AT LOS ANGELES QUARTIC SURFACES WITH SINGULAR POINTS CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, MANAGER ILonDon: FETTER LANE, E.G. EBinbutgij: I0 PRINCES STREET $rtn Horfe: G. P. PUTNAM'S SONS , Calcutta anB jUatmus: MACMILLAN AND CO., LTD. Coronto: J. M. DENT AND SONS, LTD. STokgo: THE MARUZEN-KABUSHIKI-KAISHA QUARTIC SURFACES WITH SINGULAR POINTS BY C. M. JESSOP, M.A. FORMERLY FELLOW OF CLARE COLLEGE, CAMBRIDGE PROFESSOR OF MATHEMATICS IN ARMSTRONG COLLEGE IN THE UNIVERSITY OF DURHAM Cambridge : at the University Press 1916 CambriDge : PRINTED BY THE SYNDICS OF THE UNIVERSITY PRESS Sciences f "7 ^ ' / - PEEFACE rilHE purpose of the present treatise is to give a brief account of the leading properties, at present known, of quartic surfaces which possess nodes or nodal curves. A surface which would naturally take a prominent position in such a book is the Kummer surface, together with its special forms, the tetrahedroid and the wave surface, but the admirable work written by the late R. W. H. T. Hudson, entitled Rummer's Quartic Surface, renders unnecessary the inclusion of this subject. Ruled quartic surfaces have also been omitted. For the convenience of readers, a brief summary of all the leading results discussed in this book has been prefixed in the form of an Introduction. I have to express my great obligation to Prof. H. F. Baker, Sc.D., F.R.S., who has given much encouragement and valuable #2 criticism. Finally I feel greatly indebted to the staff of the f~j University Press for the way in which the printing has been ^carried out. ro C. M. JESSOP. BH March, 1916. ? M i p CONTENTS INTRODUCTION CHAPTER I. QUARTIC SURFACES WITH ISOLATED SINGULAR POINTS. ABT. 1, 2 Quartic surfaces with four to seven nodes .... 1 Quartic surfaces with more than seven nodes ... 3 4, 5 Quartic surfaces with nodal sextic curves .... 4 6 12 Quartic surfaces with eight to sixteen nodes ... 9 CHAPTER II. DESMIC SURFACES. 13 Desmic tetrahedra ........ 24 14 Quartic curves on the desmic surface ..... 26 15 18 Expression of coordinates in terms of a- functions and resulting properties ....... 27 19 Section by tangent plane ....... 35 20 A mode of origin of the surface ...... 35 21 The sixteen conies of the surface ..... 37 CHAPTER III. QUARTIC SURFACES WITH A DOUBLE CONIC. 22 Five cones touching surface along twisted quartic whose tangent planes meet surface in pairs of conies . . 38 23 28 Expression of coordinates in terms of two parameters and mapping of surface on a plane ..... 40 29 The class of the surface ....... 46 30 The sixteen lines of the surface ...... 46 31 Origin of the surface by aid of two quadrics and a point . 47 32, 33 Perspective relationship with general cubic surface . . 49 34 Connection with plane quartic curve ..... 52 35 37 Segre's method of projection in four-dimensional space . 55 Vlll CONTENTS CHAPTER IV. QUARTIC SURFACES WITH A NODAL CONIC AND ADDITIONAL NODES. ABT. 38 Case of one to four nodes 62 39 43 Origin and nature of double points arising from special positions of base-points 64 44 46 Cuspidal double curve 69 47 Double conic consists of a pair of lines .... 72 48 53 Segre's classification of different species of the surface . 72 CHAPTER V. THE CYCLIDE. 54, 55 Equation and mode of origin of cyclide .... 86 56 Inverse points on the surface 89 57 59 The fundamental quintic and form of the cyclide . . 90 60 62 Power of two spheres : pentaspherical coordinates . . 94 63 69 Canonical forms of the equation of the cyclide ... 99 70 Tangent and bitangent spheres of surface .... 108 71, 72 Confocal cyclides 109 73, 74 Sphero-conics on the surface . . . . . . 113 75 Cartesian equation of confocal cyclides . . . . 115 76 Common tangent planes of cyclide and tangent quadric . 117 CHAPTER VI. SURFACES WITH A DOUBLE LINE: PLUCKER'S SURFACE. 77, 78 Conies on the surface 118 7981 Mapping of the surface on a plane 120 82 Nodes on surface with a double line 125 83 Plucker's surface 128 CHAPTER VII. QUARTIC SURFACES WITH AN INFINITE NUMBER OF CONICS : STEINER'S SURFACE: THE QUARTIC MONOID. 84, 85 Quartic surfaces with an infinite number of conies . . 131 86 88 Rationalization of Steiner's surface : all its tangent planes contain two conies of the surface : images of its asym- ptotic lines 134 89 Modes of origin of the surface 139 90 Quartic curves on the surface 141 91 Weierstrass's method of origin of the surface . . . 143 92 Eckhardt's point-transformation 145 93 The quartic monoid 147 CONTENTS IX CHAPTER VIII. THE GENERAL THEORY OF RATIONAL QUARTIC SURFACES. AKT. PAGE 94 The cubics which rationalize \^G(y) when Q(y) = is the general plane quartic : the surface S^ . . . 152 95, 96 The curves which rationalize \/Q (y) when O(y) = is re- spectively a sextic with a quadruple point or a sextic with two consecutive triple points : the surfaces S^ and< 3 > 155 CHAPTER IX. DETERMINANT SURFACES. 97 Sextic curves on the surface 161 98, 99 Correspondence of points on the surface . . . . 163 100 The Jacobian of four quadrics and the symmetroid . 165 101 Distinctive property of the symmetroid .... 167 102 Tangent plane of Jacobian 168 103, 104 Cubic, quartic, and sextic curves on the surfaces . . 169 105 Additional nodes on the surfaces 172 106115 Weddle's surface ........ 173 116 Bauer's surfaces 189 117, 118 Schur's surfaces 191 INDEX OP SUBJECTS 196 INDEX OP AUTHORS . 198 ADDENDA Throughout, the vertices of the tetrahedron of reference are denoted by AI, A 2 , A 3 , A 4 : seep. 50. pp. 38, 45. The oo 1 quadrics ^ + 2X^ + X 2 w 2 =0 touch the surface 2 =icV along quadri-quartics. They are the quadrics mentioned on p. 59. CORRIGENDA p. 38, line 6, for Iwty read w z \j/. line 9, for close-points read pinch-points. p. 40, last line but one, for be read be taken to be. omit foot-note. p. 76, foot-note, insert fourth edition. INTEODUCTION Ch. I. Quartic surfaces with isolated singular points. This chapter, which is based on the results of Cay ley* and Rohn, gives a method of classification of quartic surfaces which possess a definite number of isolated nodes and no nodal curves. The number of such nodes cannot exceed sixteen. Rohn has given a mode of classification for the surfaces having more than seven nodes, based on the properties of a type of seven-nodal plane sextic curves. The equation of a quartic surface which has a node at the point x = y = z 0, will be of the form 2u 3 w + u t = 0, where w 2 = 0, u s = 0, u t = are cones whose vertex is this point. The tangent cone to the quartic whose vertex is the point is therefore u 2 u 4 3 2 = 0. The section of this cone by any plane gives a plane sextic curve having a contact-conic u^, i.e. a conic which touches the sextic where it meets it. When the surface has eight nodes the tangent cone whose vertex is any one of them will have seven double edges which give seven nodes on the plane sextic. Such sextics are divided into two classes, viz. those for which there is an infinite number of cubics through the seven nodes, and two other points of the curve, and those for which there is only one such cubic. When a quartic surface is such that it has eight nodes consisting of the common points of three quadrics, the tangent cone from any node to the surface gives rise to a plane sextic of the first kind : such a quartic surface is said to be * Recent researches, etc., Proc. Lond. Math. Soc. (1869-71 Xll INTRODUCTION syzygetic : the equation of the surface is represented by an equation of the form (a$A, B, (7) 2 = 0, where .4 = 0, B = 0, (7 = represent quadrics whose intersections give the eight nodes. The second, or general kind of sextic, arises from the general type of eight-nodal quartic surface which is said to be asyzygetic. Similarly in the case of nine-nodal and ten-nodal quartic surfaces we have two kinds of plane sextics distinguished as above, giving rise to syzygetic and asyzygetic surfaces. For ten-nodal surfaces there are two varieties of asyzygetic surfaces, one of which, the symmetroid (see Ch. ix), arises when the sextic curve consists of two cubic curves. The tangent cone from each of the ten nodes of this surface then consists of two cubic cones. There are also two varieties of ten-nodal syzygetic surfaces. Seven points may be taken arbitrarily as nodes of a quartic surface, but if there is an eighth node it must either be the eighth point of intersection of the quadrics through the seven points, or, in the case of the general surface, lie upon a certain sextic surface, the dianodal surface, determined by the first seven nodes ; hence it may not be taken arbitrarily. When an eight-nodal surface has a ninth node the latter must lie on a curve of the eighteenth order, the dianodal curve. Plane sextics with ten nodes and a contact-conic are divided into three classes according as they are the projections of the intersection of a quadric with (1) a cubic surface, (2) a quartic surface which also contains two generators of the same set of the quadric, (3) a quintic surface which also contains four generators of the same set of the quadric. The first and second types of sextics are connected with eleven- nodal surfaces which are respectively asyzygetic and syzygetic; the third type gives a symmetroid with eleven nodes. A fourth surface arises when the sextic breaks up into two lines and a nodal quartic. Twelve nodes on the quartic surface give rise to eleven nodes on the sextic, which must therefore break up into simpler curves ; this process of decomposition goes on until we arrive at six straight lines, which case corresponds to the sixteen-nodal or Kummer surface. INTRODUCTION Xlll There are four varieties of surfaces with twelve nodes of which one is a symmetroid : there are only two varieties of surfaces with thirteen nodes and only one with fourteen nodes, viz. that given by the equation *Jxx' + ^lyy' + *Jzz' = 0. An additional node arises for a surface having this equation, when there exists between the planes x ... z' the identity Ax + By+Cz + A'x+ B'y' + C'z' = 0, with the condition ' = BB'=CG'. If another such relation exists between the planes x ... z', there is a sixteenth node. Ch. II. Desmic surfaces. A surface of special interest which possesses nodes and no singular curve is the desmic surface. Three tetrahedra A lf A 2 , A 3 are said to form a desmic system when an identity exists of the form oAi + /3A 2 + 7 A 3 = 0, where A; is the product of four factors linear in the coordinates. It is easily deducible from this identity that the tetrahedra are so related that every face of A 3 passes through the intersection of faces of A! and A 2 ; hence we have sixteen lines through each of which one face of each tetrahedron passes. It is deducible as a consequence, that any pair of opposite edges of A a together with a pair of opposite edges of A 2 form a skew quadrilateral ; and so for A! and A 3 , A, and A 3 . It also follows that if the edges A^A^, A-^A^, A^ 4 of Aj meet the respective edges of A 2 in LL', MM', NN' ; then A l} L, A z , L' are four harmonic points ; and so for A^MA^M', A-^NA^N'. The relationship between the three tetrahedra is entirely sym- metrical. Hence we may construct a tetrahedron desmic to a given tetrahedron A, by drawing through any point A the three lines which meet the three pairs of opposite edges of A, then if the intersections of these three lines with the edges of A be LL', MM', NN' respectively, the fourth harmonics to A, L, L' \ A, M, M'; A, N, N' will, with A, form a tetrahedron desmic to A. XIV INTRODUCTION The join of any vertex of A a and any vertex of A 2 passes through a vertex of A 3 : there are therefore sixteen lines upon each of which one vertex of each tetrahedron lies. Hence any two desmic tetrahedra have four centres of perspective, viz. the vertices of the third tetrahedron. If Aj be taken as tetrahedron of reference the identity connect- ing A!, A 2 , A 3 is given by the equation z t)(x y + z t)( x y z + t) Closely connected with the system of tetrahedra A{ is a second desmic system of three tetrahedra D t . They are afforded by the identity (a 2 - 7/ 2 ) (> 2 - t 2 ) + (a* - t*) (y* - * 2 ) + (a* - z*) (t* - y*) = 0. The sixteen lines joining the vertices of the A; are the sixteen intersections of the faces of the Di. A desmic surface is such that a pencil of such surfaces contains each of three such tetrahedra Di in desmic position. The surface has as nodes the vertices of the corresponding tetrahedra A^; hence the sixteen lines joining the vertices of the latter tetrahedra lie on the surface : along each of them the tangent plane to the surface is the same, i.e. the line is torsal ; the tangent plane meets the surface also in a conic, and hence there are sixteen conies on the surface lying in these tangent planes. There is a doubly-infinite number of quadrics through the vertices of any two tetrahedra A t -, the surface is therefore syzy- getic ; these quadrics meet the surface in three singly-infinite sets of quadri-quartics ; one curve of each set passes through any point of the surface. The coordinates of any point on the surface can be expressed in terms of two variables u, v as follows : _<7i(w) _o- a Q) _ 0-3 (M) ,_<>(). -<,&)' Py -* z (vY pZ ~a 3 (v)' Pt - since this leads to z*V) + (e s - e l ) (a?* + y 2 + (e 2 - which is one form of equation belonging to the surface. INTRODUCTION XV The three systems of twisted quartics are obtained by writing respectively v = constant, u v = constant, u + v = constant. The generators of the preceding doubly-infinite set of quadrics form a cubic complex which depends merely on the twelve desmic points ; all the lines through these points belong to the complex. Any line of this complex meets the surface in points whose argu- ments (u, v) are respectively (/3 + /*, a), (0 - p t a), (a + /*, /9), (a - /z, /3). The tangents to the three quadri-quartics which pass through any point of the surface are bitangents of the surface, and their three other points of contact are collinear. The curves u = constant, v = constant form a conjugate system of curves on the surface : the system conjugate to u + v constant is 8v n = constant ; the system conjugate to u v = constant is Sv + u = constant ; hence we derive the differential equation of conjugate tangents as dudUi + Sdvdvj^ = 0. The points of any plane section of the surface are divided into sets of sixteen points, lying upon three sets of four lines belonging to the cubic complex, where each line contains four of the sixteen points; denoting these twelve lines by a^-.a^, &!...& 4 , c l ...c i , then if G is the curve enveloped by the lines of the cubic complex in the plane, the points of contact of the lines a lie on a tangent a of C, those of the lines 6 on a tangent 0, and those of the lines c on a tangent 4- Xw' 2 ) 2 = w 2 (^ + 2X0 -f XVI INTRODUCTION and hence can be brought to the form 2 _ w zy^ where V is a quadric cone, in five ways. Each tangent plane of the cones Ff meets the surface in a pair of conies. Among the conies arising from any particular cone V 1 there are eight pairs of lines ; hence the surface contains sixteen lines. The relationship of these lines as regards intersection is the same as that of sixteen lines of the general cubic surface obtained by omitting any of its twenty-seven lines, p, together with the ten lines which inter- sect p. The coordinates of any point on the surface can be expressed as cubic functions of two parameters by the equations /**=/<(&, fc, &)> (i = l,2,3,4); so that every plane section of the surface is represented by a 4 member of the family of curves So^/i = ; where /i = 0, . . . , f t = i are plane cubic curves which have five common points; hence the surface is rational and is represented on a plane. Each of these five points, the base-points of the representation, is the image of a line of the surface. The other lines of the surface are represented in the plane by the conic through the base-points and by the ten lines joining pairs of base-points. This method enables us to determine the varieties of curves of different orders which can exist on the surface, by use of the equation N = 3n - 2of , where N is the order of the curve on the surface, n that of its image in the plane, and $ the number of times the curve on the surface meets one of the lines represented by the base-points. It is found that the sixteen lines previously mentioned are the only lines on the surface ; the only conies on the surface, apart from the double conic, are those in the tangent planes of the cones F,-. We obtain oo 2 twisted cubics on the surface, and also oo 4 quadri-quartics together with co 3 twisted quartics of the second species. It is seen that the quadrics \*UtP = touch the surface along quartics. The class of the surface is twelve. INTRODUCTION xvii The surface may also be obtained by aid of any two given quadrics Q and H and any given point 0, as follows : the surface is the locus of a point P such that the points 0, P, K, P' are har- monic, P and P / conjugate for H, and K any point of Q; P' also lies on the surface. The twenty-one constants of the surface are seen to arise from those of Q and H, and the coordinates of 0. This point is the vertex of one of the five cones Vi ; the vertices of the other four cones are the vertices of the tetrahedron which is self-polar for Q and H. The double conic is the intersection of H with its polar plane for 0. From the foregoing mode of origin of the surface is said to be a centre of self-inversion of the surface with regard to the quadric H. The surface may be related to the general cubic surface by a (1, 1) correspondence in two ways, the relationship being a perspective one in each case. The surface is connected with the general quartic curve as follows : the tangent cone drawn to the surface from any point P of the double conic is of the fourth order, its section being the general quartic curve ; the tangent planes from P to the five cones Vi, and the tangent planes to the surface at P, meet the plane of the quartic curve in lines bitangent to this curve. The other sixteen bitangents arise from the planes passing through P and the sixteen lines of the surface. The cone whose vertex is P and base a conic of the surface meets the plane of the quartic curve in a conic which has four-point contact with the quartic. The general quartic surface with a double conic is obtained by Segre as the projection from any point A of the intersection F of two quadratic manifolds or varieties P = 0, = 0, in four dimen- sions, upon any given hyperplane S 3 . Among the varieties of the pencil F + \<& = Q there are five cones, i.e. members of the pencil containing only four variables homogeneously ; each cone possesses an infinite number of generating planes consisting of two sets, and each generating plane meets F in a conic. These generating planes are projected from A upon S as the tangent planes of a quadric cone. Hence arise the five cones of Kummer, and the conies lying in their tangent planes. The double conic is obtained as the projection from A on S of XV111 INTRODUCTION the quadri-quartic which is the intersection with F of the tangent hyperplane at A of the variety which passes through A. When A lies on one of the five cones of the pencil F + \ = Q, this quadri- quartic becomes two conies in planes whose line of intersection passes through A. Hence the conies are projected into inter- secting double lines of the quartic surface. By this protective method the lines and conies of the quartic surface may be obtained, as also its properties generally. Ch. IV. Quartic surfaces with a nodal conic and additional nodes. A quartic surface with a nodal conic may also have isolated nodes, but their number cannot exceed four. Each such node is the vertex of a cone of Kummer, and for every node the number of these cones is reduced by unity. There are two kinds of surfaces with two nodes, in one case the line joining the nodes lies on the surface, and in the other case it does not. Nodes arise when the base-points of the representation of the surface on a plane have certain special positions; if either two base-points coincide, or if three are collinear, there is a node on the surface. If either a coincidence of two base-points or a collinearity of three base-points occurs twice, the quartic surface has two nodes and is of the first kind just mentioned ; if there is one coincidence together with one collinearity, the quartic surface is of the second kind. There are three nodes when two base-points coincide and also two of three collinear base-points coincide ; finally, when the join of two coincident base-points meets the join of two other coincident base-points in the fifth base-point, there are four nodes. Three coincident base-points give rise to a binode, four coinci- dent base-points give rise to a binode of the second kind, i.e. when the line of intersection of the tangent planes lies in the surface, and Jive to a binode of the third species, i e. when the line of intersection is a line of contact for one of the nodal planes. When four base-points come into coincidence in an indeter- minate manner we have a ruled surface ; a special variety occurs when the fifth base-point coincides with them in a determinate manner. The double conic may be cuspidal, i.e. when the two tangent planes to the surface at each point of it coincide ; the class of this INTRODUCTION XIX surface is six. The equation of the surface may in this case be reduced to the form t = 0. The surface has two close-points C, C' given by #! = x. 2 = 17=0. If K be any point of CG' and TT the polar plane of K for U = 0, then if any line through K meets TT in L, it will meet the surface in four points P, P'; Q, Q' such that the four points K, P, L, P' and K, Q, L, Q' are harmonic. The double conic may consist of two lines ; the necessary condition for this is that three cubics of the system representing plane sections should be au = 0, av = 0, fiu = 0, where a = 0, /3 = are lines, and u = 0, v = Q are conies. Either or both of the double lines may be cuspidal. Segre's method (Ch. in) affords a means of complete classifica- tion of quartic surfaces with a double conic, by aid of the theory of elementary factors. We thus obtain seven types, each type leading to sub- types. There exists in the case of certain of these sub-types a cone of the second order in the pencil (F, ), i.e. a cone whose equation contains only three variables, say x 1} x 2 , x 3 ; if the line x l = x z = x 3> which may be termed the edge of this cone, lies upon F, the surface is ruled. If the point of projection, A, is so chosen that the tangent hyperplane for A, of the variety which passes through A, is also a tangent hyperplane of this cone of the second order, the double conic is cuspidal. When the pencil (F, ) consists entirely of cones of the first order having a common generator, and a common tangent hyper- plane along this generator, the surface is that of Steiner. Segre's table, which distinguishes each surface that can arise, is given on pp. 82-85. Ch. V. The cyclide. When the double conic is the section of a sphere by the plane at infinity, we obtain the cyclide. The equation of the cyclide is therefore S 2 + U = 0, where 8 = is a sphere and U = is a quadric. XX INTRODUCTION This equation may be written in the form {a? + f + z* - 2X} 2 + 4 {(A, + \)x? + (A 2 + \)f + (A, + \)z 2 + 25^ + 2B 2 y + 2B s z + C- X 2 } = 0. The second member of the left side will give a cone when X is a root of the quintic jF(\) = 0, where F(\) is the discriminant of the second member. We thus obtain as in Ch. in five cones F< ; the tangent planes of each cone meet the surface in pairs of circles. There are five sets of bitangent spheres of the surface; each sphere of any set cuts a fixed sphere orthogonally, and its centre lies on a fixed quadric. The centres of the five fixed spheres are the vertices of the cones F;. These five spheres S t . . . S s are mutually orthogonal, and the centres of any four of them form a self-polar tetrahedron for the fifth sphere and its corresponding quadric Q. The equations of a pair S i} Qi are respectively where \ is one of the roots of F(\) = 0. The five quadrics Q l . . . Q 5 are confocal ; the curve of intersection of a pair Si, Qi is a focal curve of the surface. The centre of a sphere Si is a centre of self-inversion for the surface. Three of the quadrics Qi are necessarily real together with their corresponding spheres : one is an ellipsoid, one a hyperboloid of one sheet and one a hyperboloid of two sheets. The surface is also obtained as the locus of the limiting points defined by Si and the tangent planes of Q f . Taking Qi as an ellipsoid, this shows the shape of the surface to be one of the following : (i) two ovals, one within the other, when Si, Qi do not intersect ; (ii) two ovals, external to each other, or a tubular surface similar to the anchor-ring, when the focal curve (Si, Qi) consists of two portions; (iii) one oval, when the focal curve (Si, Qi) consists of one portion. INTRODUCTION XXI When (X + Atf is a factor of -F(X), one of the cones V is a pair of planes. If two roots of F '(X) are equal, one of the principal spheres is a point-sphere. In a real cyclide only one principal sphere can be a point-sphere. Real cyclides must possess at least two principal spheres which are not point-spheres. If Si = 0, . . . 8 5 = are any five spheres, there is a quadratic identity between the quantities Si ... S s , viz. that given by the equation 8, S s = 0, ? 6 TT IS -2r 5 where r l ...r 5 are the radii of the spheres, and TT^- is the mutual power of the spheres Si = 0, Sj = 0. By solution of the equations Si = x 2 + f + z* + 2/! -h 20, y + 2hiZ + d, etc., it is seen that x* + y* + z 2 , x, y, z, and unity, can be expressed as linear functions of Si . . . S 5 ; hence the equation of a cyclide S Y Sf appears as a quadratic function of --,..., which are themselves connected by a quadratic identity. This gives rise to seven chief types of cyclide, by application of the theory of elementary factors ; but only three of them give real cyclides, viz. [11111], [2111], [311]. Each of these types and the corresponding sub-types, with the exception of the general cyclide, arise as the inverses of quadrics. The sub-type [(11) 111] can be expressed in terms of three variables. It is the envelope of spheres which pass through a fixed point and whose centres lie on a conic; contact with the envelope here occurs along a circle. It has also two systems of bitangent spheres, as in the general case. A variable sphere of one of these systems makes with two fixed spheres of the first system angles whose sum or whose difference is constant. The inverse of this cyclide is a cone. The cyclide [(11) (11) 1] is known as Dupiris cyclide. There are two systems of spheres which touch the cyclide along circles ; the spheres of each system cut one of the principal spheres at j. Q. s. b XX11 INTRODUCTION a constant angle. The spheres of either system are obtained as those which touch any two fixed spheres of the other system and have their centres on a given plane. S- Denoting by x i} the equation of the general cyclide appears *< 5^ 5 as 2 ciiX? = 0, with the condition 2 x = 0. i i 5 x .* The system of cyclides = is confocal with the first i a t - + X cyclide. Three confocals pass through any point and cut ortho- gonally. The system of quadrics V= 0, where V=U+kS k?, which touch the cyclide S 2 + 4iU=0 along sphero-conics are such that two of them pass through any point, three touch any line, four touch any plane. The four points of contact of the surfaces V which touch any given plane TT are the centres of self-inversion for the section of the cyclide by TT. The locus of points of contact of common tangent planes of the cyclide and any given quadric V is a line of curvature on the cyclide. The Cartesian equation of the system of confocals is where S, Q have the same form as the Si, Qi when X is substituted for Xj. The confocals to the given cyclide S* 4- 4tU= 0, where S = a? + y* + *- 2X, may be obtained as follows : when 8 + 2L = is a point-sphere and U + L? = is a cone, the locus of the centres of these point- spheres is a cyclide confocal with S* + 4 U 0. Ch. VI. Surfaces with a double line : Pliicker's surface. The quartic surface with a double line is cut by any plane through the double line in a conic also. In eight cases this conic breaks up into a pair of lines, giving sixteen lines on the surface. There is no other line on the surface with the exception of the double line. There are sixty-two planes not passing through the double line each of which meets the surface in a pair of conies, one of whose intersections lies on the double line. By aid of one of INTRODUCTION these conies c 2 the surface may be represented on a plane ; for through any point x of the surface one line can be drawn to meet c 2 and also the double line, so that with each point of the surface one such line is associated. This line is determined as the intersection of two planes, each of whose coefficients contains linearly and homogeneously three parameters ^, % 2 , 3 . A third equation, arising from the equation of the surface, is that of a plane whose coefficients are quadratic in the &, and the inter- section of these three planes is a point on the surface ; hence we obtain a (1, 1) correspondence between the points x of the surface and the points of a plane. There are nine base-points in the plane, eight of which we represent by B l . . . B 6 ; they correspond to the points of eight non-intersecting lines of the surface, together with a point A which corresponds to any point of the conic coplanar with c 2 . These nine points cannot constitute the complete intersection of two cubic curves. To any plane section of the surface there corresponds, in the plane of , a quartic curve having a node at A and passing through the points BI, The cubic through the nine base-points corresponds to the double line. The plane image of any curve of order M on the surface is a curve of order m, where /3 being the number of times the curve on the surface meets the conic corresponding to A, and Sa the total number of passages of the image through the points Bi. By applying Rohn's method to the surface, using any point on the double line as that from which a tangent cone is drawn, it is easy to see the modifications which arise when isolated nodes exist. The section of the tangent cone, whose vertex is any point of the double line, is a sextic curve, meeting the double line in a quadruple point ; with each additional node of the surface this curve acquires an additional node : when there are seven nodes the sextic becomes a nodal cubic, meeting the double line in one point together with three lines through this point. When the surface has eight nodes, the sextic curve becomes a conic together with four lines concurring at a point of the double line. 62 XXIV INTRODUCTION In the case of seven nodes there are three torsal lines meeting the double line, and each containing two nodes; also there are four tropes meeting in the seventh node, and each containing four nodes. If there are eight nodes we have Pliicker's surface which has also eight tropes. The nodes form two tetrahedra, each of which is inscribed in the other. The nodes lie in pairs on four torsal lines meeting the double line. Through any two nodes not on the same torsal line there pass two tropes. The tropes can be arranged in four pairs so that the line of intersection of a pair meets the double line in a pinch-point. Plane sections of Pliicker's surface are represented by quartic curves having a common node and touching, at fixed points, four concurrent lines. Ch. VII. Quartic surfaces containing an infinite number of conies : Steiner's surface : the quartic monoid. The nature of the quartic surfaces which contain an infinite number of conies was investigated by Kummer. He showed the existence of the following classes : surfaces with a double conic or a double line ; ruled quartic surfaces ; the surface 2 = a/3j8, where 4> = is a quadric and a, /?, 7, 8 coaxal planes ; Steiner's surface. To these surfaces discussed by Kummer must be added the surface whose equation is [xw +f(y, z, w)] 2 = (z, w~a)'. The surface 4> a = aftyS has two tacnodes at the intersection of the common axis of the planes with ; it is birationally trans- formable into a cubic cone. The conies of the surface can be arranged in sets of four lying on the same quadric ; the quadric cone whose vertex is on the axis of the planes o . . . 8, and whose base is any conic of the surface, meets the surface in four conies. Steiner's surface is of the third class and has four tropes ; the coordinates of any point of the surface are expressible as homo- geneous quadratic functions of three variables; conversely any surface, the coordinates of whose points are so expressible, is a Steiner surface. The surface has a triple point, three double lines meeting in the triple point, and a node on each double line. A characteristic property of the surface is that its section by any INTRODUCTION XXV tangent plane breaks up into two conies. Every algebraic curve on the surface is of even order. The surface being determined by the equations pxi =/< (rh , 772, y 9 ) (i = I, 2, 3, 4), we are enabled to map the surface on the plane of the 77;. Any conic of the surface is represented on the ?;-plane by a straight line, the pair of lines representing two conies in the same tangent plane of the surface are represented by the equations r} l + m77 2 + nr) 3 = 0, 77! H 773 -f - 773 = 0. 