sides intersect in one point, the pole of ir for G^. Taking the tetrahedron thus formed as that of reference, the pole of TT being Ai, it is easy to see that the conditions just stated cause the equation of G^ to assume the form aiiPSi + aispSi + (hiP\i + a^p\ + a^p^ + a^i^^a + 2X^12^34 + 2 Jfpi3P42 + ^NpuP^ + 2E^,4JB42 + 2%2i334 + '^Tp^Pu = ; for the method of Art. 79 shows that if p'ijc is the polar line oi pik for any quadratic complex /( pi*) = 0, ^'ii is given by the equations f)f P -p'vi = icpi^ + i ST" > i^C., opu and expressing that A^A^ is the polar of AiA^, &c., we obtain the form stated. Hence, if kkis any line in the plane a^, since ^14 = ^42 = ^34 = 0, its polar line I'ik is given by the equations p . I' 12 = kIu + Llii, p . I'ls = kIis + Mli3 , p . Z'i4 = a^ilns) /•-. p.ls4= 0^2^12' P • ' 42 = ''is'lS; P . t 23 = "^'23 + -" '23 J These equations do not involve the coefficients R, S, T; hence the polar of any line of the plane system at is the same for 00 ^ quadratic complexes. The equation a23?'^23 + cinP'is + a^pSi = 0, is satisfied by the tangents of the complex conic c" of O4 ; if the lines I, m are conjugate for c^, then, since I — jMm, I + jim are tangents of c^, we have aj^m^s + ciis^ismis + aiJi^mi^ = . . . .- (ii) ; if, in addition, their polar lines I', m intersect each other, we have from (i) that NaJ,^m^ + i/aia^iaJTiio + LaJ,-am-,^ = (iii), J. 12 178 POLAR LINES, POINTS, AND PLANES [CH. IX hence, there is one line m, conjugate to I for c", such that the polars of I and m intersect. From equations (ii) and (iii) we derive {k + N) aj^m^ + {k + M) a^slisniis + {k + L) a^Jf^^m^^ = 0, i.e. I'liin'si + I'iam'ti + I'^m'^ = 0, but this is the condition, (Art. 4), that the plane through Ai and V should contain the point in which m' meets a^, hence, the plane {V, m) passes through At. The polars for C^ of the lines of the pencil (I, m) form a ruled cubic /a', (Art. 137), whose simple directrix is n the polar line of the point (Z, m) for c^ ; its double directrix d being the polar line of the plane {I, m) for the complex cone of the point (I, in). Let A and B be the points in which n meets c" ; then upon n two involutions are determined, one consisting of points P, Q which divide AB harmonically, the other of the pairs of points P, P'; Q, Q', &c., in which the two generators through the points of d meet n. Now A and B are the double points of the first involution and they form a pair of conjugate points in the second ; hence the involutions are harmonic, (Introd. v.). It follows that P', Q' divide AB harmonically; hence, if the plane of the gene- rators through P and P' meets d in L, while the plane of the generators through Q and Q' meets d in L', the points L, 11 are conjugate points of an involution upon d. If p, q are the generators of p^ through P and Q, the planes (p, n), {q, n) therefore form a pair in an involution of planes ; the double planes of this involution are «< and the plane {At, n) ; hence, the planes (p, n), (q, n) are harmonically divided by the planes Ui and {At, n). Two such polar lines p, q, will be called conjugate. If «! is taken as the plane at infinity, its pole for C is termed by Pltlcker the Centre of the complex ; and it follows from what has just been seen, that if p, q are two conjugate diameters, the planes through p parallel to q and through q parallel to p are equidistant from the centre of the complex ; and if p, q, r are three mutually conjugate diameters, {i.e., meeting the plane at infinity in points which form a self-conjugate triangle for its complex conic), the centre of the parallelepiped, which has p, q, r for non-inter- secting edges, is the centre of the complex. CHAPTER X. REPRESENTATION OF A COMPLEX BY THE POINTS OF THREE-DIMENSIONAL SPACE. 144. A (1, 1) correspondence can be established between the lines of a quadratic complex C^ and the points of space by aid of formulae due to Klein and Nother*. For if the edges A-^A,^, A-i^A^ of the tetrahedron of reference be taken to be two lines of a pencil of lines of 0^ the equation of C^ assumes the form i5i26+i'i3a + ^ = 0, where a and h are linear functions, and <^ a quadratic function, of the four other line coordinates. We may therefore write i'-Pi4=(a«3- iai^x-i., V ■Pii = {axs — bx^ x^ , v .pii = {aXi — hx^ Xs, v.'p^ = {ax3-hx^)Xi, 1/ . Pi2 = (/) . ^2 — axiXi, v.pi3 = -X3 + bxT^Xi; where in a, b and <^ we suppose p^ replaced by a;,, pi^ by x^, &c. The coordinates pne will then satisfy the identity PiiPu + PiiPii + PuPw = 0, and the equation of the complex. It is seen that each line of the complex defines in general one point Xi, and vice versd. To the lines of a congruence (L, C^) will correspond the points of the cubic surface 4 (ax, - bx^) 2 kxi + ls{24 + UiPu = 0, hence to the complex conies of A correspond conies in planes which pass through A^. * This result was discovered by Caporali from the above considerations. 182 POINT-EEPEESENTATION OF A COMPLEX [CH. X 145. The reguli of a congruence \jl, fl^- To the lines of a congruence (L, C-) correspond the points of a cubic surface a^; of the 27 lines of this cubic surface, one, r, is the common generator of p and cr*, this line r is met by 10 others pi, ... ^lo*, each of these lines Pi meets p a second time and therefore meets g° once ; every plane section of o^ through one of these lines consists of the line and a conic which meets 5' four times ; these 10 systems of 00 '- conies are therefore the 'images' of the 10 systems of 00' reguli of (i, C-). Each of the remaining 16 lines meets p twice, in points which do not lie on r, i.e. in points of g°; it therefore corresponds to one of the 16 pencils of the congruence. The lines pi form five pairs of intersecting lines ; if p, p' are such a pair, since the point (p, p') lies on o-', the conies in any Fig. 8. two planes through p and p' respectively have two points in common, and therefore are the images of reguli of two associated systems of (Z, G^), (Art. 118), one system belonging to a 2^ and the other to the connected 2/. If P, P' are the points outside r in which p and p' meet p, the line PP' is a generator of p (since it contains three points of p), hence P and P' are conjugate points of q^. Two reguli of the same triplex 1,^ belong to the same linear complex A, viz. that which corresponds to the line -y=^^= joining the vertices of the cones (or planes of the conies) which correspond to the two reguli in G/, (Art. 116); hence the two reguli belong to the same system of reguli of (A, G% it follows that the images of the two reguli are conies in planes through the same line p, i.e. * Salmon, Geom. of Three Dimensions, 3rd Ed., p. 465. 145-146] POINT-REPRESENTATION OF A COMPLEX 183 the planes of these conies pass through the same point P of q^; also the conies in planes through p', i.e. whose planes pass through P', are the images of the associated system of reguli of {A, C). Thus, in each of the oo ' planes through any point P of q^, the ao ' conies through the other four points of q^ form the images of the 00 ' reguli of a triplex 2^, while the eonics arising similarly from the conjugate point P' are the images of the reguli of 2/. Since the singular lines are the complete intersection of C^ with a quadratic complex, it follows that any pencil of C contains two, any regulus of G' four, and any ruled cubic surface of C six singular lines. Hence, of the locus in S corresponding to the singular lines, any line in S contains six points, therefore this surface is of the sixth degree. The surface has q^ as double curve, for to each point P of q^ there cori'espond two singular lines, viz. those in the pencils which correspond to P. That the fundamental curve in S is of the fifth degree may be seen directly as follows : — a linear complex A, or Sa,i^jt = 0, through the funda- mental pencil, gives rise, by aid of the (1, 1) correspondence of cosingular complexes, to a special linear complex A' with directrix a' through the pencil (P, tt) corresponding to the fundamental pencil in a cosingular complex CJ^^ . and there are five pencils of the congruence (C/j/', a') which have one line in common with (P, n), e.g. if a' passes through P the pencils are the other pencil of (C^^ a') through P together with four pencils in tangent planes to the singular surface through a ; hence there are five pencils of (C^, A) each of which has one line in common with the fundamental pencil ; i.e. the plane 2aia;i=0 meets the fundamental curve in five points. When there is a double line* the curve q^ has a double point. For, to a double line I of C^ corresponds a double line l' in C^^ ^fgj. since C^ contains a line which belongs to oo ^ pencils so also must C^^) and if A contains I then will A' contain I', hence two of the four points of intersection of a' and the singular surface coincide, similarly for two tangent planes through a' ; there- fore any linear complex through the fundamental pencil and the double line contains only four distinct pencils of C^ which possess one line of the fundamental pencil, thus any plane through the point L which corresponds to I meets q'> in only four distinct points, hence Z is a double point of q^. 146. Representation of the congruence [2, 2] by the points of a plane. The analytical basis of the representation of the lines of a congruence (2, 2) by the points of a singular plane of the congruence, depends upon the theorem of Caporali that any congruence (2, 2) is contained in a tetrahedral complex. That * See Chapter XI. 184 POINT-REPRESENTATION OF A COMPLEX [CH. X there are 40 such tetrahedral complexes has already been shown, (Art. 123) ; Caporali's proof in a modified form will now be given. If C- and A, or SaiiPi*, be the given quadratic and linear complexes which contain the congruence, it is clear that C°- + AA' = is a quadratic complex which contains the congruence, whatever the complex A' may be. Let us take as the vertex Ai and plane Ut of reference, a singular point and singular plane of the con- gruence, and choose the coefficients of A' so that the squared terms disappear from the equation G^ + A A' = 0. The last equation now becomes Pa (aiPu + (hPti + (hPn) + Pn (biPu + hPii + ^sPm) •^Pii {CiPu + C^Pu + CsPm) + diPiiPii + d^PiiPis + dsPasPis + ei^M^M + e^p3iPjg-+ e^piipu = 0. Moreover since a^ is a singular plane of the congruence, the conic of this complex in a^, which is obtained by writing zero for every pit in the equation either of whose suffixes is 4, must break up into two pencils one of which is that of A for a^. But diPiaPu + d^PuPia + d^p^pn cannot contain aiapia + ai.,^is + a^pu as a factor, unless li, = cZq = c^s = 0- Similarly since At is a singular point, e^ = 63 = gj = 0. Thus the complex C + AA' = becomes P^ (chPu + a^p-u + asp-n) + Pi, (61P14 + b.p^ + h,p,i) + Pii (Cipu + C^Pu + C3P34) = 0. Also the coefficients oip^^p^., p,^, equated to zero, give special linear complexes whose directrices lie in 0(4 ; since the edges A^Ai, A^Ai, A^A^ of the tetrahedron of reference have not yet been determined, they may be taken as these respective direc- trices, which involves that ttj = tts = 61 = 63 = Ci = C2 = ; and C' + AA' = (hpisPu + hpuPu + c^PiiPm = is a tetrahedral complex. Since ten singular points lie outside each singular plane, this gives 16 X 10 = 160 tetrahedra, but each tetrahedron being thus taken four times we arrive at 40 as the number of tetrahedral complexes which contain any congruence (2, 2). 146] POINT-REPRESENTATION OF A COMPLEX 185 The coordinates puc of a line of the congruence thus satisfy the three equations Pl2Pu+Pl3P41+PuP2S = (i), aPiiPsi + bpi3P4s + cpuP9s = (ii), 2a.*^«; = (iii). We may therefore write Pi4 = pa'u P23 = o- (6 - c) x^s,) p^ = px^, psi = (T{c-a)x^u\ (iv), P3i = pXi, pi2 = 24 : P7A ■■ P23 ■■ Psi ■■ Pli = Sxj : &2 : 8x3 : R{c — b) x^^ : R{a — c) x^x^ : R{b — a) x^x^. . .(v). Through each point P of a singular plane of a congruence (G\ A) there pass two lines of the congruence one of which passes through the pole of the plane for A, thus the point P determines one line of the congruence not in the plane, hence to the 00 ^ points of the plane correspond uniquely the co'' lines of the congruence, thus (iv) establishes a (1, 1) correspondence between any line of the congruence and the point where it meets Ki. The only excep- tions arise from the six singular points in a^, which are the vertices Ai, A^, As, the pole of A for a^, and the intersections of i? = and ;S = 0. The lines common to {C', A) and any other linear complex A' = 1a'iicPii:, form a ruled quartic of class 1, which will be denoted by p*. This quartic is represented, from (v), by the cubic curve S {a'uXi + a'aA + olsr^s) + R {(c - V) a'saX^i + {a — c) a 31X1X3 + (b — a) a'l^iX^] = 0. The directrices of p* are the common polar lines of J. and A'. By aid of A the equation of A' can be deprived of one of its terms and hence by varying A' we obtain 00 * surfaces p*. Two such surfaces intersect in four lines, since this is the number of lines common to one quadratic and three linear complexes. Four lines of (C^, A) determine one surface p\ Through the lines common to two surfaces p\ viz. {C\ A, A'), (C\ A, A"), there pass 00 ^ surfaces p*, viz, those given by C^ = 0, A=0, A' + fiA" = 0. 186 POINT-REPRESENTATION OF A COMPLEX [CH. X Since every linear complex contains one line of each pencil oi(C^,A) it is clear that each surface p^ passes through the sixteen singular points and touches the sixteen singular planes. If one of the common polar lines of A and A' meets a pencil of (C'\ A) so must the other, hence the plane of this pencil breaks off from p* and we have a ruled cubic p' of which the double directrix passes through a singular point of ((?-, A) and the other lies in a singular plane : there are thus ao '■' such ruled cubics. This method of representation may be employed in connexion with any singular plane a of the congruence. Let S be the centre of the pencil of (0-, A) in a and S^-.-S^ the other singular points in <7, (Art. 118), having a^.-.a^ as their respective planes. Since any line of nn + ^O'tmi ^n X, ... X„ where the Xi are variables not yet defined, and A(X) is the determinant \ anc + Xanc \. * See "M^moire surla theorie alg^brique dea formes quadratiques," G. Darboux, Liouville (1874). + " Zur Theorie der bilinearen und quadratischen Formen," Berliner Monats- berichte (1868). For a sketch of some methods of reduction of quadratic forms due to Kroneoker and Jordan see "The reduction of quadratic forma and of linear sub- stitution" by Prof. T. J. I'A. Bromwich, Quarterly Journal (1901). A discussion of the concomitants of linear and quadratic complexes will be found in a memoir by Prof. Forsyth, "Systems of quaternariants that are algebraically complete," Camb. Phil. Trans, vol. xiv. 190 THE GENEKAL EQUATION OF THE SECOND DEGREE [CH. XI We have, by use of partial fractions V 1 V 1 A(\) X,...X„ A'(Xi)(X-\i) ftll+^^«ll ••• -X"; X,...X„ where the Xf are the roots of A (X) = 0, supposed to be all different. dF Now let Xi = ^ ^. where F =/+ X i(^-^>|. ia-Xi) dXn together with a term whose coefficient is zero ; hence the de- terminant has the value {X — XiY U^, where Ui, which is a linear function of the variables «!...«„, does not involve X. It follows that F=-^- X,- m W /+'^^ = -^A^)^^^A'(XO' whence /=-S X,:Ue 4> = t ue A'(Xi)' ^ A'(Xi)' 149. To deal with the case in which A (X) = has equal roots, we investigate in the first place the properties of certain determinants which are of importance in the solution of the problem of the expression of a general quadratic in n variables as the sum of w squares. The determinants <^j, in question have the form O'li Orin 2/11 yip 2/21 y^ •^p — dm. 2/11 ■ ^nn yn\- ■ 2/«i . ' Vrvp i 2/lp ynv where the a^ are the coefficients of a given quadratic expression /, |, is the discriminant of /, and the y's form p sets of n variables : it is clear that ^^ is a linear function of the minors of 4>o of order 'p. Now if any quadratic form ^ (3/1 y^ is identically zero, so are all its partial derivatives, hence so also is 02/1 oym applying this result to the 'p sets of variables y in j,, it is seen that if j, is identically zero for all values of the quantities y, so Iso is a„ «!„ 2/11 Vw t-ip ^nn ym • • *^nl " . Xyip yj ■ ynp .0 192 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI now the values 1, — 1, and zero may be assigned to the variables X, y so as to make this determinant equal to any minor of 4>o of order p ; hence if ^p is identically zero, so are all these minors of order p, and conversely. Since the minors of order ^ — I are linear functions of the minors of order p, it follows that if 4>j, is identically zero so also are 'J'^-i, ^o- In <3'p let us suppose the af* replaced by aii + A-afj, then „ = A(X). Moreover let X; be a multiple root of A(\) = 0, so that 4>(| contains (X — Xi)"' as a factor ; let "J"! contain (X — Xj)"' as a factor, then every minor of the first order of o will contain (X — X;)"' as a factor ; similarly if 4>2 contains (X — Xi)''s and so on. This may be indicated by the equation (X - Xf)-« = (X - Xi)""--. (X - K)"^-"- ; so that in passing from the determinant A (X), or o, to its first minors, the factor (X — Xj)"'""' is lost, in passing from the first minors to the second minors of o the factor (X — Xf)"'"''' is lost, and so on. These factors (X — X,)""""', (X — Xj)"'""" &c. were called by Weierstrass the Elementary Divisors of A (X). 150. The following properties of the Elementary Divisors* will now be proved : (i) i/o>i'i>V2 ; (ii) I/o — J'l > l*! — J'2 To see that (i) hplds, we notice that since by hypothesis $i contains (X — Xi)"' as a factor, and hence each first minor of $„. therefore ——- must contain (X — Xi)"' as a factor, i.e. Vo>Vi. In a similar way it is seen that Vi>v2, and so on. A theorem in determinants which is also of use in the sequel will enable us to prove the second property. The theorem referred to is the following : if .4 is any determinant | aa \ dA dA dA dA , d^A = A danc dUra datg ' dark dancdan ' for it is known from the theory of determinantsf that Ctik 0,ie drk C'rr: from which the required result at once follows. Aih Ajg ; . . . =.4 X coefficient of Ark Art I in A ; * For a full discussion of the Elementary Divisors see Muth, Theorie der Elementartheiler. t Scott, Theory of Determinants, Chapter V. 149-151] THE GENERAL EQUATION OF THE SECOND DEGREE 193 dA dA dA In particular we have d'A dA ddn^i^ n— 1 df^n, n ^^n— i, n— i ^^n, -« "^^n^i, n "'^n, n-~i Applying the last result to the determinant Oil + ^a„ ai„ + Xai„ ^n t/is ffljii + A,0,ii <*7m + "•O'Tm 2/ra 2/n2 2/u 3/m 2/i2 ym we obtain (,4>2 = ^i'^2 — '^s'^j, where '^i is the result of omitting the n + lth row and column in $2 and therefore is divisible by (\-Xj)% similarly for '^^,'^3,'^^, hence Vo + Vi'^ 2vi, i.e. v„ — Vi'^Vi — v^. 151. The theorem in determinants just considered affords a means of expressing any quadratic form /= SajtaJj/rft as the sum of n squares. For this purpose two new determinants Rp and Ap are introduced, where Rp = Ctnl 2/u a 171 3/11 ■2/1 X, Ctnn ym , -Ap — yi,n~p an ttjii 2/n ' yn, n—p a,, 2/11 Ctjin 2/»i y», n—p 2/i,n-p 2/n,7i-j) 2/1 , n~p+i Vn, n—p+i " " in wbich the Xi are variable quantities not yet defined. Taking the determinant Rp-i as the A of last article, we -deduce that ^n— p-itj)— 1 — ^n—p+1-tip ~ Ap , ^whence jtCp^l tip ^. n~p+l ^n-p ^n-p^ »— p+i J. 13 194 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI by addition of all the equations so formed we have but jB„= 0*, therefore 7? " A^ Now writing for Xi the value s 3 • -'^" ^^ecomes - ^0 • /t ; therefore /=-s Ae 7 <^n_, n—i^^n—i+1 152. Consider again the expression -1 F{X) = A(\) Oil H- Xttii, ... a„i + A-ctnij -2^1 X, ... Z„ in which the Xi may be any variable quantities ; F{X) may be expressed in the form 11 (^+ ^' i k \^~^ \^~ ^) ... + {K-\ifJ if (X — \i)* is a factor of A (\). By a theorem due to Lagrange and easily proved, the aggregate of all the fractions corresponding to the root ^^ is equal to the coefficient of y in the expansion of in powers of A. Now F{X,...X^,\i + h) \ — \j — A FiX,...Xn,\ + h) = - 4>„ if in Rn and *„ we suppose the Oik replaced by a^i + (Xj + h) 0^ ; hence, by the last article. F{X, ...Xn,Xi + h) = -l ^A 'n—p+\ 1 ^p-^^p' * Since it contains a square of (n+1)^ zeros. + See Scott, Determinants, Chap. XI. 151-152] THE GENERAL EQUATION OF THE SECOND DEGREE 195 where ■^n—p+i — an + 0^ + h) dn, ■■■ a«i + (\ + h) «„i, y^ . . . y,, j,_i X, yi,p-i 2/n,j7-i 2/1,3, yn,p On making the substitution i d(f+\) it follows that F(X)=f+X(j}, so that (1 "1 p=n J 2 \ coefficient of -=- in - — ; r S "•~^+^ ) . Again, if in A,^p+i we subtract from the last column the first n columns multiplied respectively by Xi, ..., x^, the terms of the last column become ^(X_x,-A)g, ... ^(x-X,-A)^, U,,...,Up, hence An-p+i= (>^ — \ — h)Bn,p+ Cn,p, where Cn,p is the aggregate of the terms in f/j, ..., Up. Now, since the coefficients of the Ui are linear functions of the minors of order ^ — 1 of ^0, while, by hypothesis, each minor of order p — l of | aik + Xoik | contains .the factor (X, — Xi)^-!, it follows that Cn,p contains A,""-' as a factor. Similarly Bn,p is seen to contain h"? as a factor. Moreover 4>j,_i and ^j, contain h'p-^ and h^r respectivelj' as factors ; hence (\-Xi-A)pp_i (\-\i-A)A"';.+-p-iAp.Ap_i ' where ^p^h-'pAp, j,4>^i is Therefore the term _ A 2 (X-Xi-A)„*i gives a coeflScient of t of this form, introducing v^ — v^ such variables f ; the term — -A^t-i brings in i;, — v^ more such variables, &c. : the total number of variables thus introduced in connexion with the root A^- of A (\) = is therefore Vo — Vi + Vi— i'2+ ■■■ = Vo. Proceeding successively to each root of A (X) = 0, we see that, since "Ev^ = n, the total number of variahles ^ is n; also F, i.e. f+\ is equal to the aggregate of sets of terms of the type one such set being contributed by each elementary divisor {X—XiYr. Hence '} It is to be observed that lep = n. If in the case of a multiple root Xi, it is not the case that X — X^ is a factor of all the first minors of A (X), then j/, = 0, and there is only one elementary divisor connected with Xf, viz. (X — Xj)"". 153. Applying the several results thus obtained to the case of 152-154] THE GENERAL EQUATION OF THE SECOND DEGREE 197 the quadratic complex, n=6*,f=0 is the equation of the complex and (ji = co = is the identical relation ; while Se^ = 6. The first case which occurs is that in which each Cp is unity ; both / and &> then consist merely of squares ; this complex, the general case in which A (\) = has six different roots, is denoted by the symbol [111111]. The equation Sep= 6 can be satisfied by sets of positive integers Bp in eleven ways, and the corresponding complexes are denoted by [111111], [11112], [1113], [1122], [114], [123], [222], [15], [24], [33], [6] : their equations will shortly be given. Each type con- tains a certain number of sub-cases, since two (or more) numbers Cp may refer to the same root \i of A(X) = 0, (i.e. when Vi^O); thus for instance if Xi is a triple root of A (X) = 0, or $, = 0, while X — Xi is a factor of i , the elementary divisor (X — Xif is lost in passing from $,, to i; two of the numbers Cp are 2 and 1, all the other roots of A (X) = being supposed distinct from each other and from Xj; this case is denoted by [111(12)], the numbers gp which refer to the same root Xj being enclosed in a single bracket. Thus again in the complex [11112], the determinant A (X) has a factor (X — XiY, and X — X; is not a factor of all its first minors ; while in [1111(11)] (X — XiY is a factor of A(X), and X — X^ is a factor of all its first minors. 154. Arbitrary constants of a canonical form. When a root X; of the discriminant of y+Xto is connected with only one elementary divisor (i.e. v^ = 0, Art. 152), the arbitrary variables y disappear from the variables f connected with this elementary divisor. For in this case (X — Xj)«p is a factor of ^o but X — Xi is not a factor of all the first minors of „, i.e. 4>i does not contain X — Xj as a factor; thus the part of f+Xw contributed by this elementary divisor is equal to the coefficient of r in the development I Bng Y (X-\i-h) of where *o = ^^»"^o, ^71,1 = ., ,, 19a) au+(Xi+/i)au,--- 2g~ ai6+(^i + A)«i6, 2/1 Ida 2 9«e ye , *,= (hi + i\+h)an,...yi «ie + (>^+^)ai6.---2/6 2/i 2/6 * This application was made by Klein, see "Ueber die Transformation der allgemeinen Gleiohnng 2. Grades zwischen Linien-Coordinaten auf eine canonische Form," Diss. Bonn (1868) and Math. Ann. xxiii. 198 THE GENERAL EQUATION OF THE SECOND DEGEEE [CH. XI SO that -^^^^ = ^(Xi + h) = '^{Xi) + h — + ...; while, since o is zero when h is zero, Oi = {V — AkhykY, if the Aik are the first minors of $0. "1 So that ^(\^) = ^--^ -- SV-^tfe^^; where 8 is the discriminant of o), and Xj=^Xi. It follows that 2v'S. f. = ,r^, (.1-] 2W' -^Itt dco 3^* VHj {\- X;)) If several elementary divisors are connected with the same root Xi of the discriminant of f+Xw, a certain number of arbitrary constants are contained in the canonical form. If, for instance, V elementary divisors relate to Xt, the corresponding sets of variables being f, f, f", ...f'"'; then the forms of /and to are unaltered if we substitute for f^ , |'e respectively provided that + This introduces v" arbitrary constants a, /8, ... between which exist, by virtue of the last condition, — ^^-^ — - equations, leaving -^-= — ~ of these quantities arbitrary. If this occurs ^„ times the total number of arbitrary constants contained in the given canonical form is yi.^ ^ — - *. 155. Complexes formed by linear congruences. When two numbers gp, ej,- are connected with the same root X; of A (X) = 0, the equation _/'+Xia) = involves only four variables, since the variables ^^ , f'^ , do not appear in the last equation. Hence in all * Klein, " Transformation der Complexe 2. Grades," Math. Arm. Bd. xxiii. 154-156] THE GENERAL EQUATION OF THE SECOND DEGREE 199 types which involve the single bracket, the equation of the complex can be brought into a form which involves not more than four variables. Regarding for a moment these four variables as the coordinates of a point, the equation of the complex will represent a quadric ; now every quadric can be brought into the form XY-ZW=-0 where X, Y, Z, W are linear in the variables. Hence the complex may also be brought to this form, and consists of a singly infinite number of linear congruences, X = fj,Z, ,j,Y= W. If P is any point oi p a, directrix of such a congruence, the other directrix being p', the pencil (P, p') belongs to the complex and hence the lines p, p' belong to the singular surface which is therefore ruled. The complex gives rise to a correspondence* among the generators of the singular surface. 156. Double Lines. The equations to determine the four pencils of complex lines to which a line x of the complex belongs, were seen to be in the case of the complex f= 0, ta = 0, the following : "£) = »• ("!)-»• («|)./W = MArt.75). If a line x is such that each point of it is singular, it is said to be a double line of the complex, and belongs to an infinite number of pencils; x will then belong to the singular surface. The condition for the existence of a double line is therefore that the preceding four equations should reduce to three, hence there is in the case of a double line x a quantity /x such that 1 + ^3^ = ". C-l.^.-^)- Now taking / and a as being composed of groups of variables ^ where corresponding portions of/ and eo are -(^l?»+...+f«rl). * On the subject of this Article see Weiler, " Die Erzeugung von Complexen ersten nnd zweiten Grades aus linearen Congrnenzen," Zeitschrift 1882 and 1884. 200 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI the parts of the immediately preceding six equations which arise from this group of variables are The equations are therefore all satisfied by fJ- = 'K, ^i = ^i= ■■■=^,1-1 = 0, and all other variables except |„ zero; so that the line whose coordinates are all zero except f„ belongs to /, and is now seen to be a double line of the complex. Hence each group of variables f gives rise in general to one double line. In the type [111 111] in which each group consists of only one member there is no double line, in every other case a double line exists. Every double line of the complex is a double line of the singular surface. For taking the complex as /(«) = 0, (of) = 0, the tangents y of the singular surface are given by the equations ^ + fjLXi = p.yi, (i=l, 2, ...6), oxi where a; is a singular line of the complex ; if a; is also a double line, there is a value of fi for which the left-hand side of each of the preceding equations is zero, for which therefore p is zero, so that any line y which meets this double line « is a tangent line of the singular surface, hence x must be a double line of the singular surface. In the case of any complex for which three numbers are enclosed in a single bracket, as [11(112)], the singular surface is a quadric counted twice ; for if ^, r], f, X^ correspond to the enclosed numbers, the equations to determine the double lines are satisfied by equating to zero all the variables except the ^, 77, f with the highest suffixes, i.e. by the vanishing of three variables ; this gives a regulus of double lines (which may become two pencils), each line of which is a double line of the singular surface. 157. The Cosin^lar Complexes. If a tangent linear complex of / is special, its directrix touches the singular surface of y! For if y is the directrix of such a complex we have 9/ da) dp'i>p-i ' X, X, where ^6-p+i = aii + (Xi + A)a„, ai6 + (Xi + /0ai6. yn---yi,p-\^i a2i + (Xi + A)a2i, ajs + (Xi + A)a.,6, aei + (^i + /l) 061 a6C + (Xj + /l)a66, 3/51 ... 2/6, p-l-X's 2/ii 2/61 ... 3/ip .. 136 If we now take Xi to be ^ ;;— , the equation I a^ + Xoii, Xj I = 2 92/i is the equation, F = 0, of the cosingular complexes ; while A^^p+T, becomes the determinant formerly denoted by -B„_y, hence 6— J)+l (7,, + -r,,h + ...)■" X-Xi-h' <^p<^p^i X — Xi — li' h% the rji being the same functions of the yi as the ^j of the Xi. Therefore Y 1 — — ^ = - S coeff. of r in , -, ; ■ A (\) h X — Xi — h = — "E coeff. of 7 in a + h h'v + .\-Xi {X-Xif ■•> IJim + . . . + lye^^?! '?i'7e -1 + . . . + Ve-iVi ■*■ /i "^ h^ ^ ...)• So that F A(X) = -2 7?,17e„ + ■ . . + -ne-n^ '^I'^e -1 + . • ■ + % -i??! X-\i - + (X-Xi)^ -(B); where each elementary divisor (X — Xifv contributes a set of terms on the right side of the last equation. This gives the expression of the cosingular complexes in terms of the variables t), &c. which enter into the expression of the canonical form of the given quadratic complex f(x) = "ttancXiX}^ = 0. 157-158] THE GENERAL EQUATION OF THE SECOND DEGREE 203 158. Correspondence between lines of cosingular com- plexes. It was shown in Chapter VIII. that a (1, 1) correspond- ence exists between the lines of (Xa;^) = 0, {x^) = and the lines of any one of the cosingular complexes. This result will now be established for the other varieties of the quadratic complex. The algebraical theorem which follows enables us to obtain the required result. Let there be two sets of n variables ^i ... f„ and t/i ...■);„, and let us denote by X„_,., F„^ respectively the expressions Further denote by -4„_,. the expression «!«„_,.+ ... + a„_rau where the a's are constant quantities determined by the series of equations -4i = «i' = -l, 42 = aia2 + «2^i = ^, ,-— , .... ^,--(_j . These equations determine uniquely the values of Mi.. and it follows that ^,j_^ + -^„_^_i = 0, {r = 0,\, ...,n-T). Now consider the equations From these equations we deduce at once XXn + X«-i = f-^n + ^ ^„_i j r,^^ + 2 [A,,^-, + - ^„_2 j tJit;^ + . . . + 2 f ^2 - - j 77i77m_i - 277i17„ + f -4,i_2 + - An-a) 7)^+ ... - 2772 '?«-!+ .... From which we conclude that XXn + .3r„_i = —Yn (II) ; similarly, XX„_i + Xn-2 = — ^n-i , \Xi = — Fi ; 204 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI hence, - X„ = i F„ - 1 F„_, + 1 F„_, ...