1. Throughout this note Q will denote a local ring, m will denote its maximal ideal, q will denote a primary ideal belonging to m and k will denote the residue field Q/m. It will not be assumed that k is infinite, but we shall suppose that Q and k both have the same characteristic. Now let υ1, υ2 …,υd be a system of parameters contained in q, so that d = dim Q; then according to the definition given in (2) the ideal (υl υ2,…, υd) is a reduction of q if (υ1 υ2, …, υd) qm = qm+1 for at least one value of m. The use of the concept lies in the fact that such a reduction is, in a certain sense, a very good approximation to q itself; but the notion does, however, suffer from a minor disadvantage in that, if k is finite, q need not have any reductions. In §3 we shall generalize the notion of a reduction in such a way that we overcome this difficulty, and in such a way that the results concerning reductions obtained in (2) acquire some useful extensions.

It is a familiar fact that the Picard surface (or hyperelliptic surface of rank 1) admits a completely transitive permutable continuous group of ∞2 automorphisms. There are, however, other non-scrollar surfaces which possess continuous groups of automorphisms, namely, the elhptic surfaces. Every elliptic surface V2 contains a pencil of birationally equivalent elhptic curves, which are the trajectories of the group in question; it also contains a second, elliptic, pencil of birationally equivalent curves; the intersection number of the two pencils is an important character, known as the determinant of V2. Just as any Picard surface can be mapped on a multiple Picard surface of divisor unity, so V2 can be mapped on a multiple elliptic surface of determinant unity, the branch curve (if any) corresponding to a certain number of trajectories.

1. Statement of problem and the method of Barnes and Swinnnerton-Dyer. 1·1. Let ξ1, ξ2, … ξn De n real, homogeneous linear forms in the real variables x1, x2,…, xn and of determinant Δ ǂ 0. Let y1y2, …, yn be any real numbers. We identify the n-tuplet (x1x2, …, xn) with the point P, coordinates (x1, x2, …, xn), of n-dimensional space.We write P≡P0 (mod 1) if x1≡y1, x2≡y2, …, xn≡yn (mod 1), and P0 is the point (y1, …, yn).

In this paper we determine the first minimum of a class of linear forms associated with certain cubic fields that depend on a parameter.

It has been known for some time that there are sets in Euclidean space which are of infinite measure in a certain Hausdorff dimension, say α, and yet which contain no subsets that are of finite positive a measure. The properties of such a set, say X, could be developed in much the same way as those of sets which are of positive finite α measure, if there existed a Caratheodory outer measure Γ, defined over the subsets of X, which was such that ∞ > Γ(X) > 0, and for any subset Y of X, Λα (Y) = 0 implied Γ (Y) = O. The object of this note is to show that there are sets for which no such outer measure exists. It is shown that a set defined by Sierpinski (7) is one-dimensional and is such that any Caratheodory outer measure defined over its subsets either takes infinite values or is identically zero or is not zero for all subsets that consist of a single point. It has been remarked by Prof. Besicovitch that these properties imply that the set is measurable with respect to any Caratheodory outer measure defined over subsets of the plane, and in fact any subset is measurable with respect to any such outer measure.

In this paper, I prove some new results which are identities connecting groups of general infinite products, and I give two tables, calculated using EDSAC, of the function 1/II(1 − aqn−1) over the range a = − 0·90(0·05) + 0·95, q = 0·0(0·05) + 1·00, and of the function 1/n(1 − qn) over the range q = 0(0·005)+ 0·890.

This paper contains all those formulae which are necessary for the numerical evaluation of the basic confluent hypergeometric functions, including asymptotic formulae, and three skeleton tables which can be used as a basis for the study of the numerical behaviour of these functions over a wide field.

Estimators are given for the solution by Monte Carlo methods of the Fredholm integral equation of the second kind and the variances of the estimators are compared. Two integral equations arising in sequential analysis are considered in detail; some numerical examples are given.

Introduction. Many automatic computers employ magnetic drums or ultrasonic delay lines as their main store. A part of the time taken by a computation on such a machine is spent in waiting for words to become available from the store. A number of methods and devices have been used or suggested to reduce this waste of time. This paper attempts to evaluate these methods by determining on the one hand the amount of waiting time eliminated and on the other hand the design or programming complications introduced. No quantitative method for determining the amount of eliminated waiting time was known. Available estimates were based on the feeling of the programmer and varied accordingly. It was therefore necessary in the first place to develop a process by which the amount of waiting time eliminated by any given method could be determined. This process is set out while being applied to the methods discussed. The methods themselves are not new, except that closer analysis has shown that the method adopted with the ACE pilot model and known as ‘optimum programming’ is in fact not a single method but a combination of several methods with different features. Also, it appeared that, contrary to an opinion expressed elsewhere (2), there is a simple way of staggering locations in the store even with words of two lengths.

The theory of a previous paper is used to discuss Eddington's work in fundamental physics.

1. Introduction. A previous paper (2), referred to here as I, described a theory depending on an unorthodox view of the relation of mathematics to physics. This theory was begun from the study of the work of Eddington which we take to be summarized in Fundamental Theory (5). In the present paper is shown the extent to which, and the way in which, the two theories are connected.

1. Introduction. A number of authors (2,3,7) have recently quantized the motion of an inviscid fluid. The starting point has been the variational principle of Bateman which uses the Clebsch variables. The density turns out to be the canonical conjugate to the velocity potential, and the transition to the quantum theory is then made in the usual way. If the fluid is making small vibrations, ‘phonons’, which are scalar, appear as a result of the quantization.

In an attempt to understand the mechanism governing separation the possibility of there being further solutions to the Falkner-Skan equation f′′′ + ff″ + β(1 − f′2) = 0, in addition to those found by Hartree, is investigated.

It is found that when β < − 0·1988 there are no solutions satisfying |f′| ⩽ 1 for all η, and that if 0>β> − 0·1988 there are two acceptable solutions, one with f″(0) < 0. The new ones are computed to three places of decimals for various values of β and tabulated. In addition, it is shown that if − 0·5 < β < 0 there is a family of solutions corresponding to boundary layers bounded on one side by free streamlines. These are also computed and graphs of f′(0) and δ1 are displayed. The impact of these solutions on the theory of separation is discussed.

1. Introduction. The problem of determining the distribution of stress in a medium with transverse isotropy (one axis of elastic symmetry) has recently been considered by Elliott (l, 2) and Shield (3). They have obtained solutions to certain special problems both with and without rotational symmetry. Elliott (2) has employed the method of dual integral equations to solve a number of axially symmetric crack and punch problems. (This crack problem is not to be confused with the torsional crack problem treated by Weinstein (4)). Shield (3) has reduced the general crack and punch problems for a medium with transverse isotropy to corresponding isotropic problems as formulated by Green (5), and has treated a few special examples. For the axially symmetric isotropic crack and punch problems the author (6) has