772- 71 The surface contains oo B quartic curves of the second species, which are represented by the general conic in the plane of 77 ; also oo 4 quadri-quartics having a node on one of the double lines ; they lie on quadrics passing through two double lines, and are represented by conies Sa^^^O, in which two of the quantities a n , a^, #33 are equal. The conies apolar to the four conies fi = form the pencil Ua> + XV = ; the conies of this pencil are inscribed in the same quadrilateral, and form the images on the plane of 77 of the asymptotic lines of the surface. A form of the preceding property of the surface, that its coordinates are expressible as homogeneous quadratic functions of two variables, is the following : in the general quadric trans- formation P*i=fi(*n a *> s, a 4>, the locus of # is a Steiner surface when the locus of a is a plane. From this we derive the fact that Steiner's surface, and the cubic polar of a plane with reference to a general cubic surface, are reciprocal. Another mode of origin of the surface, given by Sturm, is that if a pencil of surfaces of the second class is protectively related to the points of a line in such a way that the line meets one conic c 2 of the system in a point corresponding to c 2 , and another conic c' 2 in a point corresponding to c' 2 , then the envelope of the tangent cones drawn from the points of the line to the corresponding surfaces is a Steiner surface. XXVI INTRODUCTION Weierstrass and Schroter have shown that a Steiner surface arises as a locus connected with a known theorem for the quadric. The theorem is that if through any given point A of a quadric any three mutually perpendicular lines are drawn, meeting the quadric again in L, M, N, then the plane LMN meets the normal at A in a fixed point. This theorem may be generalized as follows : if A be joined to the vertices of any triangle self-polar for a given conic c 2 in a given plane a, and the joining lines meet the quadric again in L, M, N, then the plane LMN meets the line AR in a fixed point S, where R is the pole for c 2 of the trace on a. of the tangent plane to the quadric at A. If now c 2 is a member of the oo 2 conies where U=Q, F=0, TF=0 are given conies, we have a point 8 determined for each set of values of ?7i : 772 : 773. On giving these ratios all values the locus of 8 is a Steiner surface ; for it can be shown that if the coordinates of S are ^ . . . y 4 , we have 2/i : y s : y 3 : V* =/i (?) :/ 0?) :/ 0?) :ft 0), where the f t are quadratic functions of the 77^. Properties of Steiner's surface may be deduced by aid of the transformation a*yi = p (i = 1, 2, 3, 4), applied to any plane Sa^ = 0, giving the cubic surface which is the reciprocal of Steiner's surface. Steiner's surface is one example of a type of surfaces known as monoids, viz. surfaces of the wth order which have an (n l)-fold point. The equation of the quartic monoid may be written wu a + u 4 = 0, where v s = 0, u 4 = are cones having their vertices at the triple point. The surface contains twelve lines, the intersections of w 3 = and u 4 = 0. The surface is projectively related to any plane, e.g. the plane w = 0, in a (1, 1) manner, except that every point of each of these twelve lines is represented by one point only, viz. where the line meets the plane w = 0. INTRODUCTION XXV11 The surface contains conies in planes through two of the twelve lines, and twisted cubics on quadric cones passing through five of the twelve lines. The oo l quadric cones passing through any four of the twelve lines meet the surface in quartic curves having a node at the triple point ; the oo J cubic cones passing through any eight of the twelve lines meet the surface in quartic curves without double points. If the lines corresponding to a curve of each type together make up the twelve lines, these two curves lie on one quadric. All these quartic curves are quadri- quartics. Quartic curves of the second species arise as the intersection with the surface of cubic cones having six of the twelve lines as simple lines and one of them as double line ; there are 5544 such quartic curves on the surface. The surface will have a line not passing through the triple point provided that three of the twelve lines are coplanar. The cases of the quartic monoid of special interest are those in which there are six nodes; here the twelve lines coincide in pairs six times. There are two cases of such surfaces ; in the first case the six nodes may have any positions, this surface is a special case of the symmetroid ; for the symrnetroid being the result of eliminating the Xi from the equations dS, dS 2 9$o B$ 4 _ 'te i + a *te i + 'aS + a 'Sr ' (iml - 2 ' 3 ' 4) ' where the a t are regarded as point-coordinates, the surface con- sidered is the special case in which one of the quadrics S t = is a plane taken doubly. The tangent cone to the surface whose vertex is one of the six nodes breaks up into two cubic cones. In the other case the six nodes lie on a conic whose plane is a trope of the surface. Each kind of surface has the same number of constants, viz. twenty-one. Ch. VIII. Rational quartic surfaces. The quartic surfaces with a triple point or with a double curve have been seen to be rational, i.e. the coordinates of the points of such a surface are expressible as rational functions of two para- meters. Neither has shown that there are only three rational quartic surfaces apart from them. The first of these surfaces has a tacnode, i.e. is such that every plane through the node meets the XXV 111 INTRODUCTION surface in a quartic curve having two consecutive double points at the node. The coordinates of any point x of the surface are projecti vely related to the points y of a double plane by the equations Vo (y) Vft(v) P*i = yi. /M? 2 = 2/a, P x 3 = y 3 , pXi=- -, 2/i where p^ (y) = is a conic and fi (y) = is a general quartic curve. Clebsch showed that the points y can be expressed as rational functions of new variables z t in such a way as to render Vll (y) a rational function of the Zi, viz. by equations of the form eyi=M*\ (t = l,2,3) f where the curves f t (z} = are cubics having seven points in common. The plane sections of the surface have then as their images, in the field of the z^ sextic curves having the seven points as nodes and also four other common points ; the eleven points lie on the same cubic. If the quartic surface has the equation P" = r > P" = s > e * c -> the surfaces 2 and 2' coincide; and the quantities Pi, Qi, etc. are in this case the partial derivatives of a quantity which is quadratic in the o^; if, changing the notation, we represent this quantity by Si, the last set of equations take the form on replacing X and a by a; and y respectively. Thus the surface 2 is the Jacobian J, of four quadrics. The surface A = 0, where A is a symmetrical determinant, is known as the symmetroid; if in the first set of preceding equations we replace x, X, y and a by a, x, ft and y respectively, and express that q = p', etc., these equations assume the form =* a.Sf. =* a.Sf. 2a f gi = 0, S&f^-O, (j=l,2,3,4) ...... (2). t=l CAj t = i Of/j The surface A, the locus of the points a, is obtained by eliminating the #, or the y t , from these equations. The surface J is seen to be the locus of vertices of cones of the system Wi = o. INTRODUCTION XXXI The equations (1) express that the polar planes of any point x of J, with regard to each of these quadrics $, ... 8^, are concurrent in the point y of J ; the points x, y are said to be corresponding points on J. The surface A has ten nodes ; the tangent cone of A whose vertex is any of its nodes breaks up into two cubic cones; a characteristic property of this surface. The surface J has ten lines ; every point of a line of J is associated with the same point a of A, by equations (2), which is a node of A. The tangent plane of J at any point P is the polar plane of P', the point corresponding to P, for the cone of the system Sa$ t - whose vertex is P. When x describes a line of the Jacobian, its corresponding point y describes a twisted cubic ; the point ft on the symmetroid describes a curve of the ninth order having double points at each node of the symmetroid except the one which is connected with the locus of x. As the point y describes the section of the Jacobian made by the plane a y = 0, the corresponding locus of x is the sextic = 0, which has the ten lines of the Jacobian as trisecants. The locus of the associated points a on the symmetroid is a curve of the fourteenth order, passing three times through each node ; that of the associated points ft is a sextic curve which passes through the ten nodes. To a plane section through two nodes of the sym- metroid there corresponds a quadri-quartic on the Jacobian. If the quadrics S^... S have a common point, the Jacobian has a node and an additional node arises on the symmetroid. Each additional common point of 8 l . . . S 4 will give rise to a node on both the Jacobian and the symmetroid. If there are six such common points, the Jacobian becomes the surface known as Weddle's, and the symmetroid becomes Kummer's surface. Weddle's surface has thus the six points common to $, . . . S 4 as nodes, and contains twenty-five lines, viz. the fifteen lines joining the nodes and the intersections of the ten pairs of planes through the six points. XXX11 INTRODUCTION The line joining any two corresponding points P, P' of the surface meets the twisted cubic through the six nodes in two points L, M such that the four points P, P', L, M are harmonic. It follows that this cubic is an asymptotic line of the surface. If 3 , 2 , 0, 1 be the coordinates of any point on this twisted cubic, then the coordinates of the preceding points P, P' are obtained as follows : let 0, denote the points L, M ; the coordinates of P, P' are given by the equations 6 where /(a) = II (a f ), and 6 l ...6 6 are the values of 6 relating to i the six nodes. Any two points 0, 4> of the twisted cubic thus determine two points P, P 1 on the surface ; any three points 0, , ty determine three pairs PP', QQ', RR' of corresponding points which form the vertices of a complete quadrilateral ; any four points 0, , ty, % determine twelve points which form three desmic tetrahedra, viz. PP'SS', QQ'TT', RR'UU'. If in the preceding expression of the points of the surface in terms of 0, we suppose to be constant, i.e. take all chords through a given point of the twisted cubic, the resulting locus of points of the surface is a quintic curve ; these curves form a con- jugate system on the surface. If the tangent to the twisted cubic at the point meets the surface again in the point T, then the locus of points of contact of the tangents from T to the surface is one of these curves. The surface, being defined as the locus of vertices of cones which pass through six given points, is seen to have an equation of the form .PlS P& $13 ^42 provided that four nodes are taken as vertices of the tetrahedron of reference, and p,-*. #1* are the coordinates of the lines joining any point of the surface to the two remaining nodes. This equation expresses that the lines p, q meet the faces of INTRODUCTION XXX111 the tetrahedron formed by the four nodes in two sets of four points which have the same anharmonic ratio. It can be deduced that a form of the equation of the surface is a 3 64 = 0. Any point P of the surface determines a closed set of thirty- two points on the surface as follows : if P be joined to the six nodes N-i ... N 6 , then calling the point of second intersection of with the surface (N^, etc., we thus obtain the six points . . . (N 6 ) ; secondly, by joining such a point (NJ to the nodes, we obtain five points of second intersection (N 1 N 2 ), etc.; there are fifteen such points; lastly, by joining the points (NiN 2 ) to the nodes, we obtain the points (NiN^N^ which are only ten in number, since (N,N 9 N t ) = (N 4 N 5 N S ), etc. The surface may be shown to be a linear projection in four dimensions, and therefore protectively related to a Kummer sur- face. For the Weddle surface arises as the interpretation in three dimensions of the twofold of contact of the enveloping cone of a cubic variety in four dimensions, whose vertex is any point of the variety. Now, since the intersection of this cone with any arbitrary hyperplane is a Kummer surface, we are again led to a birational transformation between the Weddle and the Kummer surface. The coordinates of any point of the surface can be expressed as being proportional to the ratios of the products of four double theta functions : viz. the substitutions = c 3 6 l '. 4 = satisfy the equation of the surface. We obtain two sets of quadri-quartics on the surface ; the first set is given as the intersection of two cones passing through the XXXIV INTRODUCTION same four nodes and having the other two nodes as respective vertices, viz. the cones Pl2p3* = ^PlsP42 , A A A Cfllt'2"o3"4 0V*VVM . C 03 f 03 (7 02 t7o4 . C 4 P 4 (7, f 02 . The fifteen other points (-^2), etc. and (N-iN^Nt), etc. are obtained by addition of one of the fifteen half-periods to the argument of u in these last expressions. The equation of a plane section of the surface, referred to the three points in which the plane of section meets the twisted cubic through the six nodes, assumes a simple form. The tangents to the curve at the vertices of the triangle of reference meet in one point ; an invariant of the curve is seen to vanish ; the curve contains an infinite number of configurations of points, each configuration being formed by twenty-five points. Bauer has investigated the surface whose equation is l # 2 # s #4 d x /d 4 its origin is as follows : a point P is joined to the vertices of a tetrahedron (taken as that of reference) and the joining lines meet the faces of another tetrahedron (whose faces are a z = 0, ft z = 0, c x = 0, d x = 0) in four points; if these latter points are coplanar, we obtain as locus of P the surface whose equation has just been given. When the two tetrahedra are in perspective, the surface is the Hessian of the general cubic surface ; it has ten nodes. INTRODUCTION XXXV When the preceding connection mentioned in the beginning of the chapter, between the points ac and y which gives rise to the surface A, reduces to a collineation, we obtain a surface, discussed by Schur, whose equation is a'/3y S' = 0, in which a ... B' are linear in the variables ; and the collineation is such as to permute cyclically the planes a ... 8 and the planes a' ... '. This surface contains thirty-two lines. If, in addition, the faces of both tetrahedra are subject to a collineation which leaves one face of each tetrahedron unaltered and permutes cyclically the other three, the surface contains fifty- two lines. CHAPTER I 1. The singular points possessed by a quartic surface may consist either of a certain number of isolated nodes or may form double curves. In the present chapter we discuss the quartic surfaces which have an assigned number of nodes, beginning with those which have four nodes, and give a definite method of classification for all the cases in which the number of nodes exceeds seven. The number of isolated nodes of a quartic surface cannot exceed sixteen; for the class of a surface of order n which has & double points is n (n I) 2 2B, since this is the number of points of intersection of the surface and its first polars for two points A and S, diminished by the number 28 of these intersections arising from each double point (a simple point on the polars of both A and 5). Hence if n is four, B cannot exceed sixteen. 2. Quartic surfaces with four to seven nodes. Since the equation of the general quartic surface contains thirty- four constants, the surface with four given nodes should contain 34 - 16 = 18 constants ; if then A = 0, B = 0, C = 0, D = 0, E = 0, F=Q are six linearly independent quadrics through the four nodes, the equation containing apparently twenty constants, is a quartic surface having the given nodes. The number of constants is really eighteen, since there are two quadratic relations between the six quadrics, as may be seen by taking the four given points as vertices of the tetrahedron of reference, in which case the quadrics may be taken to be between which there exist the identities J. Q. S. 2 QUARTIC SURFACES [CH. I For five nodes, taking A...E as quadrics passing through the given nodes, the equation containing fourteen constants, represents the general quartic having these given nodes. The general quartic with six nodes is represented by the equation where A, B, C, D are quadrics through the six nodes, and J is the Jacobian of the four quadrics. For this equation contains ten con- stants and J has the given points D 1 ...D 6 as nodes, moreover J cannot be expressed as a quadratic function of A, B, C, D. The following properties of J may be used to establish these results. The surface J=0 is the locus of vertices of cones of the system now each point of the line joining any two double points, e.g. D 1 D%, is the vertex of such a cone, hence 7 contains the join of any two double points ; also since Z^/V-.-Oi A lie on J it follows that DI is a node of 7; similarly for DZ...DQ. Again there are ten pairs of planes passing through the points Di...D 6 , and each point of the line of intersection of such a pair of planes satisfies the condition of being the vertex of a cone of the system. Hence such a line lies upon J, which thus contains 15 + 10 = 25 lines. Again, since any quadric of the system is linearly expressible in terms of any four members of the system, it is so expressible in terms of any four of the previous pairs of planes ; hence if J were expressible as a quadratic function of A, B, C and D, we should necessarily have a relation of the form J=(alaa',PP,yy', S8') 2 , in which we may take the planes a, , y to contain the line D 1 D 2 , while 8, S' do not contain it, e.g., as (A, A, A), a's(A,A,A), etc., while 8 = (A, /> 3 A), S' = (Z> 2 , A, A). Hence since J" contains A A such a relation is impossible. The general quartic with seven nodes is represented by the equation where A, B, C are quadrics through the given nodes and S is any quartic surface having the seven nodes*. * This quartic surface may also be expressed in terms of the quartic surfaces which have one of the given points as a triple point and the other six as double points ; if Tj . . . T; are these surfaces, the required general quartic surface is 2^=0, t = l,...7. 2, 3] WITH ISOLATED SINGULAR POINTS 3 3. Quartic surfaces with more than seven nodes. The equation of a quartic surface having a node at the point x = y z = will be of the form w 2 w 2 + 2u 3 w + u t = 0, where u 2 = 0, u 3 = 0, u 4 = are cones whose vertex is the node. The equation of the tangent cone drawn to the surface from this node is u 2 u t u 0. The section of this cone by any plane, e.g. the plane w = 0, is a sextic curve with a " contact-conic," i.e. a conic which touches it wherever it meets it. If the surface has any other node, the tangent cone will have a double line passing through this new node and giving rise to a node on this sextic ; we obtain the different varieties of quartic surfaces possessing nodes by consideration of all special cases of sextic curves with a contact-conic*. It is to be noted that the existence of a contact-conic u 2 = of a sextic implies also a contact-quartic u t = ; if a sextic has another contact-conic v 2 = 0, and hence another contact-quartic v = 0, an identity exists of the form W 2 M 4 W 3 2 = V 2 V 4 V 3 2 . Now by multiplying the equation of the surface by u 2 we derive (u 2 w + w s ) 2 -I- ^w 4 - u 3 2 = 0, hence in the present case (u 2 w + u 3 ) 2 + v 2 v 4 - v 3 2 = (1). Denoting by c 8 the intersection of the quartic surface with the cone v 2 = 0, it is clear that v 2 = meets the surface (1) in the curve c 8 and in the four lines w 2 =v 2 = 0; but v 2 meets (1) where it meets the tw r o nodal cubic surfaces u 2 w + u 3 v 3 = Q, u 2 w + u 3 + v 3 = 0, hence in general c 8 must break up into two quartic curves, either of which is the partial intersection of v 2 with a cubic surface which contains also two generators of v 2 . These curves are therefore quadri-quarticsf. Hence the surface contains an infinite number * This method is due to Eohn, see Die Flachen vierter Ordnung hinsichtlich Hirer Knotenpunkte und Hirer Gestaltung, Leipzig, 1886. t We denote by quadri- quartic the type of twisted quartic through which an infinite number of quadrics pass. 12 4 QUAETIC SURFACES [CH. I of quadri-quartic curves which are projected from the node into quartic curves which touch the sextic u 2 u 4 u/ at each point of intersection*. Hence if the curve u 2 u 4 u 3 2 has more than one contact- conic it has an infinite number of contact-conies. 4. Nodal sextics-f*. For the purpose of classification of nodal quartic surfaces we discuss various properties of sextic curves with a contact-conic. In the first place it may be seen that sextic curves with six nodes lying on a conic c 2 can have their equation expressed as above. For if c 3 = is any cubic through the six points, any other cubic through them is of the form c 3 + c 2 L = ; and any sextic through the complete intersection of c 2 and c 3 being ^2^4 i Qj Qs = ^> if the six points are nodes on this sextic c 4 and c s ' must be of the form c 3 M +c 2 N, c s + c 2 R respectively. Hence the required sextic takes the form c 3 2 + c 2 c 3 A + c.?B = 0, i.e. the form K 3 * -c 2 *V= 0, and hence has a contact-conic. The corresponding quartic surface is w*V+ 2wK 3 + c 2 2 = ; this has the plane w = as a singular tangent plane or trope, which touches the surface along a conic. Seoctics with seven nodes. There are two different kinds of seven-nodal sextics, viz. that for which it is possible to find a pair of points P, P' on the curve, such that through the seven nodes D^ ... D 7 and P, P' there pass an infinite number of cubics, and the one for which it is not possible ; considering the former kind, then if one such pair of points exists there is an infinite number of such pairs ; for taking c 3 and c 3 as two such cubics, then c 6 , the given sextic, since it passes through the complete intersection of c 3 and c 3 ', has an equation of the form c,r, + c s T,'=0. * For such a point of intersection P is the projection of an actual intersection Q of the quadri-quartic and the curve of contact of the tangent cone, and the tangents to these curves at Q lie in the tangent plane of the surface. t See Eohn, I.e. Now c 3 meets c 6 only in Dj ... D 7 , P, P' and two further points Q, Q', hence F/ passes through D l ...D 7 and also through Q and Q'; so that two and therefore an infinite number of cubics pass through D 1 ... D 7 , Q and Q'. By varying the cubic through the nine points D l ... D 7 , P, P' we form an involution of points Q, Q' on c 6 . If Q coincides with P, Q' will coincide with P'; therefore every cubic through the seven nodes which touches c 6 once will touch it twice. Since F 3 is seen to pass through D l . . . D 7 and since only three linearly independent cubics pass through seven points, there is a linear connection between c 3 , c s ', T 3 and F 3 ', hence the sextic which has the property considered is represented by an equation of the form MWuM* *<>, where <, ty, % are any three cubics through the given nodes. This class of sextic always has a contact-conic ; for if the sextic is c 6 = c 0>'' concur ; but the first two are the tangents at P to / and $, and the third cannot pass through P, hence / and < touch at P, and touches every sextic with the eight given nodes which pass through P. Now / and c 9 meet in 9x6 8x6 = 6 points apart from the nodes, hence every sextic with eight nodes is touched by six cubics through these nodes. If /= is any sextic with nine nodes and = the cubic through them, /+ p0 2 = is the equation of the general sextic with the given nine nodes. If there is a tenth node it will be included among the points determined by the equations f\ fz fs \ _ . 01 02 03 i The number of solutions given by these equations is thirty- nine, but each of the given nine nodes occurs as a triple solution. Hence the pencil of sextics /+/30 2 = contains twelve curves which have a tenth node (see Art. 9). The foregoing result as to contact-cubics is modified as follows : through any eight nodes of a sextic with nine nodes there pass four tangent cubics ; through any eight nodes of a sextic with ten nodes there pass two tangent cubics. 5. Sextics with ten nodes. The following result for ten-nodal sextics is important for our purpose: every plane sextic with ten nodes and a contact-conic is the projection of a twisted sextic on a quadric : for choosing any centre * As may be seen by taking any one of them as x = 0, y = 0, 2 = 0. 4, 5] WITH ISOLATED SINGULAR POINTS 7 of projection and any quadric whose section by the polar plane of for the quadric projects into the given contact-conic, the sextic cone whose base is the given sextic meets the quadric in a curve c 12 which has twenty-six actual double points, since each node of the plane sextic gives rise to two nodes on c 12 , and each point of contact of the contact-conic and the sextic is the projection of a point at which two branches of c 12 touch each other. Moreover c 12 has thirty apparent double points*, hence the projection of c 12 from any point has 30 + 26 = 56 nodes, and this is one more than can be possessed by a curve of order 12 which does not break up into simpler curves. Hence c 12 must break up into two sextic curves. There are three varieties of twisted sextics on a quadric : (1) its intersection with a cubic surface, (2) its partial intersection with a quartic surface which also contains two generators of the quadric of the same species, (3) its partial intersection with a quintic surface which also contains four generators of the quadric of the same species. The following result, which may be easily provedf, is of frequent application : through every point P of space there pass n (n 1) double secants of the complete curve of intersection of a quadric with any surface of order n ; these double secants form the inter- section of a cone of order n with a cone of order n 1, the former cone passes through the 2w intersections of the polar plane of P and this curve. Let us now consider the plane ten-nodal sextic which is the projection of the first of these three varieties. This has six apparent double points and, since its plane projection has ten * Salmon, Geom. of three dimensions (fifth ed. 1912), vol. i. p. 356. t If F=0 is the surface and U=0 the quadric, it is easy to see that the section of the curve of intersection by the polar plane of P for U is given by the equations U A 2 F where AU=I l x i '~U, i d*i ' and x{ are the coordinates of P. Relatively to its plane the equation of this curve is of the form this curve contains n(n-l) nodes which arise solely from apparent double points of the curve U=0, F=0; also v n =Q is seen to pass through the common inter- sections of F=0, 7=0, AC7=0. 8 QUABTIC SURFACES [CH. I nodes, it must have four actual double points ; by the last result six of the nodes of the plane sextic lie on a conic ; it is therefore represented by an equation of the form K 3 *-cfV=Q. (Art. 4.) The second species of twisted sextic lies on a quadric and a quartic surface, their intersection being completed by two gene- rators of the quadric. This curve has seven apparent double points * ; and therefore, to complete the number of nodes of the plane quartic, must have three actual double points. Each generator of the given species meets the curve four times. There is an infinite number of quartic surfaces passing through the sextic and any two generators of the quadric. For any quartic surface through five points of each generator and any seventeen points of the sextic will meet the sextic in 8 + 17 = 25 points, and therefore contain it altogether : it will also contain the two generators. Let us denote the twisted sextic by c 6 , its plane projection by c 6 ', and take any generator p and its consecutive generator as the pair of generators just mentioned ; then the cubic cone which contains the seven double secants of c 6 will touch c 6 ' twice f ; hence, varying p, we obtain an infinite number of cubics through seven nodes of c 6 ' and bitangent to it. In the third type of twisted sextic c 6 is the partial intersection of a quadric and a quintic, the residual intersection being formed by four generators of the quadric of the same species. Each generator of this species meets the sextic five times. It may be shown as before that there is an infinite number of quintic surfaces passing through the given sextic and any four generators of the given species. The curve c 6 has ten apparent double points. We may select the four generators as follows : let p and p' be those generators which are projected from the centre of projection into the tangents of the contact-conic of c e ' drawn from some node D of c e ' ; we then take as our four generators p, p' and the generators consecutive to them. The line OD thus meets c 6 twice, and serves as join of apparent intersections for c e , p and for c 6 , p'. The compound curve of intersection of order 10 has twenty apparent double points, of which nine are projected into Z>, viz. one point arising from c 6 , two from (c 6 , p) (c 6 , p + dp}, two from (c 6 , p') (c 6 , p' + dp') and four from p and p'. Hence the two cones of orders 4 and 5 through the double secants must * After deduction of five apparent double points arising from the two lines. f Since p gives rise to two apparent double points of the compound curve. 5, 6] WITH ISOLATED SINGULAR POINTS 9 each have a common triple edge ; we therefore obtain the following results : if D is any one of the ten nodes there exists a quartic curve which has a triple point in D and passes through the nine other nodes and the points of contact of the tangents drawn from D to the contact-conic ; also there exists a quintic curve which has a triple point in D, passes through the nine other nodes and touches the contact-conic where it is touched by its tangents drawn from D. This holds for each node. 6. Quartic surfaces with eight nodes. Returning to the sextic curve w M 4 u s 0, derived from the surface u z w z + 2u 3 w + w 4 = 0, any quadric through the node is if the quartic surface has any other node which also lies upon this quadric, since this node also lies on the surface u z w + u 3 = 0, it is clear that the curve u 2 t 2 - 2^M 3 = will pass through the resulting node on u 2 u 4 w :j 2 = or c 6 . This quartic curve passes through the points of contact of c 6 with its contact-conic u 2 , and also through the nodes of c 6 which result from nodes on the quartic surface. If therefore the surface has eight nodes we have seven nodes on c 6 : to each quartic through these seven nodes and the points of contact B 1 . . . B 6 of c 6 and u%, there corresponds one quadric through the eight nodes, and vice-versa. Now it was stated (Art. 4) that plane sextics with seven nodes form two classes ; in the more general case there is a singly infinite number of quartic curves through the nodes and B l ... B 6 , and we obtain corresponding to this case a singly infinite number of quadrics through the eight nodes. For the more special case where there is a doubly infinite number of quartic curves through the thirteen points we have a doubly infinite number of quadrics through the eight nodes, which therefore form eight associated points. Such a surface is represented by an equation of the form , (7) 2 =0. It follows that any quadric through the eight nodes meets the quartic surface in two quadri-quartic curves which are projected from any node into two of the oo - cubics which pass through the seven nodes of c 6 . 10 QUARTIC SURFACES [CH. I These two classes of quartic surfaces will be termed asyzygetic and syzygetic respectively*. The equation of the general seven-nodal surface being F=(a^A, B, 0) 2 + /oS = (Art. 2), where A, B, G are quadrics through the seven nodes, if there is an eighth node we obtain, to determine it, the equations t dF -r. dF ~ dF _^ hence the eighth node lies on the surface A ID f*i ^i i *-*! ^l A 7? C* A 75 f~1 /1 3 -^^3 ^3 = 0. The eighth node may therefore not be taken arbitrarily, as in the case of the first seven nodes. If A, B are two quadrics through the eight nodes and T any eight-nodal asyzygetic surface, the general asyzygetic surface is represented by the equation = 0. The surface J is called the dianodal surface^, and is the locus of a point whose polar planes for A, B, C and S are concurrent, and therefore also concurrent for every quartic surface with the given seven nodes ; thus if P is any point of the dianodal surface, all the quartics through P have a common tangent line thereat, which touches the quadri-quartic through P and the seven nodes, as is seen by taking as the quartic a doubled quadric through P and the seven nodes. The dianodal surface. The dianodal surface contains the line joining any two nodes DI, D 2 ; for if P be any point on this line then, since we may take the surfaces A, B, 2 which appear in the equation of the seven- nodal quartic to pass through P, they will necessarily contain the line D^D^, hence the tangent planes at P to A, B, 2 all pass through D l Dz and therefore the point P satisfies the equation of * The general syzygetic surface is the envelope of the quadrics X 2 D + \E + F=0, where D, E, F are quadrics through the eight nodes. + Cayley. 6, 7] WITH ISOLATED SINGULAR POINTS 11 the dianodal surface. This surface thus contains the twenty-one lines which join any two of the nodes D^...D 7 . Again taking 2, A and B to pass through any given seventh point of the twisted cubic determined by D l . . . D 6 , this cubic lies entirely in 2 as meeting it in thirteen points, and also on A and B as meeting them in seven points, hence the tangent planes at P to 2, A and B will meet in the tangent line at P to the twisted cubic : hence, as before, the point P lies on the dianodal surface. This surface thus contains the seven twisted cubics which pass through any six of the points D l ... D 7 . The dianodal surface contains thirty-five plane cubics lying on the planes which contain three of the given nodes ; for let L be the plane of three nodes and 8 the cubic surface which passes through these three nodes and has the four other nodes as double points ; if we then write L . S for 2 in the equation of the dianodal surface it becomes J(A, B,C,L.S) = LJ(A, B, C, S) + SJ (A, B, C, L) = 0, which clearly contains the cubic L = 0, S = 0. This shows that the lines D 1 D 2 , etc. are simple lines of the dianodal surface. The twisted sextic which is the locus of the vertices of the cones which pass through the seven given nodes, lies on the surface ; for this sextic is obtained by elimination of X, p from the equations Ai + \Bi + pCi = 0, (i=l,...4), which clearly lies upon J (A, B, C, 2) = 0. Each of the seven nodes is a triple point of J, for the lines D 1 D 2 ,...D 1 D 7 do not lie on the same quadric cone. 7. Quartic surfaces with nine nodes. From the two varieties of surfaces with eight nodes we derive two with nine nodes. Considering first syzygetic surfaces, viz. if this surface has a ninth node it must lie upon the twisted sextic = 0. This curve is the locus of the vertices of the cones of the system 12 QUARTIC SURFACES [CH. I A + \B + fj,C; hence, if a ninth node exist, there is a quadric cone K whose vertex is D 9 which passes through the points _Dj ... _D 8 . Taking D 9 as the point from which the surface is projected by a tangent cone (giving rise to the curve u^u t u./ = 0), this latter curve must have eight nodes lying on a conic, and must therefore break up into this conic and a quartic curve. Therefore K, the quadric cone whose vertex is D 9 , forms part of the tangent cone from D 9 , and touches the quartic surface along a twisted quartic. The equation of the surface is therefore of the form A 2 + pKB = 0, where A = is a quadric through the nine nodes and B a quadric through the eight associated points D l ...D 8 . There is a triply infinite number of nine-nodal syzygetic quartic surfaces. Considering next asyzygetic nine-nodal surfaces, from the equation of the general eight-nodal surface it is seen that a ninth node must lie on the curve | A....A. BI . . . X>4 = 0, T T J-i J-i which is of the eighteenth order, the dianodal curve; the ninth node being taken arbitrarily on this curve, there is a singly infinite number of surfaces with the nine given nodes represented by the equation where A is the quadric through the nine nodes and P any quartic surface with these nodes. The dianodal curve. The dianodal curve lies on each of the eight dianodal surfaces obtained from the eight given nodes; moreover the dianodal surfaces corresponding to A D 6 D 7 and A . . . D 6 D 8 intersect in the fifteen lines joining any two of the points D l ... D 6 , in the dianodal curve, and in the twisted cubic through D 1 ...D 6 . Through D n as being a triple point on each, there pass nine branches of the curve of intersection of the two dianodal surfaces, but of these, six branches arise from the lines D 1 D Z ... D l D 6 and the tangent at D l to the cubic D 1 ... D 6 ; the remaining three branches arise from the dianodal curve which has therefore a triple point in each of the eight nodes D l ...D 8 . 7, 8] WITH ISOLATED SINGULAR POINTS 13 Moreover since six of the intersections of the tangent cubic cones at A to the two dianodal surfaces lie on the quadric cone of vertex A and passing through A . . . A> it follows that the remaining three intersections must lie in a plane, hence the tangents to the three branches of the dianodal curve at A are coplanar. The dianodal curve is seen from its equation to be the locus of a point whose polar planes for A, B and T are coaxal. In its equation we may take A to be the quadric through the eight nodes and any assigned point P, then B will not pass through P, and if T is a quartic of the system which passes through P, then if P is on the dianodal curve, since the polar plane of B for P cannot pass through P, it follows that A and T have the same tangent plane at P. We may also note the following results: (1) the dianodal curve meets each of the lines A A twice, apart from A and A, (2) it meets each of the seven twisted cubics D 1 ...D e , etc. twice, apart from the nodes. For we may take the quadric A and the quartic T as passing through any point P of the line A A which will then lie on each of them, hence we have at each point of A A a (1, 1) correspondence of tangent planes which involves two coincidences, say at the points Q and Q', thus both Q and Q' satisfy the equation of the dianodal curve. Next take A and T as passing through some assigned point P of the cubic through A A; this cubic will then lie on each of these surfaces, so that they will also meet in a residual quintic curve which passes through the points A A- Now the number of points of apparent intersection of these curves is seen to be seven* and hence their actual intersections are eight in number, and deducting the six points A A we obtain two as the number of their intersections apart from the nodes ; at each of these points A and T touch, and hence each point lies on the dianodal curve. 8. Quartic surfaces with ten nodes. We have, as before, two classes of irreducible sextics with nine nodes, viz. according as the points of the curve are or are not conjugate in pairs with regard to any seven of the nine nodes. We have also the sextic arising from two cubics or two lines and a quartic. We consider in the first place these last two cases. * Salmon, Geom. of three dimensions (fifth ed.), vol. i. p. 358. 