+(- 1)"- g)" F, . . . (III). Equations (II) and (III) are thus a consequence of (I). We shall now, in these equations, take n = ep and replace X by ^ — X. Also we suppose that other sets of equations, similar to (I), are formed between variables ^\, ..., f '^ , ; ti\, ...,7]'^,, &c., then by addition of all equations of the type (II) we have 2 {K (?,|e, + . . . + ^eJO + r.f.,-: + • ■ • + V'^i} Similarly by addition of equations of the type (III) we have Xt - A. (Xf - X)' "^ ■ = - 2 (f^e, + ••• + ?., ?0- These equations show that if the quantities li. •". fej,; Ta fv'*^*^- are the coordinates of a line f, and i;,, ..., r)e ; i?'i. •■•^v'e ,, &c., the coordinates of a ii'ne t;, then f belongs to f{x) = awcZ r] to a cosingular complex F = 0. It is therefore seen that by aid of the equations (I), a (1, 1) correspondence is established between the variables f of f and t) of F which arise from the elementary divisor (\ — X^fp . This holds for each elementary divisor ; hence bj' aid of sets of equations (I), (whose number is that of the elementary divisors), a (1, 1) correspondence is established between the lines ^ of / and rj of F, such that if /(a;) = 0. expressed in the canonical variables be denoted by /(^) = 0, and F= by Y{rt) = 0, /(?) + \o,(?) = a,(7,), All the results already deduced for (Xa;^) = from the existence of this (1, 1) correspondence will therefore hold for any quadratic complex. 159. The singular surface of the complex. The equa- tions of the complex are obtained by equating to zero the aggregate of such terms as 158-159] THE GENERAL EQUATION OF THE SECOND DEGREE 205 where other portions arise from each other group of variables ■q, f, &c. It has been seen, (i), Art. 157, that if the line f, rj', 5", ... is a tangent line at P of the singular surface and ^, 97, i^, ... the singular line for P, we have {\ + O") ?» + 1^71-1 = P ■ Hn, (\i + a) ^n-i + ?7i-2 = P ■ t'n-i, (\i + 0-) ^2 + f 1 = P ■ ^2', i\ + o-)^i = p- li' ; with corresponding equations for the other sets of variables 17', ^', ... of the form (kj + cr)7]m + Vm^i = P ■ Vm, &C. Now we obtain the coordinates of a double line, (Art. 156), by taking as zero all the coordinates ^, r), f, ... except ^„, and the special complex having this double line for directrix is ^i = 0. Hence, for a tangent line of the singular surface which intersects this double line, we have ^1' = 0, and therefore X; + o- = 0. Again from the equations connecting rj and 7?' we easily find, if Ym be written for tjitjm + . . . + TjmVi and Y'm for the same function of the rj', (kj + <7f F^ + 2 (X^ + 0-) F^i + F^ = p^ . 7'™, hence "■(x^.- (|f?).+- +T^') -(=^+-> ^-+ ''"■■ in which, for a line ^', rf, f', ... which meets the double line, o- = — Xi. But since ^, r), ^,... belongs to / we have rif«-. + • • • + ?»-i^i + (>-^- - 'W) I'm + F^i + . . . = ; hence, substituting in this equation for the f , 17, f , . . . their values in terms of ^', rj', ^', ... ,yfe find This last equation thus represents a quadratic complex to which belong those tangents of the singular surface which also intersect the given double line, i.e. which satisfy ^Z = 0. The singular surface must therefore he a Complex Surface of Flilcker. In all cases, therefore, in which a group of variables ^ occurs, the singular surface is a complex surface. 206 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI If the ^ are the only group of variables, the terms of the last equation which follow the |' are composed of squares, i.e. are of the form — '- 1 — ? \- Xj — Xi Xi — Xi 160. Degree of a complex. The number of cosingular complexes which pass through any line is called the degree* of the complex. For cases in which no two elementary divisors refer to the same root Xj, (the eleven principal types), the degree of the complex is four, as may be seen from the preceding equation of the cosingular complexes. In other cases the degree is easily cal- culated, e.g. in [1113] the equation to determine X having given ^, i;, 5' ••• is of the fourth degree; but in [11(13)] the coefficient of X* vanishes identically, so that the degree of the complex is three. 161. The varieties of the quadratic Complex. We shall now investigate the dififerent varieties of the quadratic complex f; they consist as has been seen of eleven types or canonical forms, and each type contains a number of sub-cases. First canonical form. [111111]. a (x) = xi' + x^ + x^ + x^ + x^ + x^, f{x) = Xi^i^ + X^x.^ + X,x^ + \iX^ + X^x^- + X^x^^. This general form has been already considered in Chapter VI. 162. The sub-cases are [1111(11)]; X, = X„. The complex is f{x) = (X, - X,) a;,= -I- (X, - X,) a;/ + (X, - X5) x^ + (X, - X,) Xt' = 0; the lines which satisfy the equations Xi = x^ = X3 = Xi = 0, are double lines of the complex, they are the edges J.1^4, A^A^ of the tetrahedron of reference. The singular lines satisfy the equations f{x) = 0,fi(x) = 0, where /i (x) = (X, - x,y <- -t- (X2 - x,y xi + (X3 - x,Y x,^ + (x, - x,y «/. * Segre. + The classification which follows was given by Weiler, Math. Ann. vii., "Ueber die verschiedenen Gattungen der Complexe zweiten Grades." His memoir contains some inaccuracies which have been corrected by Segre, see "Note sur les complexes quadratiques dont la surface singuliere est une surface du 2' d6gr6 double," Math. Ann. xxiii. See also Segre's classification of the quadratic complex in the Memorie della R. Accad. di Torino (1883). 159-164] THE GENERAL EQUATION OF THE SECOND DEGREE 207 The singular surface is obtained by substituting \ = \ in t^^e equation given in Art. 82, and is therefore + (^3 - ^4) (Xa - X5) {^ - ^5) {yi'yi + yiyl) + 2 ((Xi + X2 - 2X5) (X3 - X5) (X, - X5) - (X3 + X4 - 2X5) (Xi - X5) (X2 - X5)} 2/12/22/32/4 = ; which is a ruled quartic, (Art. 155), possessing 2/1 = 2/4 = 0. 2/2 = 2/3 = 0, as double directrices, and hence belonging to class I. 163. [(111)111], Xi = X2 = X3. f{x) = (X4 - Xi) x^^ + (X5 - Xi) x,^ + (Xe - Xi) x,^ ; / {x) = (X4 - Xi)^ x-^ + (X, - Xi)= «/ + (Xe - ^0' x,\ The double lines are those which satisfy the equations ^4 = iCg = iCg ^ U J they are one set of generators of the quadric 2/12/3 — 2/22/4 = 0. This quadric, counted twice, constitutes the singular surface, (Art. 156), as may be seen by making Xj = X2 = X3 in the singular surface of [(11)1111]. The equations of the singular lines assume the form -^ = -^ = -4- , and therefore form four linear congruences. 164. The complex [(111)111] is one of a series of five, the others being [1(11)(111)], [(111)12], [(111)(12)], [(111)3], which are formed by aid of an involution [2] between the lines of a regulus. The involution [2] is defined by an equation of the form Lz^z'^ + Mzz' {z ■^z')^-N{z-v zj + Rzz' + /S(^ + ^') + 2'=0. . .(I). By making z = z' it is seen that there are in general four elements of the involution each of which coincides with one of its corresponding elements; if these be called "double" elements, four special cases arise : Case (i) two double elements coincide, „ (ii) three „ „ (iii) four „ „ (iv) two pairs of double elements coincide. Now we may, for clearness, regard the coordinates z, z' as defining points on a given line ; to the four double elements will then correspond four points, say P, P ; Q, Q', on this line. There 208 THE GENERAL EQUATION OF THE SECOND DEGEEE [CH. XI are two points, say A and B, which are harmonic with F and P' and also with Q and Q', i.e. A and B are the double points of the ordinary involution determined by P, P' ; Q, Qf. If a and /3 are the coordinates of A and B respectively, then substituting in (I) by aid of the equations z — a , z' — a z — p z — 13 we obtain the involution [2] expressed in terms of x and so'. The coordinates of A and B are now seen to be zero and infinity respectively; hence the points P and P' harmonic with them must have coordinates of the form + 7 ; similarly Q and Q' have coordinates + 8. So that the equation to determine the double elements must be of the form Lx' + Kx' + T = 0; where Z' = 4iV+iJ: that is to say, the coefficients of the second and fifth terms in the equation defining the involution [2] must be zero. The latter equation, therefore, is of the form Lax'x' + N{x + x'y + Exx +7 = 0; which is therefore a form by which the general involution [2] may be defined. By writing in the new equation a; = ( y 1 . m, «' = I -^ j .u', the coefficients of the first and last terms are made equal. 165. It will now be shown that the complex [(111)111] is the locus of lines which intersect corresponding lines of a regulus in the general involution [2]. For the complex may be ^v^itten \tx^ + 'KiX^ + \iXi = 0, if we replace X.^ — Xj by X.^ , etc. Any line of this complex is therefore given by the equations \/X4 Xi : VXs ajs : Vx^ x^ = fi?-l : i {/j? + 1) : 2fi ; which divide the complex into a singly infinite number of linear congruences. Also any line of the regulus «! = a^a = a;, = is given by Xi'. Xs : Xe = p^ — l : i (p^ + 1) : 2p. 164-167] THE GENERAL EQUATION OF THE SECOND DEGREE 209 All lines of such a linear cons^ruence meet lines of the regulus for values of p given by the equation (^2 _ Y){p' - 1) {fi? + 1) 0>^ + 1) ^ 4/y _ ^ Vx^ ^/\, VXe If pi and p2 are the roots of this quadratic we have _ Aim _ Dfj? + G u ^ 4 „ 1 1 „ 1 1 where A = -== , C = -7= t= , B = -^= + -7= VXs VXs VA.4 VX,5 VA,4 Hence between pj and p^ the following equation exists A^ (pi + P2f = (-(72 _ j)2)i C^ - Dpipi) (Cp,p2 - D\ which defines a general involution [2]. The complex is therefore obtained by establishing a general involution [2] between the lines of a regulus, and taking all the lines which intersect each pair of corresponding lines. 166. If X4 = X„ which gives the form [(111)(11)1], = 0, A"- and the involution becomes (pi + p2? = 7^ PiPa. which is derived from the general case (i) (Art. 164) when i = M = fif= T= ; here (i) has two pairs of coincident double elements (viz. two infinite and two zero), giving case (iv). 167. [11(11)(11)], \, = X„ X, = X,. In this case f{x) = (A-x - X,) x^ + (Xi, - \,) xi + (X3 - >^5) (a;," + ^4=). /i ix) = (X, - X,)'' x^^ + (X, - X,y xi + (X3 - '^if {jei + x^\ There are two pairs of double lines, viz. A-^Ai, A^A^; A-^A^, A2A4; which form a twisted quadrilateral. If in the singular surface of [1111(11)] we put X3 = X4, it becomes of the form y-i^yi + yiy^ = Ky-^y^y-^yi, and therefore consists of two quadrics which intersect in the four double lines of the complex. From the equations of the singular lines we deduce that they satisfy the equation /i — (X3 - Xs)f= 0. which is of the form (a?! + aiTa) (a^i — ax^) = 0, hence the singular lines consist of two congruences (2, 2). J. 14 210 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI This is one of three complexes which have for singular surfaces a pair of quadrics, the others being [1 (11) (12)] and [(12) (12)]. Writing Xj for Xj — Xs it is seen that the complex consists of the singly infinite system of linear congruences Vxi'(l + «,) Xi + i VXj (1 - «i) Xi + VTj ( — - + /i. j X, VX^ (I + K^ Xi + i VX^ (1 - Ka) a^s + "^^3 ( ' + /J'jXs + i VXa (- + fi]xi = 0; where k^ , k^ are the roots of the equation 2 1 1 1 o Xi + X2 — 2X3 /c^ + 1 + 2/c . — r = (). A,j — X2 The directrices f, p of these (special) complexes have co- ordinates P I P \/X,(1+kO>*Va,(1-«,). ^>-3(— ^+/^),»Vx,(-' + /i), 0, VXi (1 + K„X i Vx^ (1 - K^), VXs f— ^' + All i VXs (- + m). 0, 0. On varying ^, it is seen that p and p' describe two reguli, whose lines are in (1, 1) correspondence, and which have (for ^ = 0, fi= cc , respectively), the common self-corresponding lines A1A3, A^Ai] while AiA^, A^A, are seen to be common directrices of the two reguli ; hence the complex is the locus of lines which intersect corresponding pairs of lines of two reguli which are in (1, 1) correspondence, and which have two common self-corre- sponding lines. 168. [1(11)(111)], X,= X3, X, = X,=Xe. The complex is f{x) = (X, - X,) x,^ + (X, - X,) {xi + xi) = 0, while f {x) = (X, - X^)^ xi + (Xj - X,)^ {xi + xi) = 0. Hence the singular lines form the congruences a^ = 0, X2+ ix,= Q ; a^ = 0, ajj — mJs = 0. 167-171] THE GENERAL EQUATION OF THE SECOND DEGREE 211 The singular surface consists as in [111(111)] of a quadric taken doubly, while the double lines are one set of generators of this quadric together with the two lines tCi "^ wJ c/-g —— ti/R ~~ \J • 169. [(11)(11)(11)], \ = X„ X3 = X4, X, = \,. This gives the Tetrahedral Complex Xi (xi' + xi) + X, ix^^ + x/) + \, (x,^ + «/) = 0. The complex possesses thirteen independent constants, viz. twelve from the tetrahedron and one from the constant double ratio. 170. [(111)(111)]. X, = X, = X3, \, = \, = \. The equation of the complex has either of the forms xi' + xi + xi = 0, «■/ + x^ + x^ = 0. The generators of one system are Xi = x^ = Xi = 0, and those of the other are Xi = x^ = Xf,= ^; hence any tangent line of the quadric will belong to the complex, which therefore consists of the tangents of a quadric. 171. Second canonical form. [11112]. o) (x) = Xi' + xi + x^ + x^ + "i-x^SB^, f{x) = X^Xi' + X^x^ + Xixi + XiXi + S^s^siTs + x^. From the form of the identity w {x) = it is permissible to write iXi-pn—pu, iiXi=Pis-P4a, a;6 = 2^23- The singular lines are those whose coordinates satisfy the equations (Xi - X,) Xi' + (X2 - X5) x^ + (X3 - X5) a;,' + (X4 - X5) «/ + x^^ = 0, (Xi - x,)' Xi' + (X3 - x^)^ xi + (X3 - x,y xi + (X, - X,)^ xi = 0. The directrix of the special complex a;^ = U is seen to be a double line of the complex (Art. 156). It is seen from Art. 159 that the singular surface is the Complex Surface of the congruence — o 21 '•'^i ^^ I ""^ I ^^ = • Xi — Xs X2 — X5 Xj — X5 Xd — Xj and hence of the congruence xi xi . xi , X,' it-g M/3 .*'4 « / , • / f\ Xi — X5 Xa — X5 X3 — Xs X4 — X5 14—2 212 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI where x\ = x^ + \x^, x'^ = i (x^ - ^ x^), 4 and therefore Sx,- + x\^ + x'e" = 0. Thus the Singular Surface is the Pliicker surface for a general quadratic complex [111 111] and an edge of a fundamental tetrahedron. It was seen (Art. 86) that the Pliicker surface for the complex {\x') = 0, (a;-) = is {\u'){\v') - {Xuvy = 0, where u is a line through A, v a. line through B, A and B being any two points on the double line, and u, v meet in a point y of the Complex Surface. The double line being in this case the edge AoAi of the tetrahedron of reference, we may take A as A2 and B as A3, which gives as the coordinates of u and v Pl^ Pl3 Pu Pas Psi P4 u 2/1 -2/3 2/4 V 3/1 2/2 -2/4 whence 2/1 -*2/i 2/4 *2/4 -2/3 -^y3 - 2/4 - *>4 2/i - iyi y, iyi, and the equation of the singular surface is ylQ^^\) , ?/,'(X4-A-3) ys' + ■ + : y,'(\-x,) y.H^-'K) yi' + = 0, {X^-X^){\^-X^) (\3 - X5) (X.4 - As). + i - ^ ^ 1 1 I \ "1 ~ A,5 Xj — Xj X3 — X5 X4 — \. which reduces to the form {X, -X^){X,-X,) (2// + y,*) - (\, - X,) {X, - X,) (Xi - X3) (y, V 4- 2/3 V) - (Xi - X,) (X, - X,) (Xs - X4) {i/,'y^ + yiy/) + 2 [(Xj + X, - 2X5) (X3 + X4 - 2X,) - 2 {(X, - X,) (X, - Xa) + (X3 - X,) (X, - X,)}] y.'y," + 2 {(X, - X,) (X, - X,) (X3 + X4 - 2X,) - (A3 - X5) (X4 - X5) (Xi + X, - 2X5)) yiy^zyt = 0. This equation might also have been obtained by finding the locus of points y for which the complex cones of [11112] become pairs of planes. 171-174] THE GENERAL EQUATION OF THE SECOND DEGREE 213 172. There are nine sub-cases, which are as follows : [111(12)], \ = \. The singular surface becomes i\ - K) (Xs - \) (yi^ + y,*) - {X, - \) (X, - X,) (X, - X,) {y,y, - y^y,y + 2 [(^3 - \) (Xi + >^ - 2X0 -2(X,- X,) (X, - X,)] y,»2// = 0. The line y^ = y^ = {) is a double line of the surface, and any plane through it cuts the surface in two lines which meet on the double line ; the surface is the ruled quartic of class II. 173. [11(11)2], X, = X,. The complex possesses three double lines, viz. A^A^ and also AiAs, AsA^; any plane through A^A^ is seen to meet the singular surface in two lines intersecting on A^A^ and vice versa, hence A^A, and A^A^ are double directrices of the surface, while A2A3 is a double generator; the singular surface therefore belongs to class VII. 174. [11(112)], X, = X,=X,. The equation of the complex is f(x) = (X, - X3) x," + (X2 - X3) X,' + xi = 0, the singular lines being given hyf{x) = 0, and /, (x) = (Xi - X^)"* X,' + (\ - X,y a;,^ = ; thus the congruence of singular lines consists of the four linear congruences «! : ajj : «5 = i (X^ - X3) : ± (Xj — X3) : + V(Xi — Xj) (Xj — X3) (Xj - Xj). The complex is composed of the singly infinite number of linear congruences Xi + ix2='2p.Xi, Xi — ix.2 = 2cr.Xi, i.e. Pii = p-Pu, .Psi = <^-Pu; where p and a are connected by the equation (Xi - X3) (/3 + .7)^ - (X, - X3) (p - cr)^ + 1 = 0. Hence, the complex is formed by lines which meet corresponding lines of two pencils (As, Oj), (A^, ai) which are in (2, 2) corre- spondence, and which have a common self-correspanding line -4 2^4 3. The (2, 2) correspondence has two of its four double elements in coincidence. The lines Xi = X2 = Xs = 0, forming the two pencils (A^, a^), {As, a^), are double lines of the complex. The singular surface is seen to consist of the centres and planes of these pencils. 214 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI 175. [(11)1(12)], Xi=A-., \, = \. The singular surface consists of the two quadrics {X, - \) (X3 - X4) (2/1.V3 - y^ViY - 4 (^= - \) yi'y,' = 0, which touch along their common generator yi = 1/^ = 0, and which have also in common the lines yi = 2/2 = 0. 3/s = y* — 0- The complex is Xj {xi' + xf) + X^x^^ -\-x^=i) (if Xi be written for X] — X4), and consists of the congruences VXia;, (fi + - j + i VXiiTj i^ - ^ j + Vx, (A;, - 1) aJs - i (k^ +l)x^ = 0, Vx^a;, (fjL+—j+ i y/XiX^ (- - fi\ + "Jx^ (k« - I) x, - i {k^+\)x^ = 0, where ^i, k^ are the roots of the equation n (a) = 0, i.e. 4iXjc + \^{k-\f = Q. Thus each of these complexes is special; the directrices de- scribe (for dififerent values of /n) two reguli which have A^A^, A^Ai in common. Hence, the complex consists of lines which meet corresponding pairs of lines of two reguli in (1, 1) correspondence which have two common self-corresponding lines, and two consecutive common directrices. 176. [(111)12], X, = X, = X3. The complex is (X, - X,) x^^ + 2 (X5 - Xi) x,x^ + a:/ = ; the double lines are those given by Xi = Xi = Xt = 0, which form one set of generators of the quadric yiya — y^yt—O- The singular surface consists of this quadric taken doubly. The singular lines are given by the equations x/ ZXeXf! Xk {X,-X,f {X,-\)iX, + X,-2X,) {X,-X,){X,-X,)iX,-X,)' they therefore form the special linear congruence Xi = Xi= 0, and two general linear congruences. If in the equation of the complex X4 be written for X^ — Xj &c., it is easily seen that any line of the complex is given by the equation Xt:xs:xe= Vx7(2/x VXj — i) : VX4X5 : - 2^ Vx7(/t VXj - i), giving oc ' linear congruences ; the line (0, 0, 0, 2p, 2, — p') is any 175-178] THE GENERAL EQUATION OF THE SECOND DEGREE 215 line of the regulus x-^=x^= w^ = 0, and is met by the lines of the preceding congruence provided that 'JXiXsp" + 4 \fXifi. (fi VXg - i) - 2p VXj (2m Vx^- i) = 0. If pi and jOa are the roots of this quadratic it is seen that A-4 A4 This equation defines an involution [2] which has two coincident double elements, and is case (i) previously mentioned, (Art. 164). If X4 = X5, giving [(111)(12)], tAe involution has all its double elements coincident, since it is given by the equation (pi — ^2)"+ -r—^l this is case (iii) of the involution [2]. 177. [(11)(11)2], Xx = X„ X, = X,. The singular surface as derived from that of [11112] is seen to consist of the planes 3/1 = 0, y^ = and a quadric whose equation is 2/12/4 — Ky^y^ = 0. If the equation of the singular surface for [11112] be formed in plane-coordinates and if in it we put Xj =X2, X3= X4, it is seen that V2V3 is a factor of the equation, which shows that the singular surface is completed by the points A^, A3 which raise its class to four. If X3 be written for X3 — Xi &c., the equation of the complex is ^\Pl3Pil + ^X^PiiP^ +p\i = 0. It consists therefore of the linear congruences 2 VXs (X3 - Xj) At>42 + 4X5 (X3 - X5) iip^ + p.\sPu - 2 v'X3X5_p,3 = 0. The directrices of the first of these linear complexes form the pencil {A2, fli), the directricies of the second form a regulus ; the lines of the pencil and of the regulus are in (1, 1) correspondence, and have a common self-corresponding line A^Ai; hence, the complex consists of the lines which meet paired lines of a pencil and regulus in (1, 1) correspondence having a common self -corre- sponding line. 178. [(11)(112)], Xi = X3, X3 = X4 = X,. In this case f{x) = (Xi - XO (xi' + xi) + x,' = i> (Xi - \,) p,,p^ + p\,. The complex consists of the singly infinite series of congruences 2 VXi - K -Pii = H'Pii' 2 VXi - Xj .^34 = - - ^,4. 216 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI Each of these complexes is special and the directrices form the respective pencils (^3, aj), {A„, a,) which are in (1, 1) corre- spondence, he7ice the complex is the locus of lines which meet corresponding lines of two projective pencils whm'e the plane of each pencil passes through the centre of the other. From consideration of the complex [11(112)] the double lines are seen to be the pencils (A^, a^), (A3, a^), and the singular surface consists of the centres and planes of these pencils. 179. [(111)(12)], \ = -K,= \„ \ = \,. The complex is (\, - Xi) (x^^ + 2x,Xs) + a;/ = 0. The singular surface, obtained from that of [111(12)], is (2/1^8 - 2/22/4)' = 0- By comparison with [111(12)] and [(111)12] the double lines are seen to consist of one set of generators of the singular surface together with the generator y^ = y^ = 0. 180. Third canonical form. [1113]. to (x) = x^ + x^ + x^ + x^ + 2xiXe, f{x) = X^x^ + XiX^ + X^x^ + X4 {x^ + 2xiX^ -f 'iXiXs. From the form of o) {x) it is permissible to write ixi=pu-p3i, ix3=pi3— Pa , a^e = 2^23 • The singular surface is (Art. 159) the complex surface for ,^(^) = -^+-^ + -^+2^,^, = 0, x,= 0; \l -~* A/4 Ag — A.4 A-j "^ A.4 (1 7)Ai\^ 3 Mi — j =0, where the lines u and v have the same coordinates pa, as in [11112]; hence, finding the X coordinates of u and v from Art. 171, the singular surface is seen to be 2/iH^->-i) _ .y^-' _ 4^^^^ (Xi — X4) (X2 — X4) X3 — X4 ""{{^ W-X4-)(X,-X4) X3-X4^*^'^='| 178-183] THE GENERAL EQUATION OF THE SECOND DEGREE 217 which reduces to the form (K - \) iyi' + 3//) -4>(\,- X,) (Xi - \,) {y,'y, - y,y/) - 4 (Xi - Xi) {\ - Xt) (Xs - X,) (y^y, + y^y^Y _4j(X3_A,,)(X, + X,-2X,) - 2 (Xi - X,) (X.2 - X,)] y,y, (y^y, - y^y,) + 2(X, + X,-2X,)!/iV = 0- It is clear that the double line yi = y^ = 0, for which all the coordinates except x^ are zero (Art. 156), belongs to the complex (p (x) = 0. Along this double line the tangent planes to the surface coincide with those of yij/s + 3/22/4 = 0. 181. [11(13)], X3 = X,. The singular surface is (Xi - X2) (2/1* + 2/2O + 8 (Xi - X3) (Xj - X3) 2/12/4 (2/12/3 - 2/22/4) + 2 (Xi + X,- 2X3) 2/1=2/4' = 0- The line yi = yi=0 is a triple line of this surface ; and any plane through 2/1 = 0, 2/4 = 0, cuts out one line from the surface, which therefore is a ruled quartic of class XII. 182. [(11)13], Xi = X2. Here there are two double lines A^A^, A^At in addition to the cuspidal double line A^A^; the singular surface is therefore a special case of class VII. 183. [1(113)], X2 = X3 = X4. The singular surface consists of the planes 2/1 + *2/4 = 0. 2/1 - Wi = 0, counted twice, together with two points on their line of intersection. The double lines are those which belong to the three complexes •Bi = .•^4 = iTj = 0, forming two pencils which have A^A^ as common line. The complex is (X,-X^)x,''+2x,x, = Q; the lines of the complex belong to the singly infinite number of linear congruences VXi — X2 (aJs + ixi) + 2fiXi (VXi — X2 ^ — i) = 0, VXj — Xj (^5 — ixi) + 2fia)i (v Xi — X2fi + i) = 0. 218 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI Each of these complexes is special ; the coordinates of their directrices p, p are Xi ajj aj, Xi Xs Xe p i V\i - Xj VXi-X, 2ya(v'Xi-X,/u,-t') p' - i VXi - X, VXi-X^ 2/x(VX, -Xj/i + t). Hence p, p' are corresponding lines of two pencils in (2, 2) correspondence, and which have -42^.3 as common self-correspond- ing line. Corresponding lines are of the form a,: + p0i, a'i + p'^i (where /S,- is A^A^). The connexion between p and p' is given by the equation \/x,-X,(p - pj + 2(p + p') = 0. In this involution [2] three double elements coincide (they are infinite). 184. [(11)(13)], Xi = X„ X3 = X,. Putting Xi = X2 in [11(13)] the singular surface is seen to consist of the planes 2/,y4 = 0, together with the quadric 2 (Til - Xs) (y.ys - jf^y,) + y,y, = 0. The complex has three double lines. The equation of the complex is *\PiiP2i + 2p,4 (p,3 + Pa) = 0, writing X, for X] — X4. It is formed by the linear congruences VXipi2 + fj.pu = 0, VXipij + f^Pu - 2/u.Xi (2fi Vx^ps^ -pi3 - p^) = 0. The directrices of the first complex, for different values of fj,, form the pencil (A^, Oj) ; those of the second form a regulus ; the lines of the pencil and regains are in (1, 1) correspondence and have AsAt as common self- corresponding line, while the line A^A^ of the pencil is a directrix of the regulris. This gives that case of the correspondence in [(11) (11)2] in which the pencil contains a directrix of the regulus. 185. [(111)3], Xi = X2 = X3. The singular surface reduces to (yiy^ + y^y^^ = ; the double lines are its generators y^^fxy^, y^ = y^. A' The singular lines form the special congi-uence Xi = Xf = 0, together with a general linear congruence. 183-187] THE GENERAL EQUATION OF THE SECOND DEGREE 219 The equation of the complex is ^^4 {o^i + 2i»4 Xe) + 2xiXi = 0, if X4 be written for Xi-\^. The lines of the complex are given by the equations «4 : ajj : ajg = ^4^ : 2/^X4 : - 2/i (1 + /x. VX4), forming 00 ' linear congruences ; the lines of such a congruence meet the line (0, 0, 0, 2p\ 2p, — 1) of the regulus Xi^ = x^ = X3 = 0, provided that 4/i/3^(l + /i VXi) - VX4P + X4^ = 0. If the roots of this equation are pi and p^, it is seen that 4 (Pi - Pif + r-piP2 (Pi + Pi) = 0. A.4 This is an involution [2] in which three double elements coincide, giving case (ii) (Art. 164). 186. Fourth canonical form. [1122]. ft) (x) = xi' + x^ + ix^Xi + 2x!,x^, f{x) = Xi^Ji^ + X^x^' + ZXsXsXt + 2\XsXe + x^ + x^. We may write ^1 = Pn + Pu t ^3 = Pw ) ^5 = Pi4 ) 1X2 = P\2 ^34, Xi = Zpii, Xg ^ ^PlS- The singular surface is the complex surface for /v» 2 /*i 2 V'y 'T* /)« 2 a;^ = 0, iTs" + ' + ' + ^f 3«L _ ^g = 0. A-i ~~ A4 A/2 — A.4 A.3 — X4 (Xg — X4J Repeating the process previouslj' adopted, the equation of the singular surface is seen to be (if Xi be written for Xi — X4, &c.), (X, - X,) 2/^0 - 4X3= i\ - X,) (yi'y^ + y,^y,') - iWy.'y," + 4 (X^As + X3X1 - XiXi, - Xs^) 2/1=^2/4' - 8X3 {X3 (Xi + X2) - 2X1X2} 2/12/22/32/4 = 0. This is a Pliicker surface with the two intersecting double lines A2A3, A^Ai. There are six special cases. 187. [11(22)]. The singular surface is obtained from the general case by putting X3 = 0, which gives 2/1= t(Xi - X2) yi^ - 4X1X2 (2/3^ + y,')} = ; while if we find the envelope of the singular planes we obtain t;/ {(Xi - X2) v^^ - 4X1X2 « + vi')] = 0. 220 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI Hence the singular surface consists of a quadric cone and a conic whose plane passes through the vertex of the cone. The double lines are given by x^ = «, = iCg = X;; = 0, and consist of the pencil (A^, Oi). 188. [12(12)], X, = \. The singular surface is 2/1= Wyi" + 4-X3 (Xi - X3) 2//} - 4x, \3= {yiy2 + ysy^Y = o. Any plane through ^2^.3 cuts out two lines from the surface which intersect on A^Ai, while AtA^ is a double generator; this is the case VIII. of ruled quartics. It gives a case of [11(11)2]. There are three double lines, viz. A^A^ and two lines coinciding with A2A3. 189. [1(122)], X^ = X3 = X,. If in the equations of the cone and conic which form the singular surface of [11(22)] we make Xj = 0, the cone becomes a plane (counted twice), and the conic a point (counted twice). The singular surface thus consists of a plane and a point in it taken four times. This complex is a special case of [11(112)]; the double lines are those given by a;, = a^a = aij = 0, and therefore form the pencil (Ai, Oi) ; they are to be taken twice as being derived from the two pencils of [11(112)]. 190. [(11)22], Xi = X,. The singular surface is seen to reduce to a ruled cubic with AiA2 as double and .43.44 as simple directrix, together with the plane j/j = 0, and the point v^ = 0. The double lines are A1A2, AsAi, A^Ai, A2A3. 191. [(112)2], Xi = X, = X4. The equation of the complex may be put in the form 2 (X3 - Xj) x^Xi + xi + xf = 0, i.e. 4 (X3 - X4) Pi3 ^42 + jDis" + pi4= = 0. The singular surface reduces to yiyi = 0, v^v^ = ; the double lines are A^A^ together with those given by x3 = x^ = Xi = 0, i.e. the pencils {A^, a^, (J.3, a^. 187-193] THE GENERAL EQUATION OF THE SECOND DEGREE 221 The complex is formed by lines which belong to the congruence fjiXs + iCs = 0, 2/i (Xs — X4) Xi — (/i^ + 1) ajs = ; and is therefore the locus of lines which, intersect paired lines of two •pencils in (1, 2) correspondence, which have a common self- corresponding line. 192. [(11)(22)], Xi = X„ X3 = V The equation of the singular surface is yi{yi + yl) = Q, together with v^ {v^ + v^) = 0. The double lines are those which satisfy the equations Xi = X2 ^^ x^ = iZJg = u, i.e. the pencil {A^, Oi), together with A-^^A^, A^A^ The complex is (Xi - X3) {x{- + x^) + x^ + x^^ = 0, i.e. 4 (Xi - X3) p^^p^ + ^13' + Pii = ; and is formed by the lines which belong to the complexes 2 VXi - X3 P12 - /A (pi3 + ipid = 0, 2 VXi - X3P34 + - (^13 - ■i'Pu) = 0- Each of these complexes is special, the directrices p, p' having the coordinates given by P12 Pl3 Pu P23 Pzi P42 p — i/U, 2 VXi — X3 — /Li p' 2VXi-X3/i -i 1 hence p and p' form two projective pencils, the centre of the former being a point on A^A^ and its plane «], the latter having A^ for centre and A^A^O for its plane; hence, the complex is the locus of lines which intersect corresponding lines of two projective pencils in which the plane of one pencil passes through the centre of the other. 193. [(12) (12)], Xi = X3, X, = X,. The singular surface consists of the quadrics 2/1'= ± 2X1(2/12/2 + 2/32/4), i.e. two quadrics touching along -42^44, A^A^. The line ^2-43 is a "doubled" double line, as also is A^Ai. The complex is (writing X2 for Xg — Xi), X2 (x^ + ^x^x^ + Xi^ + xi = 0. 222 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI It consists of the linear congruences 2 Vx, (1 — ifj.) x^ — 2(^X3 + {fi h 2i) x^ + ^X^fix^ = 0, 2 VXa (1 + if/.) x„ + 2fiX3 + (/x 2i) x^ + 2X2^6 = 0. The directrices p, p' of these (special) complexes, as ft varies, describe two reguli. The line ^2-44 is seen to be a directrix and A^A, a common line of each regulus. Hence, the complex consists of the lines which meet conjugate lines of two reguli in (1, 1) correspondence, the reguli having two common consecutive lines and two comm,on consecutive directrices. 194. Fifth canonical form [114]. o) (x) = Xi" + x} + Ix^x,^ + ^X^Xi, f{x) = Xja^i^ + XiX^ + 2X3 {x^Xs + XiX^) + 2XiX^ + x^. We may write liTj = Pi2 Pat, X^ ^ ^Pis, Xg = ^P23- The singular surface is the complex surface for x,' X. X, = 0, 2x,x, + x," + + \ = 0. The double line oi f{x), which is A^A^, is a singular line of the last complex. We obtain, as before, for the equation of the singular surface y/ (Xi - X2) + UK-K^y,'y,' + 8(\ + X,) yi'y.y, + 4y^y,' - ^'^i^.yiyiVi + 8 (X, - Xa) ^I'ys = 0, where X,, X^ are written for Xj — X3, Xj — X3 respectively. This is a Pliicker surface for a quadratic complex and one of its singular lines. There are three special forms. 195. [1(14)], X2 = X,. Putting X2 = in the equation of the singular surface gives 2// O^y,' + %0 + sxiyi^ (2/32/4 + 2/1 y,) = 0. Any plane through 2/1 = 0, 3/4 = meets the surface in this line together with one other; the line A^A^ is therefore a double generator and simple directrix of the surface, hence we have a case of class XII. In the singular surface of [11(13)], each of 193-197] THE GENERAL EQUATION OF THE SECOND DEGREE 223 the planes 1/1 = 0, 2/4 = 0. meets the surface in the line A^A^ merely, i.e. there are two " stationary " tangent planes ; in the present case both stationary planes have come into coincidence with 2/1 = 0. 