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§ 1. Extensions of Pascal's theorem are already known if we look at the matter from certain particular points of view – an extension of the theorem from the more general point of view is still a desideratum.

The object of this paper is firstly to extend the theorem of Pascal concerning six points of a conic to sets of 2 (n + 1) points of the rational normal curve of order n in space of n dimensions; secondly to explain why a wider extension to other sets of 2 (n + 1) points in [n] must be sought; and lastly to give briefly an extension to [3] and [4] which will be further generalised in a later paper. The striking feature of Pascal's theorem—that each of the sixty ways of arranging the points in a cycle, or as vertices of a closed polygon, leads to a different version of the theorem—is retained in the following extension to [n].

An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

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We construct an Abel–Jacobi mapping on the Chow group of 0-cycles of degree 0, and prove a Roitman theorem, for projective varieties over C with arbitrary singularities. Along the way, we obtain a new version of the Lefschetz Hyperplane theorem for singular varieties.

In (10), M. S. Narasimhan and S. Ramanan proved a theorem to the effect that a certain conic bundle associated with a non-singular quadratic complex does not come from a vector bundle ((10), proposition 8·1); a similar topological result was proved in (12). In the course of attempting to extend these results to the singular case, I found that I wanted to use some results on conic bundles which were not readily available in the literature. The object of this note is to give proofs of these results; the work on quadratic complexes is still in progress and the first part will appear shortly (13). A further application will appear in (14).

In connection with the problem of Map-colouring, I incidentally gave (Proc. K.S.E. 1880, p. 502) a theorem which may be stated as follows : — If 2n points be joined by 3n lines, so that three lines, and three only, meet at each point, these lines can he divided (usually in many different ways) into three groups of n each, such that one of each group ends at each of the points.

This paper presents some new theorems about the Pascal points of a quadrilateral. We shall begin by explaining what these are.Let ABCD be a convex quadrilateral, with AC and BD intersecting at E and DA and CB intersecting at F. Let ω be a circle through E and F which meets CB internally at M and DA internally at N. Let CA meet ω again at L and let DB meet ω again at K. By using Pascal’s theorem for the crossed hexagons EKNFML and EKMFNL and which are circumscribed by ω, the following results can be proved [1]:– (a)NK, ML and AB are concurrent (at a point P internal to AB)(b)NL, KM and CD are concurrent (at a point Q internal to CD)