The roots of these equations are of importance in several theories and various authors have studied certain of their properties. Here we solve the equations in the sense that we define two numbers z(1), z(0), and a sequence {zn} which include the roots of both equations. Except for a small, finite number of values of n, we find a rapidly convergent series for zn whose terms are alternately real and purely imaginary. We give a number of expansions and a variety of practical methods which enable us to calculate the small number of remaining roots to any required degree of accuracy.

The main results of this article have been announced without proof or details in Wright 1960.

Canonical forms of the four-dimensional complex Lie algebras are obtained by considering the roots of certain well-defined vectors of the algebras. A complete set of characters of the algebras is also given, enabling any given four-dimensional complex Lie algebra to be identified with one of the canonical forms.

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.

The function ɸx = log x satisfies the functional equation ɸxy = λɸx + μɸy + ϰ, where in this case x, y are complex variables, λ = μ = i, and ϰ = 2πi, o or − 2πi according as σ < − π, − π < σ < π, π < σ, where σ = arg x + arg y. Generalizing this situation, let A be a linear algebra with basis e1, …, en over the real or complex field and let ɸx be a complexvalued function of the hypercomplex variable x = Σξiei, i.e. of the n real or complex variables ξi. Assume that the gradient ∂ɸx, i.e. the column vector of partial derivatives {∂ɸx/∂ξi}, exists at a general point of A. Then ɸx is called an entropic function if it satisfies a functional equation of the above-mentioned form and obeys certain other postulates, ϰ being a step function of the two hypercomplex variables. Values of the constants λ, μ (complex numbers, not both zero) for which a solution exists are entropic roots of A. They are usually discrete.

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 preparation of 1-benzhydrylnaphthalene, 1, 2, 3, 4-tetrahydro-1-benzoylnaphthalene, and 1-diphenylmethylenetetralin is described.

Thanks are expressed to the Department of Scientific and Industrial Research for a maintenance grant (to D. K.); to the Royal Society of London for a grant (to A. J. P.); to Miss E. W. Robertson for help with the experimental work; and to the Education Committee of Stirlingshire for permission to use laboratory facilities.

The synthesis of 7-bromo- and 7-chloro-fluoranthene from the readily accessible 1, 2, 3, 1ob-tetrahydro-3-oxofluoranthene is described. Oxidation of the chloro-compound yields 1-chlorofluorene-9-one-8-carboxylic acid and thus opens up a method for the preparation of 1, 8-disubstituted fluorenes and fluorenones.

Thanks are expressed to the Department of Scientific and Industrial Research for the award of a Maintenance Grant to one of us (D. A. C).

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.