196. [(11)4]- Putting X, =X2i we obtain as the singular surface 2/i (4X1^2/12/3" + 4Xi2/i2/32/4 + y^y.^ - 2Xi^y,2//) = 0. Hence the singular surface consists of a ruled cubic together with the plane 2/1 = 0. The line 2/1 = 0, 2/2 = is the simple directrix ; 2/3 = 0, 2/4 = the double directrix. The plane 2/ = meets the surface in ^3-44 and in the two coincident generators A^A^ ; hence 2/1 = is a cuspidal tangent plane of the surface. Using plane coordinates we find v^ as a factor of the left side of the equation of the singular surface. The cuspidal point A^, therefore, completes the singular surface. 197. [(114)]- The equation of the complex may be put in the form and hence consists of the singly infinite number of congruences ^x.i = tiXt, 2a;3 + /iX = 0; that is pi4 = /ipis, /i"p42 + 2pi4 = 0. The directrices of these special complexes form the pencils (J.2, Hi), {A,, a^, which are thus in (2, 1) correspondence and have A^As as common self-corresponding line. In a (1, 2) correspondence, which is given by an equation of the form X {ay"" + hy + c) + alf + h'y + c' = 0, the two values of y which correspond to any value of x form an involution ; in the present case, the involution formed in the pencil (^2, fti) has ^a-^s as a double line. The complex is therefore formed as follows : in two pencils having a common line a, connect linearly the pairs of lines ^1, j32 of an involution in one pencil with the lines 'p of the other pencil so that a is a double line of the involution and a self- corresponding line; tlxe lines which intersect p and pi, p and p^ form the compleu;. The planes of the pencils and their centres, each taken doubly, form the singular surface. 224 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI 198. Sixth canonical form [123]. o) {x) = x^ + ^x.x^ + 2XiXe + x^, f{x) = X^Xi^ + 2X,_x^Xs ■hx.:^ + \ {ixtx^ + a;/) + 2xtX,. We may write ixi=pii-psi, a;s=^42, Xe = 2p^. The singular surface is the Complex Surface for a;4 = 0, 2x^x^ I ^'' I ^'^"^^ — ; = 0. This gives as the equation of the singular surface (writing Xi , Xj tor Xi — X3, Xj — X3), 2/1' - A-i V (2/1^2 - yzViY + 4^iyi'2/3 - (^1 - 'K) ViVi - 2X2 (Xi - X2) 2/1 y, (2/,ys. + 2/S2/4) = 0. The line y^ = y^= 0, is double ; 2/1 = ^4 = 0. is cuspidal. There are four special cases. 199. [1(23)], X, = Xa. Putting X2 = in the last equation, we derive as the equation of the singular surface in point coordinates 2/1' (2/1' + 4^1 yi 2/3 - ^i2//) = 0, in plane coordinates v^ {vi + ^XiV^Vi — XjVs^) = ; thus giving a cone and a conic whose plane touches the cone, while the vertex of the cone lies upon the conic. 200. [2(13)], Xi = X3. The singular surface is yi [y^ + ^ViVi + 2X2=^2/4 (2/1^2 + 2/3^4)} = 0. This is a ruled cubic (Cayley's) together with a plane of its bitangent developable and a point upon it. 201. [(12)3], Xi = X,. The singular surface is Vi - ^^ (2/12/2 - 2/32/4)' + 4X12/1^2/3 = 0- Any plane through A^A^ intersects the surface in two lines which meet on 42-44, while A^A, is a double generator along which the tangent planes of the surface coincide ; hence the surface belongs to class VIII. with a cuspidal generator. 198-203] THE GENERAL EQUATION OF THE SECOND DEGREE 225 202. [(123)]. The complex has as its equation xi + 2XiX^ = 0. The singular surface consists of a plane and a point in it, each counted four times. 203. Seventh canonical form [222]. (O (x) = XiX2 + X^Xi + XsX^, f(x) = 2XiXiX^ + 2X2X3Xi + 2\sXsXg + x^ + x^ + x^. There are three double lines, viz. (010000), (000100), (000001), which meet each other ; when these double lines are coplanar we may write Xi=Pyi, ^3 ^ Pis, ^s'^Pui The singular surface is seen to be* yi [y,' - (^2 - \y yi - (^i - X3)' yi - (Xi - x.)^ y,''] - 2 (Xi - X2) (Xj - X3) (x« - Xa) 2/22/31/4 = 0, which is Cayley's cubic surface of the fourth class, together with the plane yi = through the three double lines. Secondly, when the double lines are concurrent, we may write «1 = ^34, »3 = ^42. «6 = P23, x-i =Pi2, Xi = pij, ajj '^ Pif The singular surface is in this case, (writing Xj, Xj for Xj — X3, X2 — X3), V - (y^ I y^"^ w^v^ (Xi -x.)^ . ^i-^2 _Q. ^'nxi^'+vy ^'^' Xi%^ ^y^y^y^y* w ^' i.e. an equation of the form y3''y4' + y/y4' + y^'ys' = ly-^y^y^yi. The latter equation is one of the forms to which Steiner's quartic surface of the third class can be brought. The singular surface is completed by the point of intersection of the three double lines. * This equation is. most easily derived as follows; Let j/,-, y^ be two points on a line of the complex, where y^ lies in oj , so that Pvj.= - yi -2/1. Pj3= - Vs -Vi' Pu=^- Vi • Vi ' &<=. ; then for any point y^ of the singular surface the locus of 3// is a pair of lines; from which the given equation follows at once. J. 15 226 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI If for yi in the second singular surface, plane coordinates «,■ be written, we obtain the equation of the first singular surface in plane coordinates. There are two special cases. 204. [2(22)], X, = \3. In the first case the singular surface consists of a quadric cone and a pair of points, reciprocally in the second case we have a conic and a pair of planes. 205. [(222)]. In the first case of Art. 203 and the complex consists of the lines which meet the conic 2/1 = 0, y,= + 2/3= + 2/,»=0. In the second case and the complex consists of the tangents to the cone 1)1 = 0, vi + v,^ + v^ = 0. 206. Eighth canonical form [15]. o) {x) = x^ + 2x2Xe + 2a;3a;5 + x^'. f(x) = \xi^ -i- X, (^x^Xg + 2x3X1 + x^) + ix^Xf + 2x3Xi. Here we may write i«>l=Pl2+PM< ""i^Pu, a^3=/'i3, Xt=pi2 Pu, x^ = jip^, x^^ Zpi^. The singular surface is the complex surface for x^ aJa = 0, (j) = x^X + 2a;3a;e + 2xtXi + - — ^ = ; A,i — A, X = being any linear complex. It is easy to see that the double line of the complex surface (all of whose coordinates are zero except x^ is a singular line of ^, and also that each tangent line y of the singular surface of ^ given by the equations ^ + fj.^ =p . ~ , (Art. 157), belongs to <^, if X is the double line. Hence the singular surface of /(«)= is the complex surface for the general quadratic complex, in which the double line is a singular line of the second order. The equation of the singular surface is i'K - ^) iiy' - y.yzT - ^yi'y^y*} + ^yiy^ + y^'y, - y^y.y, = o. 203-210] THE GENEBAL EQUATION OF THE SECOND DEGREE 227 The line y^ = y^ = is the double line, the point 2/1 = 2/3 = 2/4 = is a triple point. 207. [(15)], X, = X,. The singular surface is seen to be Cayley's ruled cubic, together with a cuspidal plane and a point, as in [(11)4]. 208. Ninth canonical form [24]. ft) {x) 'E.XiX^-'r X^Xg + XiX^. f{x) = 2\iXiX^ + xi' + X2 {2x^X6 + 2XiX^) + 2X3X11 + x^. As in the form [222] there are two reciprocal cases. Case (i). x^ = p^ , a;, = p„ , «4 = i^ia , The singular surface is, writing \i for Xi — Tij, 2/1 (^i2/3 + 2/4)' + -22/. (2/1' - V2/4') = 0, together with y, = 0. This is a special case of Cayley's surface, and belongs to the species VIII. of Schlafli*, (a cubic of the 6th class with three proper nodes). Case (ii). !«i=P3i, «:s=Pn< ^4=Pi2, ^2— Put ^S~Plit ^5—Pl3- The singular surface is one of the third class and fourth degree, a special case of Steiner's surface, together with the point of intersection of the double lines. 209. [(24)]. If Xi = 0, the equation of the singular surface is in case (i) 2/1H2/4'' + ^2/12/2) = 0' ^^ point-coordinates, and v^{v^+vf) = in plane-coordinates ; hence, the singular surface consists of a cone and a pair of points on a generator of the cone : in case (ii) we have, reciprocally, a conic and a pair of planes whose line of intersection touches the conic. The complex is a special case of [(22)11]. In each case the complex has A^A^, AsA^ as double lines. 210. Tenth canonical form [33]. ta {x) = 2Xi,Xi + x^' + 2xiX^ + x^^, f(x) = X, (2a;ifl;3 + ajj^) + 2x^X2 + Xj (2a;4a;6 + ajj^) + 2a;4a;5. * See Schlafli "On Surfaces of the third order," Phil. Trans. 1863 ; also Cayley "A Memoir on cubic surfaces," Phil. Trans. 1S69. 15—2 228 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI Here we may write Xs = -2ip^, ixs=pn-Pn, a;3 = 2j)42. The equation of the singular surface is found by the usual process to be yi' (2/3 + 3/4 + \y^) - 3\iyi''2/sy4 + \' (y^yz + yzy*)" = 0. It possesses the two cuspidal lines yi = 0, 3/3 = ; 2/1=0,3/4 = 0. 211. [(33)]. If we take Xj as zero in the previous equation, it is seen that the singular surface consists of the plane yi — O triply, and the plane ^3 + ^4 = 0, together with the point A^ triply, and one other point in 2/1 = 0. The complex is x^x^ + XtX, = 0, or, Pis (p,2 +pst) + pu ( j)i2 - JD34) = 0. It consists therefore of the lines of the 00 ' congruences Pu'^fJ-ipK-Pu), Pu = -f^(Pii+Psi); i.e. Pis + Pii+2/j,p^-=0, p,3-p,4-2/ip,j = 0. These complexes are special, their directrices p and p' having the coordinates P12 Pu Pu P23 Pm Pta p p 2/. 1 -1 1 -2^ 1 Thus p and p form two projective pencils, the plane aj of the latter passing through A^ the centre of the former, while the centre of the latter lies upon A^A^, and the line OA^ corre- sponds to the line of intersection of the planes of the pencils. The complex is therefore the locus of lines which intersect corre- sponding lines of two projective pencils, in which the plane of one passes through the centre of the other, and the intersection ■ of the planes corresponds to the line joining the centres. 212. Eleventh canonical form [6]. (o (x) = 2xjXe + 2X2X1 + 2x3Xi, f{x) = \ {2xiXe + 2x2X5 + 2x3Xi) + 2x1X5 + 2x2X1 + x^. There are two reciprocal cases : Case (i). 3^1 = ^14, X2 = pn, a;s = Pi3. Xs = p23, X5 = pn, Xi=Pti. 210-213] THE GENERAL EQUATION OF THE SECOND DEGREE 229 The equation of the singular surface is 2/1 [yiyi + m - 22/1^32/4} = 0, and therefore consists of the plane 3/1 =? together with a surface of the third degree and fourth class. This surface is the complex surface for the congruence a;i = 0, ^ = x-i^X + 2x2X^ + 2x3X^-^x^ = 0, where X = 'laiXi is any linear complex ; and it is easy to see, if x is the double line of the complex surface, that the members of the ., 9tf) dm i.e. the lines for which 2/5=1. y^= a^ + ji, yi=yi = yi = yi = 0, are all singular lines of and any line, a line of (j), a singular line of (p, of 2nd order, both double, [(41)1] 14 Complex Surface for general quadratic com- plex (j> and a tangent of I one cuspidal, 1 its Kummer Surface. [ I The complex surface has two cuspidal. > twodoublelines which are' Cayley's surface of 3rd There are three degree and 4th class double lines and the plane through which are co- the three double lines, planar or Con- or reciprocally, ciu'rent;in[42] Steiner's Roman Surface two coincide; and the point of inter- in [6] all three section of the double coincide, lines. Complex Surface for a general quadratic complex (f) relative to a singular line of the 3rd order. Class I. of ruled quartics (two double directrices). Class VII. of raled quartics (double directrix). The same with a cuspidal generator. Ruled cubic with two directrices together with a point on the double directrix, and a plane through the single directrix. The same when the point and plane are cuspidal. Class II. of ruled quartics. Class VIII. of ruled quartics. The same with a cuspidal generator. Class XII. of ruled quartics. Ruled cubic (Cayley's) with a point and a plane as in [(11)22]. A case of Class XII., the stationary tangent planes coincide. 214] THE GENERAL EQUATION OF THE SECOND DEGREE 231 [(51)] Number of Constants 13 Degree 3 [(22)11] [1(23)] 14 13 2 2 [(22)2] 13 2 [(42)] 12 2 [(33)] 11 [(n)(n)ii] 15 2 [(11X11)2] 14 2 [(21)(11)1] 14 2 [(12)(12)] 13 2 [(22)(U)] 12 1 [(31)(11)] [(ii)(n)(ii)] [(111)111] 13 13 14 2 1 2 [(iii)(ii)i] 12 1 [(111)21] 13 2 [(111)(21)] 11 1 [(111)3J 12 2 Singular Surface Cayley's ruled cubic and a point and plane as in [(11)4]. A quadric cone and a conic. Quadric cone and conic through its vertex, the plane of the conic touches the cone. Quadric cone and pair of points or recipro- cally a conic and a pair of planes. Two planes and a conic touching their intersection or reciprocally a cone and two points on one of its generators. A triple plane and a triple point together with another plane and a point on it. (The complex is the locus of lines which meet corresponding lines of two projective pencils, in which the plane of one passes through the centre of the other, and the line joining the centres corresponds to the intersection of the planes.) Two quadrics meeting in a twisted quadri- lateral. One quadric becomes two tangent planes of the other. Two quadrics touching along a generator and having in common two generators of the other system. Two quadrics touching along two inter- secting generators. Two planes with a third plane taken twice together with two points and a third point taken twice. (The complex is the locus of lines which meet corresponding lines of two projective pencils in which the plane of one pencil passes through the centre of the other.) As in [(11)(11)2]. Tetrahedron. A quadric taken twice (the complex is the locus of lines meeting paired lines of a regulus in involution [2]). As in the last case. (The involution has two pairs of coincident double elements.) As above. (The involution has two coinci- dent double elements.) As above. (The involution has all its double elements coincident.) As above. (The involution has three of its double elements in coincidence.) 232 THE GENERAL EQUATION OF THE SECOND DEGREE [CH. XI [(211X11)] 11 [(211)2] [(311)1] [(411)] [(221)1] [(123)] [(222)] 12 12 Number of Constants Degree Singular Surface [(211)11] 13 2 Two planes and on their intersection two points, all taken doubly. (Formed by aid of two pencils in (2, 2) correspondence having a self-correspond- ing line.) 1 As before. (The complex is locus of lines which meet corresponding lines of two projective pencils having a common line.) 2 As before. (The pencils are in (1, 2) corre- spondence.) 2 Two planes and on their intersection two points all taken doubly. (The complex is formed by the lines which meet corresponding lines of 2 pencils in (2, 2) correspondence and having three coincident double elements.) 2 As in [(211)2]. (The common line of the planes is double in the involution of one pencil.) 1 A plane and a point on it each counted four times. 1 As in [(221)1]. Complex consists of tangents of a cone or of lines which meet a conic. Complex consists of the tangents to a quadric. There are thus 49 distinct species of quadratic complexes, if the reciprocal cases which occur in [222], [42], [6] and their sub- cases, be considered as of the same species ; if considered as forming different species, we obtain 55 species of quadratic complexes. 11 11 10 [(lll)(lll)] 9 CHAPTER XII. CONNEXION OF LINE GEOMETRY WITH SPHERE GEOMETRY. 215. The fact that a line and a sphere in space of three dimensions have the same number of coordinates suggests the existence of a connexion between the geometry of the line and that of the sphere. Such a connexion was discovered by Lie*, and is discussed in the present chapter. The equation of any sphere in Cartesian coordinates is - 2ttos - 2^y -2y2 + x^+y'' + z'' + G = 0, where C=«^ + /S= + 7=-i?' (i), R being the radius of the sphere. The quantities a, /3, 7, C regarded as its coordinates, determine the sphere, and if a fifth coordinate R be employed, the equation (i) holds between these five coordinates. The last equation may be written C'=(a+/3t)(a-/3z)-(ie + 7)(iJ-7) (ii), by comparison with the equation which holds for the five co- ordinates r, s, p, o", rj of any line (Art. 2), which is 7i = r = ; hence, the spheres of a linear sphere-complex intersect a fixed sphere at a constant angle. Again, since all the lines which belong to two given linear complexes intersect two given lines, it follows that the spheres which belong to two linear sphere complexes touch two fixed spheres. To the table of correspondence lately given must there- fore be added the following : Space A Space Z any point, a point and a line of ?- + o- = through it, the points of a given line p, the points oip' the polar of p, a linear lino complex, a linear line congruence, a regulus. a minimal line, a minimal line and a point on it, one set of minimal lines of a given sphere, the other set of minimal lines of the sphere, a sphere complex, composed of spheres intersecting a given sphere at a constant angle, the spheres touching two given spheres, the spheres touching three given spheres. 218. Surface Element. The association with any point, of an indefinitely small area containing the point, gives rise to the idea of a "surface element"; with each point of space are connected 00 " surface elements, and in space there are oc ' surface elements. With any surface are connected 00 '' surface elements, the plane of each surface element being a tangent plane of the surface. If with each point P of any given surface S we associate in some definite way a plane ir through P, and for the lines p of the pencil (P, tt) take the polar lines p' with regard to a given linear complex G, the lines p form a pencil whose plane tt' and centre P' are respectively the polar plane of P, and the pole of tt, for C. If now 7r is the tangent plane of S at P, then, by Art. 44, the locus of P' is the polar surface S' of S, and the tangent plane to S' at P' is tt' 217-219] CONNEXION OF LINE- WITH SPHERE-GEOMETRY 237 219. Corresponding surfaces in A and 2. In each pencil of tangent lines of S there is one line which belongs to the complex (7, or r -f- cr = ; thus to each point of S one line of C is assigned, and thereby one point Q is determined in S. Through each point of S there passes one complex curve k of C, and on S there are ao ^ such curves k. To a point P of such a curve k and its tangent at P, correspond a minimal line q and a point Q on q. Thus to the locus of tangents and points of k correspond a minimal curve the locus of Q, and its tangents; to S, the locus of the curves k, corresponds therefore a surface T formed by oo ' minimal curves. Again consider the surface S' which is the polar of S with regard to C, the Hue PP' touches at P' one of bo ^ complex curves k' of G on S', to these curves k' correspond the other set of oo ' minimal curves on T. If p is any tangent line at P to S, and p' its polar line, which therefore touches S' at P', to p and p' will corre.spond the same sphere in X ; this sphere contains the minimal lines cor- responding to P and P', and these lines meet in the point Q which corresponds to the line PP' of C ; hence this sphere touches T at Q. To the pencil of tangent lines to /S at P (or to S' at P') correspond the oo ^ spheres which touch T at Q. Thus each surface element of S determines one surface element of T. The connexion between surface elements of S, S' and T admits of simple analytical expression. Per the polar plane of {xyz), or P, in the complex r+(T=0, is where (^ijf) are current coordinates. Since this is the tangent plane to S' at P\ if the surface element of P' be {x', y', z'; - 1, m', n'), where the quantities - 1, wi', n' are proportional to the direction cosines of the normal at P, it is clear that m'=0, n'=-y, and, from the symmetry of the relation between S and S', m^z', n= -y', while since the polar plane of P passes through P' x! = x-nz — my ; thus the surface element at P' is determined. To determine the surface element of T at Q, we observe that the coordinates a, /3, y of § satisfy the equations (vi), also y^r, where r is a coordinate of the line PP', hence x — x' my + nz , ... y=»-= ',=-" (vn). ' z-z z-m 238 CONNEXION OF LINE- WITH SPHERE-GEOMETRY [CH. XII Again since the direction cosines of the two minimal lines through Q are from (vi) proportional to l-z\ -i{l+z^), 2z; ].-z'\ -i(l+0) 22', respec- tively, the direction cosines of a line perpendicular to their plane are proportional to 1 - zz\ - i {zz' -\-\),z + z', hence if the surface element of T at § is (a, ft y ; - 1, mj, Jij) we obtain i(zz' + V) z+z' ' 1 — Z2 ' zz -V ilzm + \) z+m , .... ' \—zm zm-l The equations (vi),(vii),(viii) completely determine the surface element at Q. 220. Principal tangents and principal spheres. It has been seen that to a tangent line p oi S &t P there corresponds a sphere touching T at Q, i.e. having with T a common surface element at Q ; if the tangent to (S is a principal tangent, i.e. if a consecutive surface element of S passes through p, the corre- sponding sphere must have in common with T a consecutive surface element, i.e. it touches T at a consecutive point as well as at P, and is therefore a principal sphere at P; hence, to a principal tangent curve on S corresponds a line of curvativre on T. If (S is a ruled surface, any one of its generators p has oo ' surface elements in common with S, thus T is touched along a line of curvature by the sphere corresponding to p. This line of curvature k is a circle ; for let Q be a point on k, and Q' a consecutive point on the second line of curvature through Q, through Q there passes a consecutive line of curvature k' , and the tangent plane at Q' passes through Q; it follows that Q and Q' are equidistant from the centre of the sphere which touches T along k', therefore k lies upon this consecutive sphere, and the two consecutive spheres intersect in k, which is therefore a circle : T is the envelope of 00 ' spheres. To a ruled surface, therefore, corresponds a surface which is the envelope of c» ' spheres and of which one set of lines of curvature are circles. To a quadric corresponds a surface which is the envelope of two sets of 00 ' spheres, and of which all the lines of curvature are circles, i.e. a Dupin's Gyclide ; each sphere of one set touches each sphere of the other set. 221. Pentaspherical CoordinateB. The analogy between line geometry and point geometry with pentaspherical coordinates will now be investigated* * For full discussion of pentaspherical coordinates see Darboux, La TMorie generate des surfaces, prem. partie, p. 213. 219-221] CONNEXION OF LINE- WITH SPHERE-GEOMETRY 239 The equation of any sphere may be written in the form 2ax + zpy + 272 -h 6 ^—5 1- le ^ — ^ = 0. We find for the coordinates x^, y^, z„ of the centre, the following expressions -aR _ -l3R _ -yR ^'-B + ie- ^"'S + ie- ^»~8 + ie' the radius p is equal to S; — ' . If the sphere is b + ie '^ not a point we may take a'' -I- /S'' -1-7^4- 8^ 4- e^= 1, and then p = K — - . Taking a second sphere {x^yoZg p') we easily find that {X, - xjy + (2/„ - yo'y + {z, - z,y -p^- p'' and hence that the spheres are orthogonal if aa' -I- /3|8' + 77 + BB' + ee' = 0. Now consider five mutually orthogonal spheres S, = 0,...,S, = 0, of radii p^, ...,^5; the equation of any one of them is 2a*^ + 2^,2/ + 2y,z + S, ^^ + 2/' + ^'-^^^ ^ . x' + y' + z^ + R' - . , where we have by hypothesis and ctkUi^ + ^k^kf + ykykf + BA + €k€if = ] these two groups of equations are those of linear orthogonal substitutions in five variables and we infer therefore that a,2 + a," + a,' + a/ + «,' = 1, | tti/Si + a^A + ttsA + «4/34 + Hsft = 0, &C.J ^^^' Now if Sk is the "power" of any point in reference to the o sphere 8k = 0, — is the first member of (a) ; denoting it by Xk we ^ Pk see from (c) that 240 CONNEXION OF LINE- WITH SPHERE-GEOMETRY [CH. XII The quantities x^ are called the pentaspherical coordinates of the point with reference to the given five mutually orthogonal spheres, and the fundamental relation between them has been shown to be 2a;t* = 0. The following results proceed from this definition of the quantities xt: (i) Any linear equation in x, as Sa^xt = 0, represents a sphere. (ii) Two spheres Sa^ajj; = 0, ^a'^x^ = 0, are orthogonal if Saia'ifc = ; for Sat^ri = 2*5:0^0^ + SySai^* + 2ztakyk a^ + y^ + z" - E" ^ . .a^ + y^ + z^ + R"^ + ^-^^ ^at:6k + I ^— ^ Zaiet, and this sphere is orthogonal to Sa't^t = if lakttkta'kak + = 0; i.e.: by (6), if SofcCi't = 0. (iii) The radius of Sa^aJt = is seen to be Pk If therefore Sa*' = we have a point, if 2 — = a plane ; in the former case the quantities aj, are the pentaspherical coordinates of this point {x'y'z'), and SatiKifc ^ a {x — x"^ + y — y'^ ■{■ z — z'^]. (iv) If Xk and x'^ are any two points, the condition 1,xjcx\ = states that each of the points lies on the sphere whose centre is the other point and which has zero radius*; this may be expressed by saying that each point lies upon the null-sphere of the other. (v) If lajcXk = is any sphere and x^ any point, the coordinates of its inverse point with regard to this sphere are x'k where Gx\ = Xk%a^ - la^lakXt ; for if P and P' are inverse points with reference to any sphere, and Q is any point on the sphere, we have where m is a constant. * This does not involve the coincidence of the points. 221] CONNEXION OF LINE- WITH SPHERE-GEOMETRY 241 Hence each point Q of the given sphere which lies on the null-sphere of P will also lie on the null-sphere of P' ; so that if P is xjc and P' is x^', then any point fj; which satisfies the equation ^(^hSk = and either of the equations 2^fta;j;=0, Sfta;*' = 0. will satisfy the other ; hence a-xic' = Koo^: + \aje , (^ = 1, , 5) ; from which, by squaring each side and adding, the result follows. The connexions between line geometry and point geometry with pentaspherical coordinates, derived from these results, are set forth in the following Table. Line Geometry Point Geometry with pentaspherical coordinates Six fundamental complexes, in mutual involution, ^i = 0, ..., x^^O. Fundamental relation between 6 coordinates of a line, 'S,xjp=0. 1 6 Linear Complex 'S,aj^ji=Q. 1 6 Special Complex, if 2a!j:^ = 0. 1 Two linear complexes (cm;)=0, {a'x) = Q) are in Involution if 6 1 Two lines xj„ s:{ intersect if 6 1 Two lines x,^, x{ are polar with regard to a linear complex 'S,a-^Xjc=Q \ix^=\a^-\-)ix^,... (k—\, ... 6). Five spheres mutually orthogonal, a'i = 0, ...,^5 = 0. Fundamental relation between co- 5 ordinates of a point, 2a;j2 = 0. 1 5 Sphere 2a|.a;ji=0. 1 5 Point Sphere, if ^a^=Q. Two spheres are orthogonal if 6 1 Each of two points is on the null- 5 sphere of the other if Sxj,x,,' = 0. 1 Two points are inverse with re- gard to the sphere Saj,x^=0 if Xi=\a^+liX^', ... {k=l, ... 5). At any point Xi of the surface /"(% ... Xs) = there are oo ^ tangent spheres, whose equation is f /- -1- fjuci) Vi = 0. 1 \dxi "^ I" This is shown just as in the case of the tangent linear complexes of fix\ . . . a^s) = 0. The similarity in form of the equations of the oo ^ tangent spheres at the point Xi and the oo '■ tangent linear complexes of the line Xi, suggests one of the most important connexions of line- and sphere-geometry. For as has J. 16 242 CONNEXION OF LINE- WITH SPHERE-GEOMETRY [CH. XII been seen (Arts. 74, 76, 115), when a tangent linear complex. is special its directrix touches the singular surface ; when it is special and bitangent (Art. 131), its directrix is a double tangent of the 6 singular surface ; while, for the quadratic complex 2 'l^oo-^ = 0, 1 these bitangents form the six congruences (Art. 83) y'=^' ^T^=^' (^+*')' (^■=i.--6). Ic \lc — ^i But a special linear complex corresponds to a point-sphere, and the centre of a point-sphere which is bitangent to a surface is a focus*, hence, to a double tangent of the singular surface of a complex there corresponds a focus of a surface. 5 The five focal curves of the surface IXiXi- = are thus seen 1 to be given by the equations y,= 0, 1^1^ = 0, (k^i); if to i the values 1, ... 5 are successively attributed. As in the case of the congruences of bitangents of the singular 6 surface of "^.X^Xi^ = 0, these curves are not affected by the 1 substitution of for X^ ; i.e. the oo ' surfaces A, + /i 5 r? 2-^^ = 1 A-i + M are confocal. Hence we have the result that confocal cyclides correspond to cosingular quadratic complexes. For the purpose of comparison we place side by side corre- sponding characteristics of cosingular complexes and confocal cyclides. 00 1 cosingular complexes, a bitangent of the singular surface, six bitangent congruences of the singular surface, four complexes of the system through any line I, the four tangent linear complexes of I with regard to these complexes are mutually in involution. 00 1 confocal cyclides, a focus of the cyclides, five focal curves, three cyclides of the system through any point P, the three tangent spheres at P of these surfaces are mutually orthogonal, i.e., confocal cyclides cut at right angles. * Salmon, Geom. of three dimensions. Third edition, p. 108. 22i] CONNEXION OF LINE- WITH SPHERE-GEOMETRY 243 The quantities at in the equation of a sphere ■ Saia;j = 0, 1 completely define the sphere : we may introduce a sixth co- /l 6 ordinate a^, where iae = \/ 'Zaf, in which case Sa/ = 0. The V 1 1 condition of (internal) contact of the spheres («o. 2/o, -^o; p) and «, y^', V; p) being (x, - w,'y + (y, - y.J + {z, - z^f -p^- p" + 2pp' = 0, is seen from the preceding to be 5 5.5 ta^a^ — Sa/ 2a/^ = 0, 1 11 6 i.e. 2 aiai = 0. 1 A complete correspondence is therefore now established between the geometry of the line in Klein coordinates and that of the sphere in the coordinates «»; in fact we have returned to the spbere-geometry of Lie which has been discussed in the preceding Articles of the present chapter, in which the intersection of lines corresponds to the contact of spheres. 16—2 CHAPTEK XIII. CONNEXION OF LINE GEOMETRY WITH HYPERGEOMETRY. 222. For five variable quantities Zi ... X^, there are oo ' sets of values of their ratios : each such set of values may, from the analogy of space of three dimensions, be said to define a " point," and the totality of such points to constitute a " space " S^ of four dimensions, which thus contains oo * points. If A and B are any two points of Si the locus of the co ' points Ai + XBi will be called a line of S^ 5 Any linear equation of the form liatXi = singles out oo ' 1 points from St, which will then form a space of three dimensions ; the locus of these oc ^ points will be called a hyperplane ; there are clearly oo ' hyperplanes in S4. Any line which does not lie in a given hyperplane will obviously meet it in one point only ; if two points of a line lie in a hyperplane the line will lie altogether in that hyperplane. Two h3rperplanes intersect in a plane ; for any line in one of them meets the other in one point; a plane in 84 is therefore defined by two linear equations in the quantities X ; there are 00 ^ planes in Sf. Three hyperplanes which have not a plane in cqmmon, intersect in one line, thus a line is determined by three linear equations; there are 00 ^ lines in Sf Four hyperplanes have in general one point in common, i.e. any two planes of S4 meet in one point. Three points not in the same line define a plane of S4, and four points not in the same plane determine a hyperplane ; hence two lines which do not intersect determine a hyperplane. A line p in one hyperplane will not in general meet a plane a in another hyperplane ; hence through any line p there pass 00 " 222-223] LINE GEOMETRY AND HYPERGEOMETEY 245 planes obtained by constructing the planes which pass through p and the points of a. 223. Equations connecting lines of A and points of S^*. Any six linear complexes ajj = Xe = being taken as those of reference, and w = la^XiXk = being the fundamental relation, then, since -D,(a) = J-iifti^ + . . . + 2J.„ar«s + ■ • •, — H (a\b) = AiT,aibi + . . . + A^ (arbg + asbr) + ..., it follows, that if any one of the complexes, say Xi = 0, is special, An = 0, and if Xi=0, Xk = are in involution, A^ = 0- If now «! = 0, ajg = 0, ajj = 0, 0:4 = are any four linear complexes in mutual involution, and Xs = 0, Xe = the two special complexes whose directrices are the two lines common to the first four complexes, then - n (a) = Audi + A^a^'' + As^a^^ + A^ai^ + 2A^a^a^ ; all the other coefficients A being zero in this case. Now since the Anc are the first minors of the discriminant of ^OLikXiXjc, the ajj; are proportional to the first minors of the discriminant of Xi(a), and therefore in the present case, and the fundamental relation assumes the form -^11 -^ii -"-33 -^144 ■'^se But we may take A -A -A -A -11:^-1 jln — .