14 QUABTIC SURFACES [CH. I It may be shown that, if the tangent cone from one node of a ten-nodal quartic surface breaks up into two cubic cones, this will also occur for each node. For let the tangent cone from D 1 break up into the cubic cones V and V, touching the surface along the curves c 6 and c 6 ' respectively, then A is a triple point on both c 6 and c e '*, and A ... Ao are ordinary points on c 6 and c 6 '. Now the cubic surface which has D l and A for nodes and which passes through A--- Ao and also through any other three points on c 6 , will meet c 6 in6 + 2 + 8 + 3 = 19 points and therefore contain c 6 ; it therefore meets the quartic surface in another curve k 6 which has A as triple point and A A Ao as ordinary points. Hence k 6 is projected from A by a cubic cone which passes through A, A, A ...Ac- In the same manner, by aid of c 6 ', we obtain another sextic curve k 6 ' which projects from A by a cubic cone. Hence the lines AA> AA-" AAo form the complete intersection of two cubic cones, so that the sextic tangent cone to the quartic surface from A has as double edges the complete intersection of two cubic cones: it must therefore break up into two cubic cones. Applying the same reasoning to each node it is seen that the tangent cone from each of them must break up into two cubic cones. This surface is called the symmetroid^. In the next place, when the sextic splits up into two lines and a quartic curve, we see that through the node x = y = z = there pass two planes, each touching the surface along a conic ; each is a trope. The equation of the surface is of the form A 2 + pxyB = 0, where A and B are any two quadrics. * Since any plane through Dj meets c e in three points apart from DI and so for c e '. t See chap. ix. It is seen from the foregoing that any cubic cone whose vertex is a node and which passes through the nine other nodes, meets the surface in two sextic curves having the vertex as triple point and passing through the nine nodes. We thus obtain ten sets of sextic curves on the surface. Since the equation of the surface may be written where F=ic 2 u 2 + 2icu 3 + u 4 , M=wu z + u s , FF' = u 2 u 4 -M 3 2 , it follows that the cubic surfaces touch F along the sextics F=0, P W+ F'=0. 8] WITH ISOLATED SINGULAR POINTS 15 The nodes lie on two conies : the tangent cone from each of the eight associated nodes breaks up into a plane and a quintic curve with four double points. We now pass to irreducible sextics, first those whose points are conjugate in pairs giving syzygetic surfaces with ten nodes. Such surfaces are represented by an equation of the form A* + pK^K 2 = 0, where K l and K 2 are cones whose vertices lie on the quadric A. Next if P = is any asyzygetic surface with nine nodes, then among the surfaces A 2 + P P = 0, there are thirteen which have a tenth node ; for such a node is an intersection of the dianodal surface of D l ... D 6 D 9 and the dianodal curve of A ... -D 8 ; there are 6 x 18 = 108 such intersections, but of these D l . . . D e being triple points on both the surface and the curve count as 9 x 6 = 54 intersections, and the points D 8 , D 7 , D 9 each count as three, also the two intersections of the fifteen lines Dj-Da, etc. with the dianodal curve give thirty points, and its two intersections with the twisted cubic D 1 ... D 6 give two more points which are not solutions; this leaves 108 - 54 - 9 - 30 - 2 = 13* solutions. * Of these thirteen solutions one gives a symmetroid ; for if P and A have the equations where DI is the point x = yz = Q, we may write the equation A 2 + pP=0 in the form w z (tf + 2pu 2 ) + 2w(t l t 2 + 2pu 3 ) + 2 2 + 2/3M 4 = ; the sextic curve is therefore 2p (w, 2 "4 - Ms 2 ) + c e = 0, where c^st^u^t^uz-^t^u^ is the projection of the curve of intersection of A and P. All these curves have as double points the projections D 2 '...D 9 ' of D 2 ...D 9 , and CQ has also as double points those in which the generators of A through D meet the plane of projection. All these curves touch c e twice. Now all sextics having as nodes D 2 '...D d ' and which touch c 6 twice must have an equation either of the form where 3 is a cubic through D 2 ...D 9 , or of the form where 3 , xs are two cubics through the nodes D 2 '...D 9 ' of c 6 which touch it. But the first form is excluded, since no doubled cubic can occur in the pencil of sextics ; and the second form shows that as one curve of the pencil we have two cubics, i.e. for one of the surfaces A 2 + pP0 the tangent cone from DI breaks up into two cubic cones, and we have a symmetroid. 16 QUARTIC SURFACES [CH. I 9. Quartic surfaces with eleven nodes. The three varieties of plane sextics with ten nodes (Art. 5) lead to three types of quartic surface with eleven nodes. The equation of the first variety was seen to be of the form u 3 * - u?K = ; this sextic arises from the quartic surface Kw 2 + 2u 3 w + u 2 z = 0. The six nodes which lie on a conic are given by the equations w = 0, u z = 0, u s = ; the plane w = is a trope. The tangent cone drawn to the surface from any one of these nodes breaks up into the plane w = and a quintic cone, the tangent cones from the remaining five nodes are irreducible. If P = be a quartic surface having the six coplanar points as nodes and also five other nodes, and A a quadric passing through four of these last five nodes and also the conic containing the six nodes, then P + pA* = is a pencil of quartic surfaces having ten nodes : the equations give forty solutions, but the given ten nodes count triply among them, leaving ten surfaces of the pencil having eleven nodes and of the type just mentioned. This surface may be called XI C . The second kind of plane sextic with ten nodes has an infinite number of bitangent cubics through seven of its nodes (Art. 5) ; the quartic surface to which it corresponds must therefore be syzygetic ; the equation of the ten-nodal syzygetic surface being A z + pKtKz (Art. 8) it may be shown that in this pencil there are twelve surfaces which have an eleventh node. It is easy to see that the equation of such a surface has the form where K l = 0, K 2 Q, K 3 = Q are cones, and such that the vertex of K! lies upon K z K 3 = 0, etc. This surface is called XI 6 . There remain two cases in which the sextic curve breaks up into simpler curves : either into two lines and a nodal quartic or into two cubic curves. 9, 10] WITH ISOLATED SINGULAR POINTS 17 In the first case the equation of the surface is A* + pKxy = 0, where A is a quadric and K a cone whose vertex lies on A. This is XI d . In the second case we have a symmetroid, hence (u 2 w + u 3 y> + v s v 3 ' = u 2 (u 2 w 2 + 2u 3 w + 1* 4 ). Here either v 3 or v 3 ' has a nodal line, arising from an eleventh node on the surface. The tangent cone from this eleventh node to the surface gives a plane sextic of the third variety. This case is XI a . 10. Quartic surfaces with twelve nodes. A surface with twelve nodes gives rise to a sextic curve with eleven nodes: this sextic must therefore break up into simpler- curves. The cases which provide eleven nodes are the following i (1) a quintic with six nodes, and a straight line, (2) a quartic with two nodes, and two straight lines, (3) two nodal cubics, (4) a cubic, a conic and a straight line, (5) a quartic with three nodes, and a conic. It may be shown that a plane quintic with six nodes and a contact-conic may be regarded as the projection of a twisted quintic on a quadric. The proof is exactly similar to that for the plane sextic with ten nodes. By addition of a generator it is easy to see that we obtain a special case of the second class of twisted sextics on a quadric*; hence the quartic surface corre- sponding to case (1) must be syzygetic. Moreover it will contain six nodes on a conic. If D^ ... D 6 ' are the intersections of the plane quintic and the line, then Z) x . . . D 6 lie on a conic. Two cases occur according as four or two of the associated nodes lie on this plane ; in the first case since four of the associated nodes are coplanar, so also are the other four, and the equation of the surface is of the form it is a case of XI^. This surface is * A quartic surface through three generators of a quadric meets it also in a quintic; each generator of this set meets the quintic four times, hence (Salmon, p. 358) H=B and therefore ft' = 6. J. Q. S. 2 18 QUAKTIC SURFACES [CH. I The surface has two tropes each containing six nodes : taking them as A--- D 6 and AAA--- Ao it is clear that the tangent cones from the points AA--- Ao break up into a plane and a quintic cone ; the tangent cones from A an d A i n t two planes and a quartic cone; the tangent cone from Ai includes K, and therefore breaks up into a quadric cone and a quartic cone with three double edges. When only two* of the points A A are among the eight associated nodes, if e.g. they are A and A, then the tangent cones from A A break up into a quadric cone and a quartic cone, but this quartic cone must consist in part of the plane A At> thus the tangent cone splits up into two cubic cones and we have a symmetroid with twelve nodes. This is XII a . The second case, a binodal quartic curve and two straight lines, leads in general to XII d , i.e. A 2 + pxyK=0, but if K breaks up into two planes we obtain the surface A* + pxyzw = ; this is a twelve-nodal surface in which the tangent cone from each node breaks up into two planes and a quartic cone with two double edges. This surface is XII C . The cases (3) and (4) lead to the surface XII a . Case (5) may lead to Xlld, but if in more than two cases the tangent cone from a node breaks up into a quadric cone and a trinodal quartic cone, we have a special case of XI 6 . In this case every tangent cone must split up into such a quadric and quartic cone, otherwise we should obtain one of the preceding cases, which are excluded. The twelve nodes form three sets of eight associated points. 11. Quartic surfaces with thirteen nodes. The plane sextics with twelve nodes divide themselves into the following classes : (1) three conies, (2) a nodal cubic, a conic and a straight line, (3) a trinodal quartic and two straight lines, (4) a cubic and three straight lines. * The case in which three of the six points belong to the associated nodes cannot occur. t The quadric cone cannot split up, as giving two tropes. 10, 11] WITH ISOLATED SINGULAR POINTS 19 Three conies u, v, w with a common contact-conic form a degenerate sextic of the first kind arising as the projection of three conies u 1} v 1} w upon the same quadric. The cone whose vertex is D^ which stands on u meets this quadric in the pair of conies u 1} u^. Similarly we have the pair v l} Vi and w lt w^' ; since Wj and v v have two apparent points of intersection, the three conies u i} v 1 and w have six which lie on a quadric cone. This applies also to the conies hence we have from the conies u, v, w four new conies upon which their twelve intersections lie by sixes. Hence there are four tropes*, and the surface is a case of XII C , viz. A 2 + pxyzw 0. The thirteenth node is one of the eight solutions of the equations A! . x = A z . y = A a . z = A t . w. The tangent cones at each of the first twelve nodes break up in each case into two planes and a quartic cone with three double edges. This surface is XIII a . The plane sextic (2) consisting of a line, a conic and a nodal cubic (all having a common contact-conic), is the projection of the complete intersection of a quadric and a cubic surface which have a line and a conic in common. Let D 2 ' be the node; DiDu'Du the intersections of the line and cubic; J) 3 'D 4 ' the intersections of the line and conic ; D & '. . . D lo ' those of the conic and cubic. Considering the three loci on the quadric it is clear, since the generator meets the conic once and the cubic twice, while the conic and cubic meet three times, that there are five apparent intersections of these curves. Let their projections from D 13 be DiD 3 'D 5 'D 6 'D 7 ' ; then these five points lie on a conic with D 2 ', hence D 1 D 2 D S D S D 6 D 7 lie on a conic. The cone joining any point to the conic on the quadric meets the quadric in another conic ; by associating this new conic with the generator and twisted cubic it is easily seen that the points AAAAAAo lie on a conic. * See the first variety of surfaces with eleven nodes. 22 20 QUARTIC SURFACES [CH. I Hence we have that DiD 2 D,D t D t Dj lie on a conic, let ac - be its plane. = 0. It follows that three tropes pass through Dj. Since A...DIO lie on two conies intersecting in A and D 2 , a quadric S through D l . . . D 10 has an equation of the form am, + yv zw = ; hence since x = 0, y = are tropes meeting the surface in two conies lying on S, the equation of the surface has the form (xu + yv zwf + 4&yV 0. But since z = is also a trope it follows that V = zw' uv ; hence the equation of the surface is o?u? + y*v* + z*w* lyzvw Zzxwu Zxyuv + ^xyzw' = 0. This may be written in the form z y u z x v y x w u v w w This surface is XIII&. The tangent cone from D a consists of three planes and a cubic cone; the cones from D 2 AA of two planes and a quartic cone with three double edges ; the cones from D 5 . . . D 18 of a plane, a quadric cone and a cubic cone with a double edge. Hence if the plane sextic consists of three lines and a cubic we have XIII& ; if it consists of two lines and a trinodal quartic we have XIII a or XIII 6 . 12. Quartic surfaces with fourteen nodes. The plane sextic with thirteen nodes is formed either by two lines and two conies or by three lines and a nodal cubic ; it will be seen that either leads to the same fourteen-nodal quartic surface. For in XIII a the tangent cone from one node splits up into three quadric cones; if there be another node one of these cones must consist of two planes a/9 which pass through the additional node D u . Since a/9 is a tangent cone it will pass 11, 12] WITH ISOLATED SINGULAR POINTS 21 through eight associated points of the nodes D 1 . . , D 12 ; if the equation of the surface is A 2 + pxyzw = 0, let these eight points be taken as the intersections of -4=0, xy = 0, zw = Q] then A = a/3 -\-pxy + qzw, and the equation of the surface is (a/3 + pxy + qzw) 2 + pxyzw = 0. But since a/3 is a pair of tropes it follows that p = 4xyzw' 13 V 'xx + ^yy' + v ' zz' 14 The same, where Ax + By + Cz + A'x' + B'y' + C'sf = 0, 15 with the condition AA = BB' = CO' The same, where an additional condition of this form 16 exists. CHAPTER II DESMIC SURFACES 13. An interesting type of quartic surface which possesses nodes but not singular curves is afforded by desmic surfaces. Desmic* surfaces are such that a pencil of such surfaces contains the special quartics formed by three tetrahedra. The equation of a desmic surface is 4 XAj + /LiA 2 + vA 3 = 0, where Aj = H (o^ -f a/# 2 + a/X + a/"^), etc., i and where an identity exists of the form aAi + A 2 + 7 A 3 = 0. Such tetrahedra are called desmic. They are shown to exist by consideration of such an identity as (a? - y 2 ) O 2 - 1*) + (a? - t 2 ) (f - z 2 ) + (a 2 - z*) (V - f) = 0. . .(1 ). Writing the preceding identity in the form it is clear that any face of Aj and any face of A2 are coaxal with some face of A 3 . Hence AJ, may be written in any one of the forms It follows that the edge (A lt A 2 ) of A 2 meets A 2 in edges of the latter, viz. at the points A l = A 2 = G t = 0, (i = l, 2, 3, 4), and two of these points are necessarily distinct since the faces of A 3 are not concurrent. These two edges of A^ do not intersect, for otherwise (Ai, A 2 ) would lie in a face of A 2 . Also since A 2 = II (C^ + K { A t ) it is clear that (A lt A 2 ) and (A s , A 4 ) meet opposite edges of A 2 , i.e. (A lt A 2 ) and (A z , A t ) meet the same pair of opposite edges of A 2 . Hence any pair of non-intersecting edges of one tetrahedron meet a pair of non-intersecting edges of either of the other two * Seoyx6j = pencil. See Humbert, Sur les surfaces desmiques, Liouville (1891). 13] DESMIC SURFACES 25 tetrahedra ; also we obtain sixteen lines through each of which a face of each tetrahedron passes. Taking A! as tetrahedron of reference and one face of A 2 as x+y+z+t 0, the identity becomes xyzt + (x + y + z + t) 2 3J B 4 + G&Cid = ; and the fact that any two opposite edges of A x meet two opposite edges of A 2 leads at once to the form of B z , B 3 and _B 4 . Finally the identity IGxyzt ...... (2) shows the form of C^ ... G t . The form obtained for A 2 shows that any edge, e.g. (xy\ of A x meets opposite edges of A 2 in two points harmonic with the points (xyz), (xyt), hence any two vertices of Aj are harmonic with the points in which their join meets opposite edges of A 2 . Hence if A x is given, a tetrahedron A 2 desmic with it is obtained as follows: if P be any point, draw through P a line to intersect a pair of opposite edges of A x and let P' be the fourth harmonic to P and the points of intersection, also let P", P'" be the two other points similarly determined, then the tetrahedron PP'P"P'" i s desmic to A x . The identity 4 O 2 + 7/ 2 + z 2 + P shows that A x , A 2 , A 3 are self-polar for the quadric 2 + f + z* + t 2 = 0. Hence since the intersection of any two faces of A 2 and A 3 lies in a face of A x , it follows by reciprocation that the join of any two vertices of A 2 and A 3 passes through a vertex of A x ; we thus obtain sixteen lines each of which contains a vertex of each tetra- hedron. Therefore three desmic tetrahedra are such that any pair of them have four centres of perspective, viz. the vertices of the third tetrahedron. Conversely if two tetrahedra have four centres of perspective they are in desmic position. For it is easy to see that two such tetrahedra have the property that each pair of opposite edges of one tetrahedron meets a pair of opposite edges of the other tetrahedron, and this necessarily involves that the 26 DESMIC SURFACES [CH. II tetrahedra are in desmic position, as may be seen by expressing the latter conditions. The identity (1) affords another system of desmic tetrahedra A> A> A closely related to that given by (2): the faces of A> A> A are respectively x y = 0, x + y = 0, z t = 0, z + t = 0; x z = 0, x + z = 0, yt = 0, y + t = 0; x- t = 0, x + t = 0, y-z=0, y + 2 = 0. The vertices of the three tetrahedra A< which arise from (2) being respectively I (0001) (0010) (0100) (1000) II (1111) (llll) (llll) (1111) III (llll) (llll) (llll) (llll) it may be observed that the preceding sixteen lines joining the vertices of Af are the intersections of the faces of two tetrahedra A (e.g. the planes x y = 0, x z = contain the three points in the first column) ; and that the join of two vertices of a A meets two opposite edges of another A in two vertices of a D. 14. Desmic surfaces. We may therefore take as the equation of the general desmic surface the equation a A + 6 A + cD 3 = 0, where A + A + A - 0- This may be written in the form (a? - s/ 2 ) (z* -P) + k (# 2 - * 2 ) (y z - V} = ; it has the twelve points I, II, III, the vertices of the A f , as nodes ; and contains each of the sixteen lines joining the vertices of Af by threes. The equation of the surface may be written in the form where = ^-y 2 , /3 = # 2 -2 2 , (w), g> (#) we obtain 1 a? e^(a?- 1 2 ) = 0. This gives on expansion which is the form of the equation of the desmic surface previously obtained. Nodes, lines and quadri-quartics of the surface. If 20!, 2&> 2 are the periods of $ (6) and if w l + &> 2 + o> 3 = 0, then since r(0 + 2w\ = crV0) ' a(Q + 2&) 2 ) = ~ ^70) ' 6tC '' rt* ?/ 2^ it follows by considering the ratios - , 7,3, that to any point of t t t the surface there corresponds an infinite number of arguments of the form eu ev + 2&&>] + 2A;'o> 2 + ^(Uj + 4A/C02 ; e = + 1. We obtain the nodes I when v has the values 0, o>i, &> 2 , a> s , II w v 0, 2ft)!, 2&> 2 > 2o> 3 , Ill u + v 0, 2ft)!, 2a> 2 , 2t 3 . The sixteen lines of the surface correspond to the equations u = 0, v = 2&&>! + 2&'&> 2 ; = &>!, v = &>! + 2A;&)! + 2&'&> 2 ; w = 2 + 2^&)! + 2A;' (u) v v (u) (v) h'w. 2 . 16. Intersection of a line of the cubic complex with the surface. Any line p of the preceding cubic complex is a chord of three pairs of quadri-quartics : if (u^) ... (u 4 v t ) are the arguments of its four points of intersection with the surface, let a pair of curves of the system II, III be v = a, v = /?, then we may take Vi = v 4 = a, v 2 = v 3 = ft ; * See Harkness and Morley, Theory of Functions, p. 315. 30 DESMIC SURFACES [CH. II the Ui are then connected by the equations u 1 + a = e 1 (w 2 + ft), u 4 + a = e z (u 3 + ft), U 1 -a = e 3 (MS - ft), 4 - a = e 4 (w 2 - /3), where e^ = + 1. Since p is any generator of the quadric containing the curves v = a,v = ft, the Ui each involve one indeterminate, hence on sub- tracting the third equation from the first and the fourth from the second and identifying the results we obtain = 6 4 , i 2> 3 4 , l 4 hence taking e t = 1 * and writing Wj = ft + //,, the arguments of the points of intersection are given as it l = ft + /ju, u t = ft-p, u 2 = + /*, u 3 = a fji; i = , ^4 = , *> 2 = /3, v 3 = ft. 17. Bitangents of the surface. The tangents to the quadri-quartics of the three systems which pass through the point (u, v) are bitangents of the surface ; for (!, vj = (u t , v t ) if p = and then (^, v 2 ) = (u 3> v 3 ) ; (i, t>i) = (MS, w a ) if a = ft ...... (u z , v 3 ) = (u t , v 4 ) ; (MJ, v^ = (u 3 , v 3 ) if a = ft ...... (w 2) ^2)5(^4, v 4 ). It also follows that the three bitangents of the surface deter- mined by the point (u, v) touch it at the points (v, u), (2v u, v), ( 2,vu, v). These three points are collinear, since the join of the points (2u u, v), ( 2v u, v) by the preceding Article meets the surface in the points (v, u), ( 3v, u). If p touches curves of the system II, III at P and Q so that P is the point (a, ft) and Q the point (ft, a), then, as Q moves to a consecutive position on the curve v = a, P takes a consecutive position on the curve u = a ; thus the tangent plane to the surface at P' passes through PQ, so that the tangents at P to the curves u = a, v = ft are conjugate, and the curves u = constant, v = constant form a conjugate network on the surface. Similarly it is seen that the system conjugate to u + v = const. is 3v u = const., and that the system conjugate to u v = const, is 3v + u = const. We can now determine the relation connecting any pair of conjugate tangents at any given point of the surface ; for if du, dv ; * Taking e t = - 1 gives the same form of result. 16-18] DESMIC SURFACES 31 du lf dv l correspond to this pair of conjugate tangents we have an involutive equation of the form dud^ +p (dudv! + du^dv) + qdvdv^ = 0, where p and q are functions of u and v. Expressing that this equation is satisfied by du = dv 1 = Q and by du dv = 3dVi + du = 0, we obtain that ^ = 0, ? = 3; and the equation assumes the form dud^ + 3dv dv! = 0. The asymptotic lines correspond to the assumption du = dui, dv = dv 1 and their differential equation is therefore du 2 + 3dv 2 = 0, whose integrated form is u + \/3iv = constant, u V3w = constant. 18. Plane sections of the surface. The plane (e 2 - e s ) o-i (a) o-! (/3) o-! (7) o-x (S) +... + ... - Oi - eg) (e 2 - e 3 ) (e s - e,) . (c') ( 7 ) + (e* - M) V (a + + 7) ov () ov () o-, (7) = 0. In the second formula let a = a + /3 + 7, 6 = a, c = & d = y, a'=0, 6'= + 7, c' = 7+a, d 7 =-( + ), a" = 7, 6"= , c"= a, d" = a+/S + 7, we then obtain i, bi and Ci respectively, then expressing that through each point cbi k there pass three lines, one of each set, we have c 4 = 0, + c s = 0, a + b s + c 2 = 0, Oj 4- 6 4 + GI = 0, Consider the determinant 2 , and has four poles in a parallelogram of periods which are congruent to hence it has four zeros congruent to Hence, expressed in terms of ^--functions, the determinant has a factor 4 a (ui + uz + us + Uj). Therefore, if Sw i =0, four points for which the v is the same i are coplanar. J. Q. S. 3 34 DESMIC SURFACES [CH. II 02 + &! + c s = 0, a 3 + &! + c 2 = 0, a 4 + 6j 4- G! = 0, 2 + 6 2 -f c 4 = 0, a s + 6 2 + c x = 0, a 4 + 6 2 + c 2 = 0, 2 + 6 3 + GI = 0, a s + 6 3 + c 4 = 0, a 4 + 6 3 + c 3 = 0, 2 + &4 + c 2 = 0, a s + 6 4 + c 3 = 0, a 4 + 6 4 + c 4 = 0, whence we deduce that 2oi = 2a 2 = 2a s = 2a 4 , aj + c^ =02 + 03, 26j = 26 2 = 26 3 = 26 4 , ^ + & 4 = & a + 6 3 , 2^ = 2c 2 = 2c 3 = 2c 4 , d + c 4 = c 2 + c,. The solution of these equations is seen to be n . of n a n' tt-j C//2 (* ~T~ Q y Cvg "" Ct "T~ "7T" j C&4 """" Ou "T" ~v~ "T~ "" j 7 11 J / 7 7 7- n iy iy n with the condition a + b + c = 0. Now the arguments of the lines a t are seen to be those of the four tangents at the points in which the tangent of argument 2a meets the curve*; similarly for the lines b i} c t , hence we have the result that if C is the curve of the third class which is touched by the twelve lines a^, 6j, Cf, the three sets of four points of contact with G of the lines ai, b{ and C; lie on three tangents to C which are concurrent. Hence we derive a desmic configuration as follows. From any point P of the plane draw three tangents to a given curve G of the third class ; each tangent meets G in four points in addition to its points of contact ; the tangents at these points give rise to a desmic configuration of sixteen points Q ; conversely if C and one of the points Q are given, the point P is uniquely determined. If P describes a straight line the points Q describe a curve K of order /*; and since two different points P cannot give rise to the same point Q, two curves K can only have in common the sixteen points Q arising from the point of intersection of the two lines which give rise to these curves; hence p? = 16, i.e. K is of the fourth order. * Clebsch, Vorlesungen uber Geometric, p. 607. 18-20] DESMIC SURFACES 35 Conversely, every quartic curve through the sixteen points of a configuration can be generated in this manner. For if P be the point of the plane which corresponds to the given sixteen points, then one line through P can be chosen such that the quartic curve deduced from the line by this method meets the given quartic curve in any assigned point of the latter ; the two quartics hence intersect in seventeen points and are therefore identical. 19. Sections by tangent planes. We now consider the form of the section of the surface by a plane which touches the surface at any point P. From P six tangents can be drawn to touch the curve of section ; it has been seen that the points of contact of three of these tangents are collinear, viz. the tangents to the three quadri- quartics through P. Hence the curve must have an equation of the form z*xy + a/37 ( ax + by + cz) = 0, where the inflexional tangents at P and the line joining the points of contact of the above three tangents form the triangle of reference. If (xyz) is a point near P, we may (Art. 17) take x Bu i V3 Bv, y = Su + i V3 8v, and, since the directions of a, ft, y are respectively given by Bv = 0, Su Bv = 0, 8u + Bv = 0, it follows that x y w = - Hence the equation of the curve of section by a tangent plane is + (a? T/ S ) (ax + by + cz) = *. 20. If p, q, r are three lines of a cubic surface, forming a triangle, any three planes through p, q, r respectively meet the surface also in conies which lie on a quadric. Cremona has shown that the locus of the vertices of such of these quadrics as degenerate into cones is a desmic surface. This will now be proved. The points of contact of the other three tangents from P to the curve are 32 seen to lie on the line ax + by + ^z = 36 DESMIC SURFACES [CH. II For let the cubic surface be ft + xyz = 0, then, if x = x at, y' = y fit, z = z yt, the equation of the surface may be written t{f+ ayz + $zx + yxy afizt - ayyt - fiyxt + aj3 -^r = w = are here close-points on the double conic. At each of them the two tangent planes of the surface coincide with the tangent plane of = 0. This equation may be written (0 + Xw 2 ) 2 - w 2 (i/r + 2X0 + X 2 w 2 ) = 0, where X is arbitrary. The system of quadrics ty + 2X0 + X 2 ^ 2 includes five cones ; every tangent plane of each cone meets the quartic surface in a pair of conies : for each generator of such a cone is bitangent to the quartic surface, and hence any one of its tangent planes meets the surface in a quartic curve having four nodes, viz. two on the generator of the quadric cone and two where this tangent plane meets the double conic ; this quartic curve, therefore, breaks up into two conies, two of whose intersections are collinear with the vertex of the cone. This may also be seen analytically : for if B 2 A C = is the equation of one of these cones V 1} the equation of the surface is where ZTj = + XjW 2 . * Kummer, Ber. Akad., 1863. 22] QUARTIC SURFACES WITH A DOUBLE CONIC 39 It is seen that the equation of the surface involves twenty-one independent constants. The surface arises as the intersection of two corresponding members of the pencils of quadrics U l wB = pwA, U l + wB = -- wC. P Each of these quadrics passes through the double conic ; the quadrics therefore intersect in another conic, whose plane a is given by p*A + 2pB + (7=0, and this plane is tangent to Fj. The surface is also generated as the intersection of the quadrics H! wB = - wC, U-i + wB = pwA, giving the other conic in the plane a. Hence the tangent planes of V l meet the surface in pairs of conies. A similar result arises in connection with each of the cones F 2 . . . F 5 . Thus the surface contains five sets of l pairs of conies. The conies which lie in the tangent planes a of F x belong to two classes, viz. those given by = 0, U l = w(B and those given by the equations It is clear that two points of intersection of the conies in the plane a lie on the double curve ; the other two points lie on the line a = B + pA = 0, hence they lie on the generator along which a. touches F,. By considering the conies in two different tangent planes a, y8 of F! it is seen that the conies of the same class do not intersect, and that therefore two conies of different classes inter- sect twice in points lying on the line (a, /3). Among each class of conies which lie in the planes a there are four pairs of lines, arising from those planes a which touch U l -w(B + pA) = and U 1 + w(B + pA) = Q respectively. For the condition of tangency gives a quartic for p in each case. Hence the surface contains sixteen lines. Each cone F 2 ... F 5 gives rise to eight pairs of lines, but as will be seen, these sets of sixteen lines are the same as the foregoing but differently arranged. 40 QUARTIC SURFACES WITH A DOUBLE CONIC [CH. Ill 23. Expression of the coordinates in terms of two parameters. If the cone V z has as its equation B' 2 A'G' = 0, we obtain as before two classes of conies on the surface, viz. the intersection of the plane a' or + \^w 2 . There cannot be more than one point common to a conic in the plane a and a conic in the plane a', and therefore each conic in a meets each conic in a! in one point. For instance, a point common to the conic a = p* and to the conic a = a*A' + Za-B' + C" = 0, U 2 = w (B' + aA), must also lie in the plane (\-\ 2 )w = B-B' + pA- &, &), where the curves fi have five points in common, we can show that it possesses a double conic ; for these equations establish a correspondence of such a character that to a plane section there corresponds a plane cubic curve, and since the deficiency of the plane cubic is unity, so also is that of the plane section; the surface, therefore, possesses a double curve of the second order, which must be a conic, since, if it were a pair of non-intersecting straight lines, the surface would be ruled}. To each of the five common points of the cubic curves, the base-points of the representation, there corresponds a line on the surface ; to the points of such a line correspond the points indefinitely near to its corresponding base-point; hence these five lines cannot intersect. * Otherwise p=oo , ; L 6 there correspond Thus the relationship of the sixteen lines on C 3 as regards intersection is the same as that of the corresponding lines on (7 4 , being deduced in both cases from the relationship of the base- points and lines in the plane. Now the sixteen lines of C 3 are obtained by omitting from its twenty-seven lines, 7r 6 and the ten lines which meet 7r 6 , hence the sixteen lines of the quartic surface are obtained by omitting from the twenty-seven lines of a general cubic surface any one of these lines and the ten lines which intersect it. 31. Determination of the surface by aid of two quadrics and a given point. The surface may be obtained by aid of any two given quadrics and a given point, a^. For let P, P' be two points #;, x{ collinear 48 QUARTIC SURFACES WITH A DOUBLE CONIC [CH. Ill with <*i and conjugate for a given quadric H, let K be the fourth harmonic point for <%, P, P / ; then if K is the point y t we have . * dH a i x i> where Aff = Sot; ^ . i OXi These equations lead to pyt^vtbH-otH, (af + bz 2 ) + a^ = 0. This is a general quartic cone, having the planes ay 2 -1- bz* = 0, the tangent planes to the surface at P, as bitangent planes. Hence any plane quartic may be regarded as the "projection" of a nodal quartic surface from any point on the double conic. * Sulle superficie di quarto ordine con cornea doppia, Ann. di Mat. n. xiv. (1887). 33, 34] QUARTIC SURFACES WITH A DOUBLE CONIC 53 Now having given any pair of bitangents ay 2 + bz 2 = of a quartic curve, the equation of the curve may be written in the form (ay* + bz 2 ) aft = V 2 in five ways*, giving a group of six bitangents, and in consequence the equation of the surface may be written in the form a ( + xLJ = z* (yt + x 2 ). The equation of the preceding tangent cone of vertex P is then yt(z 2 -L 2 ) = p ........................ (1). Now the cone p 2 y(L + z)- < 2p4>-t(L-z) = ............... (2) touches the cone (1) along four lines, and the plane p 2 y + 2px -t = meets the cone (2) in a conic which is seen, by elimination of p, to lie on the quartic surface. Regarding the equations (1) and (2) as representing curves, it is seen that the four-point-contact conies (2) are the projections from P of a system of conies of the surface. The other system connected with this cone of Kummer gives rise to the four-poiut-contact conies Theorems relating to the four-point-contact conies of a quartic curve are thus connected with theorems concerning this quartic * See Salmon, Higher Plane Curves. 54 QUARTIC SURFACES WITH A DOUBLE CONIC [CH. Ill surface; e.g. take the theorem: the eight points of contact with the quartic of any two conies of such a system lie on one conic*. We obtain the theorem for the quartic surface : the two pairs of principal tangents at a point P of the double conic and the points of contact with the surface of the two planes through P which touch the same cone of Kummer, lie on a quadric cone^. Again in Art. 31 it was seen that the intersection of the tangent planes at P to the surface and the vertices of the five cones of Kummer lie on a quadric cone whose vertex is P ; hence we derive the result for quartic curves that the six intersections of pairs of bitangents of a group lie on a conic. It has been seen that the group of six pairs of bitangents determined by the tangent planes to the surface at P gives four- point-contact conies which are the projections from P of the conies of the surface. It will now be shown that the other four-point- contact conies are projections of cubics on the surface. Refer the surface to coordinate planes consisting of the plane of the double conic and three tangent planes of a cone of Kummer of which one, x, contains two lines of the quartic surface; the equation of the surface is then of the form {AB + z(y-t) + xLY = z- {x 2 + y z + t 2 - 2xy - 2xt - 2yt}. The equation of the quartic tangent cone whose vertex is P is then {y (z + L) + 1 (z - L) + AB} 2 = kABt (z - L) ; of which a four-line-contact cone is p*At + p {y (z + L) + 1 (z - L) + AB} + B (z - L) = 0. This meets the cubic surface ZAztp = (L-z) {x (L + z} -f AB} in the line L + z=Q, pt + B = 0, which passes through P, and also in a quintic curve having a triple point at P and which lies on the quartic surface. Hence the preceding quadric cone also meets the quartic surface in a cubic curve passing through P. * For the quartic (z 2 - L 2 ) \fs = 2 may be written (z 2 - i 2 ) (X V + 2\ + 2 2 - L 2 ) = (\ +z n -- L 2 ) 2 . The points of contact are given as the intersections of the conies moreover the conic X/u^ + (X + p.) + z 3 - L 2 = passes through them and also through the four points similarly obtained on replacing X by p. t The principal tangents being z 2 -L 2 =0 = 0, and taking ij/syt and X = oo, X = successively. 34, 35] QUARTIC SURFACES WITH A DOUBLE CONIC 55 35. Segre's method of projection in four-dimensional space. Segre has shown* that if F=Q, = are two quadratic manifolds or varieties in flat space of four dimensions S t , the projection upon any hyperplane S 3 of their intersection F, is a quartic surface with a double conic. For if A, or x', be any point of $ 4 the substitution of #/ + pxi for Xi in F=0 gives the two intersections of the line (x, x) with F. The elimination of p between the equations x = 0, gives the "cone" joining A to the points of F. The intersec- 5 tion of this cone with the hyperplane S 3 or S o^ = 0, gives a i surface in S 3 represented by the equation - (3>DF - FD3>)(F'D3> - &DF) = I onaa = 0. i Taking F to be/, that member of the pencil (F, <3>) which passes through A, since f(x')= 0, we may write as the equation of the projected surface /f - Df(fD - !