Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric is

Two proofs of this result are given, in §§ 4 and 5.

Let G be a group, written additively with identity 0, but not necessarily abelian and let S be a semigroup of endomorphisms of G. The set for all is a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings are general, for if N is any zero-symmetric near-ring with identity then there exists a group G and a semigroup S⊇ G such that For background material and definitions relative to near-rings in general we refer the reader to the book by Pilz [7]. For material on centralizer near-rings we refer the reader to [4] and [6].

1. To designate the line which bisects at right angles the join of two points A, B, the term axis of A, B is proposed. Reasons:—

(1) brevity; (2) avoidance of the suggestion that the line joining AB is necessary in constructing it (important in teaching Geometrical Drawing); (3) two points, like any other pair of circles, have a radical axis which is the line in question.

Let Δ be the Laplace operator on ℝd and 1 < δ < 2. Using transference methods we show that, for max {q, q/(q – 1)} < 4d/(2d + 1 – δ), the maximal function for the Schrödinger group is in Lq, for f ∈ Lq with Δδ/2f ∈Lq. We obtain a similar result for the Airy group exp it Δ3/2. An abstract version of these results is obtained for bounded C0-groups eitL on subspaces of Lp spaces. Certain results extend to maximal functions defined for functions with values in U M D Banach spaces.

In a recent number of the Annals of Mathematics I have shown that the asymptotes of a conic circumscribed about a triangle are isotomic lines with reference to the triangle. By means of this theorem it is easy to find the locus of the centre of a conic, circumscribed about a given triangle, when one of the asymptotes passes through a fixed point.

The problem “to inflect a straight line between two sides of a triangle so that the intercepted portion is equal to the segments cut off” has been discussed in the third volume of the Proceedings.

If we discuss the same analytically; taking CB and CA as axes of x and y (Fig. 1) and calling each segment k, the equation of the line considered is

A group G is said to be quasi-injective if, for each subgroup H of G and homomorphism θ:H→G, there is an endomorphism such that . It is of course well known that the category of groups does not possess non-trivial injective objects and so we consider groups satisfying the weaker condition of quasi-injectivity.

By assuming that multiplication by a line is the true operation corresponding to the passing from space of n dimensions to space of n+1 dimensions we may arrive very simply at certain well-known results in geometry of higher dimensions.

Let n be a positive integer. We give an elementary construction for the nth variation, Vn(f), of a real valued continuous function f and prove an analogue of the classical Jordan decomposition theorem. In fact, let C[0, 1] denote the real valued continuous functions on the closed unit interval, let An denote the semi-algebra of non-negative functions in C[0, 1] whose first n differences are non-negative, and let Sn denote the difference algebra An - An. We show that Sn is precisely that subset of C[0, 1] on which Vn(f)<∞. (Theorem 1).

In many branches of applied mathematics there exists a class of problems which depend for their solution upon the integration of a set of simultaneous linear partial differential equations subject to certain boundary conditions. In all but the simplest cases it is not practicable to deal with these equations by standard methods. For problems involving infinite regions, solutions can often be found by the use of integral transforms. However, in many problems we are concerned with media of finite extent, so that if we are to make a direct application of this method, we shall have to use finite transforms, and under certain conditions these are much more difficult to apply than transforms over an infinite range.

Mr H. Poincaré, in a recent work (la Valeur de la Science) speaks of the interest attaching to proofs of theorems in pure geometry by arguments derived from mechanical or physical considerations. The following is a case in point.

To find the centre of gravity of a circular arc by reference to a principle of dynamics.

Let A be a commutative Banach algebra with identity 1 over the complex field C, and let d0 be a character on A. We recall that a (higher) point derivation of order q on A at d0 is a sequence d1, …, dq of linear functionals on A such that the identities

hold for each choice of f and g in A and k in {1, …, q}. A point derivation of infinite order is an infinite sequence {dk} of linear functionals such that (1.1) holds for all k. A point derivation is continuous if each dk is continuous, totally discontinuous if dk is discontinuous for each k≧1, and degenerate if d1 = 0.