Writing of his now well-known Interpolation Formula, Professor Everett said, “The only novelty in the formula is the simplicity of its form.… The best known formulae for interpolation by central differences are difficult to carry in the memory on account of their unsymmetrical aspect, one law being applicable to the odd and another to the even terms. … This disadvantage does not apply to the formula proposed,” viz., that which now goes by Everett's name.

In (2) Hickin and Phillips establish various results connecting the ideas of local systems and serial subgroups in group theory. In (4) Petty shows that if a group has a local factor system of -groups, then it may be embedded in an ultraproduct of -groups and he uses this result to extend some local theorems which were originally due to Mal'cev (3) and at the same time providing an elementary proof. We shall show that Petty's method may be used to prove results about seriality and local systems.

Einstein has recently adopted a new set of field-equations in his Unified Field-Theory of Gravitation and Electricity, the so-called theory of parallelism at a distance or Teleparallelism, and has given a solution of these equations with spherical symmetry, corresponding to the field of a charged mass-particle. In the present paper we discuss the solution of these equations with axial symmetry, which corresponds to a statical field whose field-variables depend on a single coordinate only, viz. the coordinate which is measured along the axis of symmetry.

1. A given function f(x) is said to be represented operationally by another function φ(p), if

provided that the integral converges.

A bounded linear operator T on a complex reflexive Banach space is said to be well-bounded if it is possible to choose a compact interval J = [a, b] and a positive constant M such that

for every complex polynomial p, where ‖p‖J denotes sup {|p(t)|:t ∈ J}. Such operators were introduced and first studied by Smart (4). They are of interest principally because they admit (and in fact are characterised by) an integral representation similar to, but in general weaker than, the integral representation of a self-adjoint operator on a Hilbert space. (See (2) and (4) for details.) It is easily seen, by verifying (1) directly, that T is well-bounded if it is a scalar-type spectral operator with real spectrum.

The object of this note is to draw attention to a simple extension of a well-known theorem concerning corresponding points on confocal ellipsoids–an extension which seems to have escaped notice. It has been familiar to me for years as an illustration of the quaternion treatment.

This note is intended to be supplementary to the paper, by Mr Muirhead, on “The dissection of any two triangles into mutually similar pairs of triangles.” The constructions given there, for the general case of this problem, yield no real solution if one angle of one triangle be greater than the sum of any two angles of the other triangle. For this particular case, the following constructions supply the necessary requirements; the first leads to a division of the triangles into three parts, the second to a division into four parts.

Let A be a finite-dimensional Bernstein algebra over a field K with characteristic not 2. Maximal subalgebras of A are studied, and they are determined if A is a genetic algebra. It is also proved that the intersection of all maximal subalgebras of A (the Frattini subalgebra of A) is always an ideal. Finally the structure of Bernstein algebras with Frattini subalgebra equal to zero is described.

In this paper we generalize recent results of Kreer and Penrose by showing that solutions to the discrete Smoluchowski equations

with general exponentially decreasing initial data, with density p, have the following asymptotic behaviour

where J = {j: cj(t)>0, t>0} and q = gcd{j: cj(0)>0}.

§1. The general theorem underlying the subject of this paper is as follows:—

If through c1, c2, c3, … points on AB, a side of ΔABC, rays be drawn from two vertices O1, O2, the former meeting AC in b1, b2 b3, …, and the latter meeting BC in a1, a2, a3, …, then the lines a1 b2, a2b2, … envelope a conic touching the sides AC and BC. This follows since the ranges a1 a2 a3 …, b1 b2 b3 …, are homographic.

In (6) Taylor has introduced the notion of a convolution measure algebra. In the same paper he constructed a canonical embedding of an arbitrary, semisimple commutative convolution measure algebra A into the algebra M(S) of all bounded, regular Borel measures on a compact semigroup S. This embedding has the properties that A is σ(M(S), C(S)-dense in M(S), that if μ is in A and ν is absolutely continuous with respect to μ, then ν is in A and that the set A^ of non-zero complex homomorphisms of A can be identified with the set S^ of continuous semicharacters of S where h ∈ A^ is identified with χ ∈ S^ by the equation

We consider nearly-Euclidean balls of the shape

where ε is a small positive number, and n is even. If ε is small enough, then the maximum lattice-packing density of this body is essentially greater than the Minkowski-Hlawka bound for large n.

The spectrum σ(Tφ) of a Toeplitz operator Tφ on the open unit disc D for a unimodular symbol φ is studied and many sufficient conditions for σ(Tφ)⊆∂D or σ(Tφ) = are given. In particular if φ is a unimodular function in H∞ + C, then σ(Tφ)⊆∂D or σ(Tφ) =