>/) = I eta* = 0; that is (2/f - Df. DW - (D/) 2 {(Zty) 2 - 4^'} = I * = f. This is a quartic surface with the nodal conic It is seen that the double conic is obtained as the intersection of f and Df, since the only cases in which the line joining A to any point x meets F in two points are when the foregoing quadratics in p become identical, we then have F x -3> x - ^F x = 0, F'D3> - 3>'DF = 0. These equations represent respectively the variety / through A and its tangent hyperplane Df. Their intersection gives a quadric cone in three dimensions which meets any variety of the pencil * Surfaces du quatrieme ordre a conique double, Math. Ann. xxiv. For many details the reader is referred to this important memoir. t Compare with Art. 31. 56 QUARTIC SURFACES WITH A DOUBLE CONIC [CH. Ill (F, ), and therefore F, in a twisted quadri-quartic k*. This quartic k* is projected upon S 3 as a conic ; any generator of the cone meets k* in two points P, Q ; the tangent planes to F at P and Q are projected into the tangent planes of the quartic surface at a point of this conic. Among the generators of the cone (/, Df) there are in general four which touch k* (viz. at the points where the plane Df= 0, D = meets F). It follows that there are four pinch-points on the double conic. There are in general five cones in the pencil (F, <). For if F and 4> are not specially related to each other we may take i the pencil therefore contains the five cones Ol) #3 2 + (4 - l) #4 2 + (5 - l) 5 2 = 0, etc. If f is a cone, i.e. if A lies on one of the cones of the pencil (F, <1>), we have two double lines instead of a double conic. For the hyperplane Df meets f in two planes*, the intersection of these planes with F will consist of two conies having two common points lying on a line through x'. These conies are projected from x' into two intersecting double lines of the quartic surface. Any one of the five cones of the pencil (F, <>) may be represented 4 by an equation of the form 2 dixf = 0, whence by comparison with the general three-dimensional quadric it is seen that this cone possesses two sets of generating planes, each generating plane of one set meets each generating plane of the other set in a line, the two planes therefore lie in the same hyperplane, while two generating planes of the same set intersect only at the vertex of the conef. 4 * For we may take the cone / to be 20f^sO, the tangent hyperplane to this i 4 cone at a point x' is Zo^s/scO; interpreting these equations to represent a i quadric and its tangent plane at x', since the plane meets the quadric in two lines, the hyperplane Df will meet /in two planes whose intersection contains x'. t It will be seen hereafter (see Art. 49), that the pencil (F, ) may contain, in certain cases, a cone of the second species, i.e. a cone whose equation contains only three variables, e.g. x l , x 2 , x 3 ; in this case the generating planes consist of a simply infinite set of planes passing through the line xi = x z =x a = 0. 35] QUARTIC SURFACES WITH A DOUBLE CONIC 57 Each generating plane of a cone meets F in a conic; con- versely each conic, c 2 , of F lies in a generating plane of a cone of the pencil (F, <1>); for the variety of a pencil which passes through any point P in the plane of c 2 and not upon c 2 , must contain the plane entirely, and a variety which contains a plane is necessarily a cone*. Hence F contains oo 1 conies belonging to five sets, each set con- taining two classes (corresponding to the two systems of generating planes of a cone). The hyperplane through any generating plane a of a cone and A meets the cone in another generating plane a' ; for taking the cone as x^x z # 3 # 4 = 0, and the generating plane as #! fJLX 3 = [JLXv # 4 = 0, the hyperplane is X 4 X 1 ' - iXs = ...... 1 . By comparison with the three-dimensional quadric it follows that this hyperplane also contains another generating plane of belonging to the other system. Since a and ' belong to the same hyperplane (through A), it follows that they are projected from A into the same plane @ of S 3 , and in /3 there lie two conies of the quartic surface. The envelope of /? is seen from (1) to be a quadric cone whose vertex is the projection of the vertex (00001). Thus we regain the pair of conies in each tangent plane of a cone of Kummer; the points of intersection of such a pair lying on the generator of the cone along which the plane touches the cone. Other leading properties of the quartic surface considered are readily obtained by the method of Segre. We obtain the sixteen lines of the surface as follows : The surface F is determined by the equations by a change of the coordinate system these equations may be replaced by X 1 X 2 -X 3 Z 4 = ..................... (a) * Since its equation is expressible in the form xiA +x 2 B = Q, if %i = x z = is the given plane. 58 QUARTIC SURFACES WITH A DOUBLE CONIC [CH. Ill Every plane X t = \X Z , X 4 = \X 2 (a generating plane of (a)), will meet (6) in a conic, which reduces to two lines, if (a&X lt X 2 , X 3> Xrf is reduced to a perfect square ; this leads to a biquadratic in X. Hence four generating planes of this system meet T in two lines, and similarly four generating planes of the other system meet F in two lines ; this gives sixteen lines on T, and therefore, by projection, on the nodal quartic surface. It follows also that eight tangent planes of each cone of Kummer contain a pair of these lines. Each of the sixteen lines p on F lies on each of the five cones of the system ; the plane through p and a vertex of one of these cones is a generating plane of that cone and therefore meets F in another line, hence each of the sixteen lines is met by five others. Cubics and Quartics on the surface. Any hyperplane through one of the sixteen lines meets F in a cubic curve, and since there are oo - hyperplanes through any line we thus obtain sixteen sets of oo 2 cubic curves on the surface. In $ 4 there are oo 4 hyperplanes and each of them meets F in a quadri-quartic, any two of these quadri-quartics intersect in four points, lying in the plane common to the two hyperplanes ; through these four points there pass oo l quadri-quartics deter- mined by the pencil of hyperplanes through the plane of the four points. Since four non-coplanar points determine one hyperplane, ifc follows that one quadri-quartic of the surface passes through any four non-coplanar points of a nodal quartic surface. Any hyperplane 2=0 cuts the quadri-quartic k* whose projection is the double conic, in four points lying in the plane of intersection of this hyperplane with Df, the tangent hyperplane at A. Let Q l . . . Q 4 be these four points and a. their plane, and Qi ... Q 4 ' the points in which AQ, etc. again meet k*. Let /3 be the polar plane of A for the system of quadrics through k* and p the line (a/3); then the planes (pA), a, /3, (pQi) are harmonic and the plane (pQi) must pass through Q 2 ' ...Q 4 ', i.e. the points Qi ...Q 4 ' are coplanar. Hence we have oo 1 quadri-quartics, arising from the hyperplanes 2 + XD/= 0, through Q^ ... Q 4 , and oo 1 quadri-quartics through Qi ...Qi. It follows on projection that each of the oo 4 35, 36] QUARTIC SURFACES WITH A DOUBLE CONIC 59 quadri-quartics of the projected surface cuts the double conic in four points of which three determine the fourth; and through four such points there pass oo 1 quadri-quartics on one sheet of the surface and oo 1 quadri-quartics on the other sheet of the surface. Among the oo 4 hyperplanes of S t , oo 3 pass through A ; these hyperplanes meet F in quadri-quartics which are projected into plane sections of the projected surface. Quadrics inscribed in the surface. Let ^=0 be any variety of the pencil (F, ); its intersection with the polar hyperplane of A for F is given by DF= 0, F=Q, and is a quadric A; the intersection of F with A is a quadri- quartic c 4 . Let X be any point of c 4 ; the tangent plane to F at X is given by the equations and the tangent plane to A at X is given by each of these tangent planes lies in the hyperplane 2Jff r = 0, OXi which passes through A since X is a point on DF=0. Hence the tangent planes to F and to A at X lie in the same hyper- plane through A, they are therefore projected into the same plane of S 3 . Thus the projection of A touches the projection of F along a quadri-quartic, the projection of c 4 . Now F is any member of the pencil (F, <3>), hence oo * quadrics touch the quartic surface along quadri-quartic curves. 36. Fundamental inversions. As in the case of the quadric in three dimensions where the points of contact of the tangent lines to a quadric < which pass through a point lie on a plane, so the points of contact of tangents passing through a point of $ 4 lie on a hyperplane, the polar hyper- plane of the point. If C is the vertex of a cone of the pencil (F, 4>), the polar hyperplane of C is the same for each member of the pencil, e.g. if the system is determined by the two equations 2^ = 0, 20^ = 0, i i the polar hyperplane of the point (10000) is x = 0, and so on. .60 QUARTIC SURFACES WITH A DOUBLE CONIC [CH. Ill Let a be the polar hyperplane of G, A the centre of projection, and f the member of the pencil which passes through A. Then any plane through the line (C, A) meets / in a conic passing through A : this conic is met by a = in two points B, B' whose join is the polar line of C for this conic; so that if any line through C meets the conic in two points Q, Q' then, by elementary geometry, {A, BB'QQ'} = - 1. Now there are two generators of the cone whose vertex is G which lie in the plane of this conic ; each of these generators meets the conic in two points of F, since the points lie both on /= and on the cone, and the conic is projected from A into a line of $ 3 , hence denoting by a the quadric which is the projection of the quadric a = 0, /= 0, and the projection of C by C" (which is the vertex of a cone of Kummer), it follows that any line through C" meets a- in two points B l} BI and the projected quartic surface in two pairs of points Q lt Q/; R lt jR/ such that both Q l} Q/ and jRi, RI are harmonic with regard to B l} B^. Hence C' is said to be a centre of self-inversion of the projected quartic surface*. 37. Plane representation of the surface. To represent F, and therefore the projection of F, upon a plane, we take the oo 2 planes through a line p of F, any one of these planes meets any two varieties of the pencil (F, <) in p and two other lines respectively, the intersection, Q, of these latter lines lies on F ; hence the plane through p meets F in one other point, viz. Q. Moreover it meets any given plane K in one point Q', thus there arises a (1, 1) correspondence between the points of F and K. The five lines of F which meet p have as images the five base-points; if q be one of the ten lines which do not meet p, the hyperplane through p and q meets K in a line, and since this hyperplane meets F in two non-intersecting lines p and q, it must also meet it in two other lines which meet both p and q. Hence the image of q is a line passing through two base-points. If x l = 0, # 2 = are tangent hyperplanes to F at any two points of p, and # 3 = 0, x t = the tangent hyperplanes to <$> at these points, the tangent plane to F at any point of p is repre- sented by x l + \x 2 = 0, x 3 + X# 4 = 0. * The quadric a is the quadric H of Art. 31. 36, 37] QUAETIC SURFACES WITH A DOUBLE CONIC 61 As \ varies, the intersection of this plane with the given plane K is clearly a conic ; hence the points of T contiguous to p are represented by the points of a conic which passes through the five base-points, i.e. the image of p is this conic. Again the oo 2 hyperplanes through p meet F in oo 2 cubics and K in oo 2 lines, i.e. the lines of K are the images of oo 2 cubics of T. Any hyperplane meets F in a quadri-quartic and also meets each of the five lines which meet p, moreover it meets any cubic of F in three points, hence the image of the section of F by this hyperplane is such that it is met by any line of K in three points ; it is therefore a cubic which passes through the five base-points. The oo 3 hyperplanes through A which give rise to the plane sections of the projected surface (Art. 35) meet F in quadri-quartics such that through any three points of F there passes one such quadri-quartic, hence among the oo 4 cubics of K through the base-points there are oo 3 cubics forming a net or linear set ; these are the images of the plane sections of the projected surface. CHAPTER IV QUARTIC SURFACES WITH A NODAL CONIC AND ALSO ISOLATED NODES 38. A quartic surface with a nodal conic may have in addition one or more isolated nodes ; such a node is the vertex of a cone of Hummer, for taking the node as a vertex of the tetrahedron of reference, the equation of the surface is where A = 0, U are cones whose vertex is the node, and L = is a plane through the node; we may write this equation hence the node is the vertex of a cone of Kummer. This result may also be seen from the fact that any tangent plane drawn to the surface from the node meets the surface in a quartic curve with four nodes, and if the surface is not ruled this section must consist of two conies. The sextic tangent cone whose vertex is the node D, here consists of the cone V of Kummer of vertex D and the cone U (counted twice); the latter cone meets the surface in the double conic and in the four lines given by A = U = 0. The surface contains twelve lines; for if the foregoing four lines meet the double conic in P^ ... P 4 , through the line DP we can draw two tangent planes to V each of which meets the quartic surface in two conies, and in each plane there is therefore one other line in addition to DP^. similarly for the tangent planes drawn to V through the lines DP Z , DP 3 , DP 4 . Hence we have in all 4 + 8=12 lines on the surface. There are only three cones of Kummer in addition to V, for if we take the vertices of the triangle self-polar for the sections of 38] SURFACES WITH NODAL CONIC AND ISOLATED NODES 63 U and V by # 4 as vertices of reference, the equation of the surface may be written (ax? + bzj + ex? + 2xi (#! + /&c 2 + 7# 3 ) + 2X# 4 2 j 2 = 4# 4 2 [x* (1 + Xa) + x? (1 + X6) -f x? (1 + Xc) and the values of X for which the quadric on the right is a cone are given by the cubic equation 1+Xa 1+X6 1+Xc' If there is a second node D', then if the cone V contains D' it will have a double edge and therefore consist of two planes, and the equation of the surface is If V does not contain D' then U must contain it, and since the line DD' meets the double conic it therefore lies on the surface. The equation of the surface may be written in either of the forms where D, the vertex of V, lies upon U=0, and D', the vertex of V, lies upon U' = 0. In this case two of the lines DP . . . DP must coincide, since otherwise jy could not be a double point of the curve of intersection of U and the quartic surface, consisting of four lines. In fact the tangent plane at any point of DD meets the surface in a section which contains four nodes lying on DD, hence the section consists of the line DD' taken doubly together with a conic. The tangent plane at any point of DD' is the same since otherwise this line would be a double line of the surface. The line is torsal. If there are three nodes the section of the surface through these nodes contains five double points and therefore consists of two lines and a conic ; one line joining a pair of nodes does not lie on the surface, whose equation may be written in either of the forms (w 2 + V - pqf = 4w 2 F, (V-pq- w 2 ) 2 = 4>w 2 pg, where the vertex of Flies upon w* pq = 0. The lines joining the vertex of V to the two nodes each meet the double curve, hence the plane through these two lines meets the surface in each of them doubly. If there is a fourth node two lines joining a pair of nodes do not lie on the surface, and four lines joining pairs of nodes lie on the surface, whose equation is therefore (w 2 + rs pqf = 4wVs. 64 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV If the nodes on the lines (p, q] and (r, ) are DI, Z) 2 , and D 3 , D it is clear that the lines DiD 3 , D^D^ D 2 D 3 , D^D^ lie on the surface. There cannot be more than four nodes, for if D be the node which is the vertex of a cone V, then V cannot contain more than one other node, hence the remaining nodes must lie on U, and it was seen that each node on U causes the coincidence of a pair of the lines _DP a . . . DP 4 : hence U cannot contain more than two nodes of the quartic surface apart from its vertex D*. 39. Special positions of the base-points. It will now be shown that singularities of the surface arise from special relative positions of the base-points. If a node exists, any line through it meets the surface in two points apart from the node, hence any two cubic curves f t which correspond to plane sections of the surface through the node meet in two variable points only. In order that this may be possible one of the two following cases must arise : Either, in the lirst case, these cubics must have a common node and intersect in three other fixed points, e.g. if pa; 3 = f,Zr, + 2 3 , p# 4 = s LI + 2 4 , where the Li are quadratic in , 2 and the 2 f cubic in 15 2 . The point (1000) will then be a node to which the point = f 2 = will correspond. The system of cubic curves will touch at the point = 2 = 0, so that two base-points coincide. Thus the coincidence of two base-points leads to a node on the quartic surface. The four lines through the node correspond to the following : the point consecutive to ^ = 2 = upon A = 0, and the joins of this point to the three other base-points. Or, in the second case, three base-points are collinear, and we may take as equations of Clebsch where u z = Q,u t = 0, u t = are conies having two common points, which also lie upon yi = 0. The base-points are then given by yj = a = and the two other common points of the system. * These surfaces have been investigated by Korndorfer, Die Abbildung einer Flache vierter Ord., etc., Math. Ann. i. and n. 38, 39] AND ALSO ISOLATED NODES 65 The point (1000) is seen to be a node, for if be any line through it, the points in which this line meets the surface have as their images the intersections of the conies % Us tf A~B~ C' which are two in number, apart from the two base-points. Hence if three base-points are collinear the surface has a node. The image of the node is here the line a = 0. Two nodes on the surface may arise in three ways : first if the base-points are doubly collinear, e.g. when the join of the base-points 1, 5 meets the join of the base-points 2, 4 in the point 3; secondly when two base-points are coincident and three are collinear ; thirdly if there is a double coincidence of two base- points. Considering the first case, let a = 0, /3 = be the lines (1, 5) and (2, 4) ; the equations of Clebsch are here px 1 = CLU, px 2 = ajSL-L, px 3 = a/3L 2 , px t = /9-y, where u = 0, v = 0, are conies through two base-points ; L and L 2 = are any lines. Thus as in the case of one node the points A l} A 4 are nodes, the line @ corresponds to A and a to J. 4 ; to the point a = /3 = corre- sponds the line A^ A 4 which lies on the surface. There are nine lines on the surface whose images are the base- points and the lines 12, 45, 14, 25 ; those which correspond to the base-points 1, 3, 5 pass through one node and those to 2, 3, 4 through the other node. There are three sets of pairs of conies : first those which have as their images the pencils of lines whose centres are the base-points 1 and 5, these conies pass through a node of the surface which is the vertex of a cone of Kummer; secondly those correspond- ing to the pencils of lines whose centres are 2 and 4, these conies also pass through a node which is the vertex of a cone of Kummer : lastly those which are represented by the conies through the base-points 1, 2, 4, 5 and the pencil of lines whose centre is 3. j. Q. s 5 66 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV That there are only three cones of Kummer may also be seen thus : referring to the equation of Art. 38 ; if D is the vertex of the cone and D' the vertex of .r 2 (l+Xa)+y 2 the line DD' is g(l+Xc) and since DD' meets the double conic we have aa 2 6/S 2 cy 2 _ (l+Xa) 2 + (1+X6) 2 + (l+Xc) 2 ' which is the condition that the cubic equation for X should have a pair of equal roots. 40. When three base-points are collinear and two are coin- cident the appropriate equations are where the Li are quadratic and 8 cubic in 2 > & Here the node 6f corresponds to the line x = 0, and the node a* to the point 2 = 3 = 0. Thus the join of the nodes does not lie on the surface. The number of lines of the surface is easily seen to be eight. There are three cones of Kummer, as is seen by forming the discriminant of the surface being U z = 4 7) an( l having a as its tangent at ( 15 2 ), the system of cubics intersect in three consecutive points at (&, 2 ) and pass through two other fixed points. Here, therefore, three base-points are coincident. Among the curves of the system appear (i) (3 3). and where u passes through the point (g lt 3 ). The five base-points consist of the points (u, v) and the point ( n s ). The case of additional nodes arises as in the case of the surface with a nodal conic. Either or both of these lines may be cuspidal. The equation of the surface in the latter case is {x z x - Xi (ax l + bxz)} 2 = x^x z . An additional node exists if 4a6 = l. 48. Classification of quartic surfaces with a nodal conic. The method of Segre (Art. 35) affords a means for the classifi- cation of quartic surfaces with a nodal conic. The two four- dimensional varieties F=Q, = are reduced by the method of Elementary Factors* of Weierstrass to their canonical forms, * See Quadratic Forms, etc., Bromwich, Camb. Math. Tracts; or the Author's Treatise on tJie Line Complex. 47,48] AND ALSO ISOLATED NODES 73 leading to various types and each type to sub-cases. Each pair of forms thus arising affords one species of the quartic surface considered. An elementary factor (\-\i) e p of F+\<& gives rise, if e p is greater than unity, to a group of terms in e p variables, viz., \\ e . . . ep\> in F and <> respectively, so that F + \i<& is a cone of the pencil (F, ) whose vertex lies on each variety of the pencil ; at this point the varieties have a common tangent hyperplane (K 1 ; this point is therefore a double point of F. We now consider the principal types, indicating them as in Segre's notation by {11111}, {2111}, {221}, {311}, {23}, {41}, {5}. The surface which is the projection of T is denoted by [11111], etc., but if the point of projection A lies on a cone of (F, <>) the projected surface is represented by [11111], and so on. The general type [11111] has been already considered in the preceding chapter ; we may find its class by aid of this method. Taking F = S ar f a , 3> = S a^ 2 , i i the required class is equal to the number of tangent planes of the projected surface which can be drawn through any line p of S t ; but if the projection of a plane IT from A on 8 3 passes through p, then A, p and IT must lie in the same hyperplane ; our problem is therefore to find the number of hyperplaries through the plane ( A, p) which contain tangent planes of P. If the plane (A, p) is given by the equations the condition requires that where < = Ai + \B^. Thus we obtain Oi), (i = 1 , 2, 3, 4, 5), 74 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV whence and therefore These equations show that the last equation, considered as a quartic in - , has equal roots ; and forming its discriminant, which is of degree six in a; 2 , we obtain an equation of degree twelve in X. The class of the projected surface is therefore twelve. We now proceed to consider the remaining six principal types. The canonical forms corresponding to the type {1112} are -t* = &'i -J- X% -f- 3/3 -f- ^X^X^, At the point (00001) which is a double point of T, F and 4> have the common tangent hyperplane # 4 = 0. The tangent cone of F at this point is #4 = (! ^4) #1 2 + ( 2 4 ) #2 2 + (3 ^4) #3 2 = 0. This cone contains the four lines a 3 # 3 2 = 0, which also belong to F. If there is any additional line on F, the plane through it and the vertex of the cone a t F must lie on this cone. Now through any generating line of a cone in 8 t we can draw two generating planes (one of each set), Art. 35, and hence through each of the preceding four lines ; such a plane meets F in a conic which therefore reduces to two lines. Hence corresponding to each of the four lines through the double point we have two other lines of F. Therefore the surface [1112] has a conical node with four lines passing through it, and eight other lines. The class of the surface is ten, being diminished by two from that of [11111] owing to the additional node. In [1112] the same applies, the double conic being here two intersecting lines. In [1112] the point of projection lies on the cone 4> a 4 F; the double point of F is projected into the intersection of the two 48] AND ALSO ISOLATED NODES 75 double lines, and this point is now triple*; the tangent cone at it consists of the plane of the double lines and the projection of the tangent cone of F at its node. If T is of the type (122}, F and 3> have the forms Jf = X^" -|- AX^XS ~r X 4 X$) Here F has two nodes, viz. (00100), (00001) ; the tangent cones thereat being #2 = (i a) x? + 2 ( 3 a 2 ) x 4 x s + x? = ; and The line joining the nodes belongs to F, and along this line the plane x z = # 4 = touches both F and and therefore F. Through the first point there pass the two lines # 2 == xf + 2x 4 (K 5 = c^a?! 2 + 2a 3 # 4 # 5 + # 4 2 = ; similarly two lines pass through the second point. As in the case {1112} each of these additional four lines gives rise to a line of F; hence F contains nine lines in all. The nature of the surfaces [122], [122], [122] is therefore determined. For the type {113} we have _* i^ t//j r" ^2 l" t *-'4 ~T" ^'"-'S'^'Sj The point (00001) is a double point of F; the tangent cone at it, which is represented by x 3 = 0, (a x a 3 ) x? + (a 2 a 3 ) x a s F = 0, of the same system as /^ ; * For any plane a passing through A and the vertex K of the cone corresponding to 2 meets that cone in two lines, and each of these meets T in one other point giving two points Q, R of F on a. The plane a is projected from A on S 3 into a line r passing through K' the projection of K and Q, R are projected into the two other points in which r meets the surface. If however A lies on the cone, one of the two previous lines must pass through A, and r thus meets the surface in one point only (apart from K') ; the point K' is therefore triple. 76 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV and so for the three other lines r/, r, 2 , r 2 '; hence each of these four planes meets F in an additional line, giving rise to four new lines *1> *1 > ^2> ^2 Hence applying to the surface [113] we have a surface of the ninth class* which has a biplanar node and contains eight lines. For the type {23} we have F = 2x^ 2 + 2x 3 x 5 + # 4 2 , = 2o 1 # 1 # 2 + Xj 2 + a 2 (2x s x 5 + x?) + 2x 3 X4. From consideration of the cases {1112}, {113} it is seen that F possesses a conical node at D and a biplanar node at D' at which the tangent cone breaks up into two planes /^ and yu, 2 . The line Diy is given by x l x 3 = x^ = ; and the plane /^ is # t = x 3 = ; this touches F along the line DD'. As in {113} there are two lines r 2 , r 2 ' in the plane /JL 2 . The section of F by its tangent hyperplane a^ at D is $7j - tmiOuQtJC^ ~\ OC^ - CC^CU^ \) . and is therefore the line DD' together with one other line. The two generating planes of the cone whose vertex is D which pass through the latter line, meet the surface in two new lines. These six lines constitute all the lines of the surface. The nature of the surfaces [23], [23], [23] follows immediately; they are of the seventh class, the first has a conical node and a biplanar node, the second has a conical node and a triplanar point, the third has a biplanar node and the intersection of the double lines as a triplanar point. For the type {14} we have F = x? + 2x 2 x s + 2x 3 X4, = a^! 2 + 2a 2 (x 2 x 5 + # 3 # 4 ) + 2x 2 x 4 + x/. The double point D, or (01000), is here biplanar, and the two nodal planes intersect in a line which lies on F ; the biplanar point is therefore of the second kindf. The nodal planes meet F also in two lines r lf r 2 through D. In the two other generating planes of the cone whose vertex is D which pass through TI and r 2 respectively there are two other lines * A biplanar node of the first kind reduces the class of the surface by three. Salmon, Geom. of three dimensions, p. 489. t See Salmon, I.e. 48, 49] AND ALSO ISOLATED NODES 77 of F (say) s z and s 2 , which meet r^ and r 2 respectively ; and since the line of intersection of these latter planes meets s : and s z it must therefore meet them in the same point. Hence we have four lines on F forming a skew quadrilateral, together with another line through D. The preceding defines the surfaces [14], [14], [14] which are of the eighth class. For the type {5} we have F = 2# 1 #, + 2# 2 # 4 + # 3 2 , The (one) cone of the pencil meets x l in the two planes #! = # 2 = ; x l = x 3 = ; and these planes meet in a line r of F. The first plane touches F along r ; thus since one of the nodal planes touches F along r, the biplanar point is of the third species. The other nodal plane meets F in a line r, through D. Another generating plane of the cone passes through r' which meets F in a line s. The lines r, r' and s are the only lines on F. The properties of the surfaces [5] and [5] follow ; they are of the seventh class ; the latter has two double lines meeting in a triplanar point. 49. Cones of the second species. In the preceding types the pencil (F, ) contains cones, the equation of each cone being expressible in terms of four variables. When, however, two elementary factors are equal, the equation of the corresponding cone a,iF= contains not more than three variables and the cone is said to be of the second species ; e.g. in {(11) 111} we have - OvF = (a, - xf + (a 4 - a x ) x? + (a 5 - Oj) x s *. This cone has oo 1 generating planes through the line QC% #/4 =: t27g == \) In the previous types there were seen to be two systems of generating planes given as the intersection of a cone with its tangent hyperplanes. In the present case we have one system of generating planes obtained as the intersection of the cone with its tangent hyperplanes which all pass through the line 78 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV the edge of the cone. In each generating plane there is one conic ofr. " Each of these conies passes through the two points of inter- section of the edge of the cone with F. At either of these points there is a tangent hyperplane common to the pencil ; these points are therefore double points of F. If the group considered is (11) there are two such double points; if it is (21), (31), (41) the points coincide, as is seen by reference to the corresponding forms. In the cases {1 (22)}, {(23)} the edge itself is seen to lie on F ; in these cases since each generating plane of the cone meets F in the edge and one other line, there arise oo 1 lines on F, which is therefore a ruled surface, having the edge as a double line. We can easily determine the number of lines on F in the other cases ; for any line of F must lie on this cone of the second species and therefore meet the edge in one of the double points of F upon it. The tangent hyperplane at either of these double points meets the pencil (F, ) in a pencil of ordinary quadric cones having the double point as vertex ; the lines of intersection of two of these cones will be the lines of F through the point. Hence these surfaces cannot have more than eight lines. Projecting F on S 3 gives us the surface we are investigating. In this case, however, the point of projection A may lie on a cone of the pencil (F, 3>) of the second species. Here only one generating plane of this cone passes through A, which cuts F in a conic which is projected from A on S 3 into a line which will be a double line of the projected surface. Reference to Art. 35 shows that if y= is a cone of the second species, the double line of the projected surface, given by /= Df= 20^:= 0, is therefore to be regarded as arising from the coincidence of two double lines*. This line contains two triple points (distinct or coincident), the projections of the two double points of F which lie on the edge a of F. For the generating planes of f cut F in conies through the two double points, hence their projections from A meet 8 3 in conies through the projections of these points which are therefore triplef. Each of the oo 2 hyperplanes through the edge of the cone meets the cone in two planes ; each tangent hyperplane of the * Segre calls this line bidouble, see page 70. + Since any line through one of these points on the projected surface meets the surface in one other point only ; see page 75, footnote. 49, 50] AND ALSO ISOLATED NODES 79 cone meets it in a generating plane counted twice, and hence touches F in a conic. If / = is the variety of the pencil (F, <1>) which passes through A, and c x = any hyperplane, the inter- section of / and c x is projected from A upon S 3 into the quadric 5 5 c x Dfc X 'f=0, ]y*i#; = 0; where 0^ = represents 8 3 . This i i quadric passes through the double conic /= Df= 0. Now let c x be one of the preceding hyperplanes through the edge of the cone of the second order; we obtain on projection oo 2 quadrics through the double conic and the two double points ; each meets the quartic surface in two conies. If c x is one of the oo ] tangent hyperplanes of the cone of the second species we obtain on projection oo * quadrics touching the quartic surface along a conic and passing through the double conic. Through any point of $ 4 there pass two of these tangent hyperplanes, hence through any point of S 3 there pass two quadrics containing the double curve which touch the quartic surface along a conic. Thus the quartic surface is the envelope of a system of quadrics simply infinite and of the second order which pass through the double conic and the two double points of the quartic surface*. The existence of this set of quadrics is peculiar to those surfaces which have a cone of the second species. For such a quadric is the projection from A of the intersection of some hyper- plane c x with/, the variety through A. This hyperplane therefore touches F along a conic, and hence c x meets the pencil (F, 3>) in a pencil of quadrics which touch along this conic; among these quadrics is therefore included the plane of the conic counted twice ; if c x d x = is this plane, it is seen that among the varieties of the pencil (F, ) there is one of the form d x 2 + c x e x , and this is a cone of the second species. 50. Quartic surfaces with a cuspidal conic. It was seen (Art. 35) that if /= is the variety of the pencil (F, 4>) which passes through A (or x'}, and any variety of the pencil, the equations of the projection of F from A on S 3 are = (Dfy cci = 0. * For the surface 2 4w^pq, the quadrics are p?wp + /j.(p + wq = Q. We have also, as in the general case, the quadrics \ 2 u> 2 + \ +pq = 0, which touch the surface along quadri-quartics. 80 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV Let x be any point on the double conic and x + f a point on the surface contiguous to x ; substituting x + g for x and retaining only terms of the second order in , we obtain as one of the equations of the two tangent planes to the surface at x, where L and M are the terms of the first order in f< arising from /and Df respectively. If these planes coincide we have ' = 0. This equation together with /= 0, Df= 0, 2 = from x' contains the three-dimensional cone /= 0, Df= 0. Hence we have an identity of the form i.e. 4f -Af = (D(f>)* + XDf. This shows that the pencil must contain a cone of the second species. Thus having given a pencil (F, <) which contains a cone ty of the second species, the surface F projected from A on S 3 has a cuspidal conic provided that A is so chosen that the tangent hyperplane at A of the variety through A is also a tangent hyperplane of ty. The equations of the surface given at the beginning of this article may therefore, when a cuspidal conic exists, be written in the form , = 0, (2/f - DfDtf = (Z>/) 2 {Af+ XDf} ; i the latter equation is A where ^ = + TV//, and L is linear in the variables. This is the TrQ) equation obtained in Art. 44. The close-points. The two intersections of the edge of a cone of the second species with F were seen, in the general case, to give rise to two nodes on the projected surface ; when a cuspidal conic exists, since 50-52] AND ALSO ISOLATED NODES 81 the tangent hyperplane of the variety through P passes through the edge of this cone, these two intersections are therefore pro- jected into two points on the cuspidal conic; they are the two close-points. Quartic surfaces with a bidouble line. If A (or x') lies on a cone of the second order i/r = 0, then D\jr = touches ty along a plane TT, also IT meets (any variety of the pencil) in a conic c 2 on F. The tangent plane to F at any point x of c 2 is given by the equations 3 If we suppose, as is permissible, ty to be of the form Sa<|i 2 = 0, it is seen that the first of these hyperplanes is identical with Zhjr = 0, since for each point x of TT we have 1 2 3 iCj $?2 X Hence the tangent plane of F at any point of c s lies in the fixed hyperplane D\|r = 0, and is therefore projected from A into- the same plane of S 3> viz. Zty = 0, Scc^ = 0. The pair of tangent planes at each point of the bidouble line coincide. 