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.

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.

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

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 general paraxial particle-optic properties of the hexapole field are summarized and lens formulae derived. Formulae are given for the transmission as a function of particle velocity and system geometry for geometries appropriate to the case of neutral atom beams and of slow neutron beams respectively. These formulae are used to evaluate the hexapole magnet as a velocity selector and polarizer for atomic beams and as a spin polarizer for neutron beams. Some experimental observations on potassium beams are quoted in support of the theory.