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.

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.

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.

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.

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.

The properties of (1) consistency and (2) asymptotic normality of maximum likelihood estimates of one unknown parameter, for independent observations, have been discussed rigorously by Cramér ((1), p. 500) and Doob (2). These properties were first stated by Fisher (3).

The ‘phase-space’ method in quantum theory is used to derive exact expressions for the transition probabilities of a perturbed oscillator. Comparison with the approximate results obtained by perturbation methods shows that the latter must be multiplied by an exponential factor exp (− ∊/ℏω), where ∊ is the non-fluctuating part of the work done by the perturbing forces; as long as ∊ is small, exp (− ∊/ℏω) ˜ 1 and only dipole transitions have an appreciable probability. As the perturbation energy increases, however, this is no longer true, and multipole transitions become progressively more probable, the most probable ones being those for which the change in energy is approximately equal to the work done by the perturbing forces.

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.

In many compressible fluid flow problems the classical solution breaks down completely owing to the formation of regions of infinite acceleration in the flow field. The actual behaviour of the fluid in such cases does not seem to have been investigated mathematically and this is largely due to the difficulties which enter with non-uniform shocks. It is with these difficulties that this paper is principally concerned.

The way in which discontinuities may arise mathematically in a flow field is first discussed. The equations governing the one-dimensional motion of a gas due to an accelerating piston are then set up. It is shown that when allowance is made for varying entropy conditions due to the presence of non-uniform shocks these differential equations reduce (outside the shock wave) to three first order quasilinear ones.

The initial solution breaks down when a point of infinite acceleration occurs in the flow field. From this point onwards a shock wave grows in the fluid and behind it three different sets of characteristics are required to describe the flow. By working in the plane of two quantities that are constant along two different sets of characteristics, we can use the shock-jump conditions to determine the equation of the shock-line in this plane and to reduce the equations of the characteristics to three differential equations, which would be linear if the relation between the entropy and other flow variables were known.

In the case of a constantly accelerating piston a first approximation is found by neglecting reflexions and entropy variations behind the shock. Using this as a basis we then find a second approximation for the entropy in the neighbourhood of the initial portion of the shock-line and show that the problem reduces to the solution of a second order linear partial differential equation. The introduction of a Riemann function and the satisfaction of the boundary conditions at the shock lead to an integral equation whose solution enables us to determine the position of the shock as a function of the time. The solution is in the form of a power series and is valid provided the shock wave does not become too strong.

Finally, it is shown that if the piston is given a constant terminal velocity a reflected wave from the shock is reflected again from the piston and eventually overtakes the shock and reduces its velocity to a final steady value which is in agreement with the value arising from an impulsive start.

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.

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.

1. The forces acting on the two-dimensional aerofoil in a bounded uniform stream have been found for a variety of cases, and in the present paper an attempt is made to extend the theory to include linear shear-flow. The special case of the symmetrical Joukowsky aerofoil in unbounded shear-flow has been solved by Tsien (6) using real-variable theory. A more satisfactory method is using complex-variable technique indicated in § 3, and is applicable to a more general shaped aerofoil. The effect on the lift and moment when a plane boundary is present is then considered. When the aerofoil is not too near the boundary, the lift and moment can be expanded in powers of the ratio of a typical length in the aerofoil to the height of the aerofoil above the boundary, by following exactly the technique used by Green (1) in a recent paper dealing with the same problem but with a uniform fluid flow. In the last section the limiting case of the flat plate touching the boundary with its trailing edge is discussed.

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.

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

A modification of the MO method is described which leads to the same polynomial secular equation as does the LCAO method, but via a simpler secular determinant. The basic idea is to divide the whole molecule into parts, calculate the MO's for these by the usual method, and then use linear combinations of these MO's as trial eigen-functions in a variational treatment of the whole molecule.

Equations are developed for a number of cases, and it is shown that the computations of energy levels and MO's from them is often much simpler than from the corresponding LCAO equations.

The use of artificially pulsed sources of particles in experiments in nuclear physics has already given rise to powerful methods of investigating short-period phenomena and of studying the effects of ‘slow’ neutrons of different energies. On the other hand, the use of such sources introduces complications in counting experiments which are not present when steady sources are employed and it is possible to assume that the time distribution of ionizing events in any counter is a strictly random one. Again, sources which are effectively steady in respect of one counting arrangement are not necessarily so in respect of all. Ordinary cyclotron sources (i.e. cyclotron sources which are not ‘artificially’ pulsed) belong to this class; they have justifiably been considered as steady in the majority of experiments hitherto performed, but the flux of primary particles reaching the target is definitely periodic in time (with the period of the R.F. Voltage applied to the dees), thus a cyclotron will appear as ‘naturally’ pulsed whenever it is used with a counting system for which the effective resolving time is sufficiently short (say < 10−7 sec.).

The object of the paper was to find the forces acting in bubble rafts and to use them for a description of the mechanical properties of these rafts. The first section is based entirely on Nicolson's work and gives expressions for the cohesive forces in the lattice. The second and third sections deal with the repulsive forces arising when a bubble is compressed uniaxially, and are really the basis of the paper. The fourth section introduces the idea of ‘interaction’ between compressions in the three close-packed directions—a conception which explains the breakdown of Cauchy's relations. The conditions of equilibrium in the lattice are investigated in § 5, and in § 6 all these forces are combined to give a complete account of the elastic properties of the rafts.

The remaining section gives a brief account of the experimental checks of the theory; it appears to be a reasonable approximation for small bubbles, and, indeed, we could certainly not expect it to apply to large ones, since for them not only are all the calculations of the attractive potentials invalid because of the large local slopes of the surface, but also the ‘flats’ and ‘caps’ become so large as to interfere with one another.

The forces are considered well enough supported to justify their further use in an investigation of the plastic behaviour of the bubble rafts—an investigation which has explained several interesting experimental facts but brought us little nearer an understanding of the low yield strains of metallic single crystals (see (2)).