In § 1 we introduce our system and prove a theorem about its syntax. In § 2 we recall some stock results about the propositional calculus. In § 3 we consider a method of deriving an expression from a given expression and a real number. In § 4 we use this to derive a sequence of expressions from a given expression. In § 5 this sequence is shown to be just all the terms of a conjunctive normal form of the given expression. In § 6 we note that we may not need to produce all of these terms. In § 7 we describe a practical method (suitable for a binary digital electronic computer) which results from all this, and in § 8 we attempt to explain just why this is so.

A set is said to be σ-finite (with respect to Λs-measure) if it can be expressed as a countable sum of sets of finite Λs-measure. I have proved(1) that every non σ-finite analytic set in a Euclidean space contains a closed set of infinite measure, and Prof. Besicovitch asked me whether the closed subset could itself be chosen to be non σ-finite. In this paper an affirmative answer is given.

In the following pages there is developed an abstract theory of infinitely near points for two-dimensional regular local rings which have infinite residue fields. The scope of the theory may be indicated by the fact that it contains generalizations of Noether's formula for intersection multiplicities (Theorem 8), the theory of proximate points (§ 8) and the conditions for a set of integers to be curve multiplicities at a given finite sequence of consecutive points (§ 10). To this extent our account resembles that given by van der Waerden ((6), Chap. 9), but the starting point and (with the exception of the concluding section) the type of reasoning employed are very different. There is also a marked contrast with the methods used by Zariski (7) in an account of related topics.

Let | Si | (i = 1, …, k) be k linear systems of hypersurfaces on an algebraic variety Vd, and let

The purpose of this note is to prove that the Jacobian of these systems is given by the equivalence

where Kh is Eger's operator, defined as follows:

In Schubert's book Abzählende Geometrie(3) there are listed, with no formal verifications, eleven first-order degenerations of the twisted cubic curve in S3. The curve, in this connexion, was seen by Schubert as a union C = (CL, CE, CT) of three simply infinite element systems, the locus CL of its points, the envelope CE of its osculating planes and the system CT of its tangent lines, the complete twisted cubic so envisaged being regarded as a single geometric variable of freedom 12, and any degeneration (projective specialization) of it, C¯ = (C¯L, C¯E, C¯T), being said to be of the first order if it has freedom 11. Schubert used the symbols λ, λ′, k, k′ w, w′, θ, θ′, δ, δ′, η to denote the eleven first-order degenerations in question, the dashed symbol denoting always the dual of that denoted by the undashed. In this paper we exhibit analytically the existence of the above degenerations and indicate further how degenerations of higher order can be systematically investigated. We use for this purpose the analytical theory of complete collineations which has just recently been developed (1,4), and also some of the ideas contained in van der Waerden's earlier development of his general method of degenerate collineations (5).

The equations to be considered are of the form

where Xk is to be determined as a function of t, is its ith derivative with respect to t, Fj is a given analytic function of t, a given constant, and i = 0, 1, …, n; j, k = 1,…, m, and they represent a system of simultaneous linear differential equations of order n in m functions Xk (k = 1,…, m).

The solution of a system of simultaneous homogeneous linear differential equations with constant coefficients has been obtained (1) in the form

where the elements of the vector Z are the k functions determined by the system and their derivatives of order up to m – 1, where m is the order of the system. If Zn is the value of Z at t = n, then

The paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue Λ(h) of the corresponding finite difference equations towards the eigenvalue λ of the differential equation itself and it is shown that

where h is the size of the interval of the grid covering the range of the independent variable; the constant ν is usually a negative number and consequently Λ(h) may well be a lower bound to λ. This convergence property is used in the numerical calculation of λ by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.

Suppose that x1, …, xn are jointly distributed in a multivariate normal distribution, f(x1 …, xn), with a given correlation matrix (ρij). Then the evaluation of

is a problem which has been much discussed and has many applications (see David (1)). When n = 3

Many of the statistical properties of a random noise function f(t) have been derived by Rice (4); for example, the mean frequency of zero crossings and the statistical distribution of maxima and minima. These distributions have been shown to apply quite accurately to physical phenomena; in particular, when f(t) represents the height of sea waves at a certain point or the pressure at a fixed point on the sea bed (1, 3, 5).

A correction is made to a previous paper(2) on clumps formed by random placing of laminae, a useful property of such placings is deduced and formulae are found for clumps formed when convex bodies are randomly placed in space and for clumps formed by the orthogonal projections of such placings.

The Tamm-Dancoff theory of meson-nucleon scattering (pseudoscalar coupling) leads to a series of singular integral equations. The asymptotic behaviour of the solutions of these equations is obtained here for all scattering states. For attractive states there is a critical coupling constant beyond which no normalizable solutions exist for these equations; for repulsive states another critical coupling constant appears, beyond which the solutions oscillate infinitely often but are still normalizable. It is concluded that the renormalization procedures proposed previously (3) are consistent and successfully define finite vertex-functions, and that the equations obtained for these vertex-functions have satisfactory properties for numerical work.

The exact solution for a trapezoidal potential barrier is given in terms of Airy functions and trigonometrical functions. Numerical values are found for the ratio of the amplitudes of the transmitted and reflected waves. These are compared with those given by the WKBJ approximation.