^22 — -^33 — -0-44 — 2 ~ ' in which case the fundamental relation becomes a?!^ -I- «2' + ooz + x^ - n^iHHs = 0. If we now write p.Xj^ = XiXs, p.Xs = X^, ^ 4 p.X^ = X^Xs, p.Xe = 'S,Xi .(i), p.X3 = XaXs, p.Xi = XiXs, * The theory here given is due to Klein, see "Ueber Liniengeometiie und metrische Geometrie," Math. Ann. v. 246 LINE GEOMETRY AND HYPERGEOMETRY [CH. XIII a (1, 1) correspondence is established between the lines of the three dimensional space A and the points X of S^. Exceptional elements, however, occur in this correspondence ; for denoting by s the directrix of the special complex a;,, if x^^O then is also X5 = 0, and therefore one solution is a;i = a;2 = ^s = ^4 = 0, which gives s; so that s corresponds to any point of the hyperplane 4 Z, = 0. If, however, in addition to X, = we have SX/ = 0, then must p = 0, axid the lines corresponding to a given point X whose coordinates satisfy the last two equations, are determined by ^__^__^_:^ r-n Cii"* — Y — Y — Tr I ■''5 — "i \^^J- 1 ^2 -0.3 ^4 4 The locus Xs = 0, SXi' = is a quadric surface contained in 1 the hyperplane X^ = 0, it will be denoted by ; thus to any point X of the oc ' lines given by equation (ii) correspond ; these lines are those of a pencil which includes s ; for they are determined by the equations a;i = pXi, {i=l, 2, 3, 4), hence if a is any line, the equation 'S.Xi 3— = gives one value for -*, i.e., one of these lines meets any given line. Hence the lines determined by (ii) pass through a point P of s and lie in a plane tt through s ; all the lines of such a pencil correspond to the same point of . To the » ^ lines of the complex ajj = there correspond the x ^ points of . If two lines x, x', corresponding to different points X, X' of ^, 4 , .. * intersect, Sa^tO;/ = ; hence it follows from (ii) that 1,XiXi' = 0, 1 1 4 4 which, together with the equations 'S,X' = 0, 2Xi" = 0, shows 1 1 that each point of the line Xi + X,X/ lies on <1>, XX' is therefore a generator of 4> : hence it follows, firstly, to each of the 00 ' pencils through s of centre P corresponds a point on a generator o-j of <1> ; secondly, to each of the oc ' pencils through s whose plane is tt corresponds a point on a generator a^ of . Now these two sets of pencils have one pencil in common, viz. (P, tt) ; hence cti and ffa intersect each other and therefore belong to different systems. Thus with each point of s is associated one generator o-j of , and 223-224] LINE GEOMETRY AND HYPERGEOMETRY 247 with each plane through s is associated one generator cr^; so that with each pencil containing s is associated one point of . The nature of the correspondence expressed by equations (i) is therefore the following : Space A Space Si any line, a point, exceptionally the line s, any point of ^5=0, the lines of a pencil whose centre and plane are united to s, one point on *, a linear complex through s. a hyperplane, a linear congruence through s, a regulus through s. a plane, a line. 224. Correlation of Schumacher*. The foregoing corre- spondence may be obtained as a special case of a more general (1, 1) correspondence between the lines of A and the points of S^. Having ■^rveiT'two lines p^ and p^ of Si which do not intersect, we can establish a (1, 1) correspondence between the 00 ' planes of S4 which pass through p^ and the points of any plane a^ in A, and similarly a (1, 1) correspondence between the 00" planes of Si through P2 and the points of any plane a^ in A ; provided that A is not the hyperplane 2 determined by p^ and pa. Any point P of Si determines one plane through pi and one plane through p^, and hence one point Qi in Oj and one point Q^ in Oj, thus P corresponds to the line Qi4; conversely, Q1Q2 determines a point in «! and a point in a^, and hence two planes through pi and p^ respectively, and therefore one point P in Si. The 00 " planes through pi are X, = \X„ X, = imX, (i), where Z5 = is the hyperplane containing pi and p^, and Xj = 0, X2 = are any two hyperplanes through p^. In the coUineation of the planes through p^ and the points of Oi, a plane given by (i) corresponds to the point Q^, or Xi, of ai, where K.Xi = Ci + XBi + fiAi (ii), provided that the point C is the correlative of the plane (Xj, X^), the point B of {Z5, X^), and the point A of (X5, Xi). * " Classification der algebraischen Strahlensysteme," Math. Ann. 37. 248 LINE GEOMETRY AND HYPEEGEOMETBT [CH. XIII Any hyperplane through p^ will have an equation of the form aXi + /SZj + 7X5 = (iii); and on substitution in the last equation from (i) we obtain oX + j8/x + 7 = 0. This equation connects the quantities X, fi of the planes through pi in the hyperplane (iii). It shows that the locus of the points Qi, corresponding to these 00 ' planes, is a line of a^ ; hence, each hyperplane through p^ determines a line of a^, and vice versd. In the general case the hyperplane Xs will not correspond to s (the line of intersection of a^ and otj). Similarly the ao ^ planes through p^ are Xs = pX„ Xi = a-Xi (iv), where X, = 0, X4 = are any two hyperplanes through p^ ; and the point Q2, or yi, which corresponds to this plane is given by the equations T.yi = Gi' + pBi' + aAi (v), where C" is the correlative of (Z3, X,), B' of {X^, X,), A' of {X„ X,). If ^ is the line Q1Q2, it follows from (ii) and (v) that v.pik = {cc')ik + p {cb% + a- {ca% + \ (bc')ik + p. (ac% + \p (bb% + p-a- {aa')ik + Xo- (6a'Xt + p,p {ab^ ( vi), where {cc')i}e is a Plilcker coordinate of the line CC, &c. If we now assume the collineation between the planes through Pi and the points of a^ to be such that the line s corresponds to X5 = 0, both A and B will lie on s. Similarly if also in the second collineation the line s corresponds to X5 = 0, the points A' and B' will lie on s. The quantities (66')*. (aa')i*. (6a')it, (ab% are now coordinates of the same line s ; hence taking s as one edge of the tetrahedron of reference in A, the four quadratic terms disappear from jf?t;e of the equations (vi). 5 Finally, if S,, or SafXi = 0, is any hyperplane whatever and if we substitute in its equation for the X^ by means of (i) and (iv), we obtain the equation OiX + a^p + Uap + a4or + 05 = (vii), connecting the quantities X, p., p, a. Now eliminating the latter quantities between (vii) and the previous five equations of (vi), we obtain a linear complex, which passes through s, and whose lines correspond to the points of Sj. 224-225] LINE GEOMETRY AND HYPERGEOMETET 249 Hence, if the line corresponding to X5 = is for both collinea- tions the line s, to the points of any hyperplane will correspond the lines of a linear complex through s. We observe from equations (vi) that, in the general case, to a linear complex of Q1Q2 corresponds a locus of points in 84 of the second degree. It was seen that to the points of any given line in a^ there correspond the planes through pi in a given hyperplane; thus to the lines which intersect any two given lines of Ui and Oj respectively, correspond the points of a plane in 84 which meets pi and p^ ; in particular, to the lines of a plane system in A correspond the points of such a plane in 84. When the coUineations satisfy the condition that to s the hyperplane (pi, p^) corresponds in each of them, the equations of Art. 223 may be regained by taking as coordinate complexes in A the special complex a^j = whose directrix is s, four complexes a!j = 0, cu^ = 0, ajj = 0, x^ = 0, in mutual involution to each of which s belongs, and the special complex «6=0 whose directrix is the second line common to x^ = 0, aij = 0, x^ = 0, Xi = 0. For let the hypei-planes which correspond respectively to the four complexes in involution be Xj = 0, Xj = 0, X3 = 0, X4 = 0, and let X5 = be the hyperplane which corresponds to s, then we have from (vi) «! = trXi, Xi= a-Xi, Xs^aXg, Xi = aX4, Xs = aX^, while ajj is a quadratic function of Xi . . . X5 ; but since to {x) = x^ + x^ + x^ + xl — x^x^ it follows that x^ = (Xi^ + X^ + Xs^ + X^^) -^ , and writing a- = pXs, w^e obtain the original equations. 225. Correlatives of the lines of any plane system and sheaf of A. When m (x) has the above form, the Invariant — 0(a) of any complex l,aiXi = is Oi" + aa^ + as^ + a/ — 4a5a6 ; 4 if the complex is special and contains s, then ae = 0, 1ai' = 0, and the coordinates of its directrix are (%, a^, a^, a^, 0, — ^a^); to the lines of this special complex correspond the points of the hyperplane 5 '2aiXi = 0. 1 This hyperplane 'touches' * at the point (ai, i/j, «3, ^4) 0), since it passes through this point and every point on * consecutive to it ; the hyperplane 250 LINE GEOMETRY AND HYPEBGEOMETRY [CH. XIII therefore contains the generators o-i, o-j of * through the point. If a^ vary, the directrix describes a pencil (P, n) containing s (Art. 223), where P is the point corresponding to cr^ and n the plane corresponding to o-j. 5 If 'S.hiXi = is another special complex which contains s, and 1 of which the directrix is either concurrent with s and the directrix 5 of 'ZoiXi = 0, or coplanar with them, then 1 lai' = l.aibi = h^ = (i). 11 1 Hence to the lines of the plane system, or of the sheaf, determined by 5 5 laiXi = 'ZbiXi = 0, 1 1 correspond the points of the plane 5 5 1 ,1 this plane passes through the line (Oi + Xfti, a„ + Xfcj, ffls + Xftj, ai + \bi, 0) of Si, which from (i) is seen to be a generator of ^. Hence to the lines of a plane system correspond the points of a plane through a generator o-j of , to the lines of a sheaf correspond the points of a plane through a generator cr^ of ^. If the centre of any sheaf is P, and the plane tt of any plane system meets s in P', then the line PP' of the sheaf and the intersection of tt with the plane (s, FP'') correspond to the same point of 4> (Art. 223); hence the generators o-j and a^ have one point in common and therefore belong to different systems. To sheaves whose centres lie in the same plane w through s correspond planes through the same generator o-j, to plane systems whose planes pass through the same point P of s correspond planes through the same generator If a and b are any two intersecting lines, to the lines of the pencil Oj + \bi correspond the points of a line Ai + \Bi, and since any pencil in A contains one line which meets s, the corresponding line will meet $, i.e., to the lines of a pencil in A correspond the points of a line which meets 4> ; to the '.c^ pencils which contain the line (ai, Oj, a,, at, 0, —205) correspond the 00^ lines through the 5 point (oi, Oj, as, a4_ 0) in the hyperplane 'ZaiXi = 0. 1 225-226] LINE GEOMETRY AND HYPERGEOMETET 251 226. Metrical Geometry. The locus of points in S^ of which the equation is X^" + X^ + X^ + Z/ = 0, and which corresponds to the lines of x^ = 0, forms a three-dimensional space which is met by any line of /S4 in two points ; such a space is therefore denoted by S^. The points common to S^ and any hyperplane "2.^ form a two-dimensional space which is met by any line of Sj in two points and is therefore a quadric in S3. In geometry of three dimensions the properties connected with the sphere-circle and the plane at infinity are called metrical, and a quadric through the intersection of x^ -\- y'^ -^^ z^ = 0, and the plane at infinity is a sphere ; by analogy, in four-dimensional space, the three-dimensional quadric spaces through the intersection of 8^ and ^5 = 0, i.e. which contain $, may be termed ' hyperspkeres.' 6 To a linear complex 1,aiXi=0 corresponds therefore the 1 hypersphere X,ia.iXi + a,XXi' = 0. 1 1 In the equations of connexion of A and 84 we may, to complete the analogy, take X^ as being unity, we then have ZXi TT ^2 -tr -^S XT- *^4 K 'V 2 ^6 1——, ^2="! - (x) ; they involve either a collineation or a reciprocity, in A, (Art. 40). It will now be shown that the corresponding transformations in S, are anallagmatic, i.e. change hyperspheres into hyperspheres. For any such transformation in A being iKi = a/«i'"+ + as'a-'e'. «6=/iV + +fs«'s; the connexion between the corresponding points X and X' is given by the equations a/X/ + + a/ + a/2X/^ X = 1 , e/Z/ + + e; + ee'XZ/^ 1 d^X^ + + d^ + d:^X^ ^' " e/Z/+ +e^+e:2X^' 4 //X/ + +/,'+/,' w 1 with, of course, equations of similar form having the accents only on the left. Thus in 8^ a hypersphere is changed into a hypersphere. All anallagmatic transformations are equivalent to one or a finite number of combinations of the following kind* * See Koenigs, La GSometrie riglee, p. 125. 254 LINE GEOMETRY AND HYPERGEOMETRY [CH. XIII (i) translations X]' = Xi + hi, X2 = X2 + hi, Xa = -^3 + "3) Xi = -A 4 + hi ; (ii) similar transformations Xi = mXi, X2' = mXj, Xs = mX^, X/ = mX^ ; (iii) refleasions Xi = + Xi, X2 = X X2, X3 — + Xs, Xi = + X4 ; (iv) inversions y / nrJii „, K JL^ y ' _ 3 y ' — ^ '~2Z^' ^~2Z^' "^'~"SZ^' "^^"2Z^- 228. Principal Surfaces of A and Lines of Curvature of S4. The Principal Surfaces of a complex, (Art. 132), are chiefly of interest as being the analogues of the lAnes of Curvature of a hypersurface. For, taking the hypersurface F {Xi, Zj, Z3, Xi, 1) = obtained from the line complex /(aji ... aje) = 0, to the tangent linear complexes of the latter there correspond the tangent hyperspheres of .F = 0, viz. 4 sp 4 44 ^{Yi-Xi)^ + ^ (2S YiXi -XXi-% F/) = 0. 1 O^i 1 11 It was seen, (Art. 132), that for any line x of/ there are three tangent linear complexes which touch f in x and also in three respective lines consecutive to a; ; we have therefore in Si the result that for each point of F there are three tangent hyper- spheres which also touch F in three respective consecutive points. From the analogy with three-dimensional space, these hyper- spheres are called Principal Spheres ; and we infer that there are for each point P of a hypersurface three Principal Spheres, the lines joining P to the three consecutive points of contact p, p» p//, yjg^jjg jjg^jigjj ^jjg Lines of Curvature of i^ at P. , The three lines consecutive to x, just mentioned, are in an involutory position with regard to each other, (Art. 133) ; corre- spondingly, the lines PP', PP", PP'" are mutually orthogonal*. As another instance of the analogy between A and Si, the theorem of Art. 65 leads to the following result in Si : — of any six hyperplanes in ;S'4, any four pass through one point, and by means of iive of the hyperplanes we obtain^w such points ; through each such set of five points one hypersphere passes, and hence six hyperspheres are obtained, corresponding to the six sets of five hyperplanes ; our theorem is then that these six hyperplanes have one common point. * For taking 15=1, from the equation u(dx'\cW) = Q, (Art. 133), we derive 2dXj'dXi"=0, &o. 227-230] LINE GEOMETRY AND HYPEEGEOM.ETRY 255 229. Line Geometry is point geometry of an S^^ in an Sg. Any six quantities «! ... oo^ may be regarded from another point of view as being point coordinates of a 'space' of five dimensions Ss, and when they are coordinates of a line in A they satisfy w (x) = 0, i.e. they are coordinates of a point in a ' space ' of four dimensions contained in 8^ ; moreover under- standing by a 'line' of Sj a locus of points which satisfy four linear equations 6 6 6 6 'LaiXi = Q, 1hiXi = Q, 20^ = 0, l.diXi = 0, 1111 the space represented by &» (a;) = is met by any line of S^ in two points, and is therefore denoted by S^ ; thus line geometry may be regarded as point geometry of an 'S^ in S^. To a linear 6 complex l.aiXi=Q, (Art. 225) ; and m of these m + n points lie on o-j, (Art. 223). Similarly in any plane 62 through a generator era there lie m + n points, corresponding to the lines of the congruence which belong to any sheaf of centre P and the plane system (P, s) ; of these points n lie in a^. Thus in S4 we have 00 " points, corresponding to the lines of the congruence, and such that m + n of them lie in any plane of St; these points therefore form a 'surface' in S^ which may be denoted ty Pm+n. Since any line meets a hyperplane in one point only unless it is wholly contained in it, if any point of /S4 be joined to all the points of Pto-1-71, the joining lines will project these points into c»'' points of any given hyperplane S, which does not contain 0, to form a surface /m+n : the degree of this surface is m+n; for, since 230-232] LINE GEOMETRY AND HYPERGEOMETRY 257 in each plane of St there are m + n points of Fmrrn, the plane through and any line p of S contains m + n points of Fm+n, which are projected into m+n points of/^+m lying on p ; i.e. p meets fm+n in m + M points. The surface fm+n will, ia general, possess a double curve, for any hyperplane through contains a curve of points of Fm+n, and this curve will have a certain number of "apparent double points" which are projected into coincident points of fm+n- Through will therefore pass a cone of chords of F,n+n- An exception occurs when F^+n is entirely contained in the same hyperplane. 232. Rank of a congruence. Any point on $ being taken as the centre of projection, then if (a^, a^, a,, a^, 0) are the coordinates of 0, it was seen (Art. 225) that the hyperplane s 2ajXi=0 touches and corresponds to a special linear complex 1 whose directrix d belongs to the pencil (P, tt) associated with (Art. 223). To the points of any line through in this hyper- plane there correspond the lines of a pencil containing d. Denoting this tangent hyperplane by 2, we observe that 2 meets the cone of chords of Fm+n through in A chords, if h is the degree of the cone ; now 2 contains the two generators a^ and 0-2 of at 0, while a-^ contains ^m (m — 1) pairs of points of Fm+n> and a^ contains |7i(w — 1) pairs of points of F^+n', the number of chords common to 2 and the cone, exclusive of o-i and 0-2, is therefore h — \m (m — 1) — ^n {n—V) = r. This number r is called the " rank " of the congruence, and has the following meaning for the space A; if P and tt are the point and plane of s which correspond to a-i and tr^, and d is the line of the pencil (P, tt) which is the directrix of the special complex which corresponds to 2 in A, then d lies in a pencil with two lines of the congruence r tim,es. Since now with any given congruence the coordinate systems employed are quite arbitrary, it is seen that in general two lines of the congruence will lie in one pencil with any given line d a definite number r of times. 17 CHAPTER XIY. CONGRUENCES OF LINES. 233. Order and class of a congruence. A set of oo ' lines, such that any two given conditions determine a definite finite number of lines of the set, is said to constitute a congruence. The locus of lines which belong to each of two complexes is one instance of a congruence. The requirement that a line of the system should pass through a given point is equivalent to two conditions, and the number of lines of the congruence which pass through any given point is called its (A-der ; similarly the number of lines of the system which lie in any given plane is called its class*. When the congruence is the complete intersection of two com- plexes, its order and also its class,, is equal to the product of the degrees of the complexes. It is convenient to designate a line of a given congi'uence by the term ray. A congruence whose order is m and class n is termed a congruence (m, n). A given congruence (m, n) establishes on any planes tt and ir' a correspondence, such that to each point P of tt correspond m points of tt' (where the rays through P meet tt'), and to each point of it' correspond m points of tt ; while, if P is any point on the line (tt, tt'), the pencils (P, tt), (P, tt') are so related that on any line p of one pencil and on any line p' of the other pencil .there are n pairs of respectively corresponding points, determined by the rays of the plane (p, p'). The most general congruence can be constructed as follows: it was seen (Art. 231) that the degree of a ruled surface, whose generators are those rays of a given congruence which meet any given line, is m + n; in any plane tt take any pencil of lines and construct the equation in point-coordinates of the most general ruled surface of degree 7n + n which has a line p of this pencil as m-fold directrix (i.e. such that through each point of p there * See Art. 231. 233-234] CONGRUENCES OF LINES 259 pass m generators), and which has any plane through p as n-fold tangent plane (i.e. any plane through p contains n generators); then taking X as the parameter upon whose variation p depends, express each coefficient of the equation of this surface as a poly- nomial of degree m in X ; the oo " generators of the oo ' ruled surfaces form the required congruence (m, n). The ruled surface of degree m + n formed by the lines of a given congruence which meet any given line I will be denoted by (I). 234. Halphen's Theorem. Two congruences (m, n), (m', n') have mm' + nn' rays in common ; this theorem was shown by Halphen, the following proof is due to Schubert. The class of a surface (l) of the first congruence is m, + n, for the tangent planes to (I), through any line I' which meets it in the m + n generators p, p' ..., are the planes through p, p' ... and I ; hence their number is m + n-*. Similarly the class of (l) for the second congruence is m' + n'. The class of the developable common to the two surfaces (Z) is therefore (m + ?i) (»w.' + n'), which is the number of planes through any point A passing through a generator of each surface ; but nn' of these planes coincide with the plane (A, I), hence there remain mm' + mm! + m'n planes through any point which contaiu a ray p of (m, n) and a ray p' of (m',n'), where p, p, I are concurrent. We now seek the surface which is' the locus of a line p of (m, n) such that the point of intersection P of p with a line p' of (m', n') lies in a given plane a, while the plane tt oi p and p' passes through a given point A. By taking the line I in a, it is seen, from what has just been proved, that for any line / there are mm' + mn' + m'n planes tt through A, hence the locus of P is a curve k of degree mm' + mn' + m'n ; reciprocally, the envelope of the planes tt is a cone of vertex A whose class is nn' + nm' + n'm. The degree of the required surface is therefore equal to mm' + m'n + mn together with the number of pairs p, p' for which p lies in a ; and since there are n rays p in a which are paired with the n' rays of (m', n') in the planes (A, p), the degree of the required surface is seen to be mm' + m'n + mn' + nn' = (m + n) (m' + n'). This is also the degree of the locus of the corresponding rays p'. * The degree and class of a ruled surface are in general the same. 17—2 260 CONGRUENCES OF LINES [CH. XIV If {B, y8) is any arbitrary pencil, such a ray p determines one line of the pencil, while it meets the locus of ^ in (m + n){m' + n') points the generators at which determine just as many other lines of the pencil ; thus in the pencil a {(m + n) (m' + n'), (m + n) (m'+ n')} correspondence is established. The number of coincidences is therefore 2 {m + n) (m' + n'), (Introd. xv.). Such coincidences may arise (i) from rays common to the two systems ; (ii) from rays p, p' whose plane passes through B ; this occurs riw' + mn' + m'n times, since this number is the class of the cone vertex A ; (iii) from rays p, p' whose point of intersection lies on /8; this occurs mm! + mn' + m'n times, since this is the degree of the curve in a; therefore the required number of common rays is 2 (m + n) (rn' + «') — (nn'-f- mn' ■\-m'n)—{Tnm' +m'n+mn)=m,m' -\-nn' . By taking m' = n =k it is seen that the number of rays of (m, n) which belong to a complex of degree k and meet any line is k{m, + n); hence, a complex of degree k has in common with (m, n) a ruled surface of degree k (m + n). 235. Characteristic numbers of a congruence. In addition to the order and class of a congruence, it was seen in the last chapter that there is a third characteristic number, viz. the rank* r, which is the number of times two rays belong to the same pencil with any given line I. If the system is the inter- section of two complexes of which one is linear, it is seen that r = 0, since I does not in general belong to the linear complex. Two loci are of special importance in the present theory+: (i) the locus of points of intersection of rays in a plane tt which turns about a line I, is a curve which will be denoted by \l\; r points of this curve lie on I, and in any plane tt there are ^n{n — l) points of the curve which do not lie on I, hence the order of il\ is ^n{n—l) + r: (ii) when tt describes a sheaf of centre P, the locus of such points of intersection is a surface which will be denoted by (P). This surface is seen to be the locus of points on the lines I of this sheaf at which two rays belong to the same pencil with I, it is also the locus of the oo ' curves 1 1 1 when I describes a plane * Schumacher, to whom the introduction of this characteristic is due, used the term Art. t See Art. 250. 234-237] CONGRUENCES OF LINES 261 pencil of centre P. Now since there are ^m (m — 1) pairs of rays through P, there will be that number of these curves through P, which is therefore a point of multiplicity \m (m — 1) on (P), and since any line through P meets this surface in r points distinct from P, it is seen that the degree of (P) is r + |m(m — 1). 236. Focal points, planes and surface. The ruled surface formed by rays which meet any given ray I is still of degree m + n; for I' being any line, I meets (V) in m + n points, hence (l') and (Z) have m + n rays in common. Any line p meets m + n generators of the surface ; if ^ meets I in P, then p meets (l) where it meets (i) the n — 1 rays of the plane {I, p) distinct from I, (ii) the m — 1 rays of the point P distinct from I ; hence the intersection of I with p must be counted twice to complete the number m + n of intersections of p and (Z). It follows that Z is a double generator of (Z), and there are two points upon any ray for each of which there are only m — 2 rays distinct from it ; hence, each ray is intersected by two consecutive rays. These two points on a ray are called Focal Points, (Art. 119): the locus of the ao "^ Focal Points is called the Focal Surface of the congruence. If I and Zi are consecutive rays meeting in Pi and Z/ is con- secutive to Zi and meets it in P/, then Pj and P/ are ultimately focal points, thus li (and hence any ray) touches the focal surface at each focal point. Let the focal points on any ray I be Pj and Pj, and on Zi, the consecutive ray through P„ be P/ and P/, then Pj being consecutive to Pa', the plane (Z, Zj) touches the focal surface at Pa, since it contains two tangent lines of the surface at Pj, viz. Z and F^F^. Thus if Zi and Zj be the two rays consecutive to Z which meet it, the plane (Z, Zj) touches the focal surface at Pj and the plane (Z, Zj) touches it at Pj. The planes (Z, Zj), (Z, l^ are called Focal Planes. All the rays of a congruence touch the focal surface twice, but all bitangents of the surface are not rays of the given congruence. 237. Degree and Class of the Focal Surface. Upon any arbitrary pencil of planes whose axis is Z the congruence effects an involutory correspondence [n.(m— 1)]; for, any plane tt through Z contains n rays which determine n points P on Z through each of which pass m — 1 other rays not in tt, and regarding the n (m — 1) planes through Z and these other rays as corresponding to ir, we have determined an involutory correspondence [re(m— 1)]. 262 CONGRUENCES OF LINES [CH. XIV The number of coincidences is 2n(m— 1) which may arise (i) from the coincidence of two rays through a point P ot I; such a point P belongs to the focal surface ; (ii) from the r points on I through each of which two rays lie in the same pencil with I ; in this case the plane ir through such a pair of rays p and p' coincides with two of its corresponding planes, since one of the planes determined by p is {I, p), i.e. it, and one of the planes determined by p' is {I, p), i.e. it. But in an involutory correspondence in which an element coincides with two of its corresponding elements, such an element counts for two of the coincidences (Introd. xv.) ; hence if mj is the degree of the Focal Surface, i.e. the number of points in which I meets the Focal Surface, wii = 2w (m — 1 ) — 2»'. Reciprocally, we obtain an involutory correspondence [m {n — 1)] of points on / in which two points correspond which are the inter- sections with I of two rays in one plane through I; and it follows by similar reasoning that if n^ is the class of the Focal Surface, i.e. the number of tangent planes to the Focal Surface through I, «!= 2rn(n — 1) — 2r. 238. Singular Points. If more than m rays pass through a point S, then oo ' rays will pass through S, which is called a singular point of the congruence. For taking any line I through 8, m + n of the rays which meet I also meet any line V, but if a finite number m', greater than m, of rays pass through S, then any line I' through S, such that the plane {I, I') does not contain any of these rays, will meet rri + n rays of {I), which is impossible. Thus the rays through S form a cone, say of degree h, and (I) breaks up into this cone, which will be denoted by {Sti), and a ruled surface of degree m + n — h. Similarly, if more than n rays lie in a plane there will be oo ' rays in that plane which envelope a curve. It is clear that each singular point S^ lies on each surface {I), and since {Sj^ meets I ixi h points, therefore Sji is an /i-fold point of(0. Each surface (P) passes through each singular point. In the case of a singular point S,, each line of the pencil of rays whose centre is S^ touches the focal surface ; the plane of the pencil is therefore a singular tangent plane of the focal surface, the curve of contact being of degree n{m — \)—r. 237-239] CONGRUENCES OF LINES 263 239. Expression of the coordinates of a ray in terms of two variables*. The equations of Art. 223 establish a (1, 1) correspondence which connects any line of the space A with a point of four-dimensional space St ; so that to each line of the given congruence (m, n) there corresponds one of the points of S^, and vice versd, ; the latter points being projected from a point of Si upon any given hyperplane, give rise to a surface /^H-m- Hence a (1, 1) correspondence is determined between the lines of the congruence and the points of y^^^. If u and v are the variables in terms of which the coordinates of any point oi fm+n CQ^y be expressed, and xt is any line of the congruence, this correspondence may be set forth by six equations of the form so that the coordinates of any line of the congruence may be expressed in terms of two variables. The foregoing mode of expression enables us to deduce important properties of a congruence : it was seen, (Art. 9), that if two consecutive lines x and x + dx intersect, the equation (dx^) = must hold ; if each of these lines belongs to the con- gruence, we have Edu^ + 2Fdudv + Gdv^ = 0, we infer, therefore, that any line of a congruence is intersected hy two consecutive lines of the congruence, viz. those which correspond to the two values of -r- determined by the last equation, (see also Art. 236). The equations of a linear congruence are of the form 6 6 y., = laiyi, y^=1hyi; 3 3 if the eight constants «;, hi satisfy the six equations 6 dx-i, 4 dxi dxi 4 3*i x, = taiXi, _ = 2aig^, Yv^T'^V |,, dx^ ^-, dxi 9^2 vi 9*'i x^^^h^u 9^=f6,g^, Yv^TTv' * This mode of treatment of the congruence belongs to what is known as Differential Geometry. The most celebrated memoir in this field is that of Kummer, GrelU's Journal, Bd. 57 (1860), "AUgemeine Theorie der geradlinigen Strahlensysteme. " 264 CONGRUENCES OF LINES [CH. XIV this linear congruence contains a; and any consecutive line 9a; , dx , X + ir- du + IT- av du av of the given congruence ; hence, there are oo ' linear congruences which contain any line x of a given congruence and all lines of the latter congruence consecutive to x. The constants Oi, hi are determined if the additional condition be imposed that the linear congruence should pass through any given line x ; hence, through a given line and any line x of a given congruence K, one linear congruence can he constructed which passes through aU lines of K consecutive to x. It is easily seen that there are oo " linear complexes which pass through any line x oi K and all lines of K consecutive to x. 240. As an instance of the expression of the coordinates of the lines of a congruence in terms of two variables, consider the case in which the functions fi are quadratic expressions ; i.e. The identical equation {x'')=0 gives rise to a number of equations between the coefficients, among which are the following (o2)=0, {ah)=0, (62)=0, 2(A2) + (a6)=0. The two equations which determine the rays which meet any two lines are quadratics in u and v ; hence there are four rays which meet any two lines, i.e. the sum of the order and class of the congruence i&four (Art. 231). On making the substitution nu' v' the coordinates Xi become proportional to quadratic expressions in u' and i/ ; in these expressions, the coefficient hi of iu'v' is easily seen to be nhi+fi. If therefore n be taken as either of the roots jij, n^ of the equation jii!