272 MR. HANUMANTA RAO [Dec. 6, ON THE CURVES WHICH LIE ON THE QUARTIC SURFACE IN SPACE OF FOUR DIMENSIONS, AND THE CORRESPOND. ING CURVES ON THE CUBIC SURFACE AND THE QUARTIC WITH A DOUBLE CONIC By C. V. HANUMANTA RAO. [Received October 22nd, 1917.—Read December 6th, 1917.] THE quartic surface in space of three dimensions which has a double conic was discussed by Segre (Math. Annalen, Vol. 24, 1884) as the pro- jection, from an arbitrary point, of the quartic surface F in space of four dimensions which is the intersection of two quadratic varieties in that space. When the centre of projection is taken on F itself, the surface projects into a cubic surface S3- Part I of this paper begins with a short account of some elementary properties relating to four-dimensional space and a brief re"sum6 of the chief properties of F proved by Segre. I then proceed to show how the properties of lines, conies, and cubics on Sd can be deduced from the properties of the corresponding curves on F. But it is found that the cubics on F do not exhaust the cubics on S3. In Part II, I therefore develop a theory of coresiduation on F. This part of the paper follows closely a paper by Prof. Baker (Proc. London Math. Soc, Ser. 2, Vol. 11, 1912). A good deal of this work could have been deduced as a generalisation of the work in Prof. Baker's paper, but a development from first principles was thought preferable. It 13 shown that all curves on F can be expressed in terms of six funda- mental elements. The concept of a double six on F is then introduced, and it is shown that there exist conical varieties in space of four dimen- sions with properties similar to those of Schur's quadrics, these being quadrics associated in a particular manner with the cubic curves on a cubic surface.* In Part III, the results obtained in Part II are projected, and a corresponding theory of coresiduation on the quartic surface in three dimensions which possesses a double conic is briefly sketched. It * See Baker, Zoc. cit., p. 300, and Roye, referred to below. 1917.] QlTARTIC S.URFACE IN SPACE OF FOUR DIMENSIONS. 273 is known* that the lines on this surface can be grouped in double fours; and it is here found that the double four is connected with rational quartics on the surface in the same way as cubics on S5 are connected with the double six. The work is in the first place an example of co- residuation, and in the second place an illustration of the principle of double projection which consists in a projective generalisation from the cubic surface to T, followed by a projective particularisation from V to S4, the quartic with a double conic. Throughout the preparation of this paper I have been in constant consultation with Prof. Baker. I render to him my grateful thanks. The following is a list of the more important references which I found useful in the preparation of the paper:— 1. Segre, Math. Annalen, Vol. 24, 1884, pp. 313-343. 2. Jessop, Quartic Surfaces, 1916, Chapter III. 3. Baker, Proc. London Math. Soc, Ser. 2, Vol. 11, 1912, pp. 285- 301. 4. Reye, Math. Annalen, Vol. 55, 1902, pp. 257-264. 5. Berzolari, Annali di Math., Ser. 2A, Vol. 13, 1885, pp. 102-120. PART I. 1. Any point in space of four dimensions may be represented by five homogeneous coordinates x, y, z, t, u. A locus of three dimensions represented by one rational equation is called a variety or threefold. An oo2 locus is called a surface or twofold, and is given by two equations. An oo1 locus is called a curve or onefold, and is determined by three equations. When these loci are linear, they are called hyperplane, plane, and line respectively. The general hyperplane is represented by 2, ax = 0 : a plane by two such equations, a line by three, and a point by four such equations. Two planes intersect in a line or a point only, according as they do or do not lie in the same hyperplane. A hyperplane is deter- mined by four arbitrary points and a plane by three. An algebraic variety met by any line in two points is termed a quadratic variety or simply a quadratic. Its equation is of the form F2 (x, y, z, t, u) = 0, * See Salmon, Solid Geometry, Vol. 2, 1915, p. 222. SEII. 2. VOL. 17. NO. 1319. X 274 ME. HANUMANTA RAO [Dec. 6, where F2 is a homogeneous quadratic function of the five variables. Its section by any hyperplane is a quadric surface, called here simply a quadric, and to be distinguished from a quadratic; and its section by any plane is