With every matrix representation of the (real) full linear group can be associated a multi-linear mapping of one affine space, Rn, into another, RN. This mapping is studied from the viewpoint of the geometry of numbers of convex bodies, and a general arithmetical property of such mappings is proved. The result generalizes my recent work on compound convex bodies.

The Faroe-Shetland Channel is the threshold from the north-eastern Atlantic Ocean to north-west European seas. Through it passes the main bulk of the oceanic water-mass which is the predominant influx, among several other water-masses, to these seas.

The following research into the dynamics and general hydrography of the region is based on numerous observations of temperature and salinity, from surface to bottom, taken mainly on two vertical cross-sections of the Channel between the years 1927 and 1952 inclusive, excepting the war years 1940 to 1945.

The research reveals very large scale seasonal and long-term variations in the northeastward volume-transport of oceanic water, suggests the existence on occasions of what appear to be horizontal tortional currents within the oceanic water-mass, and demonstrates (a) the intrusion of Gulf of Gibraltar (extra-Mediterranean) water into this mass over a period of years, (b) the formation of heavy oceanic water and (c) of a sub-oceanic watermass. The last-mentioned may sometimes almost entirely displace the bottom Norwegian Sea water-mass which normally underlies the oceanic mass.

One or other, or both, of two types of Arctic water may also sometimes displace bottom Norwegian Sea water as the bottom water-mass of the region, the process, like that of the above-mentioned Gulf of Gibraltar water influx, waxing and waning over a term of years and thus exemplifying the phenomenon of marine climatic change.

A Lie group is said to be metrisable if it admits a Riemannian metric which is invariant under all translations of the group. It is shown that the study of such groups reduces to the study of what are called metrisable Lie algebras, and some necessary conditions for a Lie algebra to be metrisable are given. Various decomposition and existence theorems are also given, and it is shown that every metrisable algebra is the product of an abelian algebra and a number of non-decomposable reduced algebras. The number of independent metrics admitted by a metrisable algebra is examined, and it is shown that the metric is unique when and only when the complex extension of the algebra is simple.

A number of analogues to the simple fluid compressibility equation are deduced by considering fluctuations in a binary mixture; and their simplest expressions are found to be in terms of the binary mixture direct correlation functions. The accuracy of these results is tested with the aid of the appropriate extension of the approximate Born-Green theory, which facilitates the demonstration of consistency with the first three terms of the virial expansion. The problem of the evaluation of the corresponding radial distribution functions by means of X-ray or optical scattering is taken as far as the determination, in principle, of a concentration correlation function from observations of critical opalescence.