(A2) + 2n(A/) + (/2)=0, the hi' are coordinates of a line ; while, as before, we have (0-2) =0, {a'h')=Q, (6'2) = 0, 2(A'2) + (a'6') = 0; hence (a'6')=0, and the quantities af, hi, b^ are the coordinates of three mutually intersecting lines. These lines may be either concurrent or coplanar ; in the former case from consideration of the equations P23"l-i'l3"2 = 0- These equations, since ^34=0, are seen to be equivalent to one only; hence each point of A1A2 is a singular point of the congruence. The locus of singular points breaks up into A^A^ and a conic c% in the case where a/, hi and 6/ are concurrent; and A1A2 must meet c'^, for if not, then through any point P there would pass two rays, viz. the lines joining F to the intersections of the plane (P, A-^A^) with A Similarly if ai, hi, 6/ are coplanar, the congruence consists of the lines of intersection of the planes through A-^A^ and the tangent planes of a cone which is touched by A^A^. The equations which give the points ^i of the above twisted cubic may be taken as P-li=*'> P-^2=*% P-^i=t, p.|4=l; the equations giving the coordinates pn of its chords are then which corresponds to each of the lines p, p we have two intersecting curves, hence such a point is a double point on Pm+n, (since the tangent plane is there indeterminate). The cone of chords for every point of Si passes through these r double points which lie on 4>, and such a double point is projected into a double point of G. 242-244] CONGRUENCKS OF LINES 269 Hence the number of tangents proper from any point of Si to ■'' m+n IS 2h —1r = m (m — 1) + n (?i — 1). Applying to the congruence, we obtain these results : (i) through any two lines of A there are oo ^ reguli each of which contains two rays of the congruence ; (ii) m(m — l) + ?i(»i— 1) of these reguli contain two con- secutive rays. 243. Triple secants of F. Of the chords of Fm+n through any point P, there is in general a finite number t of chords which meet F^n+n three times, therefore there is a finite number of triplets of rays which belong to the same regulus with any two given lines. This gives t triple points on the double curve. In terms of the four numbers m, n, r, t, all the characteristics of the focal surface may be calculated. 244. The Focal Surface. Since to the lines of a plane system there correspond the points of a plane tj of 8^, (Art. 231), if such a plane touches F, i.e. contains two consecutive points of F, the corresponding plane system contains two consecutive rays, and its plane is therefore a Focal Plane ; similarly to a plane e^ which touches F corresponds a Focal Point : it will now be shown by Schumacher's method that the locus of focal points is identical with the envelope of focal planes. Observe, in the first place, that the oo ^ planes through any line I of Si which meet F in two consecutive points, are projected from any point of I upon any hyperplane 2 into the tangent lines through to fm+n, being the point in which I meets 2 ; and since any hyperplane 2' through I is projected into the plane (2, 2')> such of these tangent planes as lie in 2' are projected into tangents from to the section of/m+m by (S, 2')- Now let X, X' be two consecutive points of F and 2' in such a plane, then if 2' contains another point X" of F, consecutive to X, which does not lie in the plane {XX', I), 2' will contain two consecutive planes through I each of which meets F in two consecutive points ; for since the plane XXX" is projected into (2, 2'), the latter must touch /; thus through in (2, 2') there pass two (consecutive) generators of the tangent cone from to /, which are the projections of two (consecutive) planes through I in 2', each of which meets Fm+n in two consecutive points. 270 CONGRUENCES OF LINES [CH. XIV Again through any plane, and hence through the plane XX' X ', there pass two tangent hyperplanes of 4> ; for if the plane be la.Zi = 0, ihiXi=Q, 1 1 the hyperplane 2 (rtj + Xhi) X; = 1 1 touches , if S (af + Xftj)" = ; 1 let now the points X', X" be so chosen that both XX' and XX" meet , and I be taken as the generator ) ; it follows that one of these two tangent hyperplanes contains o-j, and taking it as 2', in 2' there pass two consecutive planes through a^ , each of which meets F in two consecutive points. To %' corresponds in A a special linear complex whose directrix p meets s (Art. 225) ; to this complex belong the ray a: and the rays x', x" consecutive to x and meeting it, since XX', XX" both meet 4> ; thus p passes through the intersection of x with one of these lines and meets the other, e.g. p passes through {x, x) and meets x". Then, by what has been proved, it follows that on the line p iliere will intersect two rays conseeutive to x, x' ; hence jo is a tangent to the locus of focal points, and the plane {x, p), i.e. the plane {x, x"), is the tangent plane at the point {x, p) to this locus (since it contains two tangents to the locus at the point, viz. x and p), i.e. the focal plane {x, x") touches the locus of focal points at the point {x, x"). 245. Degree and Class of the Focal Surface. We proceed to give the application of this method to the determina- tion of the degree and class of the focal surface. The degree of the focal surface is equal to the number of times two consecutive rays meet on any line p which meets s; and to the lines which meet p correspond the points of a hyperplane 2 of /Si, tangent to ^ at some point P at which the generators of $ are (say) a^ and associated with TT (Art. 223). These planes are projected from a point of a-^ into the generators of the tangent cone V from iV^ to /; corresponding to a double point of the section of the focal surface by tt there is a plane through o-j which contains two pairs of consecutive points of F, which gives on projection a double edge oi V: a plane which contains three consecutive points of F is projected into a cuspidal edge of F; to this there corresponds in the section of the focal surface a point through which three consecutive rays pass ; such a point is a cuspidal point, and hence arise on the focal surface a double curve and a cuspidal curve. * Since a^ contains m points of F.„^^^ and o-j contains n points of F^^ (Art. 231). 272 CONGRUENCES OF LINES [CH. XIV 247. Rank of the Focal Surface. The degree of the tangent cone to the focal surface, or the class of its plane section, is called the Rank of the focal surface, and is equal to the class of /; for it has been seen (Art. 244), that to a line p which touches a section of the focal surface by a plane tt through s, corresponds a tangent hyperplane of $ which contains a tangent plane to F ; also the tangent hyperplanes of which correspond to the lines of a pencil whose plane ir contains s, themselves form a pencil, and are of the form 1 1 where 2 a<'' = 2 cnbi =lb,^ = 0, 11 1 and pass through a generator o-j of , where a« corresponds to tt ; they thus contain the same plane a through a-^, viz. kaiXi = 0, ibiXi = Q. 1 1 These hyperplanes are projected from any point of o-j into a pencil of planes in 2, whose axis is the intersection of a and 2 ; and such of them as contain tangent planes to F are projected into tangent planes to / through the line (a, 2), hence the class of a plane section of the focal surface is equal to the class off. The number of tangent planes through any line I to fm+n is equal to the number of intersections of the first polar /',„+„ of any point P on I, with the curve of contact a of the tangent cone to fm+n from any point P' of I, diminished by the number of inter- sections of a with the double curve d : we proceed to find the latter number. Of the intersections of a and d, some, are at ordinary points of d, and for such the tangent from P' meets one sheet of f and touches the other, and therefore meets f in three consecutive points; such points thus lie on the second polar/" of P', and consist of all the intersections of d and /" except those arising from multiple points of f of higher degree than the second (for through such points a will not pass in general) ; their number is therefore h(m + n — 2)— (the number of intersections of d and /' ' at higher multiple points). Now at M there are m tangent planes and hence — ^^-^ 247-248] CONGEUENCES OF LINES 273 branches of d and M is a.n m — 2-fold point of /", similarly for N, also d contains t triple points, hence the number of intersections of d with /" which lie upon a, i.e. the total number of inter- sections of d and a, is 7n + n — 2 . -, „j = ^ («m + 2r) - St. In the next place it is seen that d and a touch in 2h — 2r points, viz. at the cuspidal points of d, for at such a point every plane thiough the tangent to d at the point is a tangent plane of /, hence the element of d is also an element of the curve of contact of the tangent cone of any point. Now since the order of a is m + n.m+n — 1— 2A= 2mn — 2r, the required class of /"is (2mn — 2r) (m + n — 1) ^ {mn + 2r-)-3i-t-2(»i.m-l+w.ji — 1) which is therefore the Rank of the Focal Surface. 248. Determination of r and t for the intersection of two complexes. When the congruence is the complete intersection of two complexes of degrees ;* and v respectively, i.e. a congruence, whose order and class are fi-v, the values of r and t can be determined. To one complex corresponds a 'space' jS'^'* of degree 2^ which con- tains * fi-fold; to the other a 'space' jS^" of degree Zv which contains * v-fold ; the complete intersection of these spaces consists of * counted fiv times and a 'surface' F whose points correspond to the rays of the congruence. It has been seen that r is the number of apparent double points of the curve of points of F which lie in any hyperplane. Take therefore a tangent hyperplane 2 of * which meets it in the generators o-j", a-^ whose intersection is P; 2 meets S'^'^ in a surface f'^'^ of which S^'', diminished by the number 2A— 2r of tangents from P to C (Art. 242), which touch both .S^** and ,S^^ Now the first polar of P for iS ** contains 4>, (^t — 1) times, and in particular the generators o-/ and , is four and its class 2n; for each ray p touches the focal surface twice, and if it met the surface again, then through the latter point there would pass three rays, viz., p and the (two coincident) rays through the point ; hence the degree of the focal surface is four, and (Art. 237), 4 = 2n — 2r, hence, r = n — 2, therefore the class of the focal surface, being 4 (rt — 1) — 2r, is equal to 2n. 250. The Surfaces (P). To each point is assigned one plane by the system, viz., that of the two rays through the point ; this plane will be termed the null-plane of the point. A surface (P) is therefore the locus of points whose null-planes pass through P; every surface (P) passes through each singular point Sj^ and is of degree n—1 (Art. 235); the point P is not a singular point of (P) since here — ^-^ = 1. Any ray I through S/^ meets a surface (P) in n — h points exclusive of Sji, viz., in its intersection with the n — h rays of the plane (P, I) which do not pass through Sh', hence Sh is a singular point on (P) of order n-l-{n-h) = h-l. The line SkP meets (P) in n-h-1 * See Rummer's important memoir, ' ' Ueber die algebraiscben Strablensysteme insbesondere uber die der ersten uud zweiten Ordnung," Berliner Abh. (1866). 249-252] THE CONGRUENCES (2, n) 277 points exclusive of Sf, and P, all of whose null-planes pass through P, and since P may be any point, it is seen that for a line through a singular point Sh,r = n — h—l. 251. Each singular point of the congruence is a double point of 4>. If we form for any line I through Sji the corre- spondence of Art. 237, we obtain an involution [n — h] on the pencil of planes of axis I, disregarding the lines of (S^) ; and of the 2 (w — A) coincidences 2 (m — h — 1) are due to pairs of rays in the same pencil with I, thus leaving only two coincidences due to intersections of I with <&, the Focal Surface, apart from Sh, hence Sji must be a double point of $. Since by each point P one null-plane is determined, the equation of such a plane is P,a;,' + P,a;,' + P,a;,' + P,x: = (i), where x' is any point on the plane, and the Pj are functions of the coordinates of P. If a/ be fixed and the ajj be variable, we therefore obtain the equation of the surface (P) corresponding to «/ ; the degree of the P{ is therefore n— 1; between them the following identity exists : P^Xi + P^x^ + Psa;s + PiXi = (ii). The surface (i) contains each of the rays through P, since the null-plane of qny point on these rays passes through P; they will thus lie on any polar of (P) with regard to P, and are the intersections of the polar plane and polar quadric of (P) for P. 252. Double rays of the congruence. The null-plane of any point on the focal surface is determinate, except for a point Sh, where h>l. If two rays coincide without any definite point of ultimate intersection we have & double ray; for each point of such a double ray the null-plane is indeterminate, hence such lines do not belong to the focal surface. The points whose null-planes are indeterminate are given as the intersections of three of the surfaces Pi (through which therefore all four surfaces must pass, from (ii)) ; thus the double rays when they exist are common to the surfaces Pj and hence to each surface (P). These surfaces cannot intersect in a curve, since a curve of singular points is excluded. The curve of intersection of two surfaces (P), e.g. for the points A and B, consists of the curve \AB\ together with the double rays ; hence, the number of double rays = {n-iy-{in{n-l) + n-2}=i(n-2){n- 3). 278 THE CONGRUENCES (2, n) [CH. XV Conversely, if a line p lies upon a surface {P) and is independent of the position of P it is a double ray. A point of intersection of a ray with a double ray must be a singular point Sj^ with a cone of rays ( ?i — 1 ; thus the degree of a cone {Sh) having its vertex on a double ray cannot be less than three. For each of such a pair of cones the double ray is a double edge, and if a cone has a double edge it is a double ray. In all cases except when n. = 6 the double rays are concurrent. For if p and p are two non-concurrent double rays, while Sh^ , Sh^ are the vertices of ray-cones on p, and Sh^, Sh^ on p', (Sh,), being at least of degree three must have a double edge, otherwise it would meet p' in more than two points, i.e., there would be three singular points on p'; and this double edge passes through Sh, or Sh^ say Sh,; similarly for Sh,; thus there must in this case be four double rays and hence at least six (since the number of double rays is i{n— 2) (n — 3)), i.e. n, the class of the system, must be at least six, so that hi + h„ is at least eight. Thus one of the cones (Sh^, (Sh,) is at least of the fourth degree, and it cannot be of greater degree than the fourth, otherwise there would be more than two singular points on p', hence both (Sh,) and (Sh,) are of the fourth degree and have two double edges, similarly for (Sh,) and (Sh^). The double rays form a tetrahedron. The class of the system is six. For all other values of ?i the double rays are concurrent in a point which is necessarily a singular point, since the number of double rays, if such exist, is at least three, except when n. = 4, in which case there is only one double ray. The singular points of a congruence (2, n), therefore, in general belong to one of three classes: 252-254] THE CONGRUENCES (2, n) 279 (i) The point S through which all double rays pass. (ii) Those through which one double ray passes ; the degree of such has been shown to be at least three, and it must be equal to three, for if not, its ray-cone would either meet any double ray in more than one point besides S, or would have an additional double edge, i.e. double ray, which latter double ray would not pass through S. It follows that the degree of the ray-cone of S is n—1; the degree of the ray-cone of the other point on each double ray is three. (iii) Those through which no double ray passes ; the ray-cones of such points cannot be of greater degree than the second in order to avoid meeting the double rays in more than two points. 253. The class of a congruence (2, n) cannot be greater than seven. On the generators of a cone (S^) the vertex /S^ is one local point, the other focal points form the curve a of contact of (Sh) and ; but through a double point of a surface there pass six lines having four-point contact with the surface, hence A ^ 6, hence the degree of (Sn-i) is not greater than 6, i.e. re < 7. 254. Number of singular points. It has been seen (Art. 238), that every surface (I) has each singular point S^ as an /i-fold point, and that two such surfaces have re + 2 rays in common. Every point common to three such surfaces is either a singular point or a point where a ray common to two of the three surfaces meets the third ; this occurs '3(n + 2f times. A singular point with a ray-cone of degree h counts as h^ points of intersection of the three surfaces, hence if Uh is the number of points with a ray-cone of degree h, we have {n + 2y=S{n + 2y+a, + 2'a^ + S^(U+ (A). A curve 1 1 1, being the locus of points of intersection of rays in each plane through I, is a double curve on (l); each point of inter- section of I Z I and any surface (I') is either a singular point*, or at the point one of the rays for 1 1 \ coincides with the ray for (V), i.e. it is a point in which one of the re -|- 2 rays common to {I) and (Z') is met by one of the remaining re — 1 rays in its plane through I. A point Sji is an A-fold point on {I') and an — -^-fold point on * Since through it pass two rays for | Z | and one ray for (V). 280 THE CONGRUENCES (2, n) [CH. XV \l\, and hence counts as — ^ — - points of the intersection of | Z | and {I'), therefore {n + 2){^in-l) + n-2} = (n + 2)(n-l) + 2«2 + 9as+. It will now be shown that the equations (A) and (B) are sufficient to determine the number of singular points. For it has been seen that, except for n = 6, there is one point Sn-i through which all the double rays pass, and except for n = 4, when there are two points S^, there cannot be another S'n-i, since in that case S'n-^ S, would have to be a double ray to secure that in any plane through S'n-^Ss there should not be more than n rays. Each point Sz lies on a double ray, therefore «n-a = l, a, = i(n-2)(n-3). When w = 5 we have one point S^ ; when n = 6 we have either four points Si, and therefore no point 8^, for if S^ existed, (StSi) would be of degree at least nine, i.e. n+3; or one point S^ and therefore no point S4 as before. In no other case is there a point Si or 8^, while 7i = 6 thus gives two dififerent congruences. The equations (A) and (B)* are thus sufficient in all cases to determine the numbers olj and a^ ; solving them we obtain the results embodied in the following Table. (2,2) (2,3) (2,4) (2,5) (2. 6)1 (2, 6)11 (2.7) 0, 16 10 6 3 1 Ck 5 6 6 4 8 as 2 3 6 10 a* 1 4 «5 1 «€ 1 Sa 16 15 14 13 12 12 11 From the Table it is seen that the number of singular points is 18 — n which is the number of double points of required to reduce its class to 2k. The double points of are therefore identical with the singular points of the congruence. * A third equation 2aj.h=4(n+2), due to U. Masoni, exists between the numbers a^. di Napoli, vol. xxn. p. 145. See Reiidiconti dell' Accademia 254-256] THE CONGRUENCES (2, n) 281 255. Distribution of the singular points*. The points of contact of the rays give rise to an involutory (1,1) correspondence of points on ^. But in this correspondence, to each point Sh there will correspond all the points of the curve a whose order is 2h, in which (Sj,) touches *, (Art. 253). If Q, Q' are the points of contact of a ray with 4>, since the null-plane of Q touches ^ at Q', the points of the curve of contact a' of 'i> and the tangent cone to from any point P will correspond to the points of the curve a- which is the intersection of $ and the surface (P). Since any point Sh is of order h — 1 on (P) and 2 on , it is of order 2 (/i — 1 ) on o", also a' is the intersection of and the first polar of P with regard to <1>. Now cr passes 2 (h — 1) times through S^, hence 2 (A — 1) of the intersections of o-' and a correspond to Sh ; but the first polar of P for meets a in 6h points, of which h coincide with Sh, since Sh is an h-told point on a, (Art. 253); deducting the previous 2 (h — 1) points there remain 3A + 2 of the 6h points on a and 0-' other than ^S* which have corresponding points on cr other than Sh', moreover these Sh + 2 points as being on (Sh) have Sh also as corresponding point, i.e. each has more than one cor- responding point and is therefore a singular point on (Sh); and these oh + 2 points are the only singular points on (Sh) other than (Sh), since (Sh) and $ have only the curve a in common. In the case of a cone (Sh) with a double ray, the curve a has the other singular point on the double ray as a double point, so that this point counts as two intersections of the first polar and a; hence, since the number of double rays through Sh is easily seen from the Table to be | (h— 1) (h — 2), the number of singular points on (Sh), including >Sift, is B(h + l)-^(h-l)(h-2). Thus, for instance, each singular plane contains 6 singular points, each cone (Sa) „ 9 „ „ each cone (S^) „ 11 i.e. all except 7 —n. In (2, 7) each (S3) passes through all singular points ; in (2, 6)11 each (^4) passes through all singular points. 256. Conjugate singular points. Two singular points are said to be conjugate if the line joining them is a ray. Each singular plane, since it meets each double ray, must do so either in * For a detailed investigation of the singular points see Sturm, Liniengeometrie, Bd. II. S. 43—60. 282 THE CONGRUENCES (2, n) [CH. XV a point Ss or in the point /S'„_i, hence in each such plane cr there lies one such point, but not both, since then at each point P of the double ray SgSn-i there would be an additional ray P/STj, where Si is the centre of the pencil of rays in a, which is impossible. Each (Ss or Sn-i is conjugate to each S^ and S3, for if not, the planes Sji^^SaSs, Sn^jSsS^ would contain at least n-\-l rays, which is impossible since these planes are, by the foregoing, not singular. A plane cr, the centre of whose pencil of rays is iSi, which contains a point /S,j_, will contain ?i — 2 points S.^ and 6 — w other points Si ; for if I be any line of intersection of (S,i_i) and a; except S^Sn-i, (I) consists oi (Sn-i), the pencil (Si, a), and a quadric surface, but at each point P of I the rays are I and SjP, hence this quadric can only arise as a cone (1%) whose vertex is on I. No S^ in a can lie outside such a line I since Sn-i is conjugate to each S^; hence there are n — 2 points S^ in a, and therefore 6 — n points Si other than that for which cr is the null-plane. In the congruence (2, 3) we notice that all the points S^ are conjugate; to each of the 10 pairs of points S^ there is one Si conjugate as being required to make up the order five of {S^S^) \ thus each of the 10 singular planes includes two points S^. 257. Equation of a surface (P). If the point Sn-i of a congruence (2, n) be taken as the vertex Ai of the tetrahedron of reference, since it is a point of order w — 2 upon the surface (P) of any point, the equation of such a surface must be of the form «i<^ + A/r = 0, where + X3Xi-\lr' = 0. It is then clear that the surface contains the 2 (w — 2) lines X3 = 0, = 0; Xi = 0, = 0; (P) therefore contains at least 2 + ^{n-2)(n-3) + 2{n-2) lines. The existence of the latter lines is also shown by the fact that the plane section of (P) through a ray of P and Sn-i meets it in this ray and a curve of degree w— 2 having an n — 2-fold 256-258] THE CONGRUENCES (2, n) 283 point (at (S„_i), the curve must therefore break up into n — 2 lines through /S„_i* The rays which meet any line p through Sn-x form a ruled cubic surface of which p is the double directrix, they will therefore all meet a single directrix. This result also follows from the form of the null-plane of any point which is a;,' Pi + x^ (a;, (^2 + ^^) + a;/ {x-,^^ + i/tj) + a?/ {x-^^i + ■<^i) = 0, the null-planes of the points of any line through )S„_i are obtained by keeping x^, x.j, x^ constant and varying x^, and are therefore seen to form a pencil. 258. Tetrahedral complexes of the congruences (2, n). If in the congruence (2, 6)n the four points S^ be taken as the vertices of the tetrahedron of reference, since the surface (P) of each vertex includes the ray-cone Q of that vertex, the surface (P) of any point x/ is therefore represented by ^i«iQi + oc^ihQ-s + Xa'ctsQs + Xi'a^Qi = ; where the O; are linear in the coordinates. This equation also represents the null-plane of any point Xi ; but since the edges of the tetrahedron of reference are double rays, ajQi, ot2Q2. "sQs. "4Q4 vanish identically for any point on sueh an edge, this requires that cCi ^ a?i, ofg ^ a^2j ^3 =^ ^3j ^4 ^ ^4j and the surface (P) has as its equation a^i a^i ^Ki r a?2 x^ (jjg + a?3 x^ \j^3 -r a^4 x^ (t^^ = U. The cone Qi = has -4i^2, ^^1^3, A^A, as double edges, hence v^i ^ tt23a72 a?3 "T" 0-^Xq x^ -p {X42a74 5^2 -r x^x^x^p^ where /3 = is a plane through A^. Similarly (j/g ^ ^13*^1 "^3 + ... , Vs — Ciaa?! x^ -r ... , V4 = (Xi2a7i"a?2 + ■«. • The null-plane of any point P in the plane x^^Q is therefore ^23^2^3^! ' t'l3^ia^3^2 I Ci2a^ia^2^3 ^ ^j and this plane meets x-^ = in the line Oi3a?3a?2 "r Ci^x^x^ = U. Now the coordinates of P being (a;,, ajj, aij, 0) and the coordinates of the point in which one ray through P meets a^ being (0, a;2',a;3', a;/), we have P12 ^ a?ia?2 , ^34 ^ a;3a?4 , ^13 ^ a/ia?3 , ^42 = — a;2a74 , * See Sturm, Lin. Geom. Bd. 11. S. 48. 284 THE CONGRUENCES (2, n) [CH. XV hence each ray of the congruence belongs to the complex It follows that the congruence belongs to a tetrahedral complex for which the four points S^ form the fundamental tetrahedron. The congntence (2, 5). Take as vertices of the tetrahedron of reference the point S^ and the three points S,, it follows as in the last case that the null-plane of asi has as its equation Xi Qi + a;>2 Qi + X3X3 Q, + Xt'xt Q4 = ; where Qi = is the ray-cone of Ai, etc. ; since in this case Ai is a fourfold point on each surface (P), the cones Q^ = 0, Q3 = 0, Qi=0 each contain a^ in the first degree only (Art. 257) ; moreover since Q2 passes through ^3 it cannot contain x^^, thus the result of putting 0:4 = in Q^ gives merely a term a^x^x^; similarly from Q3 arises a^Xixi ; hence the null-plane of a point P in 3:4 = is ^2 Vl *■ X^X^X^ ^(X2^3^2 '~'~ '^3^2^S / ^ ^' The trace of this plane on a;/ = is i.e. as in the previous case, the null-planes of the points of a line of the pencil (.4i, 04) pass through a line of the pencil {Ai, a.^) and hence the rays of the congruence belong to a tetrahedral complex of which the point Si and the three points S^ form the fundamental tetrahedron. The congruence (2, 4). Take as vertices A^ and ^3 of reference the two points 183, and two non-conjugate points S^ as ^2 and At; the equation of the null-plane of any point x is then ^I'Qi + ^,'^Q, + x,'Q> + Xi'BQi = 0. Since A2 and A4 are non-conjugate points, Q2 contains a term Xi^, and Qi a term x^'; as before /8 and B pass through ^1.43, moreover /8 = a;j, S = Xi, for if yS contained a term x^, then in the identity aJiQi + «2/8Q2 -f- X3Q, + XiBQi = 0, a term a;2a;4' would arise which could not be cancelled, similarly for S. Again the term which does not involve x^ in Qj is bx^Xg, in Q3 it is aXiX^, therefore the null-plane of any point in a;4 = is The trace of this plane on Xi=0 is bxgx/ + 0x^X3 = 0, i.e. the 258] THE CONGRUENCES (2, n) 285 system is contained in a tetrahedral complex which has the tetra- hedron of reference as its fundamental tetrahedron. A pair of non-conjugate singular points S^ can be chosen in three ways ; for through either point S^ and a point 8^ there passes one singular plane cr, viz. that of the pencil required to complete the degree of (S2S3), let S-^ be the centre of this pencil ; then in cr there is one other point of the second order S2 and two points of the first order (Art. 256) ; each of these latter three points is non- conjugate to S^; also the plane a' through S^, S^ and the other point of the third order S3 is singular since it contains at least five rays ; let the centre of the pencil of rays in it be /S/, then in cr' there are two additional points of the first order each non-conjugate to 82, therefore the points non-conjugate to S^ are 82 and four points 81; thus 82 being conjugate to eight points (Art. 256), must have four points 82 conjugate to it, i.e. to each point 82 there is one other point 82 non-conjugate to it; this gives three pairs of non-conjugate points 82, hence the congruence (2, 4) is contained in three tetrahedral complexes. The congruence (2, 3). Since all points 82 are here conjugate, and each cone (82) contains eight singular points exclusive of the vertex, such a cone must contain four points S^. Any two points /S'», 1S2' have one point 8^ conjugate to each of them, viz., the centre of the pencil required to complete the degree of (8282') ; it follows that three cones 82 together contain nine points >Si, and hence that there is one point 81 non-conjugate to any three points 82. Since these three points may be chosen in 10 ways, there are 10 tetrahedra whose vertices are three points 82 and a point 81 non-conjugate to them. The null-plane of any point a; for such a tetrahedron is «i'Qi + ai/Q, 4- aja'Qs + ^/a/S = 0, where a = is the null-plane of 8^, and has the form Axi + Bx2 + Gxs = 0. Since a/3 can only contain tCi, X2, x^ in the first degree (Art. 257), it follows that /8 = ajj : the part of Q2 which does not involve Xi is 0^X3 and that of Q3 is 'bx^X2\ hence the trace of the null-plane of any point in x^ = 0, upon a;/ = 0, is ax^x^ ■¥ fc^^a;,' = 0, which shows that the congruence is contained in a tetrahedral complex whose fundamental tetrahedron is that of reference ; it follows from the foregoing that the congruence (2, 3) is contained in 10 tetrahedral 286 THE CONGRUENCES (2, n) [CH. XV complexes. It will be seen in the following chapter that the congruence (2, 2) is contained in 40 tetrahedral complexes*. 259. Non-conjugate singular points. If Sh, Sk are non- conjugate singular points, the curve of intersection of {Sh) and {Sk) meets the focal surface in singular points only, since through such a point of the focal surface there pass two non-consecutive rays ; the number of such points is 2h}i , since the order of the curve of intersection of {Sh) and 4> is 2h. Taking the two singular points as being each of the first degree, we observe that on the line of intersection of the null-planes of two non-conjugate points S^ and S^ there are two singular points Sh, Sh' (say), and the rays through any point P of ShSh' being PS^, PS/ it follows that {ShSh') consist? of {Sk), {Sh') and the pencils whose centres are S^ and S/ ; therefore h + h! = n. The points Sh, Sh' are non-conjugate, since, if ShS^ were a ray, three rays would pass through each of its points. If Sh and Sh- are any two non-conjugate singular points the surface {ShSh') breaks up into two surfaces ; for the surface {P) for Sh consists of {Sh) together with a surface Q of degree n — h — 1, and Q contains ^ {n — 2){n- S) —^ {h — l){h — 2) double rays. Similarly the surface (P) for Sh' consists of {Sh') together with a surface Q' of degree n — li— 1. In the next place we observe that the curve \l\ for ShSh' consists of the intersection of Q and Q' apart from the double rays, ^{n-2){n-3)-:^{h-l){h-2)-:^{h'-l){h'-2) in number, which Q and Q' have in common ; thus \l\ is of the degree {n-h-l){n-h'-l)-i{n-2){n-S) ,. ,. , +i(h-l){h-2) + ^{h'-\){h'-2), which is equal to i{n-h-h' + 2-l){'n-h~h' + 2-2). Now I ^ I is a double curve on {ShSh'), which latter surface is of the degree n-h-h'+2, after subtraction of {Sh) and {Sh'), but I is also a part of the double curve of {S^Sh'), i.e. this surface possesses a double curve whose order is greater at least by unity than that possible for the double curve of a surface whose degree is n-h-h' + 2. Therefore {ShSh') must break up into two surfaces. * See also Arts. 123, 146. 258-260] THE CONGRUENCES (2, n) 287 It follows from this result that for two non-conjugate points S,,, S^, such that h + h'=n, there are two points S^ conjugate to S,, and to S^' ; since {ShS^^) which is here of degree 2 must split up into two plane pencils. We notice that if two points S-^ are both conjugate to the same S,^ they must be non- conjugate to each other. 260. Reguli of the congruences (2, n). A regulus of rays is formed by such as intersect a line I which passes through a point Sn-i and lies in a singular plane ^ ; the various lines of the pencil {Sn^i, a) give rise to qo * reguli of the system of lines. In (2, 6)i the cone {S^) contains twelve, i.e. all the singular points, and hence passes through the single point S^; thus through Ss there passes one singular plane, giving oo ' reguli formed by rays of the congruence. In (2, 5) the cone (S^) passes through two of the three points Si. In (2, 4), (2, 3), (2, 2) through each point Sn-i there pass several singular planes. In the case of each system of reguli, two of these reguli pass through a given point P, viz. those determined by the lines I, I' in which the two rays through P meet the pencil ( the rays coincide, and one of them must be S^P, hence the regulus passes through S^; again each cone (Ss) passes through Sn-^ and therefore meets / in one other point, i.e. the regulus passes through each point S3; the same remark applies to each S^ not in a, and lastly the null-plane of each Si not in or the degree of the enveloping cone of $ is 12, since possesses no double curve. This cone has 24 cuspidal edgesf, and since the class of the surface and also of the cone is 2n, if S is the number of double edges of the cone, we have 12 X 11 -3 x24-28 = 2n, hence S — BO — n. Of these double edges 18 — m pass through the double points of , since the curve of contact of the enveloping cone, being the intersection of and the first polar of O for the vertex of the cone, will have two branches through each double point. Deduct- ing these double edges there remain twelve, which is therefore the number of double tangents of