51. Of the sub-types arising from the equality of elementary factors the first is {(11)111}. As stated in Art. 49 we have two nodes on F ; the line joining them does not belong to F. Hence there arises the surface [(11) 111], treated in Art. 38, possessing two nodes whose join does not lie on the surface. This includes the special case of a cuspidal double conic. Other special cases are [1(11)11], [(11)111] having respec- tively two double lines and two nodes, and a cuspidal line containing two triple points. The characteristics of the various other sub-types are given in the table at the end of this chapter*. 52. Steiner's surface. The pencil (F, 3>) may consist entirely of cones of the first order having a common generator and a common tangent hyper- plane along this generator. * For many details see Segre, loc. cit. J. Q. 8. 6 82 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV Such a system, for instance, arises from the cones =0 > .a). The line upon which the vertices of these cones lie is x l = x z = x 3 = 0, this line is a double line of F. Through A, the point of projec- tion, there passes one cone of the system, its two generating planes through A intersect on a line which meets the double line of F in the vertex of this cone. Hence the projected surface has three concurrent double lines, viz. the projection of the double line of F and the projections of the conies in the two generating planes through A. Each of the oo 1 cones has two sets of generating planes meeting F in conies, hence arise oo 2 pairs of conies in plane sections of the projected surface. Three of the points of inter- section of such a pair of conies lie on the three double lines, the fourth point is a point of contact of the plane with the surface. The surface is therefore a Steiner's surface (Chapter vn). 53. We add Segre's Table which contains a complete list of the different kinds of quartic surfaces with a double conic (including two lines or a bidouble line). Class of the Index surface Character of the surface [11111] 12 General surface [2111] 10 One node [311] 9 Biplanar point of the first species [221] 8 Two nodes ; the line joining them belongs to the surface [41] 8 Biplanar point of second species [32] 7 One node and a biplanar point of first species [5] 7 Biplanar point of third species [(11) 111] 8 Two nodes ; the line joining them does not belong to the surface [(21) 11] 8 A biplanar point of the second species [(11) 21] 6 Three nodes; the lines joining two of them to the third belong to the surface [(21) 2] 6 A node aud a biplanar point of the second species [(31) i] 6 A uniplanar point of the first species [(11) 3] 5 Two nodes and a biplanar point of the first species [(41)] 5 A uniplanar point of the second species 52, 53] AND ALSO ISOLATED NODES 83 Class of the Index surface Character of the surface [(11) (11) 1] 4 Two pairs of nodes [(21) (11)] 4 A pair of nodes and a biplanar point of the second species [(22) 1] 4 Ruled surface (class II of Cremona) [(32)] 4 Ruled surface (class IV of Cremona) Surfaces with a cuspidal conic. [(11) 111] 6 General case [(21) 11] 6 The close-points of the double conic coincide [(11) 21] 4 One node [(21) 2] 4 The close-points coincide, one node [(31) 1] 4 There is a point in which the two close-points coincide with a node [(11) 3] 3 A biplanar point of the first species [(41)] 3 A singular point of coincidence of the close-points with a biplanar point Surfaces with two double lines (meeting in a point which is not a triple point). [11111] 12 General case [1211] 10 One node [131] 9 A biplanar point of the first species [122] 8 Two nodes ; the line joining them belongs to the surface [14] 8 A biplanar point of the second species [1 (11) 11] 8 Two nodes [1 (21) 1] 8 A biplanar point of the second species [1 (11) 2] 6 Three nodes [1 (31)] 6 A uniplanar point of the first species [I (11) (11)] 4 Two pairs of nodes [T (22)] 4 Ruled surface with three double lines Surfaces with a double line and a cuspidal line. [122] 8 General case [14] 8 The close-points coincide [1(11)2] 6 One node [I (31)] 6 The preceding node lies on the cuspidal line [1(1 1) (11)] 4 Two nodes [T (22)] 4 Ruled surface with two double lines and a cuspidal generator [1 (22)] 4 Ruled surface with two coincident directrices and a double generator 62 84 QUARTIC SURFACES WITH A NODAL CONIC [CH. IV Surfaces with two cuspidal lines. Class of the Index surface Character of the surface [1J11) 2] 6 General case _[1 (31)] 6 Particular case [1 (11) (11)] 4 One node Surfaces with a triple point through which two double lines pass. [2111] 10 General case ; the tangent cone at the triple point consists of the plane of the double lines and a quadric cone [311] 9 The triple point is triplanar [221] 8 One node [41] 8 The triple point is a special triplanar point [23] 7 A biplanar point of the first species [32] 7 One node ; the triple point is triplanar [5] 7 The triple point is a special triplanar point [2_(11) 1] 6 Two nodes [2 (21)] 6 A biplanar point of the second species [3(11)] 5 The triple point is triplanar; there are two nodes Surfaces with a triple point through which there pass a double line and a cuspidal line, or two cuspidal lines. [32] 7 One double and one cuspidal line [5] 7 The close-point coincides with the triple point [3 (11)] 5 One node [3 (11)] 5 Two cuspidal lines Steiner's surface. 3 General case 3 Two of the double lines coincide 3 The three double lines coincide Surfaces with a bidouble line (containing two triple points distinct or coincident). [(IT) 111] 8 General case ; the tangent cone at each triple point breaks up into a plane and a quadric cone [(2l) 11] 8 The triple points coincide in a triplanar point [(IT) 21] 6 One node [(2l) 2] 6 The triple points coincide ; one node [(11) 3] 5 A biplanar point of the first species [(41)] 5 The double nodal plane of the triple point of the last case but one touches along a simple line [(IT) (11) 1] 4 Two nodes [(II) (21)] 4 A biplanar point of the first species [(2l) (11)] 4 The two triple points coincide; two nodes 53] AND ALSO ISOLATED NODES 85 Surfaces with a cuspidal line of the second species. Class of the Index surface Character of the surface [(21) 2] 6 General case ; the cuspidal line contains a triple point and a point of osculation of the two sheets [(41)] 5 The points just mentioned coincide in a triple triplanar point [(21) (11)] 4 One node Ruled surfaces with a triple line. [(22) 1] 4 General case (class III of Cremona) [(32)] 4 Ruled surface (special case of class X of Cremona) CHAPTER V THE CYCLIDE 54. When the double conic is the section of a sphere by the plane at infinity we obtain the surface known as the cy elide*. The equation of a cyclide in Cartesian coordinates is therefore S* + u = ; where $ = represents a sphere, and u = is a quadric. Taking the centre of S as the origin and the axes in the directions of the principal axes of u, we obtain as the equation of the surface (a? + y* + z*J + 4 (A^a? + A 2 y* + A 3 z* + 2^ + 2B 2 y + 2B 3 z As in Chapter III we may write this equation in the form O 2 + t/ 2 + z* - 2A,) 2 + 4 {(A, + X) # 2 + (A, + X) y* + (A, + X) * 2 + 25^ + 2B 2 y + 2B 3 z + C - \ 2 } = 0. The second member of the left side will be a cone, F= 0, if its discri- minant is zero : this condition may be written in either of the forms We thus obtain five values for X, giving five cones. If one such cone V be XT Z 2 =0, where L=Q is any plane through its vertex, the equation of the surface is (^ + 2/ 2 + ^-2X) 2 + 4(ZF-Z 2 )=0 ............ (1). As before (Chapter in) any tangent plane of the cone meets the surface in two circles, and every circle on the surface lies in a tangent plane to one of the five cones. * For an extensive discussion of this surface see the work by Darboux entitled Sur une classe remarquable de courbes et de surfaces quelconquet. The intrinsic interest of this surface justifies a special discussion by use of Cartesian coordinates, showing the various real forms of the surface. 54] THE CYCLIDE 87 Again the sphere t? = 2(L + \) .................. (2) meets the surface in a pair of circles lying on X = 0, Y=Q re- spectively ; the points of intersection of these circles being points of contact of the sphere and surface. Hence the surface is the envelope of these bitangent spheres. Moreover every bitangent sphere must arise in this manner ; for if a? + y 2 + z 2 = 2 (X + M ) be a bitangent sphere it will meet the surface in a pair of circles and we may take the plane of one of them to be X = 0, whence M= (L + kX\ i.e. the surface may be written in the form (^ + y * + z * _ 2X)o + 4 (XY' - M' 2 ) = 0. If L = OLX + fiy + yz + 8, the condition that L = passes through the vertex of V gives and this is the condition that this bitangent sphere should cut orthogonally the sphere whose equation is 25 ar 2B 2 ZB 3 z < + 2X=0...(3). Again since (A, + X) x 2 + (A z + X) ,v 2 + (A 3 + X) considering only terms of the second degree it follows that (A l + X) a? + (A z + X) y 1 + (A, + X) z 2 + (ax + @y + 72 must break up into linear factors, hence ff 7 2 ,i_ * ~ + + Hence the cyclide may be generated in five ways as the envelope of a sphere whose centre lies on one of five fixed quadrics Qi...Q s and which cuts a fixed sphere^ orthogonally. The quadrics Q t are seen to be confocal. At each point of intersection of a quadric Q; with the corresponding sphere Si we have a bitangent sphere of zero radius ; its centre is therefore a focus of the surface ; hence arise five focal curves. * The cone F is the reciprocal of the asymptotic coce of this quadric. t This sphere is one of the quadrics H of Art. 31 ; its centre is the vertex of the cone F. 88 THE CYCLIDE [CH. V 55. The five spheres Sj, . . . S 5 are mutually orthogonal ; for the condition that any two of them, corresponding say to X z and X,, should be orthogonal is (A, + XOCZI+A5 (A 2 + Z ^A/i Zt\in ^ \J . which follows at once from the equation Consideration of the equation F (X) = shows that it has in general at least three real roots; since, taking A l} A 2 , A 3 as in ascending order of magnitude, there lie an odd number of roots in each of the three intervals oo... A lt A 1 ...A 2 , A^...A 3 . Hence there are in general at least three real pairs Si, Qi and there may be five. Important relationships between the spheres Si and the quadrics Qi are the following : the centres of any four of the spheres form a self-polar tetrahedron for the remaining sphere and for its corresponding quadric. For expressing that the spheres corresponding to \ and X 3 are orthogonal we obtain an equation similar to the last ; subtraction, and division by Xj X 3 gives us , + 1 = 0, which is the condition that the centre of the sphere S 3 should lie in the plane B 3 z (A, + \0 (A, + X 2 ) (A 9 + X x ) (A 9 + X,,) (A, + XO (A, + X 2 ) But the last equation represents the polar plane of the centre of S 2 with regard to Q t . Similarly this plane passes through the centres of S 4 and $ 6 ; and the centres of S 2 ... S 5 form a self-polar tetrahedron with regard to Q T . Again representing any one of the spheres Si by the equation x* + y* + z* + 2/> + tyy + 2h { z + d = 0, 55, 56] THE CYCLIDE 89 we derive, from the fact that the spheres are mutually orthogonal, the equations A/a + 9i92 + Ma = ^- hence / 2 (/ -/ 5 ) + # 2 (g l - g s ) + h 2 (h, - h 5 ) = ^- 5 , which is the condition that the polar plane of the centre of $ 5 for ,, i.e. -A ( x +/0 -ffid/ + ffi) -h 5 (z + AO +A + &y + M + wherein w = 1, enables us to solve for # 2 + y 1 + z 2 , xw, yw, zw, w 2 in terms of 8 l ...S s ; this gives rise to a quadratic identity between the quantities $1 . . . $ 8 . These five quantities may be employed as coordinates to determine the position of a point, a homogeneous quadratic relationship existing between the coordinates. These coordinates are known as the pentaspherical coordinates of a point. The nature of this quadratic relationship can be most readily determined from considerations relating to the mutual power of two spheres. If two spheres of radii r 1} r 2 cut one another at an angle 6, we have 2r,r 2 cos = r x 2 + r 2 2 - (/ -/ 2 ) 2 - (g, - 2 ) 2 - (^ - A 2 ) 2 The right-hand side of these equations is real for real spheres whether their intersection be real or otherwise ; taken negatively it is known as the mutual power, 7T 12 , of the two spheres, thus Forming the product of the two determinants* 1 2/ 20, 2Ai c 1 2/ a 20 2 2^j c 1 2/3 203 2/* 3 c 1 2/ 4 20 4 2// 4 c 1 2/ 5 20 6 2/? 5 c 1 2/ 6 20 6 2A 6 c ^7 ~ ft 9^ ~ h? 1 rv / 1 ~t C\ /> /* n 7i 1 c 10 /io 7io ^10 1 Or _/ _/7 _/, 1 \J t/^j / jj //H ""11 -*- V Cij y 12 *7l2 "-12 A Lachlan, On systems of circles and spheres, Roy. Soc. Trans. (1886). 59, 60] we obtain THE CYCLIDE )8 7T 1)9 7Tj >10 7T I|U , 8 ^2,9 7T2.10 ""a, 11 95 12 T6.12 , 6' = 0. /I L<\ This equation may be denoted by TT ( ,_ " ' 9 j = 0. Now, denoting the spheres $ : and S 7 by x and t/, and supposing that the spheres S 2 . . . S 6 are respectively identical with the spheres >S^ 8 ... S 12 , we have on slightly altering the notation 'x 1 ... 5> 7T y which, expanded, is equivalent to 7T,, = .(1). If we now suppose 8 x ~S y , we obtain the relationship existing between the powers of any sphere S x with regard to five fixed spheres, viz. -2r s -2r 4 2 - 2r, If this equation is such that TT XI occurs only in the term involving ir xl 2 , the sphere $j cuts S 2 . . . S 5 orthogonally ; for the coefficients of r ^'xi' 7r x2, '- ^xi^xs a ll involve 7r 12 , ... 7r 15 linearly and homogeneously, hence if they all vanish we have either or Ti2 == 7^13 == ""14 == TIB == 0, <7r 22 TT23 ^24 ^25 7T23 = 0. TT-X 7T 2B 96 THE CYCLIDE [CH. V But the last condition cannot be fulfilled since the determinant on the left side is equal to the coefficient of TT^, taken negatively, and this is by hypothesis not zero. If the sphere S x is a point-sphere, its powers with respect to the spheres S 1 ... S 5 are obtained by substituting the coordinates of its centre (#, y, z) in the expressions S^ . . . S 5 ; hence we obtain the required identical relation between any five spheres ^ = 0, . . . S s = 0, which is therefore - 2rV 7r 12 _ 9v 2 L.TS 9r 2 ^? 4 s* -2r 5 2 = 0. It follows, as in the case just above, that if Si occurs only in the form 8f, the sphere Si cuts orthogonally the remaining four spheres. If all the quantities ir^ (i =j) vanish, the identical relation becomes and the five spheres are mutually orthogonal. By virtue of this equation, the equation of the sphere S 1 (say) may be written in the form (&-- \^* , 4- 4. + 4- + = 0. This shows that the planes of intersection of S l with $ 2 , $ 3 , 84, and >S^ 5 form a self-polar tetrahedron for S l . Now the radical plane of Si and $ 2 contains the centres of S at S 4 and S 5 , and so on ; hence we again obtain the result of Art. 55 that the centres of any four spheres form a self-polar tetrahedron for the fifth sphere. We observe that if four of the spheres S { , supposed mutually orthogonal, are real, the fifth sphere is also real but the square of its radius is negative. Also we see that on inverting from any point not upon one of the spheres S^ the form of the relationship is not altered, since ft r i 60, 61] THE CYCLIDE 97 But if the centre of inversion be a point of intersection of three spheres Si, S 2 , S 3 , they are inverted into three planes which we may take to be S- coordinate planes, and since ! 1 cos 12 cos< 5 =0. J. Q. s. 98 THE CYCLIDE [CH. V If the spheres Si ... S 6 are mutually orthogonal this reduces to 5 cos >// = 2 cos &i cos 0j. i 5 5 If the two spheres are identical we obtain 2 cos 2 ^ = l; if they are 1 i 5 5 SbiSi=0, and cut orthogonally we have from above 2a i & i r i 2 =0. i i If S is orthogonal to one sphere of the orthogonal system 4 Oi.^.S,, say to $ 5 , the equation of S is 2a;$ f = 0. In this case i the volume of the tetrahedron whose base is the triangle formed by the centres of S 2 , S 3 , S t and whose vertex is the centre of S t is 4 V / 4 4 4 I 2t<7i 1 1 1 J\ gi hi 1 1 A 92 h z 1 0, /2 #2 ^2 1 V* Za 4) f* fft h 3 1 i /3 #3 h 3 I fi 9* h t 1 / 4 #4 A 4 1 Hence if i ... t are the tetrahedral coordinates of the cenl of S with regard to the tetrahedron formed by the centres of S 1 ... $ 4 , we have that ff< = a$ . 5 7T a a . When the sphere 2a i ^ r f =0 is a point-sphere, we have 2 ^- = which is zero since 2cos 2 0i=l and r=0. In this case 2a f 2 r i 2 = 0, so that if i S- 5 x\= , the equation of a point-sphere is 20^=0, with the condition 2^ = 0. 62. Pentaspherical coordinates. It has been seen (Art. 60) that the quantities x z + 2/ 2 + z-, x, y, z and unity which occur in the equation of any cyclide may be replaced by linear functions of $ a ... $ 5 ; the equation of the cyclide then appears as a quadratic in the Si which are themselves connected by a o quadratic identity. The quantities , or x i} are termed the penta- spherical coordinates of a point. If the equation of the surface expressed in pentaspherical coordinates contains only four of the variables, say x l ... x 4 , so that 4 its equation is a = the equation of the cyclide and by O = the identical relation connecting the coordinates, we obtain by this method seven types, viz. [11111], [2111], [311], [221], [41], [32], [5]; each type giving rise to sub-types. It will also be seen that only the first three forms relate to real cyclides. Writing these forms at length, we obtain by the usual method + riinn ~ l 2 * J "(n = ! + x* + x* + x* [21111 J" 2Xl ^ 2 lil = 2x x + J [2211 T411 - 1 -* ft = 2 (^^4 = \ (2a?, x s + 2 2 ) + Zadasa + 2X 4 4 5 + x?, = A, (2a? x a; 5 + 2# 2 # 4 + x/) = Zanies + 2x3X4 + 3 Z - We now pass to consideration of the type [11111]; the form of ft shows that here the coordinate spheres form an orthogonal * For discussion of the method see Bromwich, Quadratic forms and their classification by means of invariant factors (Cambridge Tracts), or the Author's Treatise on the Line Complex. See also Bocher's Potential Theorie. 72 100 THE CYCLIDE [CH. V system: eliminating one of the variables, say C 5 , we obtain an equation of the form xjflf.-.ti ..................... (i). The surface is therefore the envelope of the spheres 4 ^CLiXi = 1 subject to the condition (2). The generating sphere is orthogonal to x 6 = ; and since the i are the coordinates of its centre for the tetrahedron formed by the centres of Si...S 4 (Art. 61), the equation (2) represents the quadric Q & . We obtain similarly the other four sets of generating bitangent spheres. Moreover, assuming the cyclide to be real, and since it was seen (Art. 54) that the only bitangent spheres are those arising from a pair Si , Qi of this cyclide, it follows that the spheres #i = can be no other than the spheres Si which a real cyclide possesses. Hence it follows that of these spheres at least three are real, while two may be either real or conjugate imaginary; so that this applies to the variables a? t - ; and if e.g. # 4 and x 5 are conjugate imaginary, so also are X 4 and X 8 ; if all the Xi are real, so also are the X f . The five focal curves are determined by the equations together with four other similar pairs of equations. It was seen (Art. 57) that if two of the principal spheres are conjugate imaginary three of the focal curves are real and consist of one portion. Consider now the case in which all the principal spheres ^ are real and let #5=0 be that sphere the square of whose radius is negative, so that ai ...a 4 are real and a 6 =a 5 r 5 is imaginary. One of the focal curves is then given by \ T N N ' N -V V J A2 ~ AI A 3 AI A^ ~~ Aj Ag ~~ Aj 5 2 This curve is then real or imaginary according as the cone 9 A2 AS . A.q AS A 4 A * s AI A 2 AI A 3 24 5 =0 does or does not contain real points apart from its vertex. 63, 64] THE CYCLIDE 101 We obtain in this way criteria for the reality or otherwise of four focal curves ; the fifth, which lies upon 20^ = 0, is of course imaginary. To discriminate in the four cases we may suppose the quantities X x ... X 4 to be in algebraic order of magnitude and moreover we have as a condition of reality of the surface that the quantities ^1 ^5j ^2 ^5> ^3~~^5' ^4 ^5 cannot all have the same sign. We may take X 5 equal to unity, in which case AI X 5 must be positive, and then the three possible distributions of signs to X 2 - 1, X 3 1, X 4 - 1 are + + -, +- -, --- Inserting these signs in the equations of the four cones obtained as above it is seen that in all cases two of them are real and two are imaginary. 64. Form of the cyclide. In the case in which the variables are all real, sc s being that principal sphere the square of whose radius is negative, the equation of the surface is If we invert the surface from one point of intersection of the spheres x lt x 2 , x 3 and take as new coordinate planes those into which these spheres are inverted, the equation of the new surface is (Art. 60) or V + (X 4 - X 5 ) S 2 = 0. We may assume X 4 X 5 to be positive ; different forms of the surface will then arise according as one, two, or three of the remaining coefficients are negative. If one of them is negative then every line through the origin and within the cone V meets the surface in four real points ; hence the surface consists of two ovals, one within each portion of V. If two of them are negative, then every line through the origin and without the cone V meets the surface in four real points; the surface is ring-shaped. If all are negative, then every line through the origin meets the surface in four real points; hence 102 THE CYCLIDE [CH. V the surface consists of two ovals, one within the other, and each surrounding the origin. Hence the form of the original surface is also determined in this manner by the signs of the quantities Xj X 5 . When all the variables are real, the inverted surface is seen to be derived from the general cyclide by taking B 1} B 2 and B 3 all zero and C positive (Art. 54). This is one form of cyclide with three planes of symmetry. The other form in which C is negative corresponds to the case when two of the variables are imaginary ; for in this case we have in which we may take a?i = & + *'&, #2 = fi - i%* ; substituting these values for x l and # 2 the identical relation assumes the form 2(fc+^-&)+i*f-0. 3 Hence + 2 and 2 are point-spheres. If we make the same substitution in the equation of the cyclide, it assumes the form (in which only real quantities occur), that is Inverting from an intersection of the spheres the equation of the inverse surface is seen to be of the form (ar 2 + y> + * 2 ) 2 + (X, + *) of + (X 4 + ) f + (X 5 + /c) 2 2 - m 2 = 0. In this case every line through the origin meets the surface in two real and in two imaginary points. 65. The type [2111]. The equations determining the second type show that it represents a cyclide which can be generated in four ways ; viz. in three ways by bitangent spheres orthogonal to three given spheres respectively, and once by a sphere passing through a given point, which is one of the intersections of the spheres x 3 , # 4 and x s . Two of the principal spheres, S l and $ 2 of the general case, here 64-66] THE CYCLIDE 103 coine into coincidence with the point-sphere x l . It follows from Art. 59 that the principal spheres are all real. That this is a degenerate case of the general case may be seen as follows : let us change the notation and write 5 5 i i and let X 2 = X 1 + e, x 2 =Xi + fX.2, where e is small. 5 Then Q If we now assume that i + 2 = 0, a 2 e = l, 0,3=0,4=0,5 !, we obtain the second type. See Bocher, Potential Theorie. The surface has a node, the centre of the point-sphere x^. If we invert the surface from this node, we obtain the quadric (Xs -\)af+ (X 4 - XO t/ 2 + (X B - X,) z 2 + h 2 = 0. Hence, if the node is isolated the surface is the inverse of an ellipsoid; otherwise it will be ring-shaped if it is the inverse of a hyperboloid of one sheet ; it will consist of two sheets united at the node if it is the inverse of a hyperboloid of two sheets. That the cyclide is the inverse of a quadric when one of the principal spheres reduces to a point, may also be seen as follows : if Q is the quadric associated with the point-sphere 0, the surface is the envelope of spheres passing through and having their centres on Q ; all the spheres whose centres are consecutive to any point P of Q will pass through the point 0' which is the image of for the tangent plane of Q at P. Hence the surface is similar to the pedal surface of Q for 0, and is therefore similar to the inverse of the reciprocal polar of Q for 0. 66. The type [311]. The equations connected with this type of cyclide show that it is generated in three ways ; twice as the envelope of a sphere cutting orthogonally two given spheres respectively ; and once as the envelope of a sphere passing through a given point; the spheres x z , x 3 of the general case come into coincidence with the point-sphere x 1 . The equation of the surface being (A 4 - X x ) x? + (X 5 - \) x s 2 + 2.r 1 # 2 = 0, 104 THE CYCLIDE [CH. V the centre of the point-sphere x^ is seen to be a node, and since the spheres x z , x 4 , x 6 pass through this point and cut orthogonally, the tangent cone of the cyclide at the point consists of two planes. Inverting the surface from the node, we obtain as the inverse surface, the paraboloid (X,- X,)* 2 + (X, - Xj) 7/ 2 + 2kz = ; which is elliptic or hyperbolic according as the biplanar node has imaginary or real tangent planes. The four remaining types give rise to cyclides which are imaginary ; for the quintic F (X) = may have either one double root or one triple root, but no other coincidence of roots can occur in a real cyclide (Art. 59)*. 67. The sub-type [(11)111]. The sub-types arising from the above three chief types, as for instance [(11) 111], are such that the equation of the surface can be expressed in terms of only three variables ; thus [(11) 111] has an equation of the form Ax? + Bx? + Cx? = 0. The common characteristic of all the sub-types is that one of the five cones V should be a pair of planes, real or imaginary. For, if in the equation (0&&,&,'4P<1 we substitute $ 2 = a 2 8 l + a, S 3 = a 3 $i + /3, where a and ft are linear in the coordinates, we obtain an equation of the form (flfi + 7) 1 = (&$,); 7 being linear in the variables. The surface [(11) 111] has two nodes which may be either real or imaginary. If the nodes are real, on inverting the surface from one of them we obtain a quadric cone. The cyclide Ax s 2 + Bx? + Cx? is the envelope of the spheres 3 # 3 + a 4 # 4 + a s x 6 = 0, subject to the condition n 2 2 n 2 C i + i + 0( i = o m A + B + C The contact of these spheres and their envelope occurs along a circle instead of at two points. * The cyclide S 2 = a/3 (Art. 59) is always expressible in the form (S l , S 2 , S 3 $a) 2 =0, where S 1 , S 2 , S 3 are three mutually orthogonal spheres ; hence it cannot belong to one of the types [221], [41], [32], [5]. 66, 67] THE CYCLIDE 105 These spheres cut both x l and x 2 orthogonally, hence they pass through the two limiting points of x 1} ac z , so that their centres lie in a plane ; since they also lie on the cone (1) they lie on a conic. This surface is therefore the envelope of a sphere which passes through a faced point and whose centre lies on a conic. Two systems of bitangent spheres coincide with this system, the other three, which may be called the proper systems, remaining as before. They are obtained by writing the equation of the cyclide in the form (B - A) x? + (G-A)x*-A (x? + xf) = 0, showing that the surface is the envelope of the spheres subject to the condition a 2 a 2 a 2 a 2 ~J + ~A T> A 7 J = " (*/ Now in the preceding equation (1) we may assume 3 2 + 4 2 + 5 2 = 1, hence it is equivalent to ,B-A G-A 4 2 - + 5 2 -=1 .................. (3). Also in equation (2) we may assume that ttj 2 + a 2 2 + 4 2 + 5 2 = 1 ; hence it is equivalent to 7? C* These equations (3) and (4) hold respectively for generating spheres of the special system and a proper system. Now take two fixed generating spheres of the special system whose coordinates are (0, 0, v 1} z lt w x ), (0, 0, v 2 , z 2 , w 2 ), and a variable sphere of the system to which (4) relates whose coordinates are (x, y, 0, z, w); then if 1; < 2 respectively are the angles at which the variable sphere cuts the fixed spheres, we have (Art. 61) COS <>x = ZZi -f WW l B B-A / G G-A ~~ /< '-A 1 V G cos d> 2 zz 2 + ww. 2 I v G-A C 106 THE CYCLIDE [CH. V Hence by virtue of equations (3) and (4) we may take fa and fa to be the angles which a variable line makes with two fixed lines in its plane ; hence fa fa = constant. Therefore the sum or difference of the angles which a variable sphere of one of the three proper systems makes with two fixed spheres of the special system is a constant. The corresponding result for the general cyclide is the following: the angles which any generating sphere of one set makes with three fixed generating spheres of another set, are equal to the angles which a variable line makes with three fixed lines. 68. Dupin's cyclide. The surface [(11) (11)1] is known as Dupin's cyclide', its equation takes either of the forms (\ x - X,) (x* + xj) + (X 8 - X,) x 5 * = 0, (X, - X x ) (# 3 2 + * 4 2 ) + (X. - XO x? = 0. It has four nodes, of which at least two are conjugate imaginary, since at least two lie upon that sphere the square of whose radius is negative. Inverting from a node (supposed real) we obtain a cone of revolution. The spheres which touch the surface along circles form two systems, one of the systems is given by the equations 5 /2_l_/y2 n 2 0; 3 hence a B is constant and equal to * r^ , so that these spheres V A. 5 A, 3 cut the sphere a? 5 at a constant angle. If they lie within a? 5 , a 6 is positive and greater than unity, say equal to sec ft ; the spheres will therefore touch each of the fixed spheres (sin ft, 0, 0, 0, cos ft), (0, sin ft, 0, 0, cos ft). Hence Dupin's cyclide is the envelope of spheres having their centres on a fixed plane and touching each of two fixed spheres. The fixed spheres are not unique, since they are any two of the singly infinite set (A 1} A 2 , 0, 0, cos ft) where A^ + A^ = sin 2 ft; whose centres lie in the radical plane of x 3 and x 4 . We obtain the same cyclide as the envelope of spheres cutting Xi and x t orthogonally ; they have their centres on a second conic 67-69] THE CYCLIDE 107 whose plane is perpendicular to the line joining the centres of # 3 and x 4 . Since the join of the centres of x l and x 2 is perpen- dicular to the join of the centres of x a and x 4 these two conies lie in perpendicular planes. These spheres form the previous set (A 1} A 2 , 0, 0, cos /3) ; for they are a. l x l + a 2 x 2 + a 5 x 5 = 0, with the condition ~ - l = 8 2 tan 2 /9. This cyclide was originally defined as the envelope of a sphere touching three fixed spheres, but such spheres form four distinct sets, each set enveloping a cyclide. The equation of the tore or anchor-ring is {a? + y 2 + z* + c 2 - a 2 } 2 = 4c 2 (# 2 + f), where c is the distance of the centre of the revolving circle from the axis of revolution, and a its radius ; inverting from any point we obtain # 5 2 + A 3 2 + xf) = 0. If c is greater than a, then x 5 is a sphere the square of whose radius is negative and the cyclide is a Dupin's cyclide with no real nodes; if c is less than a we then obtain a Dupin's cyclide two of whose nodes are real. 69. We add a list of the various distinct real types of cyclide ; the remaining forms consist merely of pairs of spheres, etc. Inverse surface Nodes [11111] General cyclide : Surface either has Cyclide with three two sheets or is planes of symmetry ring-shaped (constant term posi- tive) Surface has one Cyclide with three sheet planes of symmetry (constant term ne- gative) [(11) 111] Cone 2 [(11) (11) 1] Cone of revolution 4 (two imaginary) [2111] Ellipsoid 1 (isolated) Hyperboloid of one 1 sheet Hyperboloid of two 1 sheets [2 (11) 1] Ellipsoid or hyperbo- 3 (two imaginary) loid of revolution 108 THE CYCLIDE [CH. V Inverse surface Nodes [(21) 11] Elliptic or hyperbolic cylinder 1 [(21) (11)] Circular cylinder 3 (two imaginary) [311] Paraboloid 1 (biplanar) [3(11)] Paraboloid of revolution 3 (two imaginary) [(31) 1] Parabolic cylinder 1 (uniplanar) 70. Tangent spheres of the cyclide. 5 The equation S (c^ + X) x^ = 0, where the xi are the coordi- i 5 nates of a point of the cyclide 2 a;#i 2 = 0, represents the oo 1 tangent i spheres of the cyclide at the point #;. Hence* if 2 m^yt = is a tangent sphere of the cyclide, we have o 2 m * = Eliminating X between these equations, we have the relation fulfilled by the coordinates m t - of a tangent sphere of the cyclide. m - 2 It arises by expressing that the equation 2 l = should have a double root. The equation being of the fourth degree in X its discriminant will be of the twelfth order in the m^. Let yjr (m) = represent this equation, m^ + pra/ represents the oo 1 spheres passing through the intersection of any two spheres mi and m^, and we obtain those spheres which touch the cyclide by means of the equation fy (m + pm') = 0, which is of the twelfth degree in p. Hence through any given circle twelve spheres may be drawn to touch the cyclide. If the spheres m and m' are concentric, and the equation i/r (m + pm') = gives the twelve spheres having the given point as centre, which can be drawn to touch the cyclide, i.e. twelve normals can be drawn from any given point to a cyclide. * See Darboux, Sur une classe remarquable, etc., p. 275. S- t Since S J = constant. rt 69-71] THE CYCLIDE 109 Bitangent spheres. 5 If in the spheres ^ l (a i + \)a;iy i = we take \ to be suc- i cessively a 1 ...a s ,we obtain the five bitangent spheres which touch the cyclide at the point x ; e.g. if X + a^ = 0, the corre- sponding sphere touches the cyclide in the two points x 1 ,x 2 ...x s . 5 The Cartesian coordinates of the centre of the sphere S m^ = i are clearly equal to the expressions v m i V m tfi /S* m i V m ihi / V i . l J '-' I ** > ** / ** 5 where the point ( fi, ffi, hi) is the centre of the fundamental sphere S{. Hence the coordinates of the centres of the set of spheres m t - + \w/ are each of the form -g ~ , i.e. the cross-ratio of any M + AO four of these points is equal to the cross-ratio of the corresponding values of X. Applying this to the spheres T (a t - + X.) Xiyi= 0, and taking \ to be successively , a l ...a s , the corresponding centres are the point x and five points in which the normal to the cyclide at the point x meets the five fundamental quadrics Qi ... Q s ; it follows that the cross-ratio of any four of the following six points on the normal at any point P of a cyclide is constant, viz. the point P and the centres of the bitangent spheres which lie on the normal at P. 71. Confocal cyclides. If the bitangent sphere is also a point-sphere #, its centre is a focus of the surface. Taking one system we have for instance z l = 0, Zi = (a t - cO x i} (i = 2, ... 5), i with the condition 2 z? = 0. i 5 z The equations z 1 = 0, i - = 0, give the focal curve. 2 &i &i If - is substituted for a; in the last equation, its form is not thereby altered, since a f a l is transformed into 1 1 Oi f tti+ fJi ! + /A (Of + 110 THE CYCLIDE [CH. V Hence the equation becomes i^+^Mf.o, 2 di Oi which leads to the original equation X Hence the cyclides 2 - = are confocal with the original cyclide 2 aixf = 0. They form therefore a confocal system in which the original cyclide is included as corresponding to the value infinity for X. Through any point there pass three confocal cyclides, since the 5 x $. equation 2 ^r = 0, regarding the ar t - as given, constitutes a cubic i o>i + A, 5 in A, (since 2 x? = 0). i These cyclides cut each other orthogonally, for if \ lt X 2 refer to two cyclides through the point x, then since tangent spheres at this point to them respectively are ^ Xjyt _ Q y Xjyj _ _ "* if these spheres are orthogonal we have 2 T r-r = 0, (Art. 61). But this is merely another form of the equation ff 2 rr2 2 - 2 * = 0. The three cyclides through any real point are all real ; for the variables x? may be all real, in which case the square of one of them, say x, is negative, so that if we suppose the quantities i ... a t in order of magnitude, the cubic determining X has a root in each of the intervals a 1 . . . a 2 , a z ... a 3 , a s . . . a 4 . Again, if x l and x z , and consequently a 1 and 2 , are conjugate imaginary, the cubic has a real root in each of the intervals a 3 . . . a 4 , a 4 . . . a s , and therefore possesses three real roots. 