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Let G bea finite group, let C1, …, Cn be the cyclic subgroups of G and let 1i be the identity linear character of Ci (1 ≤ i ≤ n). It is a well-known result of Artin ((l), 39·1) that a character x of a linear representation of G over the rational numbers may be writtenwhere a1, …, an are integers. In this note, we establish an analogue of this result for characters of projective representations over the rational numbers.

The theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

1·1. The points, tangents, osculating planes, …, osculating primes of a curve may be said to form a system which is characterised by the number of these elements which are incident with a prime, …, line, point, respectively. For the normal rational quartic curve the system is (4, 6, 6, 4); projection of this system from a general point gives the system (4, 6, 6) in [3]; section by a general prime gives the system (6, 6, 4). These two systems in [3], which are the systems with which we are concerned in this paper, are duals of one another, and will be called systems of the first and second kinds respectively.

1. If the canonical series of an algebraic curve of genus p is compounded of an involution of sets of points on the curve then the involution must be rational and of order two, and the canonical model is a repeated rational normal curve of order p − 1 with 2p + 2 branch points. An analogous question suggests itself for algebraic surfaces. Under what conditions is the canonical model of a surface, whose geometric genus is not less than two, a multiple surface, and what in this case are the properties of the simple surface on which the multiple surface is based? In this paper we give particular examples of multiple canonical surfaces and attempt to go some way towards the solution of the general problem.

In a paper ‘The projective generation of curves and surfaces in space of four dimensions’, F. P. White considers the representation of prime sections of the general Bordiga surface F6 in [4] by quartic curves in the plane through ten fundamental points. We shall make use of several results from the same section of that paper, and, for convenience, list them as follows. From a point P′ of F6 can be drawn an ∞1 of trisecant lines lying on a cubic conical sheet. The curve of intersection of these trisecants and F6 is a curve C8, of order eight and with a double point at P′, which meets each of the ten lines on F6 in two points. In the plane, the representation of this curve is a curve C7, of order seven and with eleven double points, namely, the ten fundamental points and the point P which corresponds to P′. C7 is hyperelliptic, containing an involution of points Q, R corresponding to the points Q′, R′ in which a trisecant from P′ meets F6. The envelope of joints of corresponding points Q, R in the plane is a conic; this conic will continually be referred to as Σ.

The following note was suggested by an interesting paper written by F. P. White, where many references are given. It refers to a theorem given by W. F. Meyer, by whom the proof is indicated as possible by generalization of an intricate analytical proof given by him for a simple case. His result is that if on the rational curve of orderr, in space [r], say the curvecr[r], there be an involution ∞k, of sets ofmpoints, expressed, suppose, by an equationthen the spaces [r− 1], formed fromrpoints of any one of the polyhedra ofmpoints, are an aggregate ∞kof primes of this space [r], which is of class (m−k,m−r), the notation (p, q) meaning the binomial coefficientp! /q!(p − q)!. By Meyer, the conditionsk