, it is clear that 28 - JSf' is the number of double tangents of «l> which lie in any plane, excluding the lines of the singular tangent planes which are aot proper double tangents of $. Thus the complete system of iouble tangents of $ forms a congruence (12, 28 -i\^'); of this jongruence the given (2, n) forms part, leaving after its removal a jongruence of double tangents (10, 28 — iV' — n). Now since for (2, 3), (2, 4), (2, 5), (2, 6)„ (2, 6)n. (2, 7) N'= 10, 6, 3, 1, 0, 0, * Salmon, Geom. of Three Dimensions, Art. 131. t Salmon, Art. 279. 19—2 292 THE CONGRUENCES (2, n) [CH. XV we obtain in these respective cases, residual congruences of double tangents which are (10, 15), (10, 18), (10, 20), (10, 21), (10, 22), (10, 21): these will now be investigated. The origin of these additional systems of double tangents is explained in part by the following theorem : the complementary reguli p of the regidi p of (2, n) determine a congruence which has ^ for its focal surface. The truth of this appears from the fact that since each generator of a regulus p touches tv/ice, p touches along a curve k which must be of the fourth order ; thus the plane through any generator of p and any generator of p' meets h in four points of which two points lie on the generator of p, and therefore two points on the generator of p; hence each generator of p meets k twice ; at each of these latter points the generator of p lies in the common tangent plane of p and $, i.e. each generator of p touches twice. The congruence formed by the generators of the reguli p is of the second order and nth class, since as many reguli p pass through a given point as reguli p, (Art. 260); similarly the congruences formed by generators of the reguli p and of the reguli p have the same class. In this way the systems of reguli p possessed by the system (2, n) give rise to oo ^ double tangents (generators of the p), arranged as follows : for (2,6)i (2,6)n (2, 6) (4, 12) (2,5) (4, 10) (2,4) (6, 12) (2,3) (10, 15). a congruence The congruence confocal with (2, 3) is thus accounted for ; in the other cases there remain oo => double tangents of which do not belong to the given (2, n), nor are generators of p, and which form respectively, in (2, 7) the congruence (10, 21) (2,6)i (2,6)„ (2,5) (2,4) (8,15) (6,10) (6,10) (4,6). It will now be shown that these systems are formed, in all cases except (2, 6)ij, hy the single directrices V of the ruled cubics {I), where I is a line of the sheaf whose centre is S^-i. For, each generator of such a cubic surface p' touches twice, hence p^ and $ touch along a curve, so that at a point of intersection of l' and r must lie in the tangent plane to at the point, i.e. V touches <1 twice. Now the two rays of (2, n) through any point P determine ; surface p' whose double directrix is S,^iP ; hence the Jn (m — 1 263] THE CONGRUENCES (2, n) 293 pairs of rays of (2, n) which lie in any plane determine as many surfaces p^, whose single directrices I' lie in the plane, i.e., the class of the congruence of lines I' is ^n(n— 1). Again it was seen, (Art. 257), that each surface (P) contains 2(w— 2) lines through /S„_i, and the null-plane of every point on such a line passes through P, hence the single directrix of the surface p^ for such a line passes through P, i.e., through P there pass 2 (m — 2) lines of the congruence V. The oc ^ double tangents I' thus form a congruence {2 {n — 2), ^{n — 1)}, i.e. they form the residual congruence of double tangents after deduction of the given (2, n) and the generators of the reguli p'. The case of (2, 6)ii, in which there is a residual congruence (6, 10), remains to be discussed. The congruence (2, 6)ii is contained in a tetrahedral complex T^, the vertices of whose fundamental. tetrahedron are the points S^, (Art. 258); now there are oo ^ twisted cubics r passing through the points S^ all the chords of which belong to T% (Art. 95) ; these chords for any cubic r form a system (1, 3), which will have in common with the cubic complex {S„ S/) a ruled surface of degree 3 (1 + 3) = 12, (Art. 234); but this surface is in part composed of the four cones of the second degree {S^, r) ; there remains after their removal a ruled quartic ; hence each of the oo ' chord-congruences of T* contains one ruled quartic of (2, 6)ii. Each generator of such a quartic meets r twice, and through each point P of r proceed two such generators, viz., the intersections of the cone (P, r) and the cubic complex ((§4, (Sf/), excluding the four lines joining P to the points Si', hence r is a double curve of the quartic which is thus the general quartic of class III. Now since any three lines p, p', p" of T'^ determine a twisted cubic* r, we see that any three rays of (2, 6)ii determine such a ruled quartic. If moreover these three rays are coplanar, the ruled quartic possesses also a single directrix I', (Art. xvi), and is of class IV ; one plane of the pencil whose axis is l' passes through any given point A, i.e. such a quartic of class IV has three generators in the plane {l', A), and since there are oo == planes through A, there are co ^ ruled quartics of (2, 6)n which have also a single directrix l'. As before this quartic touches $ and I' touches <& twice, so that the * For three of the points S^ make the three pencils of planes whose axes are P, p\ P" projective to each other, and the locus of intersection of three cor- responding planes is a twisted cubic, (Art. xii), which passes through the fourth point Si and whose chords belong to T". 294 THE CONGRUENCES (2, n) [CH. XV required system of oo '^ double tangents is formed by the single directrices of the oo - ruled quartics of class IV which belong to (2, 6)n. If V is the single directrix determined by three coplanar rays of (2, 6)n, the surface {V) breaks up into the aforesaid quartic of class IV and another ruled quartic which must also be of class IV. Now sets of three rays can be made in twenty ways out of the six rays in any plane; hence there are determined twenty such ruled quartics but only ten simple directrices I', since each I' belongs to two quartics ; hence the class of the system of lines I' is ten. And its order must be six, for if it were less than six, one or more sheaves of double tangents would exist, which is not the case. Hence for (2, 6)n the double tangents of <1> consist of (2, &)ii, the generators of the two sets of reguli p, and the single directrices of the ruled quartics of class IV which belong to (2, 6)ii. CHAPTER XYI. THE CONGRUENCE OF THE SECOND ORDER AND SECOND CLASS. 264. The congruence (2, 2), of the second order and class, is the one of the series (2, n) which has been most fully investigated : an account of this congruence is given in the present chapter. From Arts. 249, 254 it follows that the congruence (2, 2) has a focal surface of the fourth degree and fourth class, which has 16 double points and 16 singular tangent planes, i.e. is a Rummer's Surface. It will be shown, (Art. 265), that any congruence (2, 2) is the complete intersection of a linear and a quadratic complex, and is therefore identical with the congruence (2, 2) already discussed in Chapter VIII. To each point of space one plane is assigned by the system (2, 2), viz. that of the two rays through the point, while to each plane one point is assigned, viz. the intersection of the rays in the plane, so that by the system (2, 2) an involutory reciprocity is established in which corresponding elements are united, i.e. a linear complex is determined, (Art. 37), to each pencil of which two rays belong; therefore each system (2, 2) is contained in a linear complex O, ; and only one such linear complex is thus related to any given congruence (2, 2). This result may also be seen as follows : — let S and S' be two conjugate singular points, (Art. 256), o-, o-' their null-planes, and I any line of the pencil (S', a) ; then the surface (I) consists of the pencils {S, a), (S', a'), together with a regulus p which has one generator belonging to the pencil {S, a) and one to {S', a'). If any generator of p meets a in P, the line SP, since it meets three lines of p, is a directrix of p, and therefore a is a tangent plane to p, i.e. the trace of p on a' is the line SP. Since all the rays of the system can be grouped into such reguli, together with the pencils 296 THE CONGRUENCE (2, 2) [CH. XVI {S, a-), (S', a), it is clear that they establish a (1, 1) correspondence on the lines of the pencils {S', a), (S, a-'), viz. that of pairs similar to I and SP, and in this correspondence SS' corresponds to itself ; hence the rays are included in a linear complex, (Art. 39). This may again be seen from the fact that the surfaces (P) are of the form x,'F, + x,'P^ + x,'P, + x:P, = 0, with the identity Xi Pi + ajjPj + XsPs + XiPi = ; but since the Pi are linear in Xi i.e. we must have an = 0, 0,* + at,- = 0, hence (P) has the form loik {xiXk - x-xt) = 0, which is the bilinear equation connecting two points of a line of a linear complex. 265. Confocal congruences (2, 2), The equation of any Kummer suriace has been seen, (Art. 85), to be reducible to the form A^y,*+ 2B {y,^y^ + y,^y,') + 2C (y.^y/ + yiy,^) + 2P (2/,=2// + y.^yi) + 4^ y^y^y.y, = ; while its double points and singular tangent planes form a system de.scribed in Art. 29, such that each point is the pole for six complexes mutually in involution of the six singular planes through it, and the six points in each singular plane are the poles of the plane for these six complexes ; hence the singular points and singular planes of the congruence (2, 2), being the double points and singular tangent planes of a Kummer surface, form such a system. In each singular plane there is one pencil of rays which therefore belongs to the linear complex G^ of the congruence, and thus Cj is one of the above six linear complexes in involution. Moreover the double tangents which belong to G^ belong also to a quadratic complex, (Ai-t. 83); hence, any congruence (2, 2) is the complete intersection of a linear with a quadratic complex. Now it was seen that the double tangents of a Kummer surface form six congruences (2, 2), (Art. 83), hence, as we have already seen, (Art. 126), associated with any congruence (2, 2) there are five others having the same focal surface as the given congruence (2, 2). 264-266] THE CONGRUENCE (2, 2) 297 Certain properties of the singular points and planes of a congruence (2, 2) will now be investigated, taking as starting point the fact that any congruence (2, 2) is contained in a linear complex Oi; and that its singular points and planes form a system described in Art. 29 for six linear complexes in mutual involution, of which Gi is one. 266. Distribution of the Singular Points. Any two singular points Si and S, are conjugate in two of the six fundamental linear complexes associated with the focal surface ; for jSj must be contained in one of the six singular planes tt through Si; let Si be the pole for Gi of this plane tt and S^ for Gj, then from the involution of Gi and Gj there is one plane tt' through S1S2 for w'hich Si is the pole for Gj and >S'2 the pole for Gi; thus (Sj/Sj belongs to each of the complexes Gi and Gj, and it is seen that through the join of any two singular points there pass two singular planes. Any three singular points Si, S^, S3, which are non-conjugate to each other in any particular complex Gi, must lie in a singular plane ; for let Si and S^ be conjugate in Gj and Gie, S2 and S3 in Gi and Gm, S, and Si in G^ and Cy, then all these six complexes cannot be different from C; unless at least two of them are the same, e.g. Gj = C„, here SiS^ and SiS, belong to Gj, i.e. Si is the pole of SiS^S, for Gj, hence the plane S1S2S3, or tt, is a singular plane. Moreover in this case S^ is the pole of w for Gk and S3 the pole of tt for 0^, i.e. S3 and S3 are conjugate in 0^ and Cy- It follows that there is a plane tt' through S2S3 for which S^ is the pole for C^ and S3 the pole for Gic, and Si, S2, S3 form a system described in Art. 26, in which the three singular planes through /Si/Sj, (SjiSs, S3S1 respectively, distinct from tt, meet in a singular point S^, which is the pole of these respective planes for C,, Gj, Cj. Hence the four points Si, (S'a, S3, Si are mutually non-conjugate for Gi. There is no point outside tt, except Si, which is non-conjugate to Si, S2, S3 for Gi; for if S were such a point then the plane SS1S2 would be a singular plane, i.e. S must lie in the second singular plane through S1S2, similarly it must lie in the other singular planes through S2S3, S3S1, i.e. it must coincide with Si. The number of tetrahedra whose vertices are non-conjugate to each other for Gi is 40 ; for in any singular plane three points of the system non-conjugate in Gi may be selected in 10 ways, and each selection determines a fourth point outside the plane non- 298 THE CONGRUENCE (2, 2) [CH. XVI Fig. 9. conjugate to them in Ci; thus the number of such tetrahedra is 10x16 = 160, but in this process each tetrahedron occurs four times, hence the number of the required tetrahedra is 40. 267. Every (2, 2) is included in 40 tetrahedral com- plexes*. Let the points S^, S^, S^, St be non-conjugate in C,, and denote by cr the singular plane S^S^Si having S as its pole in Ci, and by a' the singular plane S^S^St having S' as its pole in C] ; take also any line SiP of the pencil (S^, a); then of the two rays from any point of S^P one belongs to (S, cr) and the other to a regulus p. The line S'P belongs to p, hence cr' is a tangent plane of p ; also since 8^ and S^ are non-conjugate, the null-plane of S^ will meet S^P in a point different from S^, so that p passes through Si, and the trace of p on cr' is S'P and a line through S^; hence the reguli p make the pencils (>Si, a), (Sj, o-') projective, these reguli therefore belong to a tetra- hedral coTnplex. The other two vertices of the fundamental tetrahedron must be Ss and Si lies in the intersection of o-/ and 0-1, therefore S^ is the point (o-io-jV/). The four points S2, S3, St, S^ are similarly determined, and the positions of the 16 singular points are now known. To determine the five remaining planes, we observe that since S^Si is I7 and S1S3 is a/8, therefore S1S3 is Se; also S3S3 is le, hence ^1^3 is IS; similarly ^1^4 is I^, thus SiSsSt is the null-plane of S^ for Cj; this plane with the four other similar planes completes the system. Observe that the relationship of the pentagon SiSsS^S^Si to S3, ^84 in two ways giving 1x10x3x2 = 12 ways; thus the total number of pentagons is 16x12 = 192. 302 THE CONGRUENCE (2, 2) [CH. XVI 270. Regruli of the congruence. It has been seen, (Art. 264), that if I is any line of the pencil {S', give 00 * reguli which belong to T^, the first two complexes being the special ones whose directrices are two lines of the respective pencils {A^, ai), {Ai, Ui). If C='S,CikPik is the linear complex to which the congruence belongs, any of the preceding reguli of T^ which belong to C must identically satisfy the equation 2cijtpit = 0; hence substituting in the latter equation we obtain /J Ci3 + kXCi2 + pC^ = 0, Ci4 + CTC23 = 0, C42 -I- Y + ^23 = 0. These equations show that o- is a constant, and determine p and T in terms of X, and hence give the 00 ^ reguli of the system corresponding to the two pencils {Ai, Ut), {At, oLi). In a similar manner we obtain five other sets of reguli, each of which passes through two vertices of the tetrahedron and touches two faces of it : six varieties of 00 ' reguli thus arise. Again taking as the equation of T^ A (xi' + xi) -I- B {xi -t- xt) + {xi + xi) = 0, the substitution ^lA■>^p NB + p, vG+p * Compare with Art. 260. + This has been already shown in Art. 118. 304 THE CONGRUENCE (2, 2) [CH. XVI gives as the locus of y a tetrahedral complex T^ having the same fundamental tetrahedron as T'^, and whose complex cones and complex conies are "images " of reguli of T^, (Art. 116). Since each complex cone of T^^ has a generator through each vertex of the tetrahedron, and hence contains the four lines yi = ± »2/2. ys = ± ■i'Vi, 2/6 = ± iy^, (where the signs are to be taken all positive or two negative), the corresponding regulus of T"^ will also contain four lines of this description, i.e. will pass through the vertices of the tetrahedron. Similarly the regulus which corresponds to a complex conic of T,,^ will have a generator in each of the four faces of the fundamental tetrahedron ; we thus obtain oo * reguli of T^ through each vertex, and 00 ■" reguli touching each face of the fundamental tetrahedron. To a regulus of T^ which belongs to the complex C, whose equation may be taken as 1,Ci!Ci = 0, will correspond a cone or complex conic of 2'^'' which belongs to the complex Cj Cg C3 C4 V il + /i v.d + /x \/B + iJ, 'JB + /J, while the latter complex must be special, since it contains a cone or the tangents of a conic, hence Ci + Cg , Cr + C4 C5" + Cg _ A + fj, B + fj, G + /J, ~ ' The last equation gives two values of /i, having roots /ti and fji^; thus the cones or conies of 7^,^ T^/ whose vertices or planes are united to the respective lines ( , ^ ... I, ( , ' ... I, ^ W^+^' J' WA + /M, J give rise to four sets of reguli of T^ which belong to C. Thus there are in all ten varieties of 00' reguli which belong to the congruence. 272. Focal surface of the intersection of any two complexes. The focal surface of the intersection of any two complexes may be determined analytically as follows : let /= 0, =0 he any two complexes, then if a; is a ray of the system determined by them, all rays consecutive to x satisfy the equations ('l)=». (»S)-- 271-273] THE CONGRUENCE (2, 2) 305 The directrices of this linear congi'uence being z, z', it is clear that dxi dxi ' dsci dxi ' where Xj, Xj are the roots of the equation \0Xi./ dwi oxi \0XiJ The intersection of x with z and z gives the points of contact, P, P' of X with the focal surface, and the planes {xz), {xzf) are its tangent planes at P and P' ; hence if y is any tangent to the focal surface where X is either Xj or Xj. Applying to the congruence (2, 2), we have f=A (x,' + xf) + B («,= + xf) + C {x^^ + x^% = "ZciXi, and the equations for the determination of y are p-yi = itJ' + A)xi + \Cu p.ys = (fi + B)Xi + '\.Cs, p.y,=-{fi+C)x^+Xcs, p.yi={f^ + A)xn + \Cr,, p.yi = {iJ. + B)Xi+XGi, p.ye = (fi + C)xe+\Ce. The elimination between these equations and the equations y=0, <}) = 0, of the quantities Xi, X, fi, leads to the equation of the focal surface in line coordinates. 273. Double rays of special congruences (2, 2)*. If the complex (ex) = contains an edge of the fundamental tetrahedron of the tetrahedral complex, a particular case of the congruence (2, 2) arises; e.g. let G contain the edge A^A^, then we have Ci — ic^ = ; in this case A^A^ is a double ray of the congruence, since at each point of it there is only one ray, viz. the line A^A^. For those tangents y of the focal surface which meet A^A^ we have 3/1 — iy^ = 0, hence for such tangents fi + A =0 in the above set of equations, substituting for /i and eliminating x^, x^, x^, Xg and X by aid of the last four of the equations of the last Article, we find that y belongs to the quadratic complex (ciVa - Csy^Y + (Ciy4 - C4.V1)' (ciys - c^yiY H- (ciye - c^y^f ^ B-A ^ G-A and also to the linear complex 3/1 — 13/2 = 0. Hence the tangents to the focal surface in any plane through * The following classification of congruences (2, 2) which have a double ray is due to W. Stahl. J. 20 306 THE CONGRUENCE (2, 2) [CH. XVt A^A^ envelope a conic, therefore the focal surface is a Plucker's surface of which A,Ai is the double line. (ii) If the complex {cx) = also passes through the edge ^1^3, then C3 — iCi = 0, and the focal surface is a Pliicker surface in which A^A^ is also a double line : the congruence has two intersecting double rays. (iii) If the complex (ca;) = passes through J., J., and ^3-44, then Ci = Cj = 0, and tangents y of the focal surface which meet AiA^ also meet A^A,, and are therefore generators of the focal surface, since they meet it in the point of contact of x and the double lines A^A^, ^.3^14; such lines 3/ also belong to the quadratic complex Yi'+Yt'+Y,'+ ¥^' = 0, where F,, Y„ Y„ Y^ are linear functions of 3/3, 3/4, 1/5, j/e'i as is easilj' seen by eliminating irs,Xi, Xi, x^,\, p from the last four equations of the last Article. Thus these lines y are generators of a ruled quartic with two double directrices, i.e. of the class I ; this surface is here the focal surface : the congruence has two non-intersecting double rays. (iv) If {cx)=Q passes through A-^A^, .42^4,, .4.3^4, the focal surface is a ruled quartic which has .4i.42, .4,^4 as double directrices and .42.4s as double generator, i.e. belongs to class VII : the congruence has three double rays. (v) If (ca;) = passes through A^A^, A^A^, A^At, AtA-^^, the focal surface has four double lines and hence must consist of two quadrics which have two generators of each system in common. Hence the congruence (4, 4) of the double tangents of two quadrics becomes, when the quadrics have two generators of each system in common, two congruences (2, 2;. For the species (iii) the following theorem holds : any regulus through A1A2, A^A^ and one other ray of the congruence belongs entirely to the congruence ; for if X is the additional ray, such a regulus is given by the equations p.Xi = a + ^.+ Xi, p.X3 = X3, p.x^ = Xs, p.X2 = {a — 0)i + Xi, p.Xi = Xi, p.Xs = Xs. Now since it is given that C3X3 + C4Z4 + C5Z5 + CeXe = 0, {B - A) (Za^ + Z4^) +{C-A) (Z,^ + Ze=) = ; it follows that two equations of the same form as the last are also satisfied by x, i.e. the regulus of lines x belongs entirely to the congruence. Thus from the two given double rays d, and d^ and any regulus p of the system not containing di and d.^ the whole system can be constructed by forming the 00 ^ reguli determined by d^, d^ and any generator of p. CHAPTER XVII. THE GENERAL COMPLEX. 274. Many of the leading characteristics of the quadratic complex are seen to belong also to a complex f{x) = 0, of any degree n. For instance, the lines of this general complex through any point form a cone whose degree is n, and those in any plane envelope a curve whose class is n. Since, x and y being any two intersecting lines, the equation f{x + \y) = 0, when expanded becomes /(^) + XA/+^jAy+ =0, o where A = Sy^ ^— , this gives n values for \, i.e. there are n lines of OXi the complex in any plane pencil. fif The equation Af= "Eyt ;r- = 0, in which a; is a given line, is CXi said to be a linear polar complex of /= 0. If x belongs to /= 0, the equation A/=0 is one member of the singly infinite set of linear complexes ;hey are called the tangent linear complexes of /(«) = 0*. Each of these complexes contains x and every line x + dx vhich is consecutive to x in the given complex f(x) = ; since for uch consecutive lines we have X^dxi = 0, 2 X— dxi = 0. OXi OXi On any line x of /= 0, a correlation is established between its 'oints and their polar planes in a tangent linear complex for x : he same correlation is determined by each of the tangent linear * See Art. 74. 20—2 308 THE GENERAL COMPLEX [CH. XVII complexes of x, the polar plane of any point P oi x being the tangent plane through x to the complex cone of P- 275. The Sing;ular Surface. Again, as in the case of the quadratic complex, we consider such tangent linear complexes as are special; and as in Arts. 76, 157 we find as the necessary condition This becomes S|^J=0, if we take io(x)=0 as being Xa-,- = 0. Hence the lines of y(a;) = whose tangent linear com- plexes are special satisfy the equations These oo ^ lines are called, as before, singular lines of f{x) = 0. If a; be a singular line, the pencil ix, ~\ consists of directrices of special tangent linear complexes. These directrices therefore form a complex, which consists of oo = plane pencils. If all the lines of such a pencil intersect any given line a, we have ('-)=«' (4o='' •^■(^^='' ^(afy='- These equations determine, if a is given, 4w(n — l)'' singular lines X, for each of which the pencil {x,~-\ meets a. This may occur in two ways; either on account of a passing through the centre of the pencil, or on account of a Ij'ing in the plane of the pencil; from the duality of the subject, there will be as many solutions of one kind as of the other. Hence the locus of the points [ a;, ^ j is a surface of degree 2n {n — If, and the envelope of the planes ix, r^ J is a surface whose class is 2n(n — 1)'. These surfaces are identical*; for, denoting ^ by f, if P, P' are any two consecutive points of the first locus, let P be the point (x, f) and P' the point (x', f), where x' = x + dx, ^' = I + d^. Then 2«iff = 0, l.{Xi + dxi){^i + d^i) = Q, * This was shown by Pasch ; see reference on page 92. 274-277] THE GENERAL COMPLEX 309 and neglecting small quantities of the second order, since we have X^idxi = 0, therefore l,Xid^i = (), hence l,Xi^i = 0, 'Zxi^i'=0; so that the four lines x, ^, x', ^' form a twisted quadrilateral, and the point P' lies in the plane {x, ^), if small quantities of the second order are neglected. We therefore obtain one surface, the Singular Surface of the complex, which is both the locus of the singular points {x, ^) and the envelope of the singular planes (x, ^). In the tangent linear complexes of a singular line x, the plane (x, ^) is the polar plane of each point of x ; therefore the complex cones of /= 0, whose vertices lie on *•, touch (x, ^) along x. 276. If y is any line through the singular point, we have Xi/iXi = 0, lyi^i = 0, hence fix + \y) = I' l.yiy,fa + + X-f{y) ; therefore in the pencil (x, y) there are only n — 2 lines of _/ distinct from X, hence the complex cone of f for a singular point has the singular line for double edge. If y also lies on the cone of the complex 1,yiy]i;fa: = ^ for the singular point, the pencil {x, y) contains only n — 3 lines of f distinct from x, hence the complex cone of ^yiykfik — ^ for the singular point must split up into two planes ; they are the pair of tangent planes through the singular line to the complex cone of f for the singular point. Reciprocally, the complex curves in the planes through any singular line x have the singular point as point of contact with x, except in the case of the singular plane. Let y lie in the singular plane and also satisfy the equation ^yiykfik = 0, then, as before, the curve of the latter complex in the singular plane must split up into a pair of points whose centres lie on the .singular line; the complex curve of/ in the singular plane touches the singular line in these two points, i.e. has the singular line as a bitangent. 277. If ej, f2 are the two planes into which the complex cone of ^i/iyicfii,=0 breaks up for a singular point P, and a the polar plane for P of a tangent linear complex for a; of the complex i?'= 2(^7^ j =0, then ej, fj, a, and the singular plane form a harmonic pencil ; for let v be any line of the pencil (P, a), then Sw; g- =0 ; also the lines of 2yiyi/ii=0 which are contained 310 THE GENERAL COMPLEX [CH. XVH in the pencil (|, v) are obtained by substituting f+Xi; for y in the last equation, giving dF dF which, since 2«,- .5— =0, reduces to whence the result follows as stated above. Reciprocally if Fi, E^ are the points into which the complex conic of 2i/tytfa=0 breaks up for the singular plane, and A the vertex of the cone of F upon x for which the singular plane is the tangent plane, the points El, E2, A and the singular point are four harmonic points. 