Let A be a commutative, semi-simple, convolution measure algebra in the sense of Taylor (6), and let S denote its structure semigroup. In (2) we initiated a study of some of the relationships between the topological structure of A^ (the spectrum of A), the algebraic properties of S, and the way that A lies in M(S). In particular, we asked when it is true that A is invariant in M(S) or an ideal of M(S) and also whether it is possible to characterise those measures on S which are elements of A. It appeared from (2) that if A is invariant in M(S) then S must be a union of groups and that A^ must be a space which is in some sense “ very disconnected ”. In (3) we showed that if A^ is discrete then A is “ approximately ” an ideal of M(S). (What is meant by “ approximately ” is explained in (3); it is the best one can expect since algebras which are approximately equal have identical structure semigroups and spectra.) In this paper we round off some of the results of (2) and (3). We show that if A is invariant in M(S) then A^ is totally disconnected, and that if A^ is totally disconnected then S is an inverse semigroup (union of groups). From these two crucial facts it is fairly straight-forward to obtain a complete characterisation of algebras A (and their structure semigroups) for which (i) A^ is totally disconnected, (ii) A is invariant in M(S), or (iii) A is an ideal of M(S).

The aim in this paper is to indicate one way of interpreting covariants of a binary quintic F geometrically, interpretations having been found recently [7] for its quadratic covariant 2C2, called Γ in [7], and its invariant I4. The symbol dCn will denote a covariant of order n in the binary variables x, y and degree d in the coefficients of F, Id being used in preference to dI0 for invariants. The sum d + n is 4 for both 2C2 and I4, and no other covariant affords as small a sum; so it is natural to have begun by interpreting these two and to use them as auxiliaries in interpreting others.

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].

It is shown that an invertible disjointness preserving operator from a uniformly complete vector lattice onto a normed vector lattice has a disjointness preserving inverse and is necessarily order bounded.

Let α be an automorphism of a free group of rank n. The Scott conjecture, proved by Bestvina-Handel, asserts that the fixed subgroup of α has rank at most n. We give a short alternative proof of this result using R-trees.

In a recent paper I have discussed the families of quadrics in [2n] which are obtained by causing the members to have the greatest possible number of fixed [n – 1]'s or “generators.” It was found possible to fix four [n – 1]'s in general position; the family of quadrics through these possessed a “base” variety, common to all the members, which consisted of a highly degenerate Vn–1. Here I consider the same problem for quadrics in [2n + 1], find how many generators may be assigned arbitrarily and discuss the common part of the quadrics which pass through such generators.

In a recent paper the authors considered the transmission problem for the Helmholtz equation by using a reformulation of the problem in terms of a pair of coupled boundary integral equations with modified Green's functions as kernels. In this note we settle the question of the unique solvability of these modified boundary integral equations.

This is the problem of § 670 in Clerk Maxwell's Electricity and Magnetism. The author proposes to proceed by another method and to obtain the result in a different form. Let O be the centre of the spherical surface on which the shell lies and Z the point where the magnetic potential Vm is to be found. Also let ∅ be the strength of the shell (magnetic moment per unit area), α its internal, and α + δα its external radius. To represent the magnetic distribution let a layer of negative magnetic matter of density σ cover the inside face, and a corresponding positive layer the outside face. Finally, let Z be without the matter of the shell and on the positive side.

In regard to the algebra of binary forms and the theory of rational curves there exists a wide literature, with which I am not well acquainted. Very possibly the simple remark made in this note is found elsewhere. But the note is a grateful echo of recent delightful colloquy with persons of good will.

We characterize tame automorphisms of finitely generated abelian groups via a simple determinant condition.

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.