1. In his paper ‘On certain arithmetical functions' Ramanujan (23) considers the function τ(n) defined by the expansion

This function appears in the discussion of an asymptotic formula for the function

and also in Ramanujan's formula for the number of representations of an integer as the sum of 24 squares. It is also of interest as the coefficient in the expansion of g(z), which plays an important part in the theory of modular functions.

The groups of symmetries of three-dimensional lattices have been known for some time. They consist of finite rotation groups, the crystal classes, and infinite discrete motion groups, which include both rotations and translations. The general theory of the corresponding groups in higher dimensional Euclidean spaces has also been developed. This theory includes a demonstration that in Euclidean space of n dimensions the number of motion groups is finite, and leads to a method† for calculating the motion groups, the first step being to determine the crystal classes. The explicit calculation of the various groups by the general method is not simple, and has so far been confined to the case of two and three dimensions. In the special case of the crystal classes in four dimensions, however, we may make use of the results of a paper by Goursat‡. In this paper Goursat sets up a correspondence between the finite rotation groups in four-dimensional Euclidean space, and a set of groups each of which is formed by associating two of Klein's groups of linear non-homogeneous substitutions in one variable. Using this result he is able to evaluate explicitly all the proper and improper finite four-dimensional rotation groups which include the element −I, where I is the four-rowed unit matrix.

1. In this paper I consider various properties of the area of a surface including the relationship between Hausdorff measure and the Lebesgue area. The results here obtained will enable me, in a subsequent paper, to investigate the tangential properties of parametric surfaces. For simplicity of exposition I confine my attention to surfaces homoeomorphic to a disk, but the general case follows easily from cyclic element theory.

Summary of a paper presented by Dr. Bowen, as President of the Australian Institute of Navigation, to a meeting of the Institute held on 15 June 1951.

Accounts of the work described in this paper have been separately published, many of them in this Journal. The paper, therefore, restricts itself to giving a summary of this work, referring the reader to the appropriate literature for further study, and to pointing to some of the outstanding landmarks over the last few years.

The work referred to has been carried out by the Radiophysics Division of the Commonwealth Scientific and Industrial Research Organization in Australia and falls conveniently into two main headings, dealing with short-range aids to navigation and with air traffic control. It has been conducted for air navigation, but one at least of the navigation aids developed may find a useful application for maritime use.

As from 1953 the Air Almanac and the American Air Almanac will become a single publication, under the title of the Air Almanac. It will be produced jointly by H.M. Nautical Almanac Office of the Royal Greenwich Observatory, Herstmonceux, and the Nautical Almanac Office of the U.S. Naval Observatory in Washington, to meet the general requirements for air navigation in the United Kingdom, the United States and Canada. The Air Almanac will be printed and published separately in the United Kingdom and the United States but the two publications will otherwise be identical.

The field of flow due to a shock wave or expansion wave undergoes a considerable modification in the neighbourhood of a rigid wall. It has been suggested that the resulting propagation of the disturbance upstream is largely due to the fact that the main flow in the boundary layer is subsonic. Simple models were produced by Howarth, and Tsien and Finston, to test this suggestion, assuming the co-existence of layers of uniform supersonic and subsonic main-stream velocities. The analysis developed in the present paper is designed to cope with any arbitrary continuous velocity profile which varies from zero at the wall to a constant supersonic velocity in the main stream. Numerical examples are calculated, and it is concluded that a simple inviscid theory is incapable of giving an adequate theoretical account of the phenomenon. The analysis includes a detailed discussion of the process of continuous wave reflexion in a supersonic shear layer.

A common feature of all systems of hyperbolic navigation (such as Gee, Loran and Decca) is the necessity of interpreting the instrumental readings of lattice coordinates and of transforming them into position—either directly into latitude and longitude or relative to neighbouring geographical features. Several methods are in use: elaborate tables giving latitude and longitude in terms of the lattice coordinates are published for use with Loran; chartlets, on a special projection, are available for certain routes on the Decca system; but the most generally useful form of interpretation, combining both presentations, is the latticed chart. This paper is concerned solely with the construction of such charts. Although formulae and examples are given, details of the computing techniques are deliberately omitted; it is hoped to issue a fuller account with the tables that have been prepared for the use of this method.

The requirements of Kolmogoroff's theory of the equilibrium spectrum are satisfied only at very high Reynolds numbers, higher than any at which experiments have yet been done. In particular, when the theory holds, the rate of decay of the mean-square vorticity ω must be negligible compared with either its rate of increase due to the stretching of the vortex filaments or the rate of dissipation due to viscosity.

An extended version of Kolmogoroff's hypothesis may be proposed, in which the statistical properties of the turbulence in a range of wave-numbers (range of eddy sizes) depend not only on the rate of dissipation ∈ per unit volume and the viscosity ν, but also on the time rate of change d∈/dt of ∈. The result is to introduce a dependence on the Reynolds number R of the turbulence into quantities and constants which, on Kolmogoroff's original hypothesis, were independent of R. The Reynolds number R is defined from the decay law; u2t, with an origin of time suitably chosen, is a function of t, finite when t = 0, and R is defined as (u2t)t=0/ν. Lin's decay law follows logically from the extended hypothesis, according to which the rates of change of ω−1, due to the causes mentioned above, are constant during the decay; Lin's decay law would also follow if this less general part only of the extended hypothesis be assumed. The same decay law is also obtained if the similarity spectrum of Heisenberg is taken to apply not to the whole of the energy-bearing eddies, but only to the energy-dissipating eddies. But it is suggested that further generalization of the theory of the similarity spectrum, and of the decay law, is necessary; that the similarity spectrum is probably only asymptotically correct for a range of large wave-numbers, the range depending on the initial conditions and decreasing as the decay proceeds; that the general decay law is u2t = μRd(t), where d(t) is an integral function of t, such that d(0) = 1, and with an asymptotic value for large t to give correctly the law of decay in the final period; and that d(t), and the number of constants needed to specify it approximately, depend on the initial conditions. An experiment is suggested to test the dependence of the law of decay on the initial conditions. It is also suggested that the recently observed constancy of u2t in the initial period in the turbulence behind a single grid is only approximate, and this approximate constancy still needs explanation. Remarks are also included on the range of application of the equilibrium spectrum, for which some formulae are given when there is a definite cut-off in the spectrum.

The inequality is

which is established by an elementary argument and is shown to lead directly to the evaluation of the integral

and to the expression of sin x as an infinite product.

The following method of drawing a position line from a celestial observation is suggested as a possible means of simplifying a small portion of the navigator's work. It avoids the use of azimuths, thereby simplifying chart work.

The method is based on a geometrical construction whereby the position line is determined from intercepts relating to two separate adjacent Assumed Positions; this construction may prove simpler than the present orthodox method, particularly as it does not involve the measurement of angles.

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the region

has a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:

If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

The problem of how birds find their way can be examined by studying natural migration movements and also by forcing birds to undertake long journeys more or less under conditions which can be decided by the experimenter. The racing pigeon is an ideal subject for these homing experiments, but they have also been conducted with many species of wild bird and Fig. i shows some of the more remarkable flights that have been achieved with these. The use for these experiments of birds breeding in colonies has many advantages as large numbers can be caught and transported together, and their return watched for in a small area. Conspicuous birds such as gulls can be made individually recognizable by a code of plumage marks or by putting coloured leg rings on them.