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.

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.

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.

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.

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 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.

In the preceding paper of the same title (cf. [1]) I defined the notion of the principal genus GK of a finite number field K as the least ideal group, which contains the group IK of totally positive principal ideals and is characterized rationally. The quotient group of the group AK of ideals in K modulo GK is the genus group, its order (Ak: GK) = gK is the genus number, which is thus a factor of the class number hK (in the narrow sense). Associated with the genus group is the genus-field, of K, which is defined as the maximal non-ramified extension of K composed of K and of some absolutely Abelian field.

The Čech compactification of the set of integers is known [6] to have the remarkable property that it has no closed subsets of cardinal ℵ0 or c, every infinite closed subset of it having 2c points. The main object of the present paper is to investigate whether similar gaps in the cardinals of closed subsets can occur in metric spaces. We shall see that the situation there is rather different; if the generalized continuum hypothesis is assumed, there are no gaps, and in any case the missing cardinals, if any, must be big rather than small. The main results are obtained in §3; in particular, we completely determine the cardinals of the closed subsets of complete metric spaces, and also how many closed subsets of each cardinal there are. The methods depend on a study of the discrete subsets of metric spaces, which is carried out in §2, and which may be of independent interest. In conclusion, we briefly consider some fragmentary results for non-metric spaces, in §4. Throughout, we assume the axiom of choice but not the continuum hypothesis.

In the present paper we consider a one-dimensional local ring Q with maximal ideal tn and residue field K = Q/m. It will be assumed that not every element of mis a zero-divisor but no other restricting hypothesis will be made. In particular Q and K may have unequal characteristics and K may be finite.

In a recent paper Segedin [1] has derived a solution of the problem in which a perfectly rigid punch in the form of a solid of revolution of prescribed shape with axis along the z-axis bears normally on the boundary z = 0 of the semi-infinite elastic body z ≥ 0, so that the area of contact is a circle whose radius is a. Segedin solves the problem by building up the solution in a direct way which avoids both the use of dual integral equations and the introduction of an awkward system of curvilinear coordinates. By introducing a kernel function K(ξ), Segedin derives new potentials of the form

where U(r, z, a) is the solution of the simplest punch problem (namely that of a flat-ended punch) satisfying the mixed boundary conditions

on the boundary z = 0. It can then be easily shown that, under wide conditions on K, the function Φ (r, z, a) satisfies the boundary conditions

provided that K(ξ) is chosen so that

The receptance function is defined and constructions for it are given both for the general case when ω, the frequency of excitation, is not a natural frequency and for the special case when it is.

Green and Zerna [1] have given a method of determining the electrostatic potential due to a circular disc maintained at a given axisymmetric potential, their method depending on the solution of a Volterra integral equation of the first kind and being a generalization of a method given by Love [2] for the determination of the electrostatic potential due to two equal co-axial circular discs maintained at constant potentials. In a recent paper [3], henceforth referred to as Part I, the author applied this method to the corresponding problem for a hollow spherical cap and also to the determination of the Stokes' stream-functions for perfect fluid flows past a cap and a disc. The method consists of expressing the potential as the real part of a complex integral of a real variable t, the integrand involving an unknown function g(t). The boundary condition on the disc or cap gives a Volterra integral equation of the first kind for g(t), it being possible to solve this equation and hence determine the potential by integration.

In a recent paper Northcott [3] introduced the notion of the reduction number of a one-dimensional local ring, and demonstrated its importance in the theory of abstract dilatations. In the present paper we define the reduction number of an ideal which is primary for the maximal ideal of a one-dimensional local ring, and show that under certain necessary and sufficient conditions the reduction numbers can take only a finite number of values.

The two centred expansion of the Coulomb Green's function arises naturally in discussing the static interaction energy of two charge distributions ρ1, and ρ2. This is given by the well-known expression

In a recent paper Rogers [13] has discussed packings of equal spheres in n-dimensional space and has shown that the density of such a packing cannot exceed a certain ratio σn. In this paper, we discuss coverings of space with equal spheres and, by using a method which is in some respects dual to that used by Rogers, we show that the density of such a covering must always be at least