a conic. The general equation of a quadratic involves fourteen constants. But just as a quadric may degenerate into a cone or a pair of planes, so also a quadratic may degenerate into a cone of the first species, a cone of the second species, or a pair of hyperplanes, according as the equation is reducible by linear substitutions to a form involving four, three, or two variables only. The equation of the general quadratic may be brought to the form Sx2 = 0 by suitable choice of the coordinate hyperplanes. The equations of the three degenerate forms may similarly be brought to the forms It follows from the first of the above equations that there is one point intimately associated with a cone of the first species or hypercone as it may be called. Whereas an arbitrary hyperplane intersects the hyper- cone in a quadric, any hyperplane passing through this particular point meets it in a quadric cone with vertex at this point. For this reason this point is called the vertex of the hypercone. From this it follows that if we join every point of a quadric to a point 0 which does not lie in the same hyperplane. we obtain a hypercone with vertex at 0. Every tangent hyperplane to the hypercone passes through the vertex, and intersects the surface in two planes whose line of intersection is the line of contact of the tangent hj'perplane. These planes are called generator planes, since they correspond to the generators of a quadric. Also they fall into two classes such that two planes of the same class have only one point in common, viz., the vertex of the hypercone, whereas two planes of opposite classes have a line common. Thus a hyperplane can always be drawn through two generator planes, one of each class. The second degenerate form of the quadratic is the cone of the second species. Its equation being of the form *2+Z/2+*2 = 0, every tangent hyperplane passes through a fixed line which is called the vertex line. The section by any hyperplane is a quadric cone with its vertex on the vertex line. The principles of reciprocation apply equally well to the case of quad- ratics. Every point has a polar hyperplane, and every line has a polar plane. In the case of a hypercone the polar hyperplane of any point passes through the vertex. 1917.] QUARTIC SURFACE IN SPACE OF FOUR DIMENSIONS. 275 2. We consider now the intersection of two quadratic varieties. Two quadratics whose discriminant does not vanish* can simultaneously be reduced to the forms Their intersection is a quartic surface or twofold. It is this surface which Segre projects from an arbitrary point on to an arbitrary hyperplane, so obtaining a quartic surface in space of three dimensions possessing a double conic. We may call it F with Segre. The network of quad- ratics F-\-\$ = 0 contains five hypercones, one of which is {a-e)x2-\-(b-e)if-Hc-e)z2+(d-e)f = 0. The five cones give rise to ten sets of generator planes. We shall now show that each of these planes meets F in a conic. Any hyperplane meets F in a quadriquartic curve ; in particular, a tangent hyperplane to any owe of the hypercones. F lies entirely on each of the hypercones ; and we have seen that a tangent hyperplane to a hypercone meets it in two generator planes. Hence the quartic curve of intersection of F by any such tangent hyperplane must break up into two plane curves lying in the two generator planes and intersecting twice on the line common to the two planes. These plane curves must obviously be conies. Thus any generator plane meets F in a conic. Conversely, every conic on F must lie in one of the generator planes. There are further sixteen lines on F, falling into groups in a special manner with respect to each hypercone. The analytical proof is ex- tremely simple, and reference may be made to Prof. Jessop's treatise."1 The argument given by Segre himself is interesting as going to the root of the matter, and may be exhibited as follows. The vertices of the five hypercones belonging to the network F+\& = 0 form a pentahedron which is self-polar with respect to all the quadratics of the network. Con- sider one of the vertices and the hypercone through that point. The opposite hyperplane intersects this hypercone in a principal quadric and F in a quartic curve. In each of the two systems of generators of the principal quadric there are four which touch the quartic curve. Let g be one of these generators, and P its point of contact with the quartic curve.