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.

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.

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.

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.