A square matrix A, of order n, having complex coefficients can be inverted without the aid of any operations involving complex numbers. This can be done if the coefficients ars + iαrs are replaced by their matrix equivalents and the resulting 2n × 2n real matrix A1 is inverted. The inverse will be a 2n × 2n matrix of similar form in which the subsidiary 2 × 2 matrices can be replaced by the equivalent complex numbers, thus yielding the inverse of A. It is the purpose of this note to show that a similar technique can be employed to evaluate det A.

1. This paper is concerned with certain asymptotic properties of the solutions of the differential equation

where dots indicate differentiation with respect to t, k is a small parameter, and f(x, ẋ, t) satisfies certain conditions which will be formulated below. Equations of this type occur frequently in non-linear mechanics; for k = 0 a system satisfying (1·1) behaves as a harmonic oscillator. To ensure the existence and uniqueness of the solutions of (1·1) it must be assumed that the right-hand side is bounded and satisfies a Lipschitz condition, at least for finite x, ẋ and say all t ≥ 0. The parameter k may be considered as a measure of the ‘smallness’ of the upper bound, and of the Lipschitz constant, of the right-hand side, and need not have any intrinsic physical significance.

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

In a recent paper (1) I studied a class of generalized convex functions of a single real variable which I called sub-(L) functions. Given an ordinary linear differential equation of the second order L(y) = 0, a function f(x) is sub-(L) in (a, b) if it is majorized there by the solutions of the equation. More precisely, for every x1, x2 in (a, b),f(x) ≤ F12(x) in (x1, x2), where F12 is that solution of L(y) = 0 (supposed unique) which takes the values f(xi) at xi. It was found that sub-(L) functions are characterized in a manner closely analogous to ordinary convex functions.

In daylight and in good visibility no mandatory flying control is either necessary or desirable. The basic task of air traffic control is therefore (a) to give information so as to expedite landings and take-offs in good flying conditions and (b) to give information and instructions in blind flying conditions or at night.

The exact (Navier-Stokes) equations for the flow of a viscous compressible fluid are examined in a search for simple solutions, in which the equations reduce to ordinary differential equations. Such solutions are found for the uniform shearing motion in the Couette flow between both parallel flat plates and coaxial circular cylinders in relative motion. For a compressible fluid there are no simple solutions corresponding to certain other flows (such as the Poiseuille flow between fixed parallel flat plates) described by simple solutions for an incompressible fluid. A solution is found for the flow of a gas over an infinite porous plate with uniform suction through the plate (gravity neglected), and the relation of the solution to asymptotic solutions of the boundary-layer equations is discussed. Ordinary differential equations are obtained for the problems of (i) the flow due to the rotation of a horizontal disk in a gas (dissipation neglected), and (ii) the steady circulatory flow round a porous circular cylinder with uniform suction (gravity neglected). Discussion of the gas-dynamical problems is preceded by a description of the state of a horizontally stratified gas in static and thermal equilibrium.

1. Many investigations have been made to determine the wave resistance acting on a body moving horizontally and uniformly in a heavy, perfect fluid. Lamb obtained a first approximation for the wave resistance on a long circular cylinder, and this was later confirmed to be quite sufficient over a large range. In 1926 and 1928, Havelock (4, 5) obtained a second approximation for the wave resistance and a first approximation for the vertical force or lift. Later, in 1936(6), he gave a complete analytical solution to this problem, in which the forces were expressed in the form of infinite series in powers of the ratio of the radius of the cylinder to the depth of the centre below the free surface of the fluid. General expressions for the wave resistance and lift of a cylinder of arbitrary cross-section were found by Kotchin (7) using integral equations, and the special case of a flat plate was evaluated. He continued with a discussion of the motion of a three-dimensional body. More recently, Haskind (3) has examined the same problem when the stream has a finite depth.

A variety of marine requirements has prompted the Hydrographer of the Navy to design a plotting diagram printed on a transparency which is now under trial in prototype form. British Admiralty chart 5028, as this project is officially called, comprises a thin sheet of flexible and transparent material slightly frosted on one side to take ordinary black lead pencil work. The plotting area is printed on the frosted side and its circular portion has a diameter of 20 inches.

The term ‘dead reckoning’, so the Admiralty Navigation Manual (1938) says, is a corruption of the old ‘ded-uced reckoning’ or ‘position by account’, and is used to cover all positions that are obtained from ‘the course the ship steers and her speed through the water, and from no other factors.’ (The last italics are ours.) Had Master William Borough, Chief Pilot of the Muscovy Company, and presently to be appointed to Queen Elizabeth's Navy Board, come across this definition, he would have picked a two-fold quarrel with their present Lordships at Whitehall. For his use and explanation of the term is the earliest of which we have knowledge, although it does not appear even then to have been new.

In considering the equilibrium of stars of high density the effects of the Pauli Exclusion Principle must be taken into account. For large values of the degeneracy parameter, which will be denoted by λ, an explicit formula for the partition function may be obtained, from which we may easily find the pressure and density in terms of λ. When λ ≪ 1 the relations for a Fermi-Dirac gas reduce to those for a Boltzmann gas. For λ of the order of, but less than, unity, series expansions for the relevant physical quantities can be found, but for λ of the order of, but greater than, unity, a set of numerical quadratures must be performed at intervals (in λ) close enough for interpolation purposes. The method for this is discussed in § 2 and the numerical results are given at the end of the paper.

In his treatise entitled Introduction to the theory of Fourier integrals, E. C. Titchmarsh derives a Fourier transform of the solution of the integral equation

The transform of the solution given by him does not satisfy the Riemann-Lebesgue lemma, and further suffers from the defect that it has certain exponential properties which prevent the actual calculation of the function φ(ξ), since the inversion integral does not exist for all ξ > 0. We shall correct these defects and show further that the solution of (1) is particularly simple and that it can be expressed as the Legendre function of an order which depends on λ.

This paper is concerned primarily with an investigation of the loxodromic distance between two points on the spheroid. This is derived by a modification of the meridional parts method in terms of the track angle and the difference of latitude reduced by a correction which is a function of the latitudes concerned.

Since the method is unsuitable when the track angle is near to 90°, the known formula for the mid-latitude correction is examined and an alternative method suggested for use in this case.

The results obtained are compared in a practical example with those found by the usual methods, and finally means are proposed by which the refinement described above can be incorporated into current plotting practice on aeronautical Mercator charts.