By the use of an auxiliary independent variable, as well as of auxiliary functions, it may be possible to obtain tables which are very compact and in which, nevertheless, linear interpolation is adequate for most practical purposes. In the case of the Bessel functions of order zero, it is shown that by the use of 1/x2 as auxiliary independent variable, and appropriate auxiliary functions, linear interpolation in two tables of forty-one entries each is sufficient to give x½12J0(x) and x½12Y0(x) with an uncertainty of a unit in the 7th decimal, from x = 5 to ∞, and two tables of only three values each are sufficient to cover the range x = 22·5 to ∞ in the same way. Two possible forms of auxiliary functions for this purpose are considered.

The relativistic Hamiltonian H for a particle in a field of potential energy V is given, in atomic units, by the formula

where any ρ commutes with any σ, any σ anticommutes with any other σ, ρ1 anticommutes with ρ3 and

In this paper some elastic stability problems are solved for infinitely long, rectangular, isotropic plates, simply supported at the long edges. The problem is solved when the same arbitrarily prescribed distribution of normal stress is applied to each long edge, with no stresses and normal displacement at infinity. The extensional deformation of the middle surface during buckling is neglected; this leads to an underestimation (at most of the order of 10%) of the critical forces, but is unlikely to influence the general conclusions. A strain-energy method of analysis is used. The critical stress distributions are shown to be given by the eigenvalues of an ordinary differential equation, or, alternatively, of an integral equation. The general problem is then specialized to that of sets of concentrated normal forces, and to that of a normal force distributed uniformly over a finite length of each long edge; some numerical results are given for these cases. In the first of these problems there is negligible error in treating pairs of concentrated forces individually when they are separated by distances greater than twice the plate width. In the second problem there is little change in the critical stress when the length of edge stressed exceeds four times the plate width; also, if the length of edge stressed is less than one-quarter of the plate width, the distributed force required to produce instability does not differ appreciably from the concentrated force required to produce instability.

1. The quadrics of space are linearly dependent on ten among them; any ten linearly independent quadrics may be chosen to constitute the base, but it is customary in analytical work to select the four squares and the six products of pairs of four planes which are the faces of some tetrahedron of reference. This choice is adequate enough for many purposes, but it gives an unsymmetrical twist to the work; whereas four quadrics of the base are repeated planes the other six are plane pairs, and the curve common to two of the ten base quadrics can be a skew quadrilateral, a repeated line-pair, or a line counted four times. This radical defect can be mitigated by taking as base a certain set of ten non-singular quadrics, namely, the set of ten fundamental quadrics which is so prominent a feature of Klein's figure of six mutually apolar linear complexes. Every pair of such a set of ten quadrics has its two members related to one another in precisely the same way; their common curve is a skew quadrilateral and they are their own polar reciprocals with respect to each other.

1. In this note we shall prove the

Theorem 1. Letbe a linear space of (real or complex) functions f(s) defined in the interval 0 ≤ s ≤ 1 subject to the following two conditions:

(i) every function of the infinite sequence 1, s, s2, …, sn, … is an element of;

(ii) two elements, f(s) and g(s), ofare to be considered as distinct if, and only if, they differ on a set of positive measure in the interval 0 ≤ s ≤ 1.

The method of transforms (7), (2) is shown to be directly applicable to the case of axial symmetric stress distributions in hexagonal crystals. It is shown that solutions for problems of indentation of the hexagonal plane by rigid punches can be found for punches of arbitrary axial symmetric shape. Solutions are given in full for the cases of spherical, conical and circularly cylindrical punches.

The same method is used to find the solutions for a material containing disk-shaped cracks between hexagonal planes and the results for the isotropic case deduced from the general solution.

Hitherto it has been the practice to compute the altitude and azimuth of a celestial body either for a D.R. position or a chosen position in the vicinity of the D.R. position and to compare the computed altitude with the observed altitude, the difference (the intercept) being marked off along a line drawn in the direction of the computed azimuth from the position used for computing, either ‘from’ or ‘towards’ the object; a perpendicular drawn at the end of the intercept is the position line of Marcq. St. Hilaire.

