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Published online by Cambridge University Press: 24 October 2008
In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.
* Constructs of points of dimension three; linear forms we call spaces.
† Montesano, , “Una estensione del problema…,” Ann. di Mat. ser. 3, t. I, p. 321 (1898).Google Scholar
‡ Again as defined by Montesano (loc. cit.).
* Touched by each plane along a conic. See Segre, , “Sui fochi di 2° ordine…,” Rend. Accad. Lincei (5), v. 30 (1921), and references there given.Google Scholar
* “On some complexes…,” Proc. Lond. Math. Soc., vol. 23, p. 115 (1924).Google Scholar
* Namely a plane meeting the former in a line.
* “Allgemeine Anzahlfunctionen für Kegelschnitte…,” Mat. Ann., Bd.45, p. 153.Google Scholar
† I.e. for those whose conies are considered as loci of points.
‡ Not in general linear.
* “Nouvelle démonstration…,” Mat. Ann., Bd. 3, p. 150 (1871).Google Scholar
† These genera can be calculated in the majority of linear cases directly using the formulae in my paper “On the intersection…,’ Proc. Camb. Phil. Soc., vol. 21, p. 435 (1923), principally by formulae X, XI, XIII. Notice that in Form. IX of this paper a term rsh is omitted.Google Scholar
‡ The surface locus for tangents meeting a line is similarly of order (x + r + n), and the form, locus for those meeting a plane, of order (x + n).
* Namely such is the number of planes passing through a point.
* From (11) we have 4 (u + r) = (t)2 (n 2), and the general form of this result can be verified by considering the intersections of the curves representing the conics which touch two spaces in the points of their common plane, Ѽ. These represent either conies touching the plane, or degenerate conics whose vertices lie on the plane. The numerical coefficients however are not easily seen directly. We can deduce a new relation of mixed type by considering the intersection of the x–tic surfaces (9) for the spaces, consisting in fact of the sets of r and v points and the singular points in Ѽ, whence
