This paper considers the properties of the spectrum of a differential operator derived from differential expressions of the fourth order. With certain conditions on the coefficients of the differential expression the spectrum of the operator is discrete and an estimate is obtained of the number of eigenvalues lying in a given bounded interval of the real line. The results are compared with those obtained by alternative methods. Additional restrictions on the coefficients give special cases previously considered by other authors.

A new approach to the analysis of transport processes in a turbulent fluid is presented In this approach a model is used to represent the detailed fluid behaviour and it is shown that the model has similarities with a Fourier integral representation of the flow field. The model assumes that the turbulent motions can be represented by fluid entities of random size, shape and velocity, and that the large-scale transport processes are the consequences of the creation, decay and mutual interaction of the individual entities. The effects of this are analysed and it is shown how the diffusivities for vector and scalar quantities can be determined in terms of properties of the turbulence. The theory is applied in both bounded and free turbulent flows and it is shown that the predicted diffusivity ratios compare favourably with experimental data. Relaxation phenomena are also investigated and the memory function for stress and thermal relaxation is determined. It is concluded that the model provides a most useful framework for the analysis of turbulence phenomena and that its diverse and accurate predictions make it a powerful research tool.

This paper considers properties of the spectrum of differential operators derived from differential expressions of the second order. The object is to link the spectral properties of these differential operators with the analytic, function-theoretic properties of the solutions of the differential equation. This provides an alternative approach to the spectral theory of these differential operators but one which is consistent with the standard definitions used in Hilbert space theory. In this way the approach may be of interest to applied mathematicians and theoretical physicists.

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.

9-Carbamoylfluorene with lithium aluminium hydride, 9-cyanofluorene with this reagent and aluminium trichloride, or (on one occasion) treatment of the oximes of 9-formylfluorene with thionyl chloride yield 9,9′-dicyano-9,9′-bifluorenyl. 9-Bromofluorene and ethanolic potassium cyanide yield 9-cyano-9,9′-bifluorenyl, and 9-formylfluorene when kept in ether for a month gives 9,9′-diformyl-9,9′-bifluorenyl. The so-called α-oxime of 9-formylfluorene described in the literature contains about 33 per cent of the higher melting β-oxime. Reduction of the oximes with zinc and acetic acid yields di(9-fluorenylidenemethyl)amine, previously obtained by other methods. A new method for the preparation of 9-aminomethylenefluorene is described and its structure has been confirmed. Many 9-substituted and 9,9′-disubstituted fluorenes exhibit characteristic absorption at 1920–1880 and 1960–1940 cm.−1.

The number of different connected graphs (with some property P) on n labelled nodes with q edges is fnq. Again Fnq is the number of graphs on n labelled nodes with q edges, each of whose connected components has property P. We consider 8 types of graph for which . We use a known relation between the generating functions of fnq and Fnq to find an asymptotic expansion of fnq in terms of binomial coefficients, valid if (q – ½n log n)/n→∞ as n→∞. This condition is also necessary for the existence of an asymptotic expansion of this kind.

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.

Necessary and sufficient conditions for an ordered semigroup to admit an isotone homomorphic Boolean image are given together with a complete description of how all such images are obtained. Also discussed are the situations arising from a strengthening of the notion of a homomorphism.

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.

It is shown that the essential condition, energy-wise, for the emission of a light particle from a binary fission fragment, in the immediate post-scission stage, is that the fragment deformation must be capable of sudden partial collapse so as to release energy equal to the binding energy of the particle in the undeformed (unexcited) fragment together with an amount cancelling the mutual (electrostatic) potential energy of the particle and the residual deformed (excited) fragment at the instant of separation. On this basis the mean energy required for alpha-particle release in the thermal-neutron-induced ternary fission of 235U is estimated as 16·5 MeV, and it is concluded that those binary fissions which develop into alpha-particle-accompanied ternary fissions in this case are those in which, on the average, the total deformation-excitation energy of the fragments is some 10 MeV greater than the most probable value for all binary fissions. A qualitative discussion of the relative yields of other light-particle-accompanied modes is also given.

An Appendix contains a summary of comparable calculations based on the one-stage view of the ternary fission process.

In this paper, we shall be concerned with an investigation of the solution of triple integral equations involving sine and cosine kernels. These type of equations arise in the study of certain two-dimensional mixed boundary value problems in infinite planes and infinitely long strips.

A second method for the synthesis of 8-bromo-3-methoxyfluoranthene from 2-bromo-7-methoxyfluorene is described. The autoxidation of fluorene derivatives including the 9-carboxylic acids and of related compounds, particularly 4-α-cyanobenzyl-7−oxo-7H-benz[de]anthracene, is discussed. Unsuccessful attempts to prepare 3-bromofluorene-9-acetic acid in good yield are reported.

In the literature concerning matrices whose co-ordinates are elements of a Boolean lattice, one may find three different definitions for the determinant of a matrix. We shall call these the first, second and third determinant and will denote the value of the ith determinant of a matrix A by |A |i for i = 1, 2, 3. The first determinant may be defined for square matrices over an arbitrary lattice. The second and third determinants may be defined for square matrices over any lattice L with a greatest element I, a least element o and an orthocomplementation′: L→L, that is a′ is a complement of a, a = a″ and a ≤ b implies that b′ ≤ a′ for all a, b in L. In this paper we obtain some elementary properties of these determinants in this general setting and in the particular case where L is an orthomodular lattice, that is a lattice with o, 1 and an orthocomplementation' such that

A study is undertaken of the field excited in a semi-infinite cylindrical dielectric rod having small conductivity when plane harmonic electromagnetic waves are incident obliquely upon its end. All back-scattering and the complicated edge effects due to the end of the rod are neglected. Expressions for the normalized power absorbed in the rod are obtained by an approximation technique from the corresponding results for a similar rod whose conductivity is zero, assuming that the radiation through the walls of the rod may be ignored. Selected numerical results are presented and the relevance of these to models of a retinal rod of the human eye is discussed.

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.

The acyl groups in aromatic ketones with hydroxyl or methoxyl groups in the ortho or para positions are readily removed by hydrobromic acid in acetic acid. The results are correlated with the fine structure of the parent hydrocarbons. 5,8-Dimethoxy-I-tetralone and 1,3-dihydroxyxanthone fail to undergo ring-fission with the reagent. The mechanism is discussed and it seems probable that fission occurs by direct proton attack at nucleophilic carbon atoms bearing the acyl groups.

Humbert's 5-nodal plane sextic first appeared in his 1894 paper. Its canonical curve C was identified in 1951, when it was shown that the sextic is the outcome of projecting C from one of its own chords on to a plane.

In this present paper it is remarked that there are 60 chords of C such that the projection has two tacnodes, each a confluence of two of Humbert's 5 nodes, and an equation is found for this tacnodal curve.

A certain specialization permits C to be invariant for a group of 32, not merely 16, projectivities. Further specializations, described in the proper place, permit groups of orders 64, 96, 160. The resulting tacnodal sextics have groups of birational self-transformations isomorphic to these.

For a series of elements of a topological vector space, necessary and sufficient conditions are found, in terms of the set of finite partial sums, for unconditional convergence and for the corresponding Cauchy condition. The extent to which these results remain valid for topological groups is investigated. A new and direct proof, for locally convex spaces, is given of the theorem of Orlicz.

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.