This note explains a “numerical coincidence” observed by Professor A. C. Aitken in a recent article in these Proceedings.

Investigations based on gas masses, bright star counts, and luminosity-mass ratios of galaxies lead to one of two conclusions. If the galaxies are all of the same age, the faint ends of the initial luminosity functions of stars at formation differ greatly from one galaxy to another. On the other hand consistent results in the analysis are obtained with luminosity functions that are more nearly constant and ages which range from one to thirty thousand-million years. The various possibilities can be tested by observations on the Magellanic Clouds.

Equations are set up which describe, as functions of time, the integrated properties of a galaxy as a system of stars and gas.

The preparation of substituted fluoranthenes by the aromatization of hydrogenated fluoranthene derivatives is described. 8-Nitrofluoranthene, one of the nitration products of fluoranthene, has been synthesized from 2-nitrofluorene and the preparation and properties of some 2-nitrofluorene derivatives of potential synthetic value are reported.

The synthesis of 1, 2, 3, 4-tetrahydro-1, 6-naphthyridine and 1, 6-naphthyridine is described and the ultra-violet spectra of these and related substances are discussed.

Certain families of measures on coset-spaces, namely inherited, stable, and pseudo-invariant measures, were defined, and shown to exist, in earlier papers, where Jacobians and factor functions, generalizing the idea of Jacobians in theory of functions of several variables, were also denned. In this paper, the existence is established of exact Jacobians and factor functions, which satisfy certain characteristic identities exactly, without an exceptional set of measure zero. A study is made of how properties of a measure are reflected by properties of the Jacobian or the factor function. Necessary and sufficient conditions are found for a function to be an exact Jacobian for some measure.

A study of the γ-rays produced during the bombardment of a thick Be9 target by 600 keV deuterons was made to investigate the possible existence of a level at 2·86 MeV in B10, about which contradictory reports have appeared in the literature.

A spectrum of the γ-rays in coincidence with the 0·72 MeV B10 γ-ray (text-fig. 5) was obtained, and is interpreted as providing evidence for a level in B10 at 2·86 MeV. The relative intensities of the γ-rays in an ungated spectrum, and in spectra gated by the 0·72 and 1·02 MeV B10 γ-rays, were found, and a decay scheme consistent with the observations is deduced (text-fig. 6b). The relative intensities of the transitions in this decay scheme are consistent with the intensities of the neutron groups in a spectrum of the neutrons from this reaction. A spin value of 2 or 3 is suggested for the 2.86 MeV level.

Certain types of rings of infinite matrices are defined and some of their properties are discussed. The main theorems are concerned with the connections between the radical of a ring R and the radicals of rings of infinite matrices over R.

The probability that each of n equally correlated normal random variables shall not fall short of a given value h is obtained as the product of the joint density function of the variables at the cut-off point (h, h,…, h) and an infinite power series in h. The coefficients in the latter series may be interpreted geometrically as the moments of a regular (n – l)-dimensional spherical simplex with common dihedral angles arc cos –p relative to a certain plane of symmetry. These moments may in their turn be expressed as linear functions of the measures of regular hyperspherical simplices of various dimensionalities, tables of which are available elsewhere.