71] THE CYCLIDE 111 Corresponding points on confocal cyclides. The equations < = !, ...5), establish a (1, 1) correspondence between the point x on the 7/- 2 cyclide S a^- 2 = and the point y on the cyclide 2 = 0. di -f* A, 5 Denoting by /(X) the product II (X + a^), by resolving into partial fractions the expressions X 3 X 2 XI /(X)' /(X)' /(X)' it is seen that 2(^i -s* Q>i ^ Q>i ^> ,-. // , c & _// / \ ^* // / r ^ // , c v. /(-<) / ( - f) / ( - Oi) f (- at) Hence the equations S a^i 2 = 0, 2, acf = Q, are identically satis- fied by the substitutions f(-*i} These equations express the coordinates of any point x of the cyclide in terms of two parameters Xj and X 2 , so that if we take Xi = it follows that The quantities X 1} etc. are seen to be the roots of the cubic in X giving the three confocals through the point y. The above expressions for the y { in terms of \ lt X 2 , X 3 may be directly obtained by considering the cubic in the form -/2 6 11 .2 and hence it follows that four other equations of like form are obtained similarly. 112 THE CYCLIDE [CH. V The curves \ = constant, X^ = constant are the lines of curva- ture on the surface S o^; 2 = 0, for from the equation it follows that if #; + dx t be the consecutive point on the curve X, = constant, we have hence + dxi) = (di + Xj) Xi ( 1 - ' '-j , neglecting quantities of the second order; therefore a tangent sphere at asi, viz. 2 (a; + Xi)#;;z/i = 0, is also one of the tangent spheres at the consecutive point on the curve X 2 = constant, and is there- fore a principal sphere at the point x. Thus the two confocals through any point of the surface &iX? = intersect it in its lines of curvature ; which is other- wise manifest from Dupin's theorem. 72. The sixteen lines of the surface. It is known that every general quartic surface with a double conic contains sixteen lines (Art. 24). The existence of these lines on the cyclide is made evident by the equations /( V ,., /'(-en) For if we suppose X! = X 2 , we obtain for any point sc of the curve Xj = Xa the equations pxi = At\! + B it (i=l, 5) ; whence if , 77, are the corresponding Cartesian coordinates of the point x, fcgl *! + = ' ** hence the curve is a straight line. By taking all combinations of signs in the ambiguities in equations (1) we obtain the sixteen lines. These lines are all imaginary, since as in the general case of the quartic surface with a nodal conic, a line on the surface must form part of a conic on 71-73] THE CYCLIDE 113 the surface, and in the case of the cyclide all such conies are circles. 73. Centre of a cyclide. The locus of mean distances of points of intersection of a series of parallel chords with an algebraic curve of degree n is called a diameter ; when the terms of degree n 1 in the equation of the curve f(x, y} = are lacking, all diameters pass through the origin, the centre of the curve. The equation of a cyclide being written in the form 2 + 2/ 2 + s 2 ) 2 + 4t/' = 0, we proceed to consider the centres of its sections. Since the coordinate planes may have any directions, we may consider the section of the surface by the plane z = h\ it is seen that the diameter corresponding to chords of the section parallel to the axis of x is the axis of y, and vice-versa. Hence the line x = y = Q is the locus of centres of sections parallel to the plane z = 0, so that the locus of centres of sections parallel to any plane is a line through the origin perpendicular to that plane. The origin is therefore termed the centre of the surface. Sphero-conics on a cyclide*. The sphere S=2L (where L = ax + (By + yz + 8, S = x* + y 2 + z s ) meets the cyclide S 2 + 4s U= in a curve given as the intersection of the sphere and the quadric U + L 2 ; it is therefore a sphero- conic a- ; the centre of the sphere is termed the centre of X ~~ Xj X *~ X 4 But since S & is included, for X = X s , it follows that K I : K. 2 '. KS : K 4 = X 5 \i : X 8 X 2 : X 5 X^ : X 5 X 4 , r . whence we finally obtain 2 l = 0. - i X X t - The following result is given by Humbert* ; when the sphere S + 2L = is a point-sphere, and the quadric U + L 2 = is a cone, the locus of the centre of this point-sphere is a cyclide confocal with Let S = x 2 + f + z* + d, 17= a^a? + a 2 y* + a s z 2 + 2px + 2qy + 2rz + c, L = ax + $y + yz + 8. Then S + 2L = is a point-sphere if 2 +/9 2 + 7 a -d-28 = 0; the quadric U + L- = is a cone if a 2 A n + 2a/9A ]2 + ... + A = 0, where A is the discriminant of U, and A n , etc. its first minors. * Sur les surface* cyclides, Journal de l'6cole polykchnique, LIV. (1884). 75, 76] THE CYCLIDE 117 Now denoting by (x, y, z) the centre of the point-sphere so that x a, etc. the last equation becomes On inserting the value of S we obtain as the required locus 2qy Zrz 7 \ -iZ- + --- d) 2 3 / 2qy Zrz 7 \ 2 4A -iZ- --- - - - - = 0. On writing * = A t + \ p = B ly q = B 2 , r = B 3 , d = 2\, c = C- X 2 , the cyclide S 2 + 4 U takes the form given (Art. 54) and the locus is seen to be a confocal cyclide. 76. Common tangent planes of the cyclide and a tangent quadric. If we take any plane touching the cyclide at a point and an inscribed quadric F at a point P, the line PO touches a line of curvature of the cyclide at 0. For if the plane be taken as the plane 2 = and the line PO as the axis of x, the equation of the surface assumes the form (x 2 + y 2 + z z + 2ax + 2by + Icz + A;) 2 + 4 (AX* + By 2 + Cz 2 + 2Dyz + ZEzx + ZFxy kax kby + Zrz -j- J = 0, with the conditions F + ab = 0, A + a 2 = ; the second member of the left side representing V. But in the equation of the indicatrix of the surface at the coefficient of the term involving xy is F + ab, hence the line PO is a tangent to a line of curvature at 0. The tangent to the other line of curvature is OQ, where Q is the point of contact of the other inscribed quadric V which can be drawn to touch the plane (the two other inscribed quadrics which touch the plane coincide here with that which touches the cyclide at taken doubly). Thus the locus of the points of contact with a cyclide of a plane which touches the cyclide and a fixed quadric V, is a line of curvature of the cyclide, the intersection with it of a confocal cyclide. CHAPTER VI SURFACES WITH A DOUBLE LINE : PLUCKEE/S SURFACE 77. The equation of a quartic surface with a double line may be written in the form x;, A 3 A 4 ): hence we have finally three planes through p' meeting pi...p s , A 3 A 4 in points of a conic. Hence the planes meeting p, p t . . . p 5 in points of a conic envelop a surface of the third class. This surface contains each of the lines p x . . . p 5 ; for the plane through p 1 and the second transversal of p 1} p. 2 , p s , p t , meets the lines p, Pi-..p 5 in six points lying on two lines. Similarly for p 1 and the lines p 3 ,p4,p 5 , etc.; hence four tangent planes of the surface can be drawn through p 1} i.e. p lies on the surface; similarly for p 2 ... p s . For the same reason p lies on the surface. Now consider the three surfaces thus formed with P, PI, Pz, PS, Pt and p 5 , p e , p 7 respectively; applying the known results* for the intersections of three cubic surfaces which have four lines in common, it is seen that there is one tangent plane common to the three surfaces. Construct, therefore, the conic which meets the double line and also one line of each pair out of seven pairs of lines ; this conic meets the surface in nine points and therefore lies upon it. The plane of this conic meets the surface in another conic, the two conies have one intersection upon the double line ; each conic meets one of the two lines forming the eighth pair of lines. By taking all possible selections of seven lines in accordance with the foregoing method, we obtain 2 7 = 128 conies lying in 64 planes; each plane being a tritangent plane of the surface. * Salmon, Geom. of three dimensions, 5th Ed., Vol. i. p. 371. 120 SURFACES WITH A DOUBLE LINE [CH. VI 79. Mapping of the surface on a plane. Any one of the foregoing conies affords a means of representing the points of the surface upon a plane*. For if (7=0 is such a tritangent plane, the equation of the surface may be written (xj A - x 2 B) faA' - x 2 B') - C [Px* - 2Nx,x 2 + Mx^} = 0, where A, B, ... M are linear functions of the coordinates. We may therefore express any point of the surface in terms of two parameters, viz. the ratios of g l} f 2 , ,, as follows: (1) fl#l + f 2 #2 = 0, (2) &B + M+&0-0, (3) & (B& + A'& + Mtf + 2N&& + P& = ; giving a (1, 1) correspondence between any point x of the surface and a point f of any assumed plane. For any assigned point , the first two equations give a line which intersects the double line and also the conic 0=0, x l A-x 2 B = Q, its fourth point of intersection with the surface being the point x which corresponds to f. Conversely each point x of the surface determines such a line and hence one point . For any point x, however, of one of the eight lines of the surface which intersect this conic, the same point ( is determined ; we have therefore eight principal points of the correspondence which we denote by B^ ... B 8 , each of them corresponds to all the points of one of the eight lines. If in the preceding equations connecting x and we have (7 = 0, then either x 1 A-x 2 B = 0, or ^=0, 2 = 0; hence to the points of the conic in the plane (7 = which does not meet the line determined by (1) and (2), there corresponds the single point gi 0, 2 = 0, which we denote by A. To any plane section of the surface there corresponds in the field of a quartic curve : since this section meets each of the above eight lines, and also twice meets the conic just referred to, it follows that this quartic curve passes through the eight principal points and also passes twice through the point A. Hence we have a system of quartics having a common node and eight common points f. * Clebsch, Ueber die Abbildung algeb. Fllichen, Math. Ann. i. t The condition of possessing a node at a given point and eight fixed points is equivalent to eleven conditions, leaving a linear system triply infinite of quartic 79, 80] SURFACES WITH A DOUBLE LINE 121 To the conic (7 = 0, f l x l + 2 x 2 = 0, ^B + %. 2 A = there corre- sponds a quartic curve obtained by substituting for the coordinates x their values in (3), in terms of x : 2 ; hence this quartic possesses a triple point at A, The pencil of lines through the point A corresponds to the conies in sections through the double line; the cubic curve through the nine principal points corresponds to the double line. This cubic is obtained by writing ^ = a; 2 = in (2) and (3), and hence is given as the intersection of a pencil of lines x 3 K + # 4 K' = 0, and a pencil of conies # 3 U + # 4 V = ; to any given point # 3 /# 4 of the double line correspond two points P, P of this cubic collinear with the point -=J5T'=0,orQ. Writing a^ = # a = in (2) and (3) and eliminating j~ 3 we obtain (ftJBo + Mo) (,' + M.') - C (Po& + 2JVO&& + -M.& 1 ) = 0, in which B is the result of writing x l = a 4 b t = c 4 d 4 , giving two torsal lines as x i = x l a 3 x.J) z = 0, x 3 = x^ 4 x. 2 b 4 = ; the other two being 7 = = , 8 = = The above form of equation of the surface shows that the plane x 1 = touches the conies lying in sections through the double line. Hence each trope touches each of these conies. By aid of this form of the equation of the surface we can obtain the form assumed in the case of Pliicker's surface by the previous equations connecting a point x of the surface with 82, 83] FLICKER'S SURFACE 129 a point f of a plane ; for the equation of the surface is seen to be identically satisfied if we write where A, B, C, D are the results of substituting and f 2 for # 3 and # 4 respectively in a, /3, 7 and 8. The cubic corresponding to the double line is it touches the lines (7=0, D = 0, and by aid of the preceding conditions it follows that it also touches the lines = 0, 2 = 0. The quartic representing any plane section has a node at the point = 2 = 0, and touches the lines = 0, 2 = in given points. It also touches the lines (7=0, D = where they respectively meet 3 = 0. If E 1 E Z E 3 is the triangle of refer- ence in the field of and the principal points on E 3 E 2 , E 2 E l are Q! and Q 2 , those on ^^ 2 are Q 3 and Q 3 ', the correspondence between points x and f is of the (1, 1) character with the following exceptions: any point on E 3 E 2 corresponds to the node A 4 with the exception of Q 1} which corresponds to a torsal line, and the point E 3 which corresponds to any point of the conic in the trope #j; similarly for E r E 3 and the point Q 2 ] while to any point of C = 0, D = correspond the other two nodes in x respectively, but to Q 3 and Q 3 the other two torsal lines respectively correspond. The equation M = 4>m - 2/3 - 2cc as before connects the order of a curve on the surface and the corresponding curve in the plane, where ft is the number of intersections of the former curve with the conic in the plane #j exclusive of intersections at nodes in that plane. It is easily seen from this equation that no line can exist on the surface except the four torsal lines, which are represented by the four principal points Q, and the double line. To obtain the conies of the surface we may take /3 = giving m 1, 2 = 2 ; so that six conies are represented by the lines joining the principal points Q lf Q 2 , Q 3 , Q 3 ; or we may take ft = 1, giving either j. Q. s. 9 130 FLICKER'S SURFACE [CH. vi ra = 1, 2 = 0, so that the conies in planes through the double line are represented by the pencil of lines through E 3 ; or m = 2, S = 4, which gives the conic through the five principal points. If we add the conic represented by E 3 we obtain all the conies on the surface. The existence on the general surface with a double line, of conies whose planes do not contain the double line, and the fact that the surface is rational, have been shown very simply by Baker* as follows : consider the quadric cones u v /n N y=-, t ........................ (1), y w w where u = 0, v = are pairs of planes through the double line x =. z = ; and w = is a plane through it ; these cones intersect in a conic which meets the double line at one point only. The equation of the surface being if in A, S, G we substitute for y and t from (1), we obtain a sextic in acjz ; and the seven arbitrary constants in u, v and w may be so chosen as to make this sextic vanish identically. Hence the surface contains at least one conic which meets the double line once only. V i / U/\ The substitution t + r[y -- ) enables us to express y w V wj nr* rationally in terms of T and - ; and hence also t; i.e. we can express the coordinates of any point on the surface as rational functions of two variables. * Some recent advances in the theory of algebraic surfaces, Proc. Lond. Math. Soc., Series 2, Vol. xn. p. 36. CHAPTER VII QUARTIC SURFACES WITH AN INFINITE NUMBER OF CONICS : STEINER'S SURFACE : THE QUARTIC MONOID 84. The property of containing an infinite number of conies has been seen to be possessed by all quartic surfaces with a double conic or a double line; in the present chapter we consider all surfaces which have this property. The determination of the quartic surfaces which contain an infinite number of conies was made by Kummer*. The following is a brief account of his investigation. If the plane section of a quartic surface has four double points it will consist either of two conies or of a line together with a nodal cubic, in the latter case three double points are collinear. If the section consists of two conies, each of their points of intersection is either a double point of the surface or a point of contact of the surface and the plane section. Consider first the case in which no point is a point of contact. Let two of the double points be fixed f; the surface must then possess a double conic, the equation of such a surface is <|>2 = 4p2^, where = 0, M* = are quadrics. If the surface contain also two double points whose join does not lie on the surface, M* must break up into two linear factors (Art. 38), and the surface is then 2 = Qtpiqr, whose sections by planes through the line (q, r) are pairs of conies. If the surface has another pair of nodes, its equation will be (Art. 38) (p 2 + qr- st) 2 = or (p 2 - qr + st) 2 = * Ueber die Fltichen vierten Grades auf welchen Schaaren von Kegelschnitte liegen, Crelle, LXIV. t The cases in which none or only one of the double points are fixed lead only to a quartic surface consisting of two quadrics, or a cone. 92 132 QUARTIC SURFACES WITH [CH. VII In this case we have two sets of conies lying in planes whose axes are (q, r) and (s, t). If three of the double points are fixed they necessarily lie on a double line ; the sections of a surface having a double line by planes through the double line form a set of conies. If certain of the double points coincide we are led to special cases of the surface 2 = 4>p*qr, except in one case, viz. that in which the surface touches itself at two points ; any plane section through these points gives a quartic curve which touches itself twice, and therefore necessarily consists of two conies having double contact. The equation of such a quartic surface is 3> 2 = a/:ty8 where a = 0, /3 = 0, 7=0, 8 = are four coaxal planes ; the intersection of their axis with 4> = gives the two double points having the above property ; they are usually called tacnodes. Consider next the case in which one of the four double points is a point of contact of the plane and the surface ; if none of the three remaining points are fixed the surface possesses a double curve of the third order, which, when a twisted cubic, a line and a conic, or three lines, gives rise to a ruled surface, and the section by a tangent plane to a line and a cubic, excepting only in the case in which the three lines are concurrent. The surface then has the equation Aq*r 2 + Br~p 2 + Cp 2 q 2 + Zpqrs = 0. This surface is known as that of Steiner. If next one of the double points is fixed, the surface must have a double conic and one node. Its equation is then (Art. 38) <|>2 = 4^2^ where = is a cone whose vertex lies upon O = 0. This cone touches the surface along the curve (, ^), the tangent planes of M? thus meet the surface in pairs of conies. Lastly let two of the points be points of contact, the surface has a double conic, its equation may be written We may determine X in five ways so that + xo> + \Y = o is a cone V (Art. 22) ; V has double contact with the surface ; hence the tangent planes of five cones are bitangent planes of the surface and meet it in pairs of conies. 84, 85] AN INFINITE NUMBER OF CONICS 133 In the case of ruled quartic surfaces, the bitangent planes contain two generators and therefore meet the surface also in a conic. 85. The quartic surfaces which have respectively a double conic and a double line are discussed in Chapters III VI, that which has three concurrent double lines (Steiner's surface) in the course of this chapter. To return to the surface 3> 2 = 0/378, where a, fi, 7, 8 are coaxal planes ; this surface has been shown by Nother* to be birationally transformable into a cubic cone. For if the axis of the planes a, /3, 7, 8 be the line z = 0, w = 0, and the planes x = 0, y = are the tangent planes to where the line (z, w) meets 4>, then may be written xy + (z, w\af, and the surface becomes {xy + (z, w$ 2 } 2 = (z, w~$b)\ Choose as new variable w one of the factors of (z, w]&) 4 , the quartic surface becomes {xy + (z, w]a) 2 } 2 = w (z, wQb) 3 ; then by aid of the transformation x:y:z\w x'w' (z, w'^cCf : y'' 2 : y'z : y'w', x : y' : z' : w' = xy + (z, w\af : yw : zw : up, we obtain x' z w' = (z', w'~$b) 3 ; a non-singular cubic cone. The system of conies on the surface may be represented as folio wsf : Writing the equation of the surface in the form * J then if (z, wQa) 4 = a z* + a^z 3 w + a^z^w 2 + a (z, wl^by = b z 2 + b^w + 6 2 w 2 , the system of conies is with the condition b fjt - 1) \ 4 - (a 1/A 2 + b./j.) \ 3 + (a 2 (M 2 + b a p) X 2 a 4 /A 2 = 0. From these equations it is seen that the conies of the surface can be arranged in sets of four, lying on the same quadric. By * Eindeutige Raumtransformation, Math. Ann. in. t Sisam, Concerning systems of conies lying on cubic, quartic and quintic surfaces, American Jour. Math. 1908. 134 STEINER'S SURFACE [CH. vn suitable choice of b , b 1} 6 2 we may replace the first two of the preceding equations by z + \w = 0, z* + pxy = 0, and we obtain the result: the quadric cone whose vertex is any point of the line cc = y = Q* and which contains any conic of the system will meet the surface in four conies. To the foregoing surfaces described by Kummer we must add the surface whose equation is {xw +f(y, 2, w)} 2 = (z, w$a) 4 . This may be regarded as a geometrically^ limiting case of the last surface when the two tacnodes coincide in the point (y, z, w). Its sections by planes through (z, w) consist of pairs of conies. The equation of the surface may be written (xw - 7/ 2 ) 2 + 2 (xw - y 2 ) (z, wQa) 2 + z (z, w~$b) s = ; and by application of the transformation x : y : z : w = y' z + z'x' : y'w' : z'w : w' 2 , x' : y \ z' : w' = xw y* : yz : z- : zw the surface is transformed into the cubic cone z'x'* + 2x' (z', w'^af + (z, 86. Steiner's surface. The surface of the third class with four tropes was first investigated by Steiner . In accordance with this definition we may take as its equation in plane -coordinates 1111 - + - + -+- = 0. U^ U 2 U 3 U 4 * Since 2=0, w = are any two planes through the given line z = 0, w=0. t It cannot be derived from it by giving any particular values to the constants. See Berry, On quartic surfaces which admit of integrals of tJie first kind of total differentials, Camb. Phil. Soc. Trans. 1899. J This quartic surface is discussed by de Franchis, Le superficie irrazionali di quarto ordine di genere geometrico-superficiale nullo, Rend. Circ. Mat. di Palermo, xiv. It is there shown that the irrational quartic surfaces for which p g (the geometrical genus) is zero are either cones or birationally transformable into cones : they include the two surfaces last discussed, also the ruled quartic surface with two non-intersecting double lines, the surface {xio+f (y, z, w)} 2 = (z, w>$a) 4 where / con- tains y only to the first degree (this is a ruled quartic with a tacnodal line), and also two special quartic surfaces. See also Berry, loc. cit. The surface is also known as Steiner's Roman Surface. 85, 86] STEINER'S SURFACE 135 The equation of the surface in point-coordinates is seen to be The coordinates of the points of the surface may therefore be expressed in terms of the coordinates TJ of the points of a plane by means of the equations / \2 / / i i \2 By changing to a second tetrahedron of reference, desmic to the first (Chap, n), i.e. by writing 2/4 == $1 T #?2 *^3 i ^"4j we finally obtain 0-2/1 = 2772773, 0-2/2 = 277377!, 0-7/3 = 277J772, 0-7/4 = ?7i 2 + 77 2 2 + 77 3 2 . This method of representing the surface on a plane was first given by Clebsch*. Eliminating the 77^ we obtain as the equation of the surface 2/2 2 2/s 2 + 2/3 2 2/i 2 + 2/i 2 2/2 2 ~ 27/i2/ 2 2/ 3 2/4 = 0. This latter form of the equation of the surface shows the existence of a triple point A 4 (yi y 2 y3 = 0), three double lines, and three nodes. The section of the surface by any tangent plane contains four nodes, and hence breaks up into two conies', a characteristic property of this surface. There are no lines on the surface other than the three double lines ; for there is clearly no other line passing through A 4> and if there were a line not passing through A 4 then the section by the plane through this line and A 4 would possess a triple point at A 4 , i.e. would consist of this line together with three other lines through A. The correspondence between a point y t of the surface and a point rji of the 77 -plane is of the (1, 1) character, the only exceptions to this being for points of the three double lines; to such a point there correspond two points of the 77-plane, e.g. for a point of the line y 1 = y 2 =Q we have _ Q 2^4 _ ^i 2 + 7 ?2 2 . 2/3 giving two points on 773 = 0, for which the values of - - are reciprocal. * Ueber die Steinersche Flache, Crelle, LXVII. 136 STEINER'S SURFACE [CH. vn 87. Curves on the surface. To a curve <(?;) = of order n on the i;-plane there corresponds a twisted curve of order 2n on the surface, this being the number of points in which (17) meets any conic %* + 22a a i7i% = 0. To the straight lines of the ij-plane there correspond the oo 2 conies of the surface; to conies in the Tj-plane correspond twisted quartics which are either of the second species or are nodal and of the first species ; this is seen from consideration of the rank of such curves, i.e. the number of their tangents which meet any given straight line, e.g. Oy = /3 y = 0, i.e. the number of planes Oy + \/3 y = which touch the curve. Denoting the results of substitution for the y t in terms of the tji in Oy, ft y by u and v, we have to determine the number of conies u + \v which touch <, giving the equations + X - = a- (i = 1 2 3) drji drji drji The required points of contact are given as the intersections of du dv d(f> _ and are n (n + 1) in number if has no singular points. We thus obtain six as the rank of twisted quartics on the surface, which therefore belong to one of the two classes previously mentioned. 88. The equations n/r . f.(-n 'n >n \ (i 1 9 ^ 4,\ L/tASl " / 1 \ *l\ ) */2 > /3/ \ ^ ""~~ * 3 3 "> / determine a Steiners surface, if the curves fi = are conies. This may be seen as follows : The conies apolar to each of four conies form the pencil the members of which are all inscribed in the same quadrilateral ; if p n is one side of this quadrilateral then p* = is apolar to w a 2 and up 2 , and therefore belongs to the linear system (1), or 87, 88] STEINER'S SURFACE 137 hence this system includes the squares of four lines ; by proper selection of the triangle of reference these lines may be repre- sented by the equations % + *7 + *?3 = 0, -171 + 172 + ^3=0, r)l- Vz + 1)3 = , 1)i + ?? 2 - TJ 3 = 0, whence we again arrive at the equations connecting a point x of Steiner's surface and a point 77 of the plane. The conies of the pencil (2) are the images of the asymptotic lines of the surface * ; this may be seen as follows : The pairs of tangents drawn from each point P of the plane -17 to the conies (2) form an involution; if p, p are the double lines of this involution, then since they are harmonic with each pair of tangents, the line-pair pp' belongs to the system (1), hence its image is a pair of coplanar conies on the surface. These conies intersect on each of the three double lines, and their fourth inter- section Q corresponds to P. Moreover, since p, p' are the tangents at P to the two conies of (2) which pass through P, it follows that the line-elements of the asymptotic lines at Q correspond to the line-elements at P of the pair of conies belonging to (2) which pass through P. This being true of every pair of corresponding points Q, P the result follows as stated above f. * See Cotty, Sur les surfaces de Steiner, Nouv. Ann. 1908 ; Lacour, Nouv. Ann. de Math. 1898. t Analytically we may proceed as follows : let ( ^ + - % + - r] 3 } = u . v \ III it / 3 be a line-pair belonging to the system 'Zri i 2 + 2'Sa ik r] i -r] k Q ; if u + du be the line l consecutive to u passing through the point on u consecutive to (u, v) we have and hence since Si^tZr/^O, it follows that ST^W^ = so that the lines , v, du are concurrent, i.e. v i = au i + pdu i . From comparison with the values of u f , v i given above we have 1 m -- dm _ m dn = T' n n This gives m 2 - l = k (n- - 1), k being an arbitrary constant. Hence the image of an asymptotic line is a conic touching the four lines m=l, n=l. 138 STEINER'S SURFACE [CH. vn We obtain sub-cases of Steiner's surface when the conies u a 2 , Up 2 of (2) are related in certain ways, the four sub-cases which arise are the following : (i) When two common tangents of the pencil (2) coincide, i.e. the conies of the pencil touch two lines and touch a third line at a fixed point ; (ii) three common tangents coincide, i.e. the conies osculate at a fixed point and also touch a fixed line ; (iii) the conies touch two lines where a third line meets them ; (iv) the conies have four consecutive points common. The cases (iii) and (iv) lead to cubic surfaces. In case (i) take the intersection of the two lines as vertex A l of the triangle of reference, the fixed point as A s and the fourth harmonic to A^A 3 and the two lines as the third side, the equation of the pencil is then 2\uw + w* - tf - 0. The conies apolar to this pencil are Ay? + B (n + 1)./) + 2(^173 + Wr) 2 r) 3 = 0. The equations connecting a point x of the quartic surface with the point 77 of the plane are therefore Pi = V> px* - *?2 2 + % 2 , pt = fyi-nt, P#4 = 2773773 ; giving as the equation of the surface # 3 4 4a; 1 ar 2 a; s a + kxfx? = 0. The surface has a triple point through which there pass two double lines, along one of which one sheet of the surface touches the plane #j . The surface has two tropes x 2 = # 4 ; the plane # a meets the surface in the line (x l} # 3 ) alone, this line is thus torsal. In case (ii) the conies (2) are v* - 2uw + 2\vw = 0, giving as the apolar conies AK* + B (7? 2 2 + 277^3) + cv + 2D7 ?1 7 7 . 2 = o. The connecting equations are here The equation of the surface is z = 0. 88, 89] STEINER'S SUEFACE 139 The surface has one trope, the plane ac a . Along the double line x l = x 4> 0, the surface touches the plane x 1 = 0. As previously stated, when the conies of the pencil (2) touch two lines at given points, or have four consecutive points in common, we obtain a cubic surface : this is easily seen, by application of the present method. 89. Modes of origin of the surface. The connection of the previous mode of representation of the surface and the method of treatment of the surface by Reye* by pure geometry is shown as follows : If we have any quadric transformation pxi =fi (! , 2 , 3 , 4 ), 0' = 1, 2, 3, 4), it has been seen that the locus of the point x as the point a describes a plane is a Steiner surface. Now the preceding equations place two spaces 2, 2j in 4 correspondence, so that to each set of quadrics 2X-/< = in S there corresponds a plane in 2 15 to each pencil of quadrics in 2 a pencil of planes in 2 1} and to each set of eight associated points in 2 a point in 2j. From the foregoing we deduce also the following result : Steiner 's surface and the cubic polar of a plane with reference to a general cubic surface are reciprocal ; for if U = is any cubic surface, from the equations fiU" / -> r, i\ pui=^, (t- 1,2,3,4), we deduce by aid of the preceding that as x describes a plane, u' envelops a surface which is the reciprocal of a Steiner surface, and the u-i are the coordinates of the polar plane of x with regard to U. The following method of derivation of the surface is due to Sturm f. Having given a pencil of quadrics and a pencil of planes projective to it, the locus of intersection of a plane and quadric which are in correspondence is a general cubic surface. If we make the further assumptions that the axis of the planes touches two of the cones contained in the pencil of quadrics, and * Geom. der Lage. t R. Sturm, Ueler die E'dmische Flache von Steiner, Math. Ann. in. 140 STEINER'S SURFACE [CH. vn that each plane through the axis and the vertex of one of these two cones corresponds to that cone, the surface becomes the general cubic surface with four nodes. For these conditions are seen to be satisfied by assuming as equations of the two pencils where V 1} V 2 are the cones considered and where V 1 = a a + as 1 ft, V 2 =y* + x 2 S. The surface is therefore x, ( 7 2 + x 2 B) - x 2 (a 2 + a^/8) = 0, which has the four nodes a = x l = T 2 + # 2 (8 /3) = 0, 7 = # 2 = a 2 x l (B ft) = 0. Reciprocating, it follows that if a pencil of surfaces of the second order is projectively related to a row of points on a straight line, the envelope of the tangent cones drawn from the various points of the line to the corresponding quadrics is a surface of the third class with four tropes, provided that the line meets one conic c 2 belonging to the system in a point A and one conic c' 2 in a point A', and so that A corresponds to c 2 and A' to c' 2 . Curves on the surface. The following theorem regarding Steiner's surface is specially noticeable : Every algebraic curve on the surface is of even order*. Let a curve c m of order m on the Steiner surface $ 4 pass p times through the triple point and let r, r, r" branches respectively touch the double lines of S t at the triple point, and if p l} p 2 , p 3 other branches respectively touch the three tangent planes at the triple point, then p = r + r' + r" + p 1 + p 2 +p 3 . Denoting the double lines by a, a, a" it is seen that in the plane (a, a) p + r + r'+p! points of c m lie at the triple point, the other mpi r'p l points of intersection of $ 4 with this plane must therefore lie on a and a, suppose that q lie on a, and hence q' = m p r r p^ q lie on a'. Similarly on a" there lie q" = m p r r" p 2 q points. The cone which projects c m * See Sturm, loc. cit. 89, 90] STEINER'S SURFACE 141 from the triple point is of order m p and has a as (q + r)-fold line, a' as (q' + r')-fold line, a" as (q" + r")-fold line ; these lines being double on $ 4 count 2 (q + r + q + r + q" + r") times in the intersection of the cone and S 4 , hence 4 (m p) = m + 2 {q + r + q' + r' + q" + r"}, hence from the preceding ra= 2 [r+p l +p 2 + q\- If the curve c m does not pass through the triple point then p = 0, and hence p 1 =p 2 =p 3 = r = r' =r" = 0, and we have m = 2q. Thus a curve of order 2n which does not pass through the triple point meets each double line in n points. Hence a curve of the fourth order meets each double line twice ; if these points of inter- section coincide we have a quartic curve of the first species with a double point (Art. 87). 90. Quartic curves on the surface. We now consider further the quartic curves on the surface. Every quartic curve c 4 which does not pass through the triple point meets each double line twice ; through c 4 there passes at least one quadric which meets the surface in another quartic curve c' 4 which also meets the double lines twice and necessarily in the same pairs of points, excluding at present the case in which the quadric contains one of the double lines. If the conic 2o A ' 3= \Ti " *m) * 3 ' f For if e.g. a ll = a^ 2 = \ the curve c 4 will also lie on the quadric x 3 (xi + a 12 x 3 + a 13 2 + 023 Xi) + %XiX 2 (a 33 - 1) = 0. 90, 91] STEINER'S SURFACE 143 related to the residual intersections of the surface by V and V, we have from above U=c 2 .c' 2 , V=c*.c" 2 , (where in the conies the iji are replaced by the xj). Hence U + \ V = c 2 (c /2 + Xc" 2 ), so that the quadri-quartics associated with the pencil of quadrics are thus represented by a pencil of conies and therefore pass through four fixed points. This is a characteristic property of these curves. The conic through one vertex of the triangle of reference u = aasjifc* + a m i} 9 * + SSoatyifc = 0, (t 4= k), represents a quartic curve through the triple point ; it is clear that if in the product / 2a 12 2a ls \ ^ ( r}l+ '^~^ 2 + ^~ V* I u \ "22 "33 / we substitute for the % in terms of the yi we obtain an equation containing the latter variables only, and in the second degree. This quadric hence passes through a double line, a conic of the surface and a curve c 4 through the triple point ; this curve is of the second species, since one set of generators of the quadric meet c 4 in three points. There exist oo 4 curves c 4 passing through the triple point. If the representative conic passes through two vertices of the triangle of reference, i.e. is of the form 77s 2 + 2200,^% = 0, (t =f k), we obtain as a quadric which contains it the cone y\y* + 2 2/s (i22/ 3 + is2/2 + a 23 2/i) = 0. These curves are the intersection with the surface of cones through two of the double lines ; there are oo 3 such curves. Finally, the conies through each vertex of the triangle of reference correspond to the plane sections through the triple point. 91. A mode of origin of Steiner's surface is obtained, as shown by Weierstrass*, from consideration of a well-known pro- perty of a quadric. This property is the following : if through any given point A 4 of a quadric Q three mutually perpendicular lines * Schroter, Ueber die Steinersche Fldche vierten Grades, Crelle, LXIV. 144 STEINER'S SURFACE [CH. vn are drawn to meet Q again in points L, M, N; the normal at At to Q meets the plane LMN in a fixed point. The theorem may be restated in a general form as follows : if A 4 be joined to the vertices of any triangle self-polar to a conic c 2 in a given plane # 4 , and the three joining lines meet the quadric again in points L, M, N, then the plane LMN meets the line A 4 R in a fixed point S, if R is the pole for c 2 of the trace on # 4 of the tangent plane to Q at A t . If c 2 is a member of the set of oo 2 conies of = 0, where a* = ^U + r) 2 V + r) 3 W, then by giving all values to the % the resulting locus of S is a Steiner surface. Let the equation of Q be 3 x t a x + ^u ik XiX k = 0, (i =f= k). i Also let two vertices A l} A 2 of the tetrahedron of reference be self-conjugate * for each of the conies U, V , W, and therefore for a 2 ; let 333 U = JLa ik XiX k , V = %b ik xix k , W = ^c ik XiX k . 