The modified Thomas-Fermi method reported in (2) has been applied numerically to calculate the distribution of valence electrons in CC14. Slater wave functions have been used to describe the Cl inner-shell electrons, whilst a description of the C ls electrons based on the Thomas-Fermi model for atoms has been adopted. The electron distribution thus found is compared with the two approximations discussed in part I of this series (3). The first of these, in which valence and inner-shell electrons are treated in the same way, is shown to give a distribution of valence electrons in good agreement with the present results. The second approximation, that of effective nuclear charges, is less satisfactory and could only lead to a distribution of valence electrons in reasonable agreement with the present work if the effective nuclear charges were considerably reduced. This seems to throw some light on the results of Bowers for the force constants of tetrachlorides and tetrabromides.

The overall distribution of electrons obtained here seems to be the inevitable result of assuming the valence electrons to move in any plausible central field. It is, of course, by no means certain that such an approximation is near to the truth. In fact, the results of (3) and the present paper leave little doubt that the electron cloud obtained by such an approximate method will be too diffuse. This seems at least partly responsible for our finding that rather more than 20 of the 28 Cl valence electrons lie outside the sphere drawn through the Cl nuclei.

Classical dynamics is generally considered as the prototype of a deterministic theory; the equations of motion determine the coordinates q and the momenta p at any time provided they are given at an initial instant. It is pointed out that this is an unrealistic assumption, for it is inevitable that there should be small uncertainties Δq, Δp; a point of the mathematical continuum has no physical significance. The uncertainties can be taken into account without violating the deterministic equations by introducing a probability density P in phase space p, q. The function P satisfies the partial differential equation which expresses Liouville's theorem. Thus it can be shown that an initial uncertainty spreads out in phase space, so that finally all states of the system consistent with the mean initial values of the constants of the motion are equally probable. This property holds for one degree of freedom just as well as for many, and is not a consequence of our ignorance concerning large numbers of particles. The essential difference between classical mechanics and quantum mechanics consists not in the physical necessity of a statistical interpretation, but in the further introduction of a probability amplitude, the square of which is the probability density; thisimplies the restriction of the initial uncertainties as expressed by the uncertainty laws, and the phenomenon of interference of probabilities which makes necessary a revision of our concepts of physical reality. It is remarkable that the product of the uncertainties of a properly chosen pair of variables is constant also in classical mechanics, although of course its value is not a universal constant as in the quantum case.

The steady motion of a perfectly conducting sphere in an inviscid conducting fluid in the presence of a strong magnetic field is discussed. It is shown that if the fluid velocity is ultimately steady then it is two-dimensional, and a cylinder of fluid whose generators are parallel to the direction of the field moves with the sphere as if solid. The streamlines outside are straight lines if the sphere moves in the direction of the field but have to execute sharp turns if it moves at right angles to the field. The motion to be expected in practice is discussed using an analogy.

It will be shown that a vortex element has the following image system in a rigid sphere immersed in incompressible fluid:

(i) A transverse vortex element at the inverse point, equal in strength to minus the transverse resultant of the given vortex element, divided by the ratio of its distance from the centre to the radius of the sphere.

(ii) A radial vortex element at the inverse point, equal in strength to plus the radial resultant of the given vortex element, divided by the same ratio.

(iii) A uniform line vortex stretching from the inverse point to the centre of the sphere, equal and opposite in total strength to the radial vortex element at the inverse point.

The diffraction of a plane harmonic sound wave by a hollow circular cylinder of finite length is considered. The problem is treated by using Laplace transforms and is reduced to the solution of two complex integral equations. An approximate solution is obtained for these equations when the product of the wave number (k) and the cylinder length (l) is large. The resonance of the system is considered and an equation derived for the resonant lengths which is then solved approximately for kl large. An explicit expression is obtained for the end correction of an. r–resonant system (i.e. kl ≈ rπ), and also comparison is made with experimental results.

The usual linearized theory neglects small terms, of second and higher orders of the velocity perturbation, in the coefficients of the equation for the potential. In this paper these terms are approximated by constant mean values and the familiar formulae of the first approximation are then generalized. The characteristic lines of this improved linear theory thereby differ from the Mach lines of the undisturbed flow, though still being straight and parallel. The new formulae thus give a better calculation of the velocity field near bodies not slender enough for the ordinary theory to be accurate. A numerical example, for a 10° cone in a stream of Mach number about 3, illustrates this fact, and shows that a shock of constant strength may be fitted to the improved solution.

Although the improved theory as it stands must fail at large distances from the axis of symmetry, just as the first approximation fails, the tentative suggestion is made that a technique due to G. B. Whitham may be applied to the former, as to the latter, perhaps leading to an improved description of the whole field of flow.

A theory is presented of the non-linear propagation of waves and bores in channels of varying cross-section with a basic steady flow governed by frictional resistance; this corresponds to the flow in tidal rivers. The theory provides a condition on the tidal range required to produce a bore, in terms of the geometry and friction parameters of the river, and the propagation of such a bore is then described. The theory is applied to the River Severn and the results agree satisfactorily with observation. The results for the special case of waves moving into still water in a channel of varying section are also noted in detail.