278. The Principal Surfaces. The significance of the Principal Surfaces, as being the analogues of the lines of curvature of a hypersurface, has already been noticed (Art. 228). In relation to them a result may be given here which is the extension of a theorem shown in connexion with the linear complex (Art. 41). On each line x of any ruled surface of the complex /(«) = 0, a (1, 1) correspondence exists between the point of contact with the surface of any plane tt through x and the point of contact of x with the complex curve in tt ; the latter point being the pole of tt in the tangent linear complexes of x. There are, therefore, two planes tt for each of which the point of contact of TT with the surface coincides with the pole of tt in these tangent complexes. The locus of such points, of which there are thus two Q, Q' on each generator, is a curve k. If now the given surface is a principal surface, the complex /(a;) = is ' touched ' by a tangent linear complex along x and also along a generator consecutive to x (Art. 132), so that this tangent complex contains three consecutive generators of the surface ; hence, by Art. 41, since two consecutive tangents of A; at Q belong to the same linear complex, the osculating plane of A; at Q is the tangent plane of the surface at Q, i.e., the curve k is a principal tangent curve of the surface. That the (1, 1) correspondence of points upon a; just noticed is an involution may be shown as follows : — the coordinates of x are / dx\ functions of one variable 6, and the complex Sj/i ( ajj + o- ^M =0 has at each point P of a; the tangent plane ir as its polar plane ; for the lines of this complex through P intersect both x, dec the generator through P, and the consecutive generatoi- a; + 3^ dd. 277-280] THE GENERAL COMPLEX 311 Also this complex is in involution with any tangent linear complex of x ; for = 0, since /(* + 3a '^^j = ^■ Hence the (1, 1) correspondence of points is an involution, a pair of corresponding points being, as stated, the point of contact of tt with the surface and the pole of tt in any tangent linear complex of x ; the double points of the involution are the two points Q, Q' of Ic which lie upon x. 279. Independent constants of the complex. A homo- geneous equation/(a;) = of the nth. degree in six variables contains (w + 1) (w + 2) (n + 3) (» + 4) (n + 5) 5! terms ; but the complex represented by f{x) = 0, is also represented by ■\{r{x)=f(x) + Q)(a;).(j){x) = 0, where ^ is any expression of degree (n — 2) in the variables, and to(x) = is the fundamental relation. So that the complex contains (n-l)n{n + l)(n + 2)(w + 3) 5! arbitrary constants, viz. the coefBcients of 2a, 3^ dxi dxi dXi ' dxf \dxj = 0. \dxil '" dxi ' dXi On account of this equation and the equations laiXi =/= = 0, we obtain 4?nw (m + n — 2) lines X such that a meets x and a line of the congruence con- secutive to a; ; so that a either passes through the intersection of X and x + dx or lies in their plane, therefore degree of focal surface = class of focal surface = 2mn (m + « — 2). Since the order and class of the congruence are each equal to mn, if r is the rank of the congruence, we see from Art. 237 2mn (m + w — 2) = 2wiw {mn — 1) — 2r, hence r = mn (m - l)(n — 1)*. The focal surface maj' split up into two surfaces ; this will occur if dxi \dxJ dXi ' dxi. * See Art. 248. 316 THE GENERAL COMPLEX [CH. XVII is a perfect square, the two focal points being then given rationally. For instance in the case of the congruence of the singular lines the determinant of the last article becomes dxi) ( ^dxi'" dxi ' dxi \dxj ' dxi The focal surface breaks up into two surfaces, of which one is the singular surface of f= ; since, if the point lies on the singular surface, two of the singular lines through it coincide ; similarly for each tangent plane of the singular surface ; so that the equations f=F=laiXi = 'S.ai^=0 relate to the points and planes of the singular surface which are united to a ; thus the degree and class of the singular surface are seen to he 2w(?i — 1)-*. The solutions of f=F=2aiX, = ^ai^l{^) - 22aig^ 2^ ^^ = relate to the other portion of the focal surface, which is therefore of degree and class 2n. (n — 1) (on — 7) f . The lines which satisfy the equations form a ruled surface ; for each of them the focal points coincide. If, by virtue of the form of f= 0, i^ = 0, we have \dxiJ \dxiJ dXi dXi the congruence consists of the tangents to one set of principal tangent curves of a surface. 284. The ruled surface common to three complexes^. If /= 0, (j> = 0, ■\}r = be three complexes of degrees m, n and p * See Art. 275. t This surface has been termed the Accessory Surface, Voss, Math. Ann. ix. The Baok of the focal surface has been found by Voss to be 2mn {(m + n - 1)- - mn + 1} ; his investigation is too long for insertion here; see Art. 248. t See Voss, "Zur Theorie der windschiefen Flaohen," Math. Ann. vm. 283-284] THE GENERAL COMPLEX 317 respectively, the ruled surface which is their intersection is of degree and class 2mnp, since this is the number of lines whose coordinates satisfy the equations y _ ^ = x^ = 2xi' = XaiXi = 0, where «; is any line. We may take SAjj^j = 1 as an equation connecting the variables, and, for purposes of symmetry, assume XaiXi = t, where the a^ are arbitrary constants and t a new variable. The coordinates of the generator x + dx consecutive to x are then given by the equations 1,^ dxi = %7r- dxi = t^ dxi= Xkidxi = Xxidxi = ; dxi dxi oxi '2aidxi = dt ; whence where n = _9n d£ ¥. 91 9± a,. k- cu dXi axi dXi If two consecutive generators intersect, we have 'S,dx^ = (i, and expressing that the five complexes ^y'fxr'' ^y^tr'' ^^^'ir'' ^'^'^='- ^''''^' have a common line, we obtain 9/ flXi. dxi dxi dxi dxi OXi dxi dXi 9/9t dxi d d^jr dxi dXi "JL 2 dxidx; "^ dxi dXi dxi %th 2^.. OXi 1^-tk- OXi ^ 5 ki OXi tki' 2^1a=i t^^Xi ^"^Xi IkiXi OXi OXi OXi V 9/ ^ 3<^ or, since ia^j ^- = ZXi ;:r- = dxi OXi Xi dXi 2^ 5 Xi OXi OXi IXi" = 0; C^kiXiY ydx; %a; "DL = 2^.;2 = 0, this becomes OXi dxidxi dxidxi dcj) d\lr dxi dxi dxidXi \dxi) ^dfd± ^d±d± ^(d±y " dxidxi dxidXi 'dxiJ 0. 318 THE GENERAL COMPLEX [CH. XVII This determiQant is of degree 2(?n+n + p — 3); hence there are 4 mnp (m + n + p — 3) generators which are intersected by consecutive generators*. Such a generator is said to be ' singular.' 285. Rank of the surface. The Rank of a surface, being the degree of its tangent cone or the class of its plane section, is the degree of the surface in line coordinates ; i.e. the degree of the complex formed by the tangents of the surface. To find the equation of the surface in line coordinates, we have, if y is any tangent, 2y,a;f = 0, "Eyidxi = 0, hence eliminating the diiFerentials between the equations ^dxi = l, -^ dxi=1^ dxi = Ikidxi = "Zxidxi ■■ dxi we obtain i^kiXiY dxi df_ d±^ dxi ' dxi ' ^©- dxi dxi dxi dx{ dxi ■1yidxi=0, "jr. Jr. djr dxi' ^dfd± " dxi dxi OXil .^ 90 d"^ dXi dxi d ^i, Vi = 0, and squaring. dxidxi _. 8^ 9i|r dx; dxi dx, D' ^" ^yt dxi ^'•'i dXi = 0. It follows that l.kiXi divides out of the previous equation leaving an equation of degree m + n+p— 3 in a; and unity in y. Eliminating the Xi from this equation and /=0, <^ = 0, >/r = 0, txiyi = 0, 2a;/ = 0, the equation of the surface in line coordinates is seen to be of degree 2mnp {m + n+p — 3). If for any generator the equations which give the consecutive generator are not linearly independent, we take as an additional equation l,fi^dxidxk=0, thus obtaining two sets of values for the dxi ; in this case a double generator exists, for each such double generator the rank is diminished by two, since the class of any plane section is diminished by two for each double point. 286. Clifford's Theorem. Investigations into the general ruled surface are outside the scope of the present treatise, but on * Klein, Math. Ann. v. 284-286] THE GENERAL COMPLEX 319 account of their interest we add a sketch of some ideas contained in the memoir on Classification of Loci due to Prof. Clifford *- He denotes by a curve a continuous one-dimensional aggregate of any sort of elements, which includes not merely a curve in the ordinary sense (an aggregate of points), but also a ruled surface, or indeed a singly infinite system of curves, surfaces, complexes, &c. such that one condition is sufficient to determine a finite number of the elements. " The elements may be regarded as determined by k coordinates ; and then if these be connected by yfc — 1 equations of any order, the curve is either the whole aggregate of common solutions of these equations, or, when this breaks up into algebraically distinct parts, the curve is one of these parts. It is thus convenient to employ still farther the language of geometry, and to speak of such a curve as the complete or partial inter- section of A; — 1 loci in flat space t of k dimensions." " If a certain number, say h, of the equations are linear, it is evidently possible by a linear transformation to make these equations e'quate h of the coordinates to zero ; it is then convenient to leave these coordinates out of consideration altogether, and to regard only the remaining k — h—1 equations between k — h coordinates. In this case the curve will, therefore, be regarded as a curve in flat space o{ k — h dimensions. And, in general, when we speak of a curve as in flat space of k dimensions, we mean that it cannot exist in flat space oi k — 1 dimensions." The whole aggregate of linear complexes may be regarded as constituting a space of five dimensions, in which straight lines constitute a quadric locusj. A ruled surface is a ' curve ' lying in a quadric locus in five dimensions. If, however, the generators of the ruled surface belong to the same linear complex, the ruled surface is a ' curve ' in a quadric locus in four dimensions. If the ruled surface has two linear directrices, it is a ' curve ' on an ordinary quadric surface in three dimensions. So that the theory of ruled surfaces which have two directrices is identical with that of curves on a quadric. Hence, such ruled quartic surfaces correspond either to quadri- quadric curves of deficiency unity {elliptic curves whose coordinates are expressible as elliptic functions of one variable), or to the * See Collected Works, p. 305. t By a flat space is meant one which is intersected by a line which does not belong to it in one point only, t See Art. 229. 320 THE GENERAL COMPLEX [CH. XVII curves of deficiency zero which are the partial intersections of a quadric and a cubic surface. " Similar considerations apply to surfaces. B}"^ a surface we shall mean, in general, a continuous two-dimensional aggregate (which may also be called a two-spread or two-way locus) of any elements whatever." Theorem. A curve of order n in flat space of k dimensions {and no less) m,ay be represented, point for point, on a curve of order n—k-\- 2 in a plane. " The proposition is obvious when k = S. The cone standing on a curve of order n (in ordinary space of three dimensions), and having its vertex at a point of the curve, is of order ji — 1 ; if then we cut this cone by a plane, we have the tortuous curve represented, point for point, on a plane curve of order « — 1. Now this process is applicable in general. Starting with an arbitrary point P of a curve in any number of dimensions, let us join this point to all the other points of the curve ; we shall thus get a cone of order n — I. For, any flat locus of A;— 1 dimensions drawn through the point P, must meet the curve in n points of which P is one; and therefore it must meet the cone in n — 1 lines. Hence, if we cut this cone by such a flat (k— l)-way locus not passing through P, we shall get a curve of order w — 1 in flat space oi k — 1 dimensions, which is a point for point representation of the original curve. By continuing this process we may go on diminishing the order of the curve and the number of dimensions by equal quantities, until we have subtracted k — 2 from each ; when we are left with a curve of order n—k+2 in a plane." It follows, by taking k=n, that a curve of order n in flat space of n dimensions is unicursal, since it has (1, 1) correspondence with a conic. By taking k. = n — l, -we obtain the result that every curve of order n in flat space of n—1 dimensions is either unicursal or elliptic ; for it has (1, 1) correspondence with a plane cubic. Two applications of these last results will now be made : since a ruled quintic surface which ddes not lie in a linear complex, corresponds to a curve of order five in flat space of five dimensions such a surface is unicursal : similarly, a ruled sextic which does not lie in a linear complex is by the second result seen to be either unicursal or elliptic. 286-287] THE GENERAL COMPLEX 321 287. Symbolic form of the equation of the complex*. The complex /( p) = Soift, H, ■ • . PihPki . . . = is always uniquely capable of symbolical representation. For the coefficient 0^,^; ... has the property that if two pairs of suffixes are interchanged the coefficient is not altered in value, hence the sum of all the terms arising from such interchanges effected upon a given term is equal to that term multiplied by the number of such arrangements of its pairs of suffixes, hence we may write symbolically «tft,w, •.• =afA-«w and thus /( p) = (laa pih^. It should be noticed, however, that if in aik,u, •■• two suffixes- of the same pair are interchanged the coefficient is altered in sign,, hence this must also be true of the symbolic quantities ani, etc. Hence the complex is represented symbolically by the result of equating to zero the nth power of a linear expression in the line- coordinates. Moreover in the equation of the complex, which is generally of the form /+. Hence the coefficients of (j> are thus uniquely found so as to satisfy (iii), and hence to make /+ (^ . CO = {S (oih - ahbi)pa}''. This form of the equation of the complex, i.e. that for which (j> is thus determined, is the Normal Form (Art. 279) ; for becomes in Plucker coordinates (^ — 5— + 5 — 5- + 5 — 5— ) {^ K^A - aMpih]'' = 0. \dpiidpsi dp„dpt., dpudpj ^ ' That such a form can be determined, and only in one way, may also be seen as follows : denote by A the operator 3" 3^ 3^ dpiidpii dpudpti dpudp^ ' and apply it and its successive powers to each side of the equation /+ ^ . to = {S {aih - ahbi)pi],Y'; the operators A, A^ . . . reduce the right side of this equation to zero on account of the identity (0162 -0261) (0364 -aih^+ ... + ... = 0, hence <^ has to be so determined that A(/-|-(/>.(o) = 0, AH/+^.o') = 0, etc.; but we have . , , , , , , . 36 3(B dd> 3aj A () = tuA^ + (n + 1) ^, A' (<^(o) = (o6?^ + 2?iA^, A» ((^oj) = '/+ aA^ + 2nA<^ = 0, A'/4- a)A' + 3 (n - 1) A2<^ = 0, AY+ (oA'(f> + 4 (n - 2) A=<^ = 0, etc. In the last of these equations the middle term is zero, hence beginning with the last, the successive A*^ are determined, and hence, from the first equation, .Q) =f^ , where /i = is the required normal form of the complex, multiply the first, second, etc. of them by ~l.(»i + l)' 1.2 (« + !)«' ~1.2.3(n + l)n(n-l)'®*°- and add them all together, we obtain f'-^f' lT(^i) ^^^ \.1.(n^\)n ^"-^ '^ AV + 1.2.3(rn-l)?i(7i-l) -^ The right side of this equation when put equal to zero gives the normal form of equation of the given complex. 288. Symbolic forms for the Complex surface and Singular surface. As in Art. 87 the expression may be written in either of the forms ajiy — ayb^, (a, /S, x, y) ; hence the symbolic form of the normal equation of the complex is (a, 0, X, y)'"' = 0, or (axby — aybxY'=0. If we write Pa = 7r34 = UaVi — UtV3, etc., the symbolic form of equation becomes (.a„/3„ - a^^uT = 0- or (a, b, u, v^ = 0. If in either of the first two forms we consider the Xi as constant, we obtain the symbolic form of equation of the complex cone of the point x; if in either of the second the Mj be taken as constant, we obtain the complex curve of the plane u in plane coordinates. The Complex surface of a line a is the locus of intersection of consecutive complex cones whose vertices lie on a. If y + Xz is any point of the line (y, z), the equation of its complex cone is, by the foregoing, (a, b,x,y + \zY = 0, or, {{a, b, x,y) + X (a, b, x, z)Y = 0. 21—2 324 THE GENERAL COMPLEX [CH. XVII The locus of intersection of consecutive complex cones is therefore obtained by forming the discriminant of this equation for \. Now the discriminant of the expression {py + Xpz)" is equal to SCII (pp')* ; replacing (pp') by pyp^ — pzPy , and Py by (a, b, x, y) etc., we obtain as the equation of the complex surface of the line {y, z) ten [{a, b, X, y) (a', b', x, z) - (a, b, x, z) (a', b', x, y)} = 0. The number of the symbolic pairs ab, a'b', ... is equal to the degree of the discriminant of a binary form of the nth degree, i.e. 2(n — 1), while each pair enters n times into each term of the sum S, hence the last equation is of degree 2n (n — 1) in x, or, the Complex surface of a complex of the nth degree is of degree 2n(n-l). We may take as a definition of the singular surface, either that it is the locus of points whose complex cone has a double edge, or that it is the envelope of planes whose complex curve has a double tangent. The symbolic form of equation of the singular surface may be obtained as follows : — denoting symbolically a curve of the nth degree by 7x" = 7x'"= = 0, its discriminant A (whose vanishing is the condition for a double point) may be written symbolically in the form A = sen (77 V). The degree of A is 3(n — 1)^ in the coefficients of the curve, hence this must be the number of the sets of symbols 7 which appear ; each 7 must enter to the power n in each term of A, hence each such term must contain n(n — 1)" determinant factors. Thus the condition that the section by the plane u of the surface of the nth degree /=7x» = 7«'"= =0, should have a double point, is i?'=SCn(7, 7', 7", «) = 0. The last equation represents, in general, the equation of the surface in plane coordinates. If /"is a cone, this equation becomes where x is the vertex of the cone, and M involves x but not u. * See Clebsch, Vorlesungen iiber Geometric, Bd. i. 288] THE GENERAL COMPLEX. 325 To apply to a complex cone, we observe that the equation of the latter being where the yt are current coordinates, we have, writing yi = cixbi—ha'i, and noticing that jx = 0, (y. V' 7". ■") = (aa;& - 6a; a, J, 7". ■«*) = a^ (b, 7', 7", u) - &a; (a. 7'. 7". w) or, by a known theorem (Art. 91), = 7x" (a. 6. 7'> ^) - 70=' (a, &. 7". «) - ^a; (a, b, 7', 7") = -(a, 6,7', 7") Ma;, since 7a,' = 7/' = 0. Thus for the complex cone F = Ma;" <"-"" . M, where M = SCO (6, a, 7', 7"). Now if a plane not passing through the vertex cut a cone in a curve having a double point the cone must have a double edge, the condition for a double edge is therefore M = 0; each symbolic determinant factor is of the second degree in x, hence M is of degree 2n (n — 1)^ in x, i.e. the singular surface is of degree 2m (n - ly. CHAPTER XVIII. DIFFERENTIAL EQUATIONS CONNECTED WITH THE LINE COMPLEX. 289. Lie* has shown that with the line complex there is associated a partial differential equation of the first order, whose characteristic curves are principal tangent curves on Integral surfaces ; and has also investigated certain types of partial differential equations of the second order by aid of conceptions drawn from both line- and sphere-geometry. The present chapter consists of a sketch of his researches. In the partial differential equation of the first order F {x, y, z, p, q) = 0, we may consider x, y, z to be the Cartesian coordinates of a point and 'p, q, —\ to be proportional to the direction cosines of the normal to a surface element (Art. 218) of the point ; then the given differential equation selects oo * surface elements, and the problem of integration is to determine surfaces whose surface elements shall satisfy the equation F=0. Through every point there pass oo ' surface elements which satisfy the given condition ; these elements envelope a cone and if I, m, n are proportional to the direction cosines of any generator of this cone, since this generator is the intersection of the planes of two consecutive surface elements, we have lp + mq — n—0, Idp + mdq = 0, ^ dp + -^ dq =. ; , I _ m _ n ^"''^ Tp~T,~pF, + qFg- Eliminating p and q from these equations and the equation F=0, we obtain an equation of the form f{l,m,n,x, y,z) = 0, * See the memoir qnoted on p. 233. 289-290] CONNEXION with differential equations 327 which is homogeneous in the quantities I, m, n, and hence represents the cone connected with any point (x, y, z). Conversely, if the last equation be given, we obtain the corresponding equation J'' = 0, by eliminating I, m, n from the equations df ■ ^ dn dn A differential equation of the form / {x, y, z, dx, dy, dz) = 0, which is homogeneous in dx, dy, dz, is usually termed a Monge equation. It assigns to each point of space a cone of directions, and hence leads, by the foregoing process, to a partial differential equation of the first order. A curve, such that the direction cosines of the tangents at all its points satisfy this Monge equation, is called an Integral curve. 290. The characteristic curves of a partial diflferential equation. It will be convenient to recall for reference some well- known results in the theory of partial differential equations of the first and second orders. An equation of the form F (x, y, z, p, q) = has three types of solutions : — (i) A complete solution, viz. a solution of the form z=f{si:,y,a,b), where a and h are arbitrary constants. (ii) A general solution, obtained by making b a function of a and eliminating a from the equations z==f{x,y,a, hence c is a principal tangent curve on S. On any Integral surface, therefore, the characteristics of F = form one set of principal tangent curves. Conversely, if in a Monge equation the osculating plane at each point of every Integral curve touches the cone assigned to the point by the Monge equation, the latter equation is that of a line complex ; for if f{x, y, z, x', y , z) = is the given Monge equation, by the given condition we have y'z" — y"z z'x" — z"x x'y" — x"y' /x- fy' // hence a;"/^ + y"fy + z"f^ = 0. But the points of the Integral curve, being functions of one parameter t, satisfy the condition ^( = 0, hence ^'/« + yjy + ^'fz + "'"f^ + y"fv + ^7/ = 0, i.e. the equation x'fx + y'fy + ^s'/z = is satisfied by each line element of the Monge equation; this shows that the locus of points, which have a line element of the Monge equation in a given direction, is a cylinder, and therefore, by the foregoing, the Monge equation is seen to be that of a line complex. 293. Form of the partial differential equation corre- sponding to a line complexf. The differential equation of the principal tangents on any Integral surface of F= is dpdx + dqdy = ; * Compare with the result obtained for a linear complex (Art. 41). These curves have also the same torsion, see L. and S. Berilhr. S. 309. + See L. and S. Berilhr. S. 640. 292-293] CONNEXION with differential equations 331 expressing that this equation is satisfied by a characteristic of F=0 we obtain F^Fp + FyFg + F, {pFp + qF,) = as an equation which is identically satisfied by any partial differential equation ^ = thus derived from a line complex. Conversely it may be shown that the Monge equation of every non-linear equation F=0, which satisfies the above identity, is that of a line complex. For consider the points at which a generator of the cone of the Monge equation has a given direction I, m, n ; then we have by Art. 289 ¥p^Y,-pF, + qF, ^^^' the points in question form a surface which is obtained by eliminating p and q from these equations and F—0. Now the condition F^Fp + FyFg + F, (j)Fp + qF^) = becomes lF^ + mFy + 7iFs = (II); and since Ip + mq — n = 0, where p and q have values determined by (I) for each point of the surface considered, we have ox dx By oy oz dz over the surface, i.e. ^4l+^4-!.=«'^^'= • ("^)- But we may regard F(x, y, z, p,q) = as the equation of this surface, provided p and q have the values assigned to them from (I); the direction cosines of its normal at any point are proportional to thus the conditions (II) and (III) show that the normal at each point is perpendicular to the given direction I, m, n; i.e. the surface is a cylinder ; hence, as before, the Monge equation is that of a line complex. Hence, if a non-litiear partial differential equation F=0 has the property that on every Integral surface the oo ' characteristics are principal tangent curves, its Monge equation is that of a line complex. 332 CONNEXION WITH DIFFERENTIAL EQUATIONS [CH. XVIII 294. If the differential equation is linear, it has oc ' character- istics (Art. 290), which must in the present case he straight lines ; for the planes of the surface elements of each point form a pencil whose axis is that of the tangent to the one characteristic through the point ; and if each plane of the pencil is to be the osculating plane of the characteristic at the point and this is to occur for each point of the chai-acteristic, the latter must be a straight line. The Integral surfaces are therefore in this case ruled surfaces of the complex. 295. Contact transformations of space*. If we adopt a different notation in the equations (vi) of Art. 217, viz. by writing X, — Y, Z for a, y8, 7 respectively, and — z, y, x for x, y and z respectively, these equations assume the form X + iY + xZ+z = 0, \ x{X-iY)-Z-y = 0\] ^^^' The linear complex corresponding to the points (XYZ) of 2 is then xdy — ydx + dz = 0. Proceeding as in Art. 219 we find that the surface element {X, Y, Z; P, Q, —1) of ^ corresponding to the surface element (x, y, z ; p, q, — 1) of A is determined by the equations (i) together with the equations 7^ px + gy p^^i+l_ Q_ ^- ^g-i q + X ' q — X ' q — X ' By differentiation of the equations (i) we find that dz — pdx — qdy = — {dX + idY + xdZ + Zdx) -pdx -q{x. dX-idY- dZ + X -iY.dx} ^' _ I q-x q-x )' since the coefiBcient of dx is .Z-p-q(X-iY) = q[^-jr^iY) (Z + y r -7C - Hence dz - pdx - qdy = {q-x) [dZ - PdX -QdY]. If dz — pdx — qdy is zero, the two consecutive surface elements {x, y, z, p, q), {x + dx, y + dy, z + dz, p + dp, q + dq) * See L. and S. BerUhr. S. 522. t This form of the equations is used in L. and S. BerUhr. S. 463. 294-296] CONNEXION with differential equations 333 are in a ' united position,' i.e. the point of the second surface element lies in the plane of the first, and we therefore conclude that if two consecutive surface elements of A are in a united position, so are their corresponding surface elements in S ; hence, to the 00 '■' surface elements of any surface of A correspond . oC surface elements of 2 which also belong to a. surface. Bj' the transformation considered, the equation F=0 gives rise to a new partial differential equation F^ (X, Y, Z, P, Q) = ; each surface element which satisfies F=0 leads to a surface element which satisfies F^ = 0. Any two of the Integral surfaces of F which touch along a characteristic give rise to two Integral surfaces of F^ which touch along a curve which is therefore a characteristic of F-^. Thus to the characteristics of F there correspond characteristics of F-i; if therefore the characteristics of F are principal tangent curves, those of Fj are lines of curvature (Art. 220) ; to the line complex of principal tangents corresponds a sphere complex of principal spheres. 296. The trajectory circle. la the correspondence of Art. 215 between the spaces A and S, the two lines which coi;respond to a sphere of centre (XYZ) and radius H axe given by the equations X + i7 = s, ±H+Z=r, X-iY=p, ±H-Z=cr. Any equation H=F{XYZ) may therefore be taken to represent either a line-complex or a sphere-complex. The tangent linear complex contains the sphere {X^, F„, Z^, Ho) and also every consecutive sphere of the complex. It is easy to see that this complex is formed by spheres which cut the plane _^„ = g(X-Z„) + ||(F-F.) + ||(^-Z„)...(i) at a constant angle ; hence the points of contact of the sphere (Xo YAHo) with every consecutive sphere of the complex which touches it, lie on the circle which is the intersection of (i) with the sphere (Z - X„r + (Y- YoY + {Z-Z^y = Ho' (ii), where in equations (i) and (ii), the X, Y, Z are current Cartesian coordinates. 334 CONNEXION WITH DIFFERENTIAL EQUATIONS [OH. XVIII This circle is called the Trajectory Circle of the given sphere, and has important bearings on Lie's theory. The surface elements of the sphere at the points of this circle belong to different Integral surfaces of the differential equation jPi = which corre- sponds to the given sphere complex. The oo * surface elements of the complex thus consist of oo ' surface elements on each of the 00 ' spheres of the complex. If two consecutive spheres {X^Y,,Z^H^ and {X^ + dX,...) touch each other, we have as the condition dX"- + dY' + dZ' = dH' = fU? dZ + ^° dT+ ^^ dzX . \cXo 01 „ oZ^ ! This is a Monge equation whose cone for the point {X^Y^Z^ is (Z-X„)»-|-(F-F„)= + (Z-Z„)» S(^-^»)-^8-t(^-^«)+i<^-^») This cone is seen to be one of revolution and to contain the trajectory circle of the sphere (X^Y^Z^H^. It is termed by Lie the elementary complex cone of the sphere complex and gives the direction of those points consecutive to (XoY^Zo) whose complex spheres touch the given sphere. We observe that if of the family of surfaces H = constant, the one be taken which passes through (X^Y„Z„), the normal to this surface at this point, which has direction cosines proportional to dJSo dH^ dHo dx,- dZ- W,' is clearly the axis of the elementary complex cone of (Xf, Y^Z^). 297. Partial differential equations whose character- istics are geodesies. If we consider any point P, or (X„ YoZo), which lies upon a surface H=C, and take a point Q, or (Zo + dZ, Fo + dF, Z, + dZ), in which the elementary complex cone of P meets a consecutive surface H=C + dC, then since the latter surface passes through Q, we have dcJ^UX+^^dY+^^dZ 9Zo 9F„ dZo = '/dX' + dY' + dZ-' = PQ. The surface H= G + dG, therefore, cuts off from eacA generator of this cone the same length dC. 296-298] CONNEXION with differential equations 335 From the Monge equation of the elementary complex cones we derive a partial differential equation, which will shortly be given, whose surface elements touch these cones along characteristics, (Art. 290). Consider a surface element of this differential equa- tion which touches the cone of its point along PQ and Fig. 13. meets the surface H=G along PP' ; then it is clear that PQ and PP' are at right angles, since the cone is one of revolution having its axis normal at P to the surface N=C. On each surface element, therefore, of any Integral surface of this differential equation there are thus detennined two orthogonal directions, giving two families of orthogonal curves on the surface PP', QQ', ■ ■ ■ and PQ, P'Q', ...; the former being the intersections of the Integral surface with the surfaces H = C, the latter the characteristics of the partial differential equation. But since it has been seen that dC = PQ = P'Q' , etc., the curves PP', QQ, ... are parallel curves, and therefore their orthogonal trajectories are geodesies. Hence the charaxsteristics of the partial differential equation derived from the elementary complex cones are geodesies on its Integral surfaces. The differential equation in question is found by eliminating I, m, n from the equations ll^dn' ^~ dm' dn' and is therefore easily found to be (PH-^ + QH^ - H^f - (H^^ + H/ + H^^-l)(P' + Q^ + l)= 0. 298. We shall now show that the characteristic of the surface element at any point P of the trajectory circle of any sphere of a sphere- complex, is perpendicular to the trajectory circle. For since P is a point on each of the 00= complex spheres {X „ YoZ^ H o) yfhich are principal spheres at the point P, or (XYZ), these spheres are determined by the equa- tions ^^g-1^- {x-x„y+(Y- Y,y+{z-z,y-H^=o, 336 CONNEXION WITH DIFFERENTIAL EQUATIONS [CH. XVIII -ffo = F{Xo Y^Z^). The line of intersection of two consecutive surface elements at P gives the direction of the characteristic; now this line is clearly perpendicular to the line joining the centres of the two corresponding principal spheres, i.e. to the line whose direction cosines are proportional to dX^, dY^, dZ^. Hence, if I, m, n are the direction cosines of the required characteristic, we have ldX„ + mdYo + ndZf, = 0, I {X - X,) + VI {Y- F,) + n{Z-Z,) = 0. But since the consecutive principal sphere passes through {XYZ) we have (Z - X„) dZ„ + ( F - F„) dY, + {Z- Z„) dZ, + H, dH, = 0, i.e. Z-Z, + H,:^]dZ, = 0. (X-X, + H, g) dZ„ + (y-Y, + H, |^°) dF. dZ, Hence the line whose direction cosines are proportional to X — Xo + Ho^Y' ^~^ii + So~y^ , Z — Zo + Ho-^^, satisfies the conditions which determine the characteristic. Hence the tangent of the characteristic at P is perpendicular to the tangent plane of the elementary complex cone of G along GP, i.e. is perpendicular to the trajectory circle. The trajectory circle derives its name from this property. It follows that the tangent of the trajectory circle at P touches at that point a line of curvature of an integral surface of the partial differential equation connected with the sphere-complex; hence the tangent plane of the elementary complex cone of G along GP touches at G the locus of centres of curvature of an Integral surface of this differential equation. Now the differential equation whose characteristics are geodesies on Integra] surfaces was derived from the Monge equation of the elementary complex cones of H = F(XYZ); if .