A new proof of the Cayley-Hamilton theorem avoiding the use of determinants makes it possible to apply this theorem to matrices over a commutative semi-ring. The relationship of this theorem to a theorem by Lunts concerning switching matrices is investigated.

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.

It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.

For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

An attempt is made to develop the statistical mechanics of the liquid state based not on the usual concept of a “radial distribution function” but on that of a “next neighbour distribution function” which is closely linked up with Bernal's ideas on the characteristic features of liquid structure. Making certain simplifying assumptions it is indeed possible to construct a partition function for an atomic liquid in this way and from this to derive the thermodynamic properties of the system according to the principles of classical statistical mechanics. It is shown that the free energy, the equation of state, the specific heat and entropy as obtained from the theory are consistent with the expected behaviour of such liquids. It is further shown that the computed next neighbour distribution function for close packing is in good agreement with the one derived empirically from a model by Bernal and Mason.

The synthesis of 3-bromofluorenone is described and attention drawn to erroneous statements by previous workers in the field. The reduction of the ketone to 3-bromofluorene has been investigated and shown to be readily effected by heating with hydrazine hydrate in diethylene glycol. This appears to be the method of choice for reducing fluorenones to fluorenes.

The methods employed in papers I–IV of this series are modified to provide the solution of certain dual equations involving trigonometric series. It is necessary to introduce a modified form of the conventional operators of fractional integration and to discuss their relation with generalized Schlömilch series expansions of an arbitrary function. These general methods are illustrated by detailed reference to a particular special case.