In this note, we propose to enquire into how far it may in principle be possible to employ zenith-photography for astro-navigation in the air, also to touch on the question of whether the mechanical difficulties can be adequately overcome. The question of principle seems in fact quite clear, but the practical side will undoubtedly demand much care and study by engineers and by practical navigators if the method is ever in fact to be adopted; we hope to show that the method has advantages sufficient to make the problem worthy of attention.

It is generally agreed that the increasing speed of modern aircraft, together with the availability of short-range radio and radar aids, necessitates a fresh approach to astronomical methods; less accuracy can be tolerated in the calculations but greater speed is essential. There are three stages after observation : the derivation of the position of the heavenly body observed, usually from the Air Almanac; the solution of the astronomical spherical triangle, usually by means of reduction tables ; and the resulting plotting of the position lines and fix.

1. The problem of automatic synchronization of triode oscillators was studied by Appleton† and van der Pol‡; it gives rise to the differential equation

where α, γ, ω, E, ω1 are positive constants such that α/ω, γ/ω, (ω − ω1)/ω are small and dots denote differentiations with respect to t. When these conditions are satisfied, it is easy to see that

is an approximate solution over a limited time for any b1, b2 chosen to fit initial conditions on v and ṿ provided that v and ṿ are not too large. If it is assumed that b1 and b2 vary slowly compared with ω1t, so that can be neglected, and ḃ1, ḃ2 are comparatively small, the equations

where

are obtained. These are sufficiently accurate for the discussion of most of the physical phenomena, and have been used in this form (or in the polar-coordinate form obtained by putting b1 = b cos ø, b2 = b sin ø) by various authors §. Solutions of (1) with period 2π/ω1 are obtained approximately by putting ḃ1 = ḃ2 = 0 in (3). The steady-state solutions of (1) other than those with period 2π/ω1 correspond to periodic solutions of (3). All other solutions converge to one or other of these types of solution.

In a previous paper on steady rectilinear plastic flow of a Bingham solid (1), the point of view was adopted that the study of all possible velocity distributions was equivalent to a study of the geometry of velocity contours in a section normal to the flow. This led to an analysis of the differential geometry of velocity contours in rectilinear plastic flow under zero pressure gradient, and the problem of finding all possible velocity distributions was eventually reduced to the problem of finding the general integral of a single linear second-order partial differential equation with two independent variables. But the form of this differential equation (equation (25) of (1)) was such that only a very limited number of solutions could be found by standard methods. The particular solutions which were obtained corresponded to those flow patterns in which the velocity contours could be identified with a family of parametric curves in a curvilinear coordinate system obtainable by conformal transformation of a cartesian frame of reference. In effect, this means that the results so far achieved by using the new approach to the problem could have been obtained by more direct methods.

This paper is the sequel to one entitled ‘An Inequality in the Theory of Arithmetic on Algebraic Varieties’ (1). For the definition of the complexity of an algebraic point P—here denoted by C[P]—we refer the reader to that paper, where he will also find a short account of as much of the theory of ‘distributions’ as will be used here. For full details of the latter, the reader should consult A. Weil's article (2). Both this paper and the preceding one are the result of an attempt to generalize and simplify a few of the many fruitful ideas which are embodied in Weil's thesis.

The inequality which is proved here was established under different conditions in a paper entitled ‘Periodic points on an algebraic variety', which is to appear in the Annals of Mathematics. It is hoped that the new formulation will be more useful for applications. The present investigations use the rather sophisticated techniques of the ‘Theory of distributions’ which was developed by Weil (1). As the content of this theory may be unfamiliar to the reader, Weil's results are described at some length, though of course, his proofs are not given. The one result which we need and which is not given explicitly by Weil, is proved in detail.