^1 = be the partial differential equation which corresponds to this sphere- complex, it follows that each Integral surface of ^i has as its 298-301] CONNEXION WITH DIFFERENTIAL EQUATIONS 337 surface of centres of curvature an Integral surface of the differential equation whose characteristics are geodesies. 299. The preceding investigations have shown the existence of three classes of partial differential equations of the first order v\rhose solutions are mutually dependent ; they are (i) those whose characteristics are principal tangent curves on Integral surfaces, they are termed by Lie the equations D^ ; (ii) those whose characteristics are lines of curvature on Integral surfaces, or Djj ; (iii) those whose characteristics are geodesies on Integral surfaces, or Di,. 300 The complex of normals. Before leaving partial differential equations of the first order, Lie's solution of the following problem will be given, viz. the problem to determine all surfaces whose normals belong to a given liiie-complex. Let f{ydz — zdy, zdx — xdz, xdy — ydx, dx, dy, dz) = be the given line-complex ; since the lines of each complex cone are to be perpendicular to the surface elements of the required differential equation at the vertex of the cone, we must have dx : dy : dz =p : q : — 1, hence the differential equation is f{-y-zq, zp+x, xq-yp, p, q, - 1) = 0, or, since xq —yp — q{x + zp) — p{y + zq), the differential equation of the required surfaces has the form F(y + zq, x + zp, p, g) = 0. 301. Partial differential equations of the second order connected with line- and sphere-complexes. Lie has shown that the solutions of certain classes of partial differential equations of the second order are intimately associated with complexes of lines and spheres. He considers in the first place the problem to determine all surfaces of which one set of principal tangent curves belongs to a given line-complex. Let f{ydz-zdy, zdx — xdz, xdy — ydx, dx, dy, dz) = be the given complex, and z = (j>{x, y) J. 22 338 CONNEXION WITH DIFFERENTIAL EQUATIONS [CH. XVIII a surface which satisfies the given condition ; its principal tangents are given by the equation rdaf + 2sdxdi/ + tdf = (i) ; to express that a complex curve is a principal tangent curve of this surface, we substitute in /= dz = pdx + qdy, and thence determine one or more values of -J^ in the form dx -£ = N{x,y,z,p,q). Substituting this value of -~ in (i) we obtain dx ^ r + 2Ns + NH = (ii) as the differential equation whose Integral surfaces satisfy the given condition. Of this equation we know already two types of solutions : (a) ruled surfaces of the complex, (b) surfaces which at each point are touched by the complex cone of the point, i.e. solutions of the D^ connected with the given line-complex. Moreover there are no other solutions than these ; for if an Integral surface contains oo ' straight principal tangent lines, i.e. generators, which belong to the complex, it is a ruled surface of the complex ; if it contains oo ' complex curves as principal tangent curves, the osculating plane of such a curve at any point touches the surface, and we have seen that it also touches the complex cone of the point (Art. 292), so that the surface is touched at each point by the complex cone of the point, i.e. the surface is a solution of the con-esponding !)„. This differential equation of the second order is one whose two characteristics through each point of an Integral surface coincide; the characteristics are therefore identical with one set of principal tangent curves on each Integral surface. The symbol D^' is employed by Lie to denote this class of differential equations. 302. The consideration of a sphere-complex leads to the problem to determine all surfaces whose principal spheres belong to a given sphere-complex. 301-304] CONNEXION WITH DIFFERENTIAL EQUATIONS 339 The differential equation of the second order defining such surfaces is similarly found to be {rt - s') B' - [(]. +p')t- 2pqs + (1 + q') r] ^/l+f+q' .R + {l+p'' + q'f = ; wherein iJ is a function of x, y, z, p and q. The symbol D^^' is introduced to denote such equations. 303. Partial differential equations of the second order on whose Integral surfaces both sets of characteristics are principal tangent curves or lines of curvature. We now consider two types of partial differential equations of the second order; the first type is such that on each Integral surface the two sets of characteristics are the principal tangent curves ; the second type is such that the two sets of characteristics are the lines of curvature. They are denoted respectively by D^' and D^". Since the differential equation of the characteristics of rt-s^ + Ar + 2Bs + Ct = F(x, y, z, p, q) is {A+t)dy^ + '2.{s-B)dxdyJr{C + r)da?=0, if the characteristics coincide with the principal tangent curves, whose equation is rdx^ + 2sdxdy + tdy^ = 0, we must have A—B = C=0 and the equation of a B^i" is rt-^ = F{x,y, z,p, q). It has been shown by Du Bois-Keymond* that the equation of a Dffl" is ,_L±^\, + fl±i-%_,>=o, pq \ pq J where F is any function of x, y, z, p, q. It is easy to see that this equation is equivalent to the following : [:pqt - (1 + q') s\f + [(1 +^0 < - (1 + 1) ^]/+ [(1 +P0 s - Vr] = 0. where the functions / and i^'are connected by the equation F {0 + ?=)/+ pq] - {/>? +/(i +p')] = 0. 304. The curves s and o- of a D^^' . If in the equation last given for a D^' , -j- be substituted for /, we obtain the ay * See Partial Differential Equations, p. 130. 22—2 340 CONNEXION WITH DIFFERENTIAL EQUATIONS [CH. XVIII differential equation of the lines of curvature of any Integral surface ; it follows that a given D^' assigns to any surface element of space two definite orthogonal directions ; on any surface we thus obtain two sets of curves s and a which are orthogonal. A Dja" is thus an analytical expression of the problem to determine the most general surface whose directions of principal curvature depend by a given law on the position of the corre- sponding surface element. The characteristics of any particular integral of a D^" of the form f{xyzpq) = will be also characteristics of the D«^', and therefore lines of curvature of the Integral surfaces on which they lie; hence, such a particular integral must be a jDja*- If a Djj" has a first integral which is a D,2, to the latter will correspond a sphere-complex, and considering any sphere of this complex it is clear that its trajectory circle belongs to one of the two sets of curves s and a on this sphere. If we consider a singly infinite number of integrals of this kind, and therefore oo ' sphere-complexes, all spheres of space are thereby included. Hence for every sphere it follows that among its curves s, a there is included one or several trajectory circles. If finally a general first integral exists of the form '^ =f(v). where u and v are each functions of x, y, z, p, q, there are two conceivable cases, viz. among the curves s, ^o + f^) (\ + f^)- The partial differeatial equation 2 (5^) =0 of a special complex (Art. 280) becomes in terms of the variables fj,i f-MV IM + ... = 0. \dfiiJ {fii - /U2) (mi - /^s) (^1 - M4) But of this partial differential equation a complete solution is known f, viz. J ■ V/(/t3) J V/(/i4) in which a, b and G are arbitrary constants. Attributing to a, b and C any definite values, we obtain a surface whose two sets of principal tangents belong respectively to the cosingular quadratic complexes fi = a, fi = b; to prove this we have merely to show (Art. 288) that the congruence ^ = 0, i|r = is special, i.e. that the directrices of the linear congruence coincide, which occurs if \dxil \dXil \ oxi dXiJ where <^ is the special complex considered, and i/c is either of the complexes /i^ = a, 1^^ = b. Now 2(5^) is zero by hypothesis, hence we have to show , \dxi/ that dxi ' dxi * See Klein, Math. Ann. v. t See Jacobi's Vorlesungen iiber Dynamik. 346 CONNEXION WITH DIFFERENTIAL EQUATIONS [CH. XVIIl le coordins The Jast condition becomes in terms of the coordinates /if 9/il ■ 3/4i '{/Ml- fJ^) (fJ-i — /M,) i/J-i - f/.t) which is satisfied, since 9\|r dyjr dyfr 9/il 9/t2 9/X3 and ^ = ^(^^ ~ °-^(f^* ~ ^) which vanishes for fii = a or for fit = b. The value of the constant C has not come into consideration, hence there is a singly infinite number of surfaces <^ for each of which one set of principal tangents belongs to one, and the other set of principal tangents belongs to the other of two cosingular quadratic complexes. To the two quadratic complexes there correspond two partial differential equations Z),] ; these x> ' surfaces are common Integral surfaces of these two differential equations. MISCELLANEOUS EESULTS AND EXERCISES. 1. If (xyz), (x'y'z') are the Cartesian coordinates for rectangular axes of two points on a line, the Pliicker coordinates of the line are P23 = y^'-'!/'^, Psi = zx'-z'x, p^^ = xi/'-x'y. The first three coordinates are proportional to the direction cosines I, m, n of the line ; and since Pa = 2/ (s' - ») - » {y - y), Psi = z{x' -x)-x {z' - z), Pi'i=x{y' -y)-y{x'~x), it is clear that the Pliicker coordinates are proportional to the quan- tities I, m, n ; l', m, n where I, m, n are the direction cosines of the line and I' = yn- zm, m! =zl — xn, n' = xm - yl. The relations satisfied by the coordinates are l'' + m^ + n''= 1, W + mm' + mo = 0. The form of the last two equations suggests that V, m', n may be regarded as the derivatives of /, m, n with regard to a new variable t of which I, m, and n are such functions that dl , dm , dn bo that I =~r , m = -^r- , n =—j- . at at at " Thus instead of taking six coordinates to represent a line we may take three variables and their three derivatives, and giving this a kinematical interpretation, we may represent a line by a point moving on a sphere of unit radius ; the radius through the point is parallel to the line, and the three components of velocity of the point are the second set of three coordinates of the line*." * Hudson, "A new method in line geometry,'' Messenger of Mathematics, 1902. 348 MISCELLANEOUS RESULTS AND EXERCISES 2. Denoting by M tlie " mutual moment " of two lines (i, ?», n ; I, m', n') and (A., /x, v; A.', /x', v), i.e. the moment of unit force in one line about the other ; show that if {xyz) is any point on the first line and {x'y'z) any point on the other line, X — x', I, A. ; iA.' + m/x' + nv + I'k + 7ft'/x + n'v. M = y-y, »n, M 3. *The shortest distance of two lines (Imnl'm'n'), (A^avA'/iV) is {LMNL'M'N'), where J. J. T' P^^ > < I I i. L = mv — nn, etc. ; L = + mv +fn,v — nit. —n/ji,, etc., where p = l\ + mfi. + nv, v! = IX.' + I'k + m/j.' + m'/j. + nv + n'v, q^ = L^ + JIP + JVK 4. In rectangular Cartesian coordinates the line (— I, m, n, I', — m', — n') is the reflexion of {I, m, n, V, m,', n') in a; = 0, the line (— I, m, n, — l', m, n) is the reflexion of {l, m, n, I', m', n') iny = 0, z = 0, the hne {I, m,, n, — V, — m, — n) is the reflexion of (Z, m, n, V , m, n) in the origin. 5. The line [l, m, n, I', m, n) touches aa? + hy^ + cz" + dw^ = if (af + hm? + cn^) d + bel'' + cam'^ + abn' = 0. It touches 2axw + by'' + cz' = 0, if a (aP - 2bmn' + 2cnm') — bcP = 0. 6. The tangent planes drawn from the line to aa? + by'' + cz' + dv? = are i»/o + Q^lb + iJ7c + S'ld = 0, where \P =.ny — mz — I'w, Q = lz — nx— m'w, R = mx — ly — n'w, S = l'x + my + n'z. [Math. Trip. 1896.] To 2axw + by'' + cz^ = 0, they are 2PS/a+ Q'/b + R'/c^O. * Questions 3 — 6, 11 — 13 are due to Prof. Bromwieh. t P-0, Q=0, iJ = 0, S=0 are planes through the line and the vertices of the tetrahedron of reference. The plane through the line and the point (aJo^eZoa'o) is MISCELLANEOUS RESULTS AND EXERCISES 349 To they are {dbcdfgha'b'c'\xyzwY = 0, a h g a' P h h f b' Q 9 f c c E a h' c' d S P Q £ S = 0. 7. The line is a generator of the general central quadric {ahcfgh\xyzY + d = Q, if al + hm + gn = l' (A/df, hi + bm +fn = m' (A/rf)^, gl +/m + cn = n' (A/dy, where A = abc + 2/gh — a/' — hg"^ — ch?. [Bromwich, Mess. Math. 1900.] 8. The line is normal to a^ + 6y^+ cz^ = 1, if IV mm wn! „ — + —V- + — =0, abc (b — c) Im'n' + (c — a) I'mn' + {a -b) I'm'n = (b-c) (c — a) {a — b) Imnjabc. It is normal to 2ax + by^ + cz^ = 0, if a m' a n' a frt'? r^\ c n b m 2P\6 c/' 9. If all' + biriTn,' + cnn = 0, there is one and only one of the confocals a? y^ z^ a + p b + p c + p to which the line is normal ; this is given by (b — c)(a+ p) Im'n' + (a — a){b + p) I'mn + (o — 6) (c + p) I'm'n = {b — c)(c— a) (a — b) Imn. 10. Tangent planes are drawn to the confocals ^L + _^ + _^ = i a + p b + p c + p through the line (I, m, n, I', m', n), prove that the locus of their points of contact is the twisted cubic given by P[\+{a-b)r'-{c~a)q^] = ^\+{b-c)r'-{a-b)p'-] Tr /, ^ „ fi + V + m^ + n" = [l+(c-a)p^-{b-c)q^]= p^^,.^.^,. ; where p, q, r are linear functions of t such as I't + mn — m,'n ^" r + m'2 + n'^ etc. [H. P. Baker.] 350 MISCELLANEOUS RESULTS AND EXERCISES If normals are drawn at the points of contact of these tangent planes they generate the hyperbolic paraboloid (Ix + my + nz) [(6 — c) m'n'x + (c— a) n'l'y + {a — b) I'm'z] + (b — c) {c — a) (a — h) hnn = (b - c)lx [(c— a)nn' + (a — b) mm] + (c — a)my[(a—b)U' + (b—c)nn'^ + (a-b)nz[{b- c)mm' + (c—a) U']. If air + bmm + cnn = 0, the paraboloid becomes (Ix + my + nz — lninj6f = 0, •where 6 = ; = = j . [Bromwich.] b—c c—aa—b ^ 11. The intercept on the line (/, m, n, I', m, n') by the quadric (abofgMxyzf = 1 jg 2 ^/^' + m^ + ri' [{abcfghllmnf - {ABCFGHll'm'n'ff {abcfgh\lm,7if 12. The polar line with respect to {abcfgh\xyzY = 1 is given by the equations \ -V AV + Urn! + Gn al + hm,+gn " 13. Show that the polar of (I, m, n, I', m,', n') with regard to a? v^ !? ^ a b c is given by (X, ju,, ^, X', ju.', v') = {al', bvi, en, -bcl, —cam, -abn). If all' + bmm' + cnn' = 0, show that the polar lines with regard to quadrics confocal with the given quadric lie in the plane Ix + my + nz = — y (b — c) = two similar expressions. 14. If a line touches the quadric ayz + bzx + cxy + abc = 0, show that aH'" + b^m'^ + c'n- — Ibcm'n' — 2can'r — 2abl'm' = iabc (amn + bnl + dm). 15. If the line {Imnl'm'n') receives an infinitesimal rotation dd on a screw of pitch ro- whose axis is the line (abca'b'c), it becomes l + dl, ..., I' + dl', ..., where, if K'^a' + b^ + c^, , dl J dm , ^ dn ^ ^ '^' f<^-jD=d-0'n, k-^ = am-bl; , dl' , , ^, , , k ^ =bn +bn — cm—cm + w (bn — cm) ; J dm II ,, , « -jK = ''^ +cl — an -an-vmicl- an) ; J dn , « -j^ = am, + a m — bl' — b'l + zj (am — bl). MISCELLANEOUS RESULTS AND EXERCISES 351 Hence show that if a line (^, m^, n^ ; II, m^, n/) is rotated through an angle 6 about the line {I, m, n ; I', m, n), where 6 is to be reckoned positive when given by a left-handed screw in the sense (Imn), then if (?2, m^, ri~2 ; l^, m^, n^) are the coordinates of the line in its displaced position ^2 = ?i — {mn^ — m^n) sin 6 + {Ik — l^) cos 6, etc. ; l^ = Zi' + {n'm.1 — rij'm, ■*■ m'n^ — m^'n} sin 6 + (l'k — l-l + Im) ( 1 — cos 6), etc. ; "where A; = ZZi + mm, + jwii, ■SI = 11^ + mini + nn^ + Vl^ + mniy + ri'^ij , and (Imn), ilim^n-y) are the actual direction cosines of the lines*. 16. If the line (Imnl'mn) receive a small displacement given by (da, dh, dc) parallel to the axes and (dt^^, dtji^, d^ about them, find the consequent change in the coordinates of the line. Hence show that a line of the linear complex A'l + B'm + G'n + Al' + Bm + On — will be changed into a line of the same complex if da (BC - B'G) + db (CA' -G'A) + dc (AB' -A'B) = 0, 6?<^i : d^^ : (i<^3 = A : B : C, {BC - EC) d, = A(Cdb- Bdc), etc. 17. With the same notation and coordinate system show that the equation of the axis of the complex a'l + h'm + c'n + aV + hm + en' = cy — hz + a' az — cx + V hx — ay + c a b c ' The coordinates of the axis are given by the equations p.l = a, p.m = b, p.n = c, p.l' = — a' + Ea, p. m = — b' + Eb, p.7i! = -e' + Ec; E = {aa' + bb' + cc')l{d' + b'^ + - CjTJi + CjW ; the quadric to which the regulus belongs is (abcfgh \x — aw, y - ^w, z — -^vdf + v?D = 0, where D = ahe + %fgh — af- — bg^ — ch", a = aj, b = b.2, c = c^, la^C^-b^, 2/3 = a,- Ci, 2y=6i-02. 23. Show that any four points and their polar planes for a linear complex form two tetrahedra, of which each is inscribed and circum- scribed to the other. 24. There is one pair of lines which are polar for a given linear complex and also for a given quadric. 25. The polars of a line I with reference to a pencil of linear complexes form a regulus, to which I itself and the directrices of the two special complexes of the pencil belong. * Questions 19, 20 and 22 are due to Prof. Bromwich. MISCELLANEOUS BESULTS AND EXERCISES 353 26. The polars of the lines of a linear complex C with reference to another C form a third linear complex 0" which belongs to the pencil determined by G and C". 27. If Ai and k^ are the chief parameters of two complexes A and B, d the shortest distance, and ^ the inclination of their axes, , Q.{a\h) ^il {a) u{b) 28. Show that a linear complex may contain two lines of one regulus belonging to a given quadric and only one of the other. If the quadric is Scix/ = 0, and the complex 2a«Pft = 0, show that in the case considered 4C1C2C3C4 {ui^a^ + oiija^a + a-^^a^f = {e^Ciai^ + c^c^aj' + ...f. [Math. Tripos, 1898.] 29. If six linear complexes are in mutual involution, three of them are right-handed and three left-handed. 30. Three linear complexes which are not all right-handed or left-handed intersect in a real regulus. 31. The two common intersectors of any line and its polars for three linear complexes belong to the regulus common to these complexes. 32. In the system of five terms determined by the complexes A, B, C, D, E, the directrices of the special complexes form the linear complex I ««, K Ci, di, et, Xi I = 0. 33. The linear equation satisfied by the six coordinates of a line PQ which belongs to a linear complex being /i (PQ) = 0, and similarly fi (PQ) = ^! fs (PQ) = ^1 ft (PQ) = being analogous equations for three other linear complexes, show that the conditions that a plane ABC may contain one of the two lines common to the four complexes are MBG) MBO) MBC) MBG) I MCA) MCA) f,{GA) MGA) 1 = 0. f,{AB) M^B) M^B) M^B) I 34. The 00 ^ complexes \Ci -1- /tCj + vG^ = being designated a net of complexes, where Cj = 0, (72 = 0, C3 = are three given linear complexes, show that there is one complex of the net which contains any given pencil, and that every point of a given plane determines one complex of the net for which the given point is the pole of the plane. 35. The polars of a line I with reference to the complexes of a net form a linear congruence whose directrices are the two generators of the regulus common to the complexes of the net which meet I. The axes of the complexes of a net form a congruence (2, 3). * See Segre, CreUe, Bd. 99, J. 23 354 MISCELLANEOUS RESULTS AND EXEECISES 36. The complex C^ is said to be orthogonal to C when C-^ is in involution with C, where C is the locus of lines polar to those of G with regard to a given quadric. [d'Ovidio.] Prove that for each complex of a pencil of complexes there is one orthogonal complex belonging to the pencil. 37. If four tangents of a twisted cubic have their two intersectors coincident with one line p, then p belongs to the linear complex determined by the tangents of the cubic. 38. If two tetrahedra are inscribed in a twisted cubic their eight planes osculate another twisted cubic ; and there are oo ' tetrahedra which are inscribed to one and circumscribed to the other tetrahedron. [Hurwitz.] 39. If P is the pole of the sphere-circle for the ray of a linear congruence in the plane at infinity, and p the ray through P, the rays which have a given inclination to p form a ruled quartic which has the directrices of the congruence as double directrices. 40. Two lines are said to be conjugate with respect to a quadric when each intersects the polar of the other. [Schur.] If five lines meet a given line and are such that all but one of the ten pairs formed from them are conjugate with respect to a quadric, the quadric is uniquely determined and the remaining pair are also conjugate. Let now Oj be the first line and b^, b^, 63, 64, 65 the meeting lines, then dj the other intersector of 61, 62, 63, 6^ is the polar of 65. So each of the lines a,, a,, a,, a^, a^ is the polar of the corresponding b. But the five lines 6 all meet a^, hence the lines a^ ...a^ all meet the polar of aj which may be called 6,. The lines a and b thus form the doitble sixer 0102030405%] 1 °^ Schlafli. [Grace.] OiOiO^bfbsbg } It follows that opposite lines of a double sixer are polar lines with respect to a certain quadric. [Schur.] 41. Show that through any point three osculating planes can be drawn to the twisted cubic ^ = zx, z^ = uy, and that the three points of osculation are coplanar with the given point. Also show that each such point and plane are pole and polar plane for a linear complex which contains the four lines 2 = 0, u = 0; y = 0, w = 0; a; = 0, » = 0j x = 0, y = 0. MISCELLANEOUS RESULTS AND EXERCISES 355 42. On every ruled surface whose generators belong to a given linear complex the family of principal tangent curves is found by a quadrature. [Lie.] Denote by k the principal tangent curve of the surface whose tangents belong to the linear complex ; any generator meets k in two points a. and y, and any point ' on this generator is x + ity; the coordinates x^ and y^ are functions of one parameter \. The differential equation of the principal tangent curves is Pd\'' + iQd\d/i+Bdij:'=0, where p_ I djf dji 9^1 r,_\ , ?!} 3ii i!fi I B_U ?i> ?£i 8^«i I I *' d\' Bm' 5X2 |. «-| H, g^. g^. ^-^-^^ |. -«-| 2i. g^, g^, g^j |. Hence, since i2 = 0, the equation of the principal tangent curves is Pd\ + Qdii.=Q; where -P= | x^, ^,-, x: + p.y!, x^' + iiy^' \, -Q=| x^, i/j, x/, ?// |, the differentiations being with regard to X. Hence the required differential eqaation is where the JQ are functions of X. Since ;it=0 and ^=qo are solutions of this equation, we have Xi=X^=0, and the equation reduces to 43. The projection of every unicursal curve of degree m, whose tangents belong to a linear complex, on a plane perpendicular to the axis of the complex, has m points of inflexion at infinity. [Picard.] 44. A linear complex being given, any curve such that the polar plane of each of its points is normal to the curve at the point is a helix. [Picard.] 45. The lines of a quadratic complex G^ determine upon any two lines I and V a (2, 2) correspondence, the double ratio of the branch points on I being equal to the double ratio of the branch points on V. Taking I to be any line and V its polar line for C^, show thereby that the locus of singular points is identical with the envelope of singular planes. 46. Each of the two lines common to any four of the fundamental complexes of C^ is the polar of the other with regard to (7^ ; and there are no other pairs of lines so related except the 15 pairs of lines thus obtained. [Klein.] 47. Every quadratic complex through 16 given lines contains 16 other fixed lines. For let /i (x) = 0, /„ (x) = 0, /s (X) = 0, /4 (x) = be any four complexes through the 16 given lines, then any complex through these lines has an equation of the form 356 MISCELLANEOUS RESULTS AND EXERCISES Eince the constants X, p., v can be determined so as to make it pass through any other three lines. But this complex contains all the lines of intersection of the four given complexes, which are 32 in number. 48. A Kummer surface is identical with its polar surface for a fundamental complex. The polars for C of the lines of one of its fundamental complexes Cj belong to Cj. 49. Through two pencils of C" which have no common line one linear congruence passes ; this congruence intersects C^ in two other pencils each of which contains a line of the former two pencils ; thus for any two skew pencils of C" there are two other skew pencils of G^ having a common line with each of them. 50. If a line p is such that the points of intersection of p with the complex curve of C^ in any plane whatever lie upon a second complex curve, the locus of jo is a complex of the sixth degree. [Sturm.] If C^ is replaced by a tetrahedral complex p {ABCD)=K, the complex of the sixth degree is replaced by the three complexes p {ABGD) = \\ p {ABCD) = (1 - \f, p (ABCB) = (l - ^)° ■ [W. Stahl.] 51. By taking different values for the constants X, ft, v in the projective transformations we obtain a set of oo ' transformations, connecting the point {xyz) with each point of space. If the line-element (x, y, z; dx : dy : dz) is given, a definite line-element is assigned to each point {xiy^z^) of space, viz. that given by the equations dx dXi dy _ dy^ dz _ dz^ x~ x^' y ~ yi' z~ z^' These line-elements all belong to the same tetrahedral complex. [Lie.] 52. Every tetrahedral complex is invariant for reciprocation with regard to a quadric which has the tetrahedron of the given complex as a self-conjugate tetrahedron. [Lie.] 53. The normals of the quadric {ahefgh\xyzy = 1 belong to the complex A IV + Bmm + Cnn' + F {mn' + m'n) + G {nV -^ n'l) + H (Im' + I'm) = 0. 54. Any Une of the complex aW + hmm' -h cnn' = meets the quadric — + ~ + —=\ a c MISCELLANEOUS RESULTS AND EXERCISES 357 in points P and Q the normals at which intersect each other. The- coordinates of this point of intersection are given by the equations mn cn'^ + hm"' — hcP ,, etc. u oci- + cam- + aon 11 im/ni mnrt where 6 = e ' hcl" + cam? + ahn^ ' W mm nn b—c c—a a—b The line is a principal axis of the section of the quadric by the plane al' .F + bm! .Q + cn' . R = Q, where P = ny — mz — I', Q=lz-nx- ml, R = mx -ly- ri. The coordinates of the other principal axis are , It mt nt asl , asm- , csn , — , -r i — , a o c — + -r + —] + s {aP + bm'^ + en') = 0. The lines of the complex which are in the plane px + qy +rz + s = touch a parabola whose directrix is in the plane (b-c)- +(c-a)^ + (a-b)- = 0. As p, q, r and s vary, the directrix moves in the complex (6 - c) T, + (c - a) — 7 + (a - i) - = 0. ^ ' I ^ ' m n The projection of the parabola on the plane a; = is J {a -b)qy + v (a — c)rz= V(6 — c) s. [Bromwich.] 55. The lines of the complex aW + bmm! + cnn' = which meet the line P^ = 0, Q^ = 0, R^ = 0, touch the surface J{b - c) xPa + \/(c - a) yQ„ + \/(a - 6) zR„ = 0, where P„ = n„y—m„z~l^', etc. This may be put in three other forms such as J{b - c) iS„ + ^{c - a) zRf, + J {a - b) yQ„ = ; (Sf, = ?oa; +m|,y + n„»). [H. F. Baker.] 56. If a rigid body is turning about Oz, on a screw of pitch p, the lines of motion of its points belong to the complex nn =p(P + m^). This complex belongs to the species [(22)(11)]. It cuts any plane in the lines of a parabola, and if the plane is ax + by + CZ + d = 0, the directrix lies in the plane bcx + cay + jo (a" + 6^ + 2c^) = 0. [Bromwich.] 23—3 358 MISCELLANEOUS RESULTS AND EXERCISES 57. If I is any line of a complex C, then through I and its polar line for a fundamental complex there pass oo '■ reguli of C^ (not merely two). 58. Every regulus of a quadratic complex touches the singular surface four times. 59. Every regulus of C which contains a given line I of C" helongs to a tangent linear complex of I. 60. There are oo ' quadratic complexes which contain a con- gruence (C^ A) and have .4 as a fundamental complex ; they have the same fundamental complexes. 61. If two (2, 2) congruences E^ and K^ are such that a complex for which A', is focal contains K,, then a complex for which K^ is focal will contain K^. [Grace.] 62. A Pliicker's complex surface is its own polar surface for each of four linear complexes which are mutually in involution. [Klein.] 63. If a line I describes a congruence (m, n), its polar I', with reference to any given quadratic complex, will describe a congruence (2m + 3n, 3ot + 2w). 64. The double tangents of a general complex surface form four congruences (2, 2) ; if the double line of the surface belongs to the complex the double tangents form three congruences (2, 2) ; if it is a singular line they form two congruences (2, 2). [Sturm.] 65. The singular lines of a harmonic complex H' are the inter- sections of H' with the tetrahedral complex formed by the lines whose polars with reference to /, and f^ intersect each other ; where /j and /^ are a pair of quadrics which give rise to H'. 66. If a quadratic complex contains a regulus and also its com- plementary regulus it is harmonic. [Schur.] 67. If a quadratic complex contains a plane system it has three double lines in the plane. 68. The axes of the complexes of a system of four teiins form a quadratic complex. 69. Every congruence (m, n, r) possesses a ruled surface, of degree 4 (mn - r) - 2 {vi + n), formed by rays with coincident focal points. 70. The six singular points of the congruence (2, n) which lie in a singular plane are situated on a conic. 71. The tetrahedra of the 40 tetrahedral complexes which contain a given congruence (2, 2) can be arranged in 20 pairs so that the tetrahedra of a pair are inscribed and circumscribed to each other. MISCELLANEOUS RESULTS AND EXERCISES 359 72. If a line moves with two of its points A, B in two fixed planes, it describes a complex of the fourth degree. If it has three of its points A, £, C in three fixed planes it describes a congruence (6, 2). Taking the two planes in the first case as a; = 0, y = the complex is n'^ (P + m? + ■nF) = T^Pir? where k =--- AB ; if the three planes in the second case are the coordinate planes, the congruence is contained in the tetrahedral complex IcrriTnl + k'nn' = 0, where k' = AC. 73. If y= 0, <^ = are two quadrics, the complex of harmonic section determined by them is in general of the species [111111], having a tetrahedroid as its singular surface. When / and have certain projective relations to each other the complex is modified in form. By considering the elementary divisors of the discriminant of /+ iJ.'f' we arrive at the following canonical forms to which f and <^ can be reduced. 4 4 [1111], the general case; 4> = 'S,x^, /sSaj^i^ The equation to 1 1 determine the coefiicients of H- is {X2_ „ (.^ _ „)j |;^2 _ J („ _ J)} i^x^ -c{a.-c)] = 0, where a = 'S,a^, a = ai + a^, 6 = ra] + 03, c = a^ + ai. [11(11)] 05 = £14; the quadrics touch in two points and intersect in two conies. H"" is [(11)( 11)11]. [(11)(11)] «! = a^, 03 = osj ; the quadrics have four common generators forming a quadrilateral. H^ is [(11)(11)11]. [(111)1] ai = »2 = a3; the quadrics touch along a conic. H^ is [(111)(111)], i.e. consists of the tangents of a quadric. If in [1111] we have ^3+04=0, an edge of the tetrahedron of reference belongs to H^ which is [21111]. If ai + a^ = a^ + a^ — 0, two edges of the tetrahedron of reference belong to H^ which is then [(11)1111]. In the case [(11)11] if either 6 or c is zero, i.e. if/ and tfy are harmonic with their pair of planes of intersection and either cone of the pencil /+ /A(^, H^ is [(22)11]. [112] (f) = x^^ + x^^ + 2xsX^, / = a^^a^i^ + a^x^^ + ^a.^iXgXi + x.j^. The quadrics touch each other and H^ is [1122]. [(11)2] «!! = a^j ; the quadrics intersect in a conic and two generators. i?Ms[(ll)(ll)ll]. [1(12)] 022 = ^34; the quadrics intersect in two conies which touch. B^ is [(12)(12)]. [(112)] aii = a22 = «34; the quadrics touch along two generators. B^ is [(111)(111)]. 360 MISCELLANEOUS RESULTS AND EXERCISES If in [1 12] we have either On + Oji = 0, or a^ + a^t^O, i.e. when two cones of the pencil y+ /u,<^ are harmonic to/ and , H^ is [411]. [13]