111 Now if x is the point which forms with A l and A 2 a self-polar triangle for a 2 , we have #in + #3 a i3 = 0, a; 2 a 22 + ^ 3 23 = ............... (1), where a ik = rj, a ik + r) 2 b ik + ^ c ik . Let R, or y, be the pole of a x for a 2 , then 23j/3 0,3^ + 0533/2 It will now be shown that if A 4 x meets Q in x', and A 4 y meets the plane (A l A 2 x') in y' (8), then the locus of y is a Steiner surface. For, from equations (1) and (2) we obtain the x t and yi as quadratic functions of the r) { , thus yi-y*' 2/3 =/i (17) :/ 2 0?) :/ 3 (n) ; also, it is easy to see from equations (1) and (2) that =^ . #3 Ctji Ot22 * If A l , A 2 do not both lie on the section of Q by the plane of c 2 , we can replace them by the intersections of A t A lt A 4 A 2 with Q, and c- by its projection on any plane through these two new points. 91, 92] STEINER'S SURFACE 145 Again f rom > " OII1 2/3 where jf 4 (77) is a quadratic function of the %. Hence y/ : y/ y,' : y/ =/i (1?) : / (^) : /a (1?) : /* (*?)- 92. Eckhardt's method. A method of point-transformation applied by Eckhardt* leads easily to properties of the cubic surface with four nodes, and hence by reciprocation to the surface of the third class with four tropes, which is Steiner's surface. This transformation is the following : XiTJi = p, (*'=!> 2, 3, 4). By use of this method there corresponds to any given plane 2;#; = 0, the surface 2 = 0, y< which is a cubic surface having the vertices of the tetrahedron of reference as nodes. The equation of the surface in plane coordinates being S Va^Mi = 0, it is seen to be of the fourth class. The four tangent cones of the surface at the nodes are -<-:S^ = 0, ( = 1, 2,3,4). y* i y* They are the tangent cones from the nodes to the quadric which touches the edges of the tetrahedron of reference ; the intersection of this quadric with the plane * Math. Ann. v. J. Q. S. 10 146 STEINER'S SURFACE [CH. vn is a conic c 2 which lies on the cubic surface ; for by squaring and subtracting from the equation of the quadric we obtain Ml M* as a quadric through c 2 , and this quadric is easily seen to arise by combining the equation of the plane with that of the cubic surface. Hence the curve of intersection of the quadric and the cubic surface consists of three conies lying respectively in this plane and in two others of similar form. To a quadric through the vertices of the tetrahedron of reference corresponds a quadric through the same points, and if one quadric is a cone so also is the other (since the discriminant of ^a u x^x z is equal to that of ^a^x^). Now through four points two quadric cones can be drawn to touch a given plane and to have their vertices at a given point of that plane*, we therefore obtain for the cubic surface as the corresponding theorem : through any point oftlie surface two quadric cones can be drawn to have their vertices at the point and to touch the surface : otherwise, the tangent cone to the surface having its vertex at any point of it breaks up into two quadric cones. Reciprocating, we again obtain the result for Steiner's surface that its curve of intersection with any tangent plane consists of two conies. Again we have the theorem that eight quadrics can be drawn through any four given points to touch a given quadric along a conic ; for taking the given points as vertices of the tetrahedron of reference and the given quadric as Sa^^t^A = 0, the latter may be written in the form aaxtf + 22 (ctflfc - Va^-ajfcfc) x t x k = ; whence the eight planes of the conies of contact are seen to be taking all combinations of the ambiguities. If the given quadric consists of two planes, the conies of contact break up into pairs of lines, and hence the tangent quadrics must be cones whose vertices lie upon the line of intersection of the two planes; it follows therefore that through * For the cones which pass through the four points and have their vertices at a given point form the pencil V 1 + \V%=Q where V 1 and V^ are two cones fulfilling these conditions, and the cones of this pencil which touch a given plane are given by a quadratic in X. 92, 93] THE QUARTIC MONOID 147 four given points there pass eight quadric cones which touch two given planes ; by application of the transformation it is seen that if two cubic surfaces have four nodes in common there are eight quadric cones which touch each surface along two twisted cubics ; otherwise, the common tangent developable of two cubic surfaces with four common nodes breaks up into eight quadric cones whose vertices lie upon the curve of intersection of the cubic surfaces. By reciprocation we obtain that if two Steiner's surfaces have four tropes in common, their curve of intersection consists of eight conies whose planes touch the common tangent developable of the two surfaces. 93. Quartic surfaces with a triple point. The quartic surface with a triple point has been discussed by Rohn*; it belongs to the type known as the monoid, i.e. the surface of order n with an (n l)-fold point. We may take as its equation wu 3 + u 4 = 0, where u s = 0, w 4 = are cones of orders 3 and 4 respectively, having their vertices at the triple point x = y = 2 = Q. These cones intersect in twelve lines lying on the surface. These lines meet the plane w = in twelve points which we may call principal points in the representation of the points of the surface by their projections on this plane from the triple point. This gives a (1, 1) correspondence of points between the surface and the plane, in which, however, all the points of one of the twelve lines are represented by one principal point. In the general case there is no conic on the surface whose plane does not pass through the triple point A 4 ; for this would require the quadric cone of vertex A and base the conic, to contain six of the twelve lines of the surface, in order to complete its curve of intersection with the surface. There are sixty-six conies lying in the planes passing through two of the twelve lines; they are represented by the lines joining pairs of principal points. The quadric cone through five of the twelve lines meets the surface also in a twisted cubic passing through A 4 \ we obtain 792 such cubics which are represented by conies through five of the principal points. The system of oo x conies through any four * Ueber die Fldchen viertar Ordnung init dreifachem Punkte, Math. Ann. xxiv. 102 148 THE QUARTIC MONOID [CH. VII of the principal points represents a system of quartic curves on the surface, each quartic having a node at A 4 . Such a quartic must have two apparent double points, since its plane projection from any point must have zero deficiency, being in (1, 1) corre- spondence with a conic. Hence it is of the first species (a quadri-quartic with one node). The oo 1 cubics through eight principal points represent twisted quartics without a node, which pass through A 4 and have the same tangent at that point ; they have two apparent double points, their plane projections being in (1, 1) correspond- ence with non-nodal plane cubics; hence they are of the first species. Any quartic of the first type and any quartic of the second type lie on the same quadric, provided that the twelve lines with which they are associated are all different. For the conic and plane cubic respectively corresponding to them intersect in six points (none of which coincide with principal points), hence if we take that member of the pencil of oo 1 quadrics through the quartic of the second type which also contains any given point of the first quartic, it will meet the latter in 6 + 2 + 1 = 9 points, i.e. will contain it. We also have quartics of the second species, obtained as the intersection with the surface of cubic cones having six of the twelve lines as simple lines and a seventh as double line ; they are represented by the plane cubics through six principal points which have a node at a seventh principal point and are 5544 in number. These cubics pass through A t and are seen to be of the second species, since they have three apparent double points. The surface will also possess a line not passing through the double point if three of the twelve lines are coplanar. The maximum number of such lines is nineteen*. The surface may possess a node D ; in that case the line joining it to A 4 must lie on the surface arid hence is one of the twelve lines of the surface. Moreover, in this case, two of the twelve lines must coincide. For in this case any section through A t D consists of this line together with a cubic passing through D; we may take A t D as the line y = z = Q and the equation of the surface as w [y (cue 2 + ...) + z(ba? + ...)} + y (co? +...) + z (do? + ...) = 0; * See Kohn^ loc. cit. 93] THE QUARTIC MONOID 149 the condition that all such cubics should meet on y=z = Q is ad = be, which is the condition that the curves u s = 0, w 4 = should have the same tangent at the point where A 4 D meets the plane w = 0. Hence two of the twelve lines coincide. It is also easily seen that if these curves touch, there is a node at the point y = z wa + ex = 0. The surfaces with six nodes are of special interest : they are of two types ; in the one type the nodes have any position, in the other they lie on a conic. The equation of a surface of the first type, which has a triple point at A and nodes at six points which we may represent by 1, 2, 3, 4, 5, 6, is of the form where K = is a quadric cone whose vertex is A and which passes through the points 1 ... 5; F=0 is a quadric through the seven points A, 1 ... 6 ; P = is the plane through the points A, 4, 5 ; V = is the cubic surface with four nodes, viz. at A, 1, 2, 3 respectively, and which passes through the points 4, 5, 6. There remain three undetermined constants, viz. two in F, together with p. It can easily be shown that they can be deter- mined so as to give the required surface. For the conditions that the point 6 should be a node are seen to reduce to the following : <^.^ = <^.9v_8.F > av_aF < av = F- "dx ' dx "dy "by ~dz ' dz dw ' dw " K ' wherein the coordinates of the point 6 are substituted. These conditions express that F = 0, V = should touch at the point 6, and the two undetermined constants in F are thus found ; finally p is uniquely determined. We thus obtain one surface whose equation depends only on the coordinates of its singular points. A remarkable property of the surface is that the sextic tangent cone, whose vertex is any one of the six nodes, breaks up into two cubic cones, each having a double edge passing through the triple point. For the equation of the surface may be written # 2 (wA + B) + 2x (wG + D) if the node considered is the point (1000). 150 THE QUARTIC MONOID [CH. VII The tangent cone whose vertex is this point is then (wC + D) 2 = (wA + B) (wE + F) ; this is seen to have a fourfold edge passing through A 4 . It also has five double edges passing through five nodes. Now one cubic cone whose vertex is this node can be drawn to contain these five edges, to have the other edge as double edge, and to contain any other edge of the sextic cone. It therefore intersects the latter cone in 4x2 + 2x5 + 1 = 19 edges, and hence forms part of it. The foregoing property is also possessed by the symmetroid (Art. 8). It is easy to see that the surface we have just considered is a special case of the symmetroid. For the latter surface is seen (Chap, ix) to arise when from the equations expressing that the quadric ^ + @S 2 + yS 3 + B8 t = should be a cone, we eliminate the variable Xi, and regard the , ft, 7, 8 as point-coordinates. If we now take the special case in which 8j. is a plane a x 0, taken doubly, we obtain as the required surface = 0. This surface has the point (1000) as triple point, since for this point all the second minors of the determinant vanish. The surface has a node for such values of a ... B as make a pair of planes XY\ and for such points we have /3S Z + jS 3 + 88. = XY- aaj, i.e. f3S. 2 + 7$, + S$ 4 = represents a cone whose vertex lies on a x ; and there are six such cones since the vertices of all cones which pass through the eight fixed points $ = S. 2 = S 3 = 0, lie on the sextic curve >- + S^ 4 = 0, (^1,2,3,4). Moreover the preceding surface represents the most general quartic monoid with a given triple point and with six nodes. 93] THE QUARTIC MONOID 151 For the equation of such a surface involves 34 10 6 = 18 constants; and the determinant involves thirty-four constants which can be reduced to eighteen on multiplying by an arbitrary determinant of four rows. The second type of quartic monoid with six nodes is repre- sented by the equation where K is a quadric cone, F = a cubic cone with the same vertex, and P = any plane. For this surface has clearly a triple point and has the six nodes given by the equations P = F = K = 0. It contains twenty-one constants, the same number as the surface last considered* when the triple point is arbitrary. * For a full discussion of many special cases of the quartic monoid the reader is referred to the memoir by Bohn recently quoted. CHAPTER VIII THE GENERAL THEORY OF RATIONAL QUARTIC SURFACES 94. The quartic surfaces so far considered, with a double curve, have been found to be rational, i.e. the coordinates of a point on such a surface are expressible as rational functions of two parameters. Surfaces with a triple point are rational, as is seen by projecting the surface from the triple point on any plane. We shall now investigate the other types of quartic surfaces with a double point which are rational*. If the double point be a tacnode, i.e. such that every plane through meets the surface in a quartic curve having two consecutive double points at Of, the equation of the surface has the form x?x? + x 4 x l fe (a?,, # 2 , a? s ) + %4 (#1 , a , #) = 0. Projecting the points x of the surface by lines through to meet the plane # 4 in points y, we obtain where 1 (y) = fo (y)} 2 - 4% 4 (y). The points a; of the surface are thus related to points y of a double plane] we now obtain rational expressions of the ^ in terms of new variables zi which render Vll (y) rational in the 2,-J. The equation of the general quartic curve 1 (y) = may be taken as H u ia 1/13 W = 0, 'VI U 22 U 23 ^31 ^32 ^33 '41 W 42 1/43 where u ik = Ujd, the u ik being linear functions of the y^. * Nother, Ueber die rationalen Flachen vierter Ordnung, Math. Ann. xxxni. t Selbstberiihrungspunkt. J Clebsch, Ueber Fldchenabbildungen, Math. Ann. in. Hesse, Crelle's Journal, XLIX. = 0. 94] THE GENERAL THEORY OF RATIONAL QUARTIC SURFACES 153 A system of cubic curves having six-point contact with fl is W n 1ti2 W J3 Mj4 fli 11 ti *)i It ft 41 42 43 44 4 ! a, a s a 4 Denoting this system by (y, a), there are eight systems of oo 2 nodal cubics ; through every point 6< of the plane there passes one member of each system having a node at 6^. For if $(y, ) has a node at bi, the equations give eight sets of values for the a*. Again consider the quadrics F(y, X) = ^u ik XiX k = y.F, + y,F, + y 3 F 3 = ; if fi be one of the eight points of intersection of F l = 0, F^ = 0, F 3 = 0, the coordinates of the tangent planes to the quadric F(b, X) at the eight points ,form the preceding eight sets of values of the ;. For let quantities 77 1 -, 04 be connected by the equations i = ii (6) 1/1 + "is (b) f), + u is (b) 773 + un (b) 7? 4 , (i = 1, 2, 3, 4), then it is seen that $ (b, a) = - F (b, i7)fl(6). If now rji = {, the plane e^ touches -P(6, X) at the point , and since F(6, f) vanishes for all values of the b t we have , a) _ ' /_ ' Thus the eight different sets of oo 2 nodal cubics belonging to the curves (y, a) are obtained by taking for the quantities c^ the coordinates of the oo 2 planes through the eight points . We now select one of these eight points , and denote the others by ', f", .... It is seen that to each quadric F (y, X) there corresponds one point y and conversely ; and since there is one quadric F which contains a line s through , therefore to each such line s there corresponds one point y, but each point y determines one quadric F which has two generators s through , 154 THE GENERAL THEORY OF [CH. VIII i.e. to each point y there correspond two lines s. These two lines coincide if F is a cone, i.e. if we have u il X l + UfoX.,. + u i3 X 3 + 11^X4 = 0, (i I, 2, 3, 4), giving the points y of fl. Considering the points z the sections of the sheaf of lines s by the plane which is the field of y, we thus obtain the following relationship between the points y and z ; to each point z there corresponds one point y, to each point y there correspond two points z, which coincide when y lies on fl. To the quadrics F which touch a given plane a f , but not at , correspond points y lying on the curve

in three points, c must meet p in the corresponding points, i.e. k = 3. Hitherto s has been taken as a line through f which does not pass through any of the seven points f, ", etc. But if s passes through ' then to s will correspond a pencil of quadrics F which determine a line of points y. Denoting by A t the points in which ', ", etc. meet the plane of reference, then to each point A{ there corresponds a line, this line meets p' in one point, hence c passes through each of the seven points AI. Thus the curves c which correspond to lines in the field of y form a system of oo 2 cubics through seven fixed points. Such a system is represented by the equation /. + /.+#=<>. where f\,f z ,f are three cubics having seven points in common. The relationship between the points y and z is therefore expressed by the system -. If r=2 one o solution of (1) and (3) is n=6, aj = 2, a 2 =8; if n>6 there is no solution. Next let r, t and t be not all equal, then if r = t + a, s = t + j3, a =4=0, we have 3n - r - s = ai + 2a 2 + . . . + 1 a t ; hence (3n - r - s) t > ?i 2 - 2 - r 2 - * 2 , (3n - 2f - a - /3) t > n 2 - 2 - 2t 2 - 2 1 (a + /3) - a 2 - /3 2 , , n*-2-o-j8 whence t > therefore If /S=0 the numerator of the fraction is positive when n>4; if /34=0, since n>r+g it follows that n=>2 + a+/3 and the numerator is positive, i.e. r + s + 1 > n. 95] RATIONAL QUARTIC SURFACES 157 than n, by repeated application of the process we finally arrive at curves C 6 (a 1 z ... a 8 8 6 1 6 2 ) or at curves C 4 (a 2 6 1 6 2 b 10 )*. It will now be seen that the curves c 4 (a 2 6 x . . . 6 10 ) rationalize Vn when H = is a sextic curve with a quadruple point, and the curves c s (a^ ... c^&j&a) rationalize Vli when 11 = is a sextic curve with two consecutive triple points. The curves c 4 (a a &i ... 6 10 ). By hypothesis these curves form a linear system of oo 3 curves, hence there is one member of the system which has a node at &! (say), this curve cannot be irreducible, for then its genus would be unity and not two as required. Hence it consists of a cubic and a straight line ; this requires that the eleven points a^ ... 6 10 should lie on the same cubic. The system of curves consists therefore of linear combinations of the curves z l c, z^c and f, where f is any quartic of the system, c the cubic through the eleven points, and z l} z 2 any two lines through the point a. It can be shown that this system rationalizes Vfl when ft = is a sextic curve with a quadruple point. For let 1 = be such a sextic with a quadruple point at A 3) K = a conic passing through A s and having four-point contact with H, L = a cubic having a node at A 3 and passing through the four points of contact of K and 1 ; L thus contains two parameters. The curves fl aL 2 = have A 3 as quadruple point and touch K in the previous four points ; if now a be so determined that O oZ* = passes through an additional point of K = 0, it must contain K as a factor, i.e. we have = L* - KM, where M=Q is a quartic curve having a triple point at A s and touching ft in six points. Taking A 3 as the point y t = 0, y z = 0, we have where K l is linear in y l} y z , etc. Now if t = 2 1 zfa - where n X 2 , ... are the result of substituting z l} z z for y 1} y z in K lt * Nother, Ueber eine Clatse von Doppelebenen, Math. Ann. xxxm. 158 THE GENERAL THEORY OF [CH. VIII L z , . . . , we have such a transformation ; so that to the lines of the 2/-plane correspond curves c 4 (a 2 6j . . . 6 JO ) in the s-plane, the point a being A $ and the points b t the other intersections of the curves z 3 2 Ki 2,23X2 + ^ = 0, 3 2 /e a 22-3X3 + /^ 4 = 0. This transformation rationalizes VH, for if A (z) = be the curve corresponding to XI = 0, then since J (z,N, z z N, zfa - 2*3X3 + /**) = ^ A (z), it follows that A (z) = is a sextic curve. Moreover, since a(y) = 2/ 3 2 / 4 + 7/3/ s +/ 6) we have p 6 fl (y) = N* x power of A (z) ; and the transformation shows that this power is the square : hence The curves c e (a l 2 ... a 8 2 &i& 2 )- As before, since there are oo 2 curves forming a linear system /i + a /2 + /3/s = 0, the system will contain one curve having a node at &! (say), i.e. a curve of genus unity, this curve must therefore break up and consist of two cubics of which one passes through the ten points (^...ty&xfti and the other through the points b l> a 1 ...a 8 . If y is the former, and /' any cubic of the pencil through the points Oj... a a , and < any sextic of the family, the oo 2 sextics are included in the system / 2 + ff + /3. The transformation effected by means of this system is pyi =/ 2 0)> P2/2 =/(*)/' 0)> P2/3 = 0). The curve H (y) is, as before, the locus of points y for which the pairs of points z come into coincidence f. The curve A (z) which corresponds to l(y} is the Jacobian of/,/' and ; it is a curve of order 9 having a^ ... a 8 as triple points and not passing through &! or b z . Since any curve of the above system meets A (z) in 54 - 48 = 6 points, any line will meet fl (y} in six points, hence ft (y) is a sextic * If a = \2M4-^3/ lt j> 26 = K2M3-KiM4 c = \ 3 K 1 -\2K2, we find that hence p 3 \/O=p 3 \/L 2 - f Uebergangscurve, Clebsch. 95, 96] RATIONAL QUARTIC SURFACES 159 curve. Since O (y) and A (z) have the same genus, that of the former is seen to be four. Moreover, the point y 1 = 0, y. 2 = is a triple point on 11 (y), because the pencil /+ af = meets A (z) in three variable points only; and since f 2 + aff'= meets A (z) in three variable points and fixed points (corresponding to y l = 0, ?/ 2 = 0), f 2 meets A in the latter points only, hence the line y 1 = touches each of the three branches of O (y) at the triple point and hence meets fl (y) only at that point ; therefore fl (y} has an equation of the form ysy* + y?y*Q* + yiy a Q* + Q s = ; where Qi is of order i in y l , y 2 *. Applying the transformation to fl (y) we find that p 6 Q (y) is equal to/" 6 (z) x some power of A (z), and this power is seen to be the square, hence pn(y) ={/(*)} (A (*)}'. The transformation, therefore, rationalizes Vf2 (y). 96. The surfaces S 4 < 2 > and S 4 (3 >. It remains to determine the surfaces $ 4 which arise from the two preceding cases for fl (y), i.e. S 4 being #4 2 / 2 Oi> #2, # 8 ) + 2^/3 Oi, a, # 3 ) +/4 (#1, #2, afe) = ; the preceding results require that the curve li should either have a quadruple point or two consecutive triple points, where Now writing / a = a# 3 2 +^ 3 ^j + J 2 , / 3 = /3^ 3 3 + ^^ + a; 3 Bs + B 3 , / 4 = yx s * + x/C, + x/C, + x, + C 4 ; (1) If fl has #j = 0, x 2 = as a quadruple point the following identities must hold : - aC 1 - jA, = 0, 2 - a.C 2 - A& - yA, = 0, - a(7 3 - A,C 2 - A 2 G, = 0. * The absence of the term yiy^y^ gives rise to two consecutive triple points; thus making the genus of 0(t/) four, as required. 160 GENERAL THEORY OF RATIONAL QUARTIC SURFACES [CH. VIII By considering the possible values for a, /3, 7 we obtain* when a^O, ft = 0, 7= 0, the surface S t n = x* (x, + BJ* + x, (x, + BJ (A? - A,x, + 2 2 ) a surface having a tacnode in ^ = # 2 = # 4 = 0, i.e. the surface already arrived at. For a = /3 = 7 = we have surfaces with either a double line or a triple point and therefore excluded. For a = /3 0, 7 =^ we obtain either a surface with a triple point or the surface + 2x 4 B 3 + x 3 *C z + x 3 C 3 + C t = 0. (2) When fl has two consecutive triple points, and hence an equation of the form previously given, we obtain the identities /3 2 -a 7 = 0, 2/95, - aC, - yA, = 0, BS + 2{3B 2 - aC 2 - A& - yA 2 = 0, 2(3B 3 + 25^8 - aC, - A,C 2 - A.& = x*, - aC 4 - A,C, - A 2 C 2 = x^Q 9) By examining the various possible cases we are led to the one surface S< = xfx? + 2# 4 (x 3 x, D l + B 3 ) - X 3 3 X, + X?C Z + ^3^3+^4= 0. The surfaces $ 4 (1) , k 6 , c 6 ', k 6 ' ; hence if ra is the number of intersections of c e and k 6 ', we have 2m + 8 = 36 ; hence ra = 14. 98. Correspondence of points upon the surface. The preceding equations (1) and (2) establish a (1, 1) corre- spondence of points upon the surface ; for by aid of the equations ^i p x +^q x + \r x + \ 4 s x = 0, j p y + c^py + a 3 py" + ^p y '" = 0, \px + ..................... =0, *iq y + ........................ = 0, \px" + ..................... =0, a 1 r y + ........................ =0, XiK" + ..................... =0, a 1 s y + ........................ =0; and also of the equations \p 3 + ..................... = o XiP< + ..................... =O where P l = a^ + o^/ + a s p" + a^'", etc. ; having given any point x of the surface, a set of values of the X t - are determined and hence one set of values for the Oj, and finally a point y of the surface. Regarding the \i as point-coordinates, and also the Oj, it is seen that by aid of these equations we pass from a point x of A to a point A. of a quartic surface 2, and thence to a point a of a similar surface 2', and finally to a point y of A. Hence as the point A, describes a plane section of the surface S, the point x describes a curve c fi of A, and the point a describes a curve c 6 ' of 2'. The point y describes on A a curve which, as seen in the next Article, is of the fourteenth order. If A is a symmetrical determinant, i.e. if p' = q, p" = r, p'" = s, etc., the surfaces 2, S' coincide, A becomes the surface known as the symmetroid (Art. 8), and 2 the Jacobian of four quadrics. 99. Trisecants of c 6 *. Effecting any linear substitution for the \i merely alters the form of the p x , etc., hence any curve c 6 will be represented by the * Schur, Math. Ann. xx. 112 164 DETERMINANT SURFACES [CH. IX curve obtained by taking A, 4 = as the linear relation connecting the \ ; this gives the sextic curve P -P = 0. r ... r Now the three planes MJ/+ = w v + . . =0 .(4) will be coaxal if equations (3) are satisfied with X 4 equal to zero ; and since c e may be written in the form p' p" p'" M + = 0, the axis of the three planes will intersect c 6 in three points, viz. where it meets the cubic surface | p'q"r'" | = 0. This axis meets A in a fourth point y for which, in addition to equation (4), we have the equation So that any point x of c 6 determines one set of values (X x , X 2 , X 3 , 0), and hence one set of values for the f which makes the planes (4) coaxal, and therefore one trisecant of c 6 ; this line meets A in a fourth point y, viz. the point which corresponds to x. As x describes c 6 , these trisecants form a ruled surface of the eighth order whose intersection with A is c 6 counted thrice, together with a curve of the fourteenth order, the locus of the points y. For two of the planes (4) being their intersection will meet any line p^ if where the o f which occur in the P f , Qi satisfy the equation Q R = 0. 99, 100] DETERMINANT SURFACES 165 Hence the points a which give the number of trisecants which meet the line p^ are apparently twelve in number, but of these points four are those determined by the equations P Q = o, and these points do not in general satisfy the equation and hence must be excluded. The ruled surface is therefore of the eighth order. It meets A in c 6 , and also in a curve a- whose intersection with any plane A y = is equal to the number of intersections of P Q = 0, and | PQSA \ = 0, excluding again the four points P Q since the four planes 444 = 0; in which the coordinates of one of these latter four points o are substituted in the P i} R i} Si, do not in general concur. Hence the order of o- is 18 4=14; and c e is a triple curve on the surface formed by the trisecants. 100. The Jacobian of four quadrics and the sym- metroid. The determinant A becomes a symmetrical determinant if between its constituents there exist the identities The surface S' or | PtQ+RfSi = 0, is the Jacobian of four quadrics, for it is seen to be a consequence of the above identities that Pi, Qi, Ri, ^ are the respective partial derivatives of a quadratic expression in the quantities a. The surface S' is easily seen to be identical with 2. By change of notation, replacing \ and Of by 166 DETERMINANT SURFACES [CH. IX Xi and yi respectively, the preceding equations (3) may therefore be written, if P f = ^, Q; = r*, etc., *=i they express that the polar planes of the point Xi for the four quadrics S lt 8 2 , S 3 , 84 meet in the point y, and reciprocally; we have therefore determined on the Jacobian of four quadrics a connection between pairs of its points; they are termed corre- sponding points on the Jacobian. The previous equations (1) and (2) may be then written, replacing x it \, y { , ^ by Of, x i} &, yi respectively, 2 -L /Q 3 -L /Q 4 Regarded as arising from these last equations the Jacobian may be defined as the locus of vertices of the cones included in the 4 set of oo 3 quadrics 2i$i = 0. The surface A arising from elimination of the Xi (or yi} is called the symmetroid*; its equation may be written in the form Jll Jl-2 /13 JU / 12 /22 ./ 23 /24 y is jw j33 y 34 f f f f JU J2A JM J44 being derived from the last equations, which may be written /3)=0, (j = l,2,3,4). = 0, It is to be noticed that these equations establish a (1, 1) correspondence between points a, /3 of the symmetroid through the intervention of a pair of corresponding points, x, y, on the Jacobian. The Jacobian has ten lines and the symmetroid ten nodes. To show that this is the case, if we write ^0*$ = ^a^xix^, the * Cayley. 100, 101] DETERMINANT SURFACES 167 condition that ^c^Si should be a pair of planes requires a threefold condition between the coefficients a^, and the number of solutions in the quantities ^ is equal to the number of pairs of planes. Establishing between the a ik six arbitrary linear relations gives a ninefold relation sufficient to determine the quantities a ik . Taking these six relations to express that the quadric should pass through any six given points, the problem is reduced to determining the number of plane-pairs which pass through six given points and this is clearly ten*. The axis of such a plane-pair clearly lies on the Jacobian, hence this surface contains ten lines. For such a point f the four planes or 2,^/^ = 0, i are coaxal, hence all the first minors of | f& \ vanish for this point, which is therefore a node on the symmetroid f; hence the symmetroid has ten nodes ; to each node a one line of J corresponds. 101. Distinctive property of the symmetroid. The tangent cone of the symmetroid whose vertex is at a node splits up into two cubic cones. For taking a node as the vertex A^ of the tetrahedron of reference for the a { , the equations giving the surface may without loss of generality be taken to be - {! (x* The equation of the surface is then a t S 4 ] = 0, (i = 1, 2, 3, 4). J\2 --L ' J "ft J "" ./ * Jl3 J23 J33 J34 f f f f /14 JlA JU /44 I wherein the /# are linear functions of the coefficients of S 2 , S 3 , S 4 and the variables 2 > a s> 4- This is, when expanded, /33 /34 /34 /44 * Cayley. t Since the tangent plane of the surface at the point is seen to be indeter- minate. 168 DETERMINANT SURFACES [CH. IX where A is the determinant | /# | and Fq is the coefficient of /^ in A. The tangent cone from A l is therefore ys4 / but from a known property of determinants / f fu fu hence the tangent cone is and thus consists of two cubic cones*. It was seen that if the tangent cone whose vertex is one node of a ten-nodal quartic surface breaks up into two cubic cones, then the tangent cone for every other node will also break up into two cubic cones (Art. 8). In forming the Jacobian surface determined by any four quadrics we may suppose these quadrics replaced by any four pairs of planes belonging to the system ; and the general Jacobian surface is formed by aid of any four pairs of planes. The surface there- fore contains twenty-four constants ; hence so also does the symmetroid. The number of constants determining the sym- metroid is also seen to be twenty-four from the fact that this is the number of arbitrary constants remaining after expressing that the surface has ten nodes. 102. Construction for the tangent plane at any point of the Jacobian of four quadrics. The vertices of the cones included in the system 2;$; are given by the equations a as 1+ as 2+a d_s, + a as 4=0 l da; 1 9#! a 'dac 1 4 9a? 1 as, , te, + * Cayley, Coll. Math. Papers, vn. 101-103] DETERMINANT SURFACES 169 Let y be the point corresponding to x on the Jacobian; differentiate these equations and multiply the results respectively by y^ ... 2/4, then by addition we have, since i=l which is seen to be the same as But the polar plane of y for the cone of the system whose vertex is x is ^,. dSi Vf . dS 2 _->,. 9^ ^,,. dS 4 ..Sf^ + ^Sf,- +a>S f,_ +a.Sf <3 - = 0, it passes through a?, and, from the preceding equation, through every point on the Jacobian consecutive to x ; it is therefore the tangent plane to the Jacobian at the point x*. Hence, the tangent plane of the Jacobian at any point P is the polar plane of P', the corresponding point, for the cone of the system of quadrics whose vertex is P. Two geometrical definitions of the Jacobian of four quadrics have been already obtained : since the line joining two corre- sponding points is divided harmonically by any quadric of the system, then assuming arbitrarily any six pairs of corresponding points, the surface may also be defined as the locus of vertices of cones which divide harmonically six given segments f. Two other definitions arise as interpretations of the equations viz. that the surface is the locus of points of contact of quadrics of the system, or that it is the locus of points which have the same polar plane for any two quadrics of the system. 103. Cubic and quartic curves on the Jacobian. When the point x describes any line of the Jacobian surface, its corresponding point y describes a twisted cubic on the surface : * See Baker, Multiply Periodic Functions, p. 68. t Cayley. 170 DETERMINANT SURFACES [CH. IX for let Xi = at + pb i} and P^, P a 2 , P a 3 be the polar planes for any point a of three quadrics of the system ; the locus of y as given by the preceding equations is derived from giving a twisted cubic : this cubic will not intersect the locus of x but is seen to intersect any other line on the surface twice*. There are ten of these cubics ; they are connected with the preceding (1, 1) relationship between points a, /3 of the symme- troid, which is seen to have exceptional points in that to each node QLi of the symmetroid there corresponds a curve the locus of fti, which is of the ninth order and has double points at each of the other nodes. For, taking the node as the vertex A r of the tetrahedron of reference, to A l there corresponds a line on the Jacobian, to this a cubic on the Jacobian, and finally to the latter a curve passing through each of the other nodes twice. To find the order of the curve, the locus of ft, we may take its section by the plane /3 X = ; the number of points of section is equal to the number of intersections of the cubic curve i = 4 3$. x t = a { + pb t , 2 *, p = o, (j = 2, 3, 4), i=\ vyi with the sextic curve > *r~ o > l 9 2/2 92/3 _ / _ o Q /n "> \* * "> */ ' and these are seen to be the nine points of intersection of this cubic curve with the cubic surface = 0, (i,j=2, 3,4). Another set of cubic curves on the Jacobian arise as corre- sponding to plane sections of the symmetroid through three nodes ; these curves intersect the three corresponding lines on the Jacobian twice ; there are thus 120 cubics of this kind. To a plane section through two nodes of the symmetroid correspond on the Jacobian two lines and a twisted quartic, inter- secting the two lines twice. These quartics may be determined analytically as follows : taking the plane-pairs which respectively * This is seen by taking the quadric S 1 as the pair of planes intersecting in another line of the surface. 