On page 179 of the first volume of the Journal, Commander J. Middendorp, R.N.N.R., discusses the effect on the observed altitude that arises when the sextant is tilted out of the vertical plane, and concludes that the error is a maximum for altitudes in the region of 45; it seems to me that the conclusion is incorrect. I have no quarrel with Commander Middendorp's mathematics, but with his fundamental hypothesis, which is that when a sextant is tilted it is rotated about the line of sight to the star. I believe on the contrary that a sextant tilt is a rotation approximately around a horizontal axis.

Some new measurements of isotropic turbulence produced behind a biplane grid have been made at high Reynolds numbers, and these results are compared with the predictions of the theory of local isotropy developed by A. N. Kolmogoroff. The transverse double-velocity correlation has been measured at mesh Reynolds numbers up to 3·2 × 105, and the observed form agrees well with the predicted form. Measurements of the skewness factor of velocity differences over finite intervals have also been made, and the factor is nearly constant and equal to −0·38, if the interval is small compared with the integral scale. The invariance of dimensionless functions of the velocity derivatives has been confirmed for the flattening factor of ∂u/∂x, namely,

which is nearly constant over a wide range of conditions. It is concluded that the theory of local isotropy is substantially correct for isotropic turbulence of high Reynolds number.

1. Let

where a > 0, be an indefinite quadratic form, so that d = b2 − 4ac > 0. A classical theorem of Minkowski states that, if (x0, y0) is any pair of real numbers, there are numbers (x, y) congruent (mod 1) to (x0, y0), such that

and, more recently, Davenport has shown that this theorem can be sharpened for certain special f, for instance that it is always possible to satisfy

The paper is divided into two parts. The development of ship-borne compasses is reviewed in Part I; in this section some of the difficulties due to bad siting, &c., are mentioned and certain solutions such as the use of repeating compasses and improved methods of correction are described. Part II surveys the development of the aircraft compass and shows how its phases of development have corresponded to those of the marine compass. Some recent forms of aircraft compass and methods of development are also described.

In a previous joint paper (‘The dissection of rectangles into squares’, by R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte, Duke Math. J. 7 (1940), 312–40), hereafter referred to as (A) for brevity, it was shown that it is possible to dissect a square into smaller unequal squares in an infinite number of ways. The basis of the theory was the association with any rectangle or square dissected into squares of an electrical network obeying Kirchhoff's laws. The present paper is concerned with the similar problem of dissecting a figure into equilateral triangles. We make use of an analogue of the electrical network in which the ‘currents’ obey laws similar to but not identical with those of Kirchhoff. As a generalization of topological duality in the sphere we find that these networks occur in triplets of ‘trial networks’ N1, N2, N3. We find that it is impossible to dissect a triangle into unequal equilateral triangles but that a dissection is possible into triangles and rhombuses so that no two of these figures have equal sides. Most of the theorems of paper (A) are special cases of those proved below.

The purpose of this lecture is to review briefly the many methods at present available for the reduction of astronomical observations, and to make suggestions for the future. The main conclusion will be that altitude-azimuth tables provide the simplest and quickest method of reducing a sight.

It should be made clear that this is not a comprehensive survey setting out to examine and criticize every method; the bibliography of papers and tables (itself incomplete) published by H. Bencker in The Hydrographic Review, which contains some 1200 references, indicates that such a task would demand months of research. Attention is directed towards the principles of the various methods, and the limitations of each principle are discussed independently of the actual tables existing for the application of the method.

Several methods have been developed in recent years in the Cavendish Laboratory for the absolute measurement of fast neutron flux. All depend on observing the effects of elastic collisions between neutrons and light nuclei. The methods fall into two categories according as individual recoil nuclei are counted (1, 2), or the total ionization current due to the recoils is measured (3). In order completely to interpret the experimental results, it is necessary to know the cross-section for scattering and the angular distribution of the recoil nuclei. The total number of recoil nuclei and their energy distribution are then determined for a known incident neutron spectrum. For precise neutron flux measurements, recoil protons are invariably studied, as the neutron-proton scattering cross-section has been measured over a wide range of energies (4, 5), and the angular distribution of the recoils is isotropic in the centre of gravity space for neutrons of energy less than about 10 MeV. (6, 7). This makes the reduction of the experimental results particularly simple and certain.

When impact occurs between clean metal surfaces, plastic flow of the metal usually takes place at the points of real contact, so that the pressures developed are as high as the dynamic yield pressure of the metal concerned. Early experiment show that if the surfaces are covered with a thin film of a highly viscous liquid, the pressures developed and transmitted through the liquid film may be sufficiently great to produce plastic deformation of the metal, even though no metallic contact occurs through the film (1). The existence of these high pressures in the liquid layer means that extremely high rates of flow and shear may be developed in the liquid film, and that the energy dissipated in overcoming viscous flow may lead to an appreciable temperature rise in the liquid. Even in much gentler impacts, where plastic deformation of the metal surfaces does not occur, very high pressures, rates of flow and shear, etc., may be developed in the liquid film. These effects are of great interest in any study of collisions through liquids; they are of particular significance in the study of the mechanism of detonation of liquid explosives by impact.

Recent studies by Debye (1), Bloch (2), Stoner (3) and Guggenheim (4) expressing somewhat divergent views suggest that there is still some confusion and uncertainty in the method of application of the thermodynamic argument to magnetic phenomena. The first effective contribution to the subject was made by Larmor (5) over fifty years ago, and developing this further he finally proposed in 1929 a sufficient basis for a general theory of the whole range of phenomena. At about the same time (1927), and apparently in ignorance of Larmor's earlier work, Debye(1) also gave a brief outline of the thermodynamic aspects of the subject, starting, however, from much more tentative magnetic ideas and formulae which are now known to be without adequate physical justification. In 1933, Bloch (2) elaborated the argument still further, using, however, basic ideas more akin to those employed by Larmor, but derived by a very specialized argument. Then in 1935, using the same fundamental equation as Debye, Stoner revived the discussion with a systematic survey of the theoretical results bearing on the thermodynamic aspects of the subject, attempting later (6) to justify the basic magnetic ideas by a discussion which he now admits is not entirely satisfactory.

The questionnaire shown below was circulated to all members of the Institute. In the following analysis of the replies, forms completed by members with no experience or direct concern with surface navigation have been disregarded. The substance of replies received from members in the Royal Navy has been given separately. It has not been possible to detect any marked unanimity in the answers given by Examiners, Navigation School instructors, Marine Superintendents,&c., and these have been included under the heading of Merchant Navy.

The detailed comment given is taken from the discussion at the Meeting and from letters and memoranda sent in to the Institute. In some cases it refers (e.g. layout, &c.) to matter not reproduced in the Journal. For convenience of reading reference numbers have been given to the authors of comment.

The secondary emission electron multiplier is chosen to illustrate the phenomenon of ‘cascade multiplication’. A method is given for deriving the semi-invariants of the probability distribution for the number of output electrons after any number of identical stages of multiplication, in terms of the corresponding semi-invariants for a single stage. The output distribution is not, in general, either of the Poisson or Gaussian types, though it tends to a limiting shape as the number of stages becomes very large. The special case in which each stage replaces a single primary electron by a Poisson distribution of secondaries is considered. The overall output distribution after many stages is still not Gaussian unless the mean amplification per stage is large compared with unity.

The basic principles of Consol have been explained in a previous article; the factors affecting the range and accuracy of the system are here discussed. A summary is given of the results obtained practice, and the article concludes with a brief discussion of developments which will lead to improved performance.