Recently a power series representation of Hypergeometric functions with matrix argument has been established. This representation involves a special type of spherical function from the theory of semi-simple Lie groups, called the zonal polynomials. A general theory of these polynomials is well established; however an explicit representation of them is lacking. This paper considers two integrals which are related to this explicit representation. The final paragraph considers a third integral which gives an application of a result from a previous paper of the author.

Unsuccessful efforts to synthesize benzo[a]biphenylene and dibenzo-biphenylenes by the dehydrohalogenation of suitable intermediates are described. 2-o-Chlorophenyl-2-phenylpropionitrile with potassamide in liquid ammonia failed to give 8-cyano-7-phenylbicyclo[4,2,o]octa-I,3,5-triene, while diethyl 2-bromobenzylphenylmalonate with sodium in dry ethanol yielded I-phenylisochroman-3-one after hydrolysis. Bromination of 3,3′,4,4′-tetrahydro-2,2′-binaphthyl was accompanied by spontaneous dehydrobromination to yield 2,2′-binaphthyl, and interaction of phenyl-o-chlorophenylbromomethane with phenyllithium followed by chloracetic acid gave 1,2-di-o-chlorophenylethane.

It is the custom of our Society that the President of the day should at some period in his term of office address the Fellows. The manner in which he interprets this charge will, of course, vary according to individual preference, but for obvious reasons, he usually elects to give his personal interpretation of that area of science in which he is himself most deeply concerned.

Our Society was founded in the days before science was split into highly specialized sections and long before the advent of the specialist societies restricted to one scientific discipline, such as the Chemical Society, the Biochemical Society or the Society for Experimental Biology. This, in the opinion of many of our Fellows, is its main strength, since, as a general Society concerned with all aspects of science, it can exert an integrative function; in fact, it serves many of the purposes of the Academies which are so well known in other parts of Europe, as is made clear in the Letters Patent granting our Armorial Bearings.

By means of a generalized ring of quotients multiplicative ideal theory is studied in an arbitrary (associative) ring. A suitable generalization of the concept of maximal order is given and factorization theorems are obtained for the nonsingular (two sided) ideals, which generalize the theorems of Artin and E. Noether.

This paper studies two particular cases of the general 2-parameter eigenvalue problem namely

where A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.

On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

The Langevin equation for the harmonic oscillator is solved by a different method from that normally used. The approximate solution for the case of the slightly anharmonic oscillator is then obtained by an iterative procedure and the results are illustrated by a numerical example based on a simple model of a crystalline solid.

Let A be a subset of a compact metric space Ω, and suppose that A has non-σ-finite h-measure, where h is some Hausdorff function. The following problem was suggested to me by Professor C. A. Rogers:

If A is analytic, is it possible to construct 2ℵodisjoint closed subsets of A which also have non-σ-finite h-measure?

At this level of generality the problem, like others which involve selection of subsets, appears to offer some difficulty. Here we prove two results which were motivated by it.

Truesdell and Noll [1; sections 22, 27, 34] have discussed the concepts of material uniformity and homogeneity in continuum mechanics. A body is said to be materially uniform if, roughly speaking, all the particles composing the body are of the same material and homogeneous if there exists a global reference configuration which can be taken as a natural state for the whole body. To make the ideas precise for elastic materials, consider a small neighbourhood of each particle X and suppose that a reference configuration κ is chosen for each . Then during the motion, the deformation gradients may be calculated at each point X relative to the local reference configurations k. The stress at X is a function of these deformation gradients and if the stress relation does not depend explicitly on X the body is said to be materially uniform. If each local reference configuration κ can be taken as the configuration of its associated set of particles in some global reference configuration for the whole body, the body is said to be homogeneous. In general, however, the configurations κ need not fit together to form a global reference configuration. The body is then said to contain a distribution of dislocations.

Let Pn (n ≥ 0) be an n-polytope, that is, a convex polytope in n-dimensional euclidean space (Grünbaum [5], 3.1), and for 0 ≤ j ≤ n − 1 let be its j-faces. If Pn itself and Ø (the empty set) are also allowed to be faces of Pn, of dimensions n and − 1 respectively, then the set of faces of Pn forms a lattice partially ordered by inclusion ([5], 3.2). Two polytopes P1n and P2n are said to be combinatorially isomorphic, or of the same combinatorial type if their respective lattices of faces are isomorphic; that is, if there is a one–to–one correspondence between the set of faces of P1n and the set of faces of P2n which preserves the relation of inclusion ([5], 3.2). Similarly, any permutation of the set of faces of Pn which preserves inclusion will be called a (combinatorial) automorphism; it is clear that the set of automorphisms of Pn forms a group Γ(Pn), called the automorphism group of Pn.

There are many convenient ways in which a plane triangle can be defined and given projective coordinates. It can most simply be treated as an ordered triad of points (A, B, C) or dually as an ordered triad of lines (a, b, c), but it may seem more natural to regard it as a triad of points and an associated triad of lines which together satisfy the familiar incidence conditions. Again, the triangle for which Schubert [1] developed a calculus was a septuple, but Semple [2] has shown the advantages of a calculus for a triangle defined as an octuple.

Let θ1, …, θk be k real numbers. Suppose ψ(t) is a positive decreasing function of the positive variable t. Define λ(N), for all positive integers N, to be the number of solutions in integers p1 …, pk, q of the inequalities

and