103, 104] DETERMINANT SURFACES 171 meet on the two lines as uv, u'v' and 8 3 , S 4 any two quadrics of the system, the Jacobian is derived from the equations dS 3 ds 4 W + a 3 ^- 3 + a,~ = 0, du du <, 4 ,,/ + , +,_=(). If in addition we have the relation 4 = ka s , then writing 9 9 5, ,9 , 9 ., v a -- u 57 = $ * 5I> tt 5T> s * 9# 9w> 9v 9w the foregoing may be written BS 3 + &SS 4 = 0, ZS 3 + kS'Si = 0. This gives the Jacobian as the locus of oo J quadri-quartics, each of which twice meets the lines (u, v) t (u r , v') (and no other line of the Jacobian). Any quadric through such a quartic meets the Jacobian in another quartic which twice intersects each of the remaining lines of the surface. In this manner we obtain forty-five pairs of systems of quartics on the surface. 104. Sextic curves on the surfaces. The points yi of the section of the Jacobian by the plane a y = have as corresponding points Xi, the points of the curve j 9$ 2 9$ 3 dS 4 / . 1 O Q /1\ , ^ , = , , at 0, (i = 1, 2, 6, 4). i dtti 6xi oxi This curve has the ten lines of the surface as trisecants*. The locus of associated points o^ on the symmetroid is a curve of the fourteenth order passing three times through each nodef. For the number of points of section of this curve by any plane l> a = is the number of intersections of the preceding sextic with the sextic { i , ' = 0, (. = 1,2,3,4); * This is easily seen by taking S 1 to be the pair of planes which intersect in one of the ten lines. t See Art. 99 for the case of CH in the general surface A. 172 DETERMINANT SURFACES [CH. IX and the number of intersections of these curves was seen to be fourteen (Art. 97). Since the sextic curves lie on the same cubic surface, the latter sextic does not meet any of the ten lines. Again the curve = may be represented by the equations (Art. 99) dS 2 dS 4 _ da 3 -- h ...... + 4 ^- =0, 3# 4 das 4 hence it is the locus of vertices of cones of the system i.e. the locus of vertices of cones which pass through eight associated points. The locus of the points & when a y = 0, is the sextic II Jilt /ifci Ji3> Jilt a i J v, which passes once through each of the ten nodes. On the sym- metroid the curves c e and k 6 are of the same kind, each passes through the ten nodes, they therefore intersect in four other points. 105. Additional nodes on the symmetroid. If (Sij, $ 2 , $ 3 and $ 4 have a common point, by taking it as a vertex of the tetrahedron of reference we may write On = On = C u = (In in the equations of the respective quadrics, so that the highest power of Xi involved in the Jacobian is the second, hence this point is a node on the surface. Moreover f u = in the equation of the symmetroid, so that each term of the equation of this surface contains as factors two of the expressions f lz , f l3 and f u ; hence the intersection of these planes is an additional node on the symmetroid. Similarly if Si ... $ 4 have two, three or four points in common we have additional nodes arising on the symmetroid. Take the case in which the quadrics have two points in common ; if they are the 104-1 06] DETERMINANT SURFACES 173 vertices A l} A 2 of the tetrahedron of reference we have/ u =f w = 0, and it is seen that the plane / 12 = is a trope of the symmetroid, also that the line joining the two nodes on the Jacobian lies on the surface. Hence if the Si have k common points (k = 1, 2, 3, 4) k (k 1) the Jacobian has additional lines and the symmetroid *(*-!) , ^-5 tropes. M If k = 4 the equation of the symmetroid assumes the form v/2/34 + VAA + v/^3 = o. The condition that this surface should have an additional node was seen to be the existence of an identity of the form Af a + A'f* + Bf n + B'f^ + Cf u + C'f u = 0, where AA' = BK = GC' (Art. 12). This condition may be written in the form A : A' : B : B' : C : G' = c^ : C 3 c 4 : c^Cg : C 2 c 4 : crft : C 2 c 3 , the Ci being constants. On reference to the values of the f ik , if we take S l = ^ t a ik x i a; k) i ^ k, etc., the preceding identity is seen to lead to the equation ^12^1^2 i ^34^04 T C^sCjCj + tt24C2C4 + Q'ltCiCi ~T d^sC^Cs ^ U, with three others obtained by writing respectively b ik , c ik , d ik for afc ; and these equations express that the quadrics Si . . . S t have an additional point d in common; hence the Jacobian has an additional node. If the number of common points of the Si is six, the symmetroid has sixteen nodes and is therefore a Kummer surface ; the Jacobian has then twenty-five lines in all, viz. the original ten and the joins of the six additional nodes (Art. 2). 106. Weddle's surface. The Weddle surface is the locus of vertices of cones of the 4 system Sa{$i = 0, where $! = 0, ..., $ 4 = are quadrics having six points in common. Hence the surface is the locus of vertices of cones which pass through six given points. From the definition it is clear that the surface contains the fifteen lines joining the six points in pairs and also the intersections of the ten pairs of 174 DETERMINANT SURFACES [CH. IX planes which can be drawn through the given points. The surface therefore contains twenty-five lines *. Since through each of the six given points five lines of the surface pass, each of these points is a node of the surface. Since the quadrics have six common points, there are three linearly independent quadrics containing the twisted cubic through the six points ; through P any point of the surface draw a chord of the cubic meeting it in L and M, the chord will meet the polar plane of P for each of these three quadrics in the same point P', viz. the fourth harmonic to P, L and M ; hence the line joining two corresponding points (Art. 100) P and P' of the surface is a chord of the twisted cubic and is cut harmonically by it. Since any chord of the cubic is cut harmonically by the surface, any tangent to the cubic meets the surface in three consecutive points, and hence the cubic is an asymptotic line of the surface. 107. Parametric representation of the surface. The coordinates of any point on the twisted cubic may be represented in terms of a parameter by the relations x l : x z : ac 3 : x 4 = 6 s : 6 Z : 6 : 1. If Ay B are any two points on the twisted cubic having parameters 6, , then if L, M are the two corresponding points of the surface on A, B their coordinates are given by the relations x l : ac 2 : x s : x = md 3 + n : m n"\ ; since L, M divide A, B harmonically. Let Saijfe#i#jfc = be any quadric through the six points, then L, M are conjugate points for this quadric; expressing this fact we at once derive the equation m 2 (a u 6 + 2a 12 8 +...) = n>(a u 6 + 2a 12 < 5 + ...); also if B! . , . # are the parameters of the six points, then o n 1 6 + 2a 12 1 5 +... = 0, with five similar equations ; it follows that n* -f(0) ' where /(a) = (a - 0,) (a - 2 ) (a - 0,) (a - 0.) (a - 0.) (a - 6 ). * See Art. 2. t Richmond. 106-108] DETERMINANT SURFACES 175 Hence the points of the surface are parametrically represented by the equations CO i . tX/2 I QG * (K^ 11 108. Systems of points on the surface -f^. If we represent the quantity (18 s + ra0 2 -f n9 + pflf(6) \)jF(6), then F(6) F() = is the tangential equation of a pair of corre- sponding points on the surface. Let 6, , i/r, ^ are parameters of any four points on the cubic, we obtain six pairs of points aa' = F(((>)-F(^), axe' = F (ff) - F( X \ ftff = F(V) - F(e\ yy' = F (#,) - F( x ), 77' = F (6) - F(4>), zz' = FW - F( X ), whence arises the relation aaxx' + @/3'yy' + yy'zz' = 0, showing that the tetrahedra whose vertices are (a, a', x, x'), (0, /3', y, y'\ (7, 7', z, z') respectively form a desmic system (Art. 13). * For another method of obtaining these equations see Bateman, Proc. Land. Math. Soc., Series 2, Vol. in. p. 227. t See Bateman, loc. cit. 176 DETERMINANT SURFACES [CH. IX Conjugate quintic curves on the surface. Let S, S' be two consecutive points on the twisted cubic through the six nodes, R any other point on this cubic; then the sides of the triangle RSS' will meet the surface again in three pairs of points PQ, P'Q', TT' lying by threes on four lines. Hence PP' and QQ' which are ultimately tangents at P and Q intersect in a point T on the surface, and since the corresponding point T' ultimately coincides with S, the polar planes of S with regard to quadrics through the six nodes meet in the point T which lies upon the tangent at S. As R moves along the cubic the point T remains fixed, the points P, Q describe the curve of contact of the tangents from T to the surface. Again if U be the point derived from R in the same way that T was derived from S, and R is fixed while S varies, the points P, Q will describe the curve of contact of the tangent cone from U ; TP, TQ are the tangents at P and Q to this curve of contact. Now UP, UQ are generators of this cone; hence PU, PT are conjugate tangents to the surface at P; thus the curves obtained by keeping one point on the cubic fixed form a conjugate system. To find their order we insert the coordinates of P in any plane f = ; if 6 be constant we obtain (c^fl 8 + 0,20* + a s + a 4 ) 2 which is a sextic in , but rejecting the solution = we obtain five as the number of points of intersection with any given plane. As in the case of the Jacobian of any four quadrics the tangent plane at any point P of the surface is the polar plane of the corresponding point P' for the cone of the system whose vertex is P. It is determined analytically as follows : the plane &B! + mx 2 + nx 3 + px t = will pass through the point (#, <) on the surface if 16 s + mfr + n6 + p _ l 3 + mty + n+p It will pass through the consecutive point (6 + SO, + S(f>) for all values of 86 : 8 provided that 108] DETERMINANT SURFACES 177 1. (W 3 + m ^ 2 + ne + P\ _ o 80V VTPT / A fty 3 + ra 2 + ft(ft + A _ 8^V V/(^) /" and will then be the tangent plane at (6, <). These equations show that if 6 has a given value and varies, the tangent plane always passes through the point _ 8 / & A \ 8 f_B a \ 3 * 4 ~ : : If in the preceding, S is the point 6, then the coordinates just given are those of the point T. It is easily seen that the locus of T is a rational curve of the seventh order. It follows from the preceding equation of the tangent plane that the equation of Weddle's surface in plane coordinates is obtained by expressing that the equation (W s + m<9 2 + nd + p) 2 - kf(0) = should have two pairs of equal roots for some value of k. The differential equation of the asymptotic lines may be arrived at in the following manner : the tangent plane at (6, ) will pass through a consecutive point (6 + 80, ) if g &_ \ 80' I Also, since (Imnp) is a tangent plane, we may write (la? + mo? + nx +p\ 2 _ , _ \(a>- 0)* (x - ) 2 (x - a) (as - /3) t jjije) ) ' /(> Differentiate this equation twice with respect to x and then write x = 6; since d_ {IB 3 + mB* + uB + p] _ w\ ~3}\6) r we obtain W 3 + mB 2 + n0+p 8 2 {16 s ~~ together with a similar equation in < : hence the differential equation of the asymptotic lines becomes -/3) rf ,, J. Q. S. 12 178 DETERMINANT SURFACES [CH. IX where a and ft may be regarded as defined in terms of 6 and $ by the fact that kf(as) + \(x- 0) 2 (x - ) 2 (x - a) O - /3) is a perfect square. 109. Forms of the equation of the surface. The surface being defined as the locus of the vertices of cones through six given points, let p ik denote the coordinates of the line joining x to the point a, and g# the coordinates of the line joining x to the point b, the other given points being the vertices of the tetrahedron of reference. Then since the six lines (x, a), (x, b), (x, AI) ... (x, A 4 ) lie on a quadric cone, the anharmonic ratio of the pencil formed by the planes (p, A-^)...(p, A t ) is equal to the anharmonic ratio of the pencil (q, A^) ...(q, A t ). But these two anharmonic ratios are determined by the ratios PvPu. : _Pl3^>42 : PuP'23 and j~f ( /v i nt* i~i ( "7 1 I T* // " A/J J..1, l w ^ vt/2 -*- \** / / l * / 3 ** / 4 5 finally the point (A 1} A 2 , A 3 ) has the coordinates x l 'H(x) : x^H(x) : x 3 H(x) : x 4 . Moreover the line (A^, x) meets the surface in a point whose coordinates are -'-.'. as. '. H(r\ fcC^ */2 *^S 4 yw / Hence (^ 2l 2 , 4 3 )= (a, 6, J. 4 ), since H(x} H(x'} = 1. We thus arrive at a closed system of thirty-two points on the surface, from any one of which the others may be derived. 111. Cartesian equation of the surface *. If we take four nodes as being situated at the origin and at the points at infinity of the (Cartesian) axes of coordinates, the others being A and B, the equation of the surface assumes the form (iib 1 a. 2 bz u 3 b 3 = 0. If the point P, or x, be joined to the points at infinity on the axes and to the origin, these joining lines will, as has been seen, meet the surface again in the points (!&! rr/ , ) ~H(x), x 2 , SB,}, ( X l ) Let X be the point in which the line PB meets the surface again, then transferring the origin of coordinates to A, the new coordinates of x, 0, and B respectively are * Baker, Elementary note an the Weddle quartic surface, Proc. Lond. Math. Soc., Ser. 2, Vol. i. (1903). 182 DETERMINANT SURFACES [CH. IX Hence, for the former origin, the coordinates of the point in which OX meets the surface again will be i _ ' T O'i (x- \ ^-d i J = d i (d i -b i )- ) from which we find Now denoting by 6(x), <(#), -^r(x) the three points derived from Xi by the transformations it is seen that these points all lie on the surface, and the eight points derived from P by its projection from the nodes 0, A and B form the four couples x (Oab) or (ba) (0) *() abH ab ID x H x (aO) (x a) (a) (60) x b ab a(x 6) X x a, These show, as above, that the point (0, a, b) is identical with (P, Q, R), where the latter point is that obtained by successive projections of x from the points P, Q, R, at infinity on the axes. 112. Geiser's* method of obtaining the surface. Let H! = 0, . . . u 6 = be the tangential equations of the six given nodes, then the six quadrics Ur? 0, . . . u/ = are linearly inde- pendent and are apolarf to any quadric through the six points. Hence the general equation of a quadric apolar to the system of quadrics through the six points is * Geiser, Crelle's Journal, LXVII. (1867). t Two quadrics whose equations in point and plane coordinates are 2a ik x i x lc = 0, l = are said to be apolar when the invariant Za^a^ is zero. When the second equation represents two points, it easily follows that they are conjugate. Ill, 112] DETERMINANT SURFACES 183 When this equation represents two points they are conjugate for all quadrics through the six points and are therefore corre- sponding points on the Weddle surface. We then have an equation of the form LL' = 2 kin*. Now let M = 0, iV=0 be two points which divide the points L, L' harmonically, hence an identity exists of the form It is easily seen that this is the necessary condition in order that any quadric through seven of the eight points M, N,UI ...u 6 should pass through the eighth point*; hence every quadric through M! . . . u 6 and M will pass through N, and every pair of points on LL' which possesses this property divides LL' harmonically. Such a pair of points can only coincide at one of the points L, L'. It is therefore seen that the Weddle surface arises as the locus of points M such that the point conjugate to M in this manner for the six given points Ui coincides with M"\. From this point of view the surface has been shown as a linear projection in four dimensions I; and projectively related to Rummer's surface. For if we write a be, a = b'c, ft = ca, ft = c'a, 7 = ab', y = a'b, the equation of the general quadric surface through the six nodes of the Weddle surface in Cartesian coordinates (Art. Ill), wherein we write a, b, c for a 1} a 2 , a 3 ; 1, 1, 1 for b ly b 2 , 6 3 ; also x, y, z for #1, %2, #3 and x, y, /, a, b', c' for 1 x, 1 y, 1 z, 1 a, 1 6, 1 c, is NOW | + 7?+ =' + ,/ + ', so that if we interpret (, rj, , f ', 77', ^) as homogeneous point- coordinates in four dimensions, we have a (1, 1) correspondence between the points of our original space and those of a cubic variety in four dimensions. Again those of the quadric surfaces * Serret, Geometric de Direction, Nouv. Ann. iv. (1865). t See Bateman, loc. cit. p. 228. Baker, Zoc. cit., also Hudson, Rummer's Quartic Surface. 184 DETERMINANT SURFACES [CH. IX which pass through a seventh point (X, y lt z^ or P l} have as their equation {A K' - a'f ) + B W - /3'r,) + C - 7'0} (|^ - ^\ \p p 7 7 where = ^ */, . . . ' = ar/yi These quadrics all pass through an eighth point (x, y, z) or P, such that 7-7 These three equations determine the four- dimensional line joining (a/3, . . . ) to (^i%, . . . ); the remaining intersection of this line with the cubic variety is the point (a + Xf,, ...) where A, is given by the equation x (eftfcgi + 0ri& + 7^171 - v&' - ^Ci 7 ?/ - y? iV) + f^7 + ^i7a + i - ^'/SV - 77/7'a' - f/a'yS' = and corresponds to the eighth intersection P of the quadrics. The points P, PI therefore coincide when this straight line touches the cubic variety, this requires that X should be infinite, so that if we insert = yi^i, etc. we obtain another form of equation of the Weddle surface. This surface thus arises as the interpretation in three dimen- sions of the twofold of contact of the enveloping cone of a cubic variety in four dimensions, whose vertex is an arbitrary point of the variety. It has been shown* that the intersection of this cone with an arbitrary planar threefold in space of four dimensions is a Kummer surface. We are therefore led to a birational trans- formation between the Weddle and Kummer surfaces in the form of a projection ; the point (x, y, z) of the Weddle surface being birationally connected with the point (, 77, , ', 77', ') of the twofold of contact which is projected into a point of the Kummer surface. * Richmond, Quarterly Journal, xxxi., xxxiv. 112-114] DETERMINANT SURFACES 185 113. Sextic curves on the surface. Any quadric through the six nodes meets the Weddle surface W in an octavic curve. This quadric corresponds, as just seen, to a planar threefold and hence the octavic curve to a plane section of the Kummer surface K. If the quadric is a cone its vertex P lies on the Weddle surface W, hence the octavic has a node at P, and therefore the plane section of K is a tangent plane whose point of contact Q corresponds to P. It was also seen that a system of quadrics through the six nodes and one other point corresponds to planar threefolds passing through a line, and hence to plane sections of K through a fixed point A (the intersection of this line with the planar threefold containing K}. Hence the sextic curve which is the locus of vertices of cones of the system will therefore correspond to the curve of contact of the tangent cone from A to K. Since to any two quadrics S, 8' through the six nodes there correspond two planes in the space in which K exists, it follows that the vertices of the four cones determined by 8 and 8' corre- spond to the points of contact of four coaxal tangent planes of K. When the quadric contains the twisted cubic through the six nodes, the octavic breaks up into this cubic and a quintic curve. If the quadric is a cone these quintics become identical with those discussed in Art. 108*. Another set of sextic curves is seen to arise as the inter- section with W of any cubic surface having nodes at four nodes of W and therefore containing the lines joining those nodes in pairs; for the curve of intersection consists of the six joins of the four nodes and a sextic curve. 114. Expression of the coordinates as double Theta functions. The coordinates of any point on a Weddle surface can be expressed in terms of double Theta functions f. For the equation of the surface (Art. 109) is satisfied by the substitutions x- : x : x : # 4 = c ol B 01 3 0^0 O ' c 2 #<9## : C ^^^^ : CQQ * This is seen at once since the point whose coordinates are - =t / , etc., Jm for 6 constant, lies on tlie cone (x. 2 - Ox.^ 2 = (xj - 8x 2 ) (x 3 6x 4 ). t Caspary, Ueber Thetafunktionen mil zwei Argumenten, Crelle, xciv. 186 DETERMINANT SURFACES [CH. IX where c is the result of attributing zero values to the variables in # (u\ etc., as will now be shown. In the first place we see that the coordinates of the point x' or - - are derived from those of x by increasing the argument by Xi ^(rc + d)*; since this interchanges O lt # 01 ; # 2 > #02, etc. Again, as before, let p ilc denote the coordinates of the line (a, x) and q ik those of the line (b, x) ; we find on substitution PwP&l _ VC5C 12 t7oit/02 C C 34 17 j l/ 2 ) ( C 5 C 34 U 03 C/ 04 C Cjg "3 6/4) C 23 C J4 ,.. , P\zP#l (Cl2Cl4 C'o! #3 C-23 C M C7 03 C/j) (C 12 C^ C/ 4 C/02 C 14 C 34 #2 #04) Cfl C 5 If gs-fc denote the line joining x to a, it is easily seen that 13 5^ 42 79 2 t^ by and the latter ratio is formed, as stated above, from increasing the argument of the #'s by ^(rc + d). It will now be shown that this change does not affect the right side of equation (1). For, as is well known, the determinant 5 5 000 C 4 GO C -C forms an orthogonal matrix, from which we may therefore derive the equations C Cj " C7j 2 C C M l/o C/ 34 = C 4 Cfl 3 "4 t .(2). C 12 C] 4 C7 1 2C'14 ^23 C M t/23 " 34 = C 2 C 4 U. 2 "4 Ci2 C2S t/i2 C/23 C 14 C 34 C/ 14 1/34 = C i Cos "oi "o Increasing the arguments in each of these equations by the respective half-periods (rd + 6), (TO, + a), ^ (TO. + &), (ra + a), the left sides are transformed into the quantities which appear on * Using the notation of Hudson, Rummer's Qnartic Surface, p. 178. We com- pare for convenience Hudson's notation with that just given it /3/3/) 4 Q O . * ) f) A f fi ft A ^Ol U 03^2^4 ^o P" "3 "to t/03 PflS "02 ^04 ^4"4 u s"02) since #j divides out. The other five points (N{) and the ten points (^N^N^) are derived from (N-i) by the addition to its argument of the fifteen half-periods*. * Weddle's surface is a case of a class of surfaces investigated by Humbert, Theorie generate des surfaces hyperelliptiques, Journal de Math., serie 4, t. ix. (1893). These surfaces are termed hyperelliptic surfaces, the coordinates of any point are uniform quadruply periodic functions of two parameters ; see also Hudson, Kummer's Quartic Surface, pp. 182-187. Baker has shown that the coordinates of any point on the Weddle surface may be expressed as derivatives of a single variable (Multiply Periodic Functions, pp. 39, 40, 77). 188 DETERMINANT SURFACES [CH. IX 115. Plane sections of the surface. The equation of the plane section of a Weddle surface may be simply expressed. Take as triangle of reference the three points in which the given plane meets the twisted cubic through the six nodes. Each side of this triangle meets the curve of section in two vertices and also in two points harmonic with these vertices. Hence we obtain as the equation of the surface a 2 x?y + a 3 a?z + b s y s z + b^x + dz s x + c 2 z s y + Sxyz (lx + my + nz) = 0. Also, since the three pairs of points lying on the sides of the triangle of reference lie by threes on four lines (Art. 108), we have the condition = 0. From the last condition we infer that the tangents at the vertices of the triangle of reference are concurrent. If we form for this quartic the invariants A and B* we find A = 12lmn +12 (Ib^ + wc 2 a 2 + na 3 b s ), I a 3 a. 2 6 3 m &! C 2 G! n Hence for any plane section we have the invariant condition A 2 + 144 = 0. An infinite number of configurations of points can be obtained on the plane section as follows : let the Weddle surface be deter- mined by four quadrics Si, $ 2 > S 3 , S t , of which we may suppose the first three to contain the same twisted cubic. Then the section considered contains the following set of twenty-five points, viz. the fifteen points in which the join of two of the nodes N! ... N 6 meets the plane, and the ten points in which the ten lines (NiN^Ns, N t N 5 N 6 ) meet the plane. The first set of fifteen points lie by threes on twenty lines, viz. the intersection with the given plane of the planes (NiN 2 N a \ etc. Now consider the Weddle surfaces formed by aid of S l} S 2 , S 3 and S 4 + \a?, where a = is the plane of the section. These surfaces form a pencil whose nodes lie on the same twisted cubic, and all containing the same section lying in a = ; from each surface * Salmon, Higher Plane Curves, 3rd Ed. p. 264. 115, 116] DETERMINANT SURFACES 189 one configuration, of the kind just mentioned, arises, have an infinite number of such configurations *. Hence we 116. Bauer's surfaces. If in the foregoing four collinear systems (Art. 97) each plane system reduces to a sheaf, and is such that each plane joining the centres of three sheaves is a self-corresponding plane for three systems, we obtain the surface discussed by Bauerf. The equation of such a surface is accordingly b t ' 2 ~6, /* O/Q X.i x. This equation may also be written in the form = 0. T I 1 T~ d x wherein a x , b x , c x , d x are linear functions of the coordinates. The foregoing equation may also be obtained as follows : a point P (or x) is joined to the vertices of a given tetrahedron A (taken as that of reference) and the joining lines PA 1} etc., meet the faces of any other given tetrahedron A' (whose faces are a x = 0, . . . d x = 0) in points Qi ... Q t ', then if the points Q t are coplanar the locus of P is the surface just given. For the coordi- nates of Qj are seen to be x l -- -, x, x z , ac 4 , and expressing that the O.J points Qi are coplanar, we obtain the foregoing equation. The second form of equation of the surface shows that the edges of A' lie on the surface and also the intersections of corre- sponding faces of A and A', as x = 0, a x = 0, etc. ; the vertices of A' are seen to be nodes of the surface. The surface therefore possesses ten lines and four nodes. Denoting the lines (#1, a x ), etc. by p lt etc., if two lines p * See Morley and Conner, Plane sections of a Weddle surface, Amer. Journ. of Math. xxxi. f Bauer, Ueber Flachen 4. Ordnung deren geom. Erzeugung sich an 2 Tetraeder knilpft, Sitz. d. Konig. Akad. d. Wiss. Miinchen, 1888. 190 DETERMINANT SURFACES [CH. IX intersect, their point of intersection is seen to be a node of the surface ; if each line p meets every other line p, then the lines p lie in one plane, say the plane z = 0, also each edge of A meets the corresponding edge of A' and the two tetrahedra are in per- spective. In this case the equation of the surface assumes the form x b x c x ] = a x b x c x d x , where the Xf are constants. For in this case we may write b x , etc. The surface is the Hessian of the general cubic surface ; it has ten nodes of which six lie in z = 0. Let now an edge of A intersect the edge of A' opposite to the corresponding edge, e.g. let the line (x 1} x 2 ) intersect the line (c x , d x ) ; in this case it is easily seen that (x lt x 2 ) lies on the surface ; if this occurs in every case the surface will contain also the six edges of A and have the vertices of A as nodes *. Lastly we may assume that both sets of conditions are satisfied, viz. that each edge of one tetrahedron intersects a pair of opposite edges of the other. The tetrahedra are then in desmic position (Art. 13). The equation of this surface, viz. . x l + - - - + -- - + 1=0, X 3 + X t a?j + may be reduced either to the form Z (XiX z X 3 + X z X s %t + X s X t X l where z = "S,Xi, or to the form VJ 1 + VF 2 + where Z l = (x l + # 2 ) (x 3 + x t ), * The equation of this surface may be written in the form AZ* + BZ 2 * + CZ 3 2 + DZ^t + EZ^Z 3 + FZ 3 Z l = 0, where the Z i are pairs of planes through opposite edges of A'. 116, 117] DETERMINANT SURFACES 191 117. Schur's surfaces. A particular case of the surface A arises when the foregoing correspondence between points x, y of A is reduced to a collinea- tion*. It has been seen (Art. 99) that as x describes a curve c the corresponding point y describes a curve which is the locus of the fourth intersection with A of the trisecants of c 6 , but since the points x, y are to be in this case linearly connected, y must also describe a curve of order six, hence the intersection with A of the surface formed by the trisecants must include eight of these trisecants ^...ag, in order to complete the order, 14, of the complete intersection of the surfaces apart from c 6 . Similarly every k 6 has eight trisecants b^ ... b s which lie upon A. The lines a and b are distinct and no two lines a intersect each other ; similarly no two lines b intersect. For, in this case, to the point x of c 6 which gives rise to a line a there corresponds an infinite number of points y, viz. the points of a t - ; hence the four planes a 4 p'" = 0, =0, =0, must be coaxal ; by effecting a linear transformation of the o^ we may take four of these points a as vertices of the tetrahedron of reference, in which case the four planes p, q, r, s are coaxal ; similarly for the four planes p', q', r', s', etc. The eight lines b arise from such values of the \i as make the following four planes coaxal : \i> + \2j the quadric 2^^ = 0, where i jj = C\ T X ~\~ ^iX^X^, &2 ^= ^ \X^ = x. it is clear that the planes Oj ... 4 are the polar planes of the vertices A^ ... A 4 of A for the quadric ^> 1 . Two tetrahedra such that the faces of one are the polar planes for a quadric of the vertices of the other, may be termed conjugate. Again it is easily seen that A and A' are conjugate for the quadric and hence for each of the quadrics 3 = u^s + u 2 St + u 3 S l + u 4 S 2 = 0, w 2 $! + u 3 S 2 + uS 3 = ; since 3> 3 , 4 are the quadrics obtained by submitting ^ and 3> 2 to the given collineation. Hence A and A' are conjugate in four ways. Now it can be shown that when the tetrahedra A, A' are j. Q. s. 13 194 DETERMINANT SURFACES [CH. IX conjugate, four faces of A meet four faces of A' in four lines which belong to the same regulus* of a quadric. Now since A, A' are conjugate with regard to each of four quadrics, it occurs four times that four intersections of their faces are co-regular. But if a quadric 2 meet a non-ruled quartic surface F 4 in four lines of a regulus, it will meet F* in four other lines of the complementary regulus ; since if c 4 be this residual curve of intersection, from each point of c 4 a line can be drawn to meet the four given lines, this line therefore lies upon F', hence c 4 must consist of four lines of the other regulus of S. Therefore corresponding to each way in which A, A' have four intersections co-regular we obtain four lines of A, giving, in addition to the sixteen lines of intersection of A and A', sixteen other lines upon the surface. The existence of these thirty-two lines upon the surface may also be seen from the expression of the surface in the form pqY's'" = p'"qr's" ; for this shows the existence of eight lines not included in the eight lines a or the eight lines b, e.g. p = r' = 0, etc. If we had started with the other four lines a and b we should have obtained a second form of the equation of the surface in the form A^A^U, where A l5 A/ are two new tetrahedra; they again yield eight lines not included in the eight lines a and b. * For A being the tetrahedron of reference, and the quadric with regard to which A and A' are conjugate being 2a ac x i x k =Q, the four lines just referred to are j=:0, Oj 2 a: 2 H- 0,3X3 + a 14 x 4 = 0, etc. If the join of two points A', Y meets this line we have hence if p ac =X i Y k - X t Y it it follows that The conditions that p ik should meet the other three lines are seen to be, similarly, 42 +1>43 43 = 5 and since jp ......... (II), it is seen that the surface Ax : x z x z x 4 + (mx-L + # 2 + x 3 + x 4 ) (x^ + mx. 2 + x 3 + # 4 ) (...)(...) = is unaffected by each collineation. There are six planes each containing two intersections of faces of A and A' and two other lines, viz. the planes #1 + #2 + m (x 3 + # 4 ) = 0. The surface therefore possesses 16 + 12 = 28 lines; it has six 4 coplanar nodes, viz. the points x l = x. 2 ^Xi = 0, etc. i Next consider the collineation pXi = V2, px 2 = -y*, px = y4, p^ = yi ...... (Ill), the collineation (II) being as before. The surface Aae^assXi + (x 2 + x 3 + # 4 ) (x l x 2 + x 3 ) (x l -x s + # 4 ) (#] + x z x 4 ) = is unaffected by the collineations (II) and (III). The tetrahedra A, A' are conjugate in nine ways; in six ways arising from the six quadrics # 3 2 x? + 2#i (x 3 + &' 4 ) 2 2 (#3 #4) + 2# x # 2 = 0, etc., and in three ways arising from the three quadrics #i 2 # 2 2 + #s 2 #4 2 + 2# 1 # 2 + 2# 3 # 4 + 2x 2 x s 2#!# 4 = 0, etc. ; each manner in which A and A' are conjugate gives rise to four lines on the surface, which is thus seen to possess 16 + 9x4 = 52 lines in all. Each of the tetrahedra A, A' is inscribed in the other. INDEX OF SUBJECTS The numbers refer to the pages Anchor-ring 107 Apolar conies 136; quadrics 182 Asyzygetic surfaces 10 Base-points of representative cubics 41 Bauer's surfaces 189 Bidouble line 70, 78 Biplanar nodes of three species 67 Class of surface with double conic 46 Close-points 70 Coincident base-points 64 Collineations which leave two tetrahedra unaltered 193, 195 Cones of first species in four dimensions 56; of second species 77 Confocal cyclides 109, 115 Conies on surface 4> 2 = 0,875 133; on surface with double conic 38; on surface with double line 119, 130 Conjugate tetrahedra 193 Constants, number of in quartic surface with a double conic 39 Contact-conic 3 Correspondence of points upon the sur- faces A 163; the Jacobian of four quadrics 166 Cremona transformation 156 Cubic complex connected with desmic surface 27 Cubic curves on surface with double conic 43; with double line 124 Cuspidal double conic 69 Cyclide, canonical forms of equation 99 ; Cartesian equation of 86 ; con- jugate points on 115; shapes of 91, 101; lines of curvature of 117 Cyclides, confocal 109, 115, 116; cor- responding points on 111 Desmic surface 26 ; conies on 37 ; plane sections of 31-35 ; quartics on 27; section by tangent plane 35; sixteen points, group of 32 Desmic tetrahedra 24 Determinant surfaces 161 ; correspond- ence of points 163 ; sextic curves on 162 Dianodal surface 10; curve 12 Double plane 152 Dupin's cyclide 106 Focal curves of cyclide 87, 91 Fundamental inversions of quartic sur- face with double conic 59 Fundamental quintic for cyclide 85, 93 Hessian of cubic surface 190 Inverse of cyclide 107 Inverse points on cyclide 89 Involutory property of the surface =0 70 Jacobian of four quadrics 2, 165; cubics on 169 ; quartics on 171 ; sextics on 171 ; ten lines of 167 ; tangent plane of 169 Rummer's surface 22, 173, 184 Monoid, the quartic 147; twelve lines of 147; with six nodes 149; cubics and quartics of 147 Nodal sextic curves 4 Nodes, maximum number of 1 Normals to cyclide from external point 108 Orthogonal system of five spheres 97 Pentaspherical coordinates 98 Perspective relation of surface with double conic and general cubic sur- face 49 Pinch-points on surface with double conic 38 INDEX OF SUBJECTS 197 Pliicker's surface 128 Power of two spheres 94 Principal spheres of a cyclide 87 Quadrics, five associated with cyclide 87; inscribed in surface with double conic 59, 79 Quartic curves on surface with double conic 38, 45, 58; with double line 125 Quartic, plane, and surface with double conic 52 Eational surfaces 152 Rationalization of surface with double conic 40 ; with double line 120 ; Pliicker's surface 128; Steiner's sur- face 135 Ruled surfaces, arising in special case 68 Schur's surfaces 191 Sextic curves, plane, with ten nodes 6 ; with seven nodes 4 Sigma functions, in desmic surfaces 27 Sixteen lines of surface with double conic 39, 46; with double line 118 Sphero-conics on cyclide 113 Steiner's surface 81, 132, 134; asym- ptotic lines of 137; conies on 35; quartics on 136, 141 Symmetroid 14, 166 ; ten nodes of 167 ; sextic curves on 171 ; curves of ninth order 170; curves of fourteenth order 171 Syzygetic surfaces 10 Table of forms of surfaces with double conic 82-85 Tacnode 132, 152 Ten lines of Jacobian of four quadrics 167 Ten-nodal sextic curves 6 Torsal line 37, 63, 127 Trisecants of sextic curves on surfaces A 164 Trope 4 Uniplanar node 68 Weddle's surface 173 ; asymptotic lines of 177; conjugate quintics 176; con- nection with Kummer's surface 184; expression of coordinates as double Theta functions 185 ; forms of equa- tion 178; group of thirty-two points 180 ; octavic curves on 185 ; plane section of 189 ; septic curve on 177 INDEX OF AUTHORS The numbers refer to the pages Baker 130, 169, 181, 183, 187 Bateman 175, 183 Bela Totossy 70 Berry 134 Bioche 37 Bobek 48, 51 Bocher 99, 103 Bromwich 72, 99 Caspary 185 Cayley 10, 166, 167, 168, 169 Clebsch 34, 41, 120, 135, 152 Conner 189 Cotty 137 Cremona 35, 49 Darboux 86, 108 de Franchis 134 Eckhardt 145 Geiser 46, 49, 182 Harkness 29, 31 Hesse 152 Hierholzer 178 Hudson 183, 186, 187 Humbert 24, 32, 113, 116, 187 Korndorfer 64 Kummer 38, 131 Lachlan 94 Lacour 137 Morley 29, 31, 189 NSther 133, 152, 157 Eeye 139 Eichmond 174, 184 Kohn 3, 147 Schroter 143 Schur 163, 191 Segre 55, 72, 78, 82 Serret 183 Sisam 133 Sturm 139, 140 Weierstrass 143 Weiler 49 Zeuthen 52 CAMBRIDGE : PRINTED BY THE SYNDICS OF THE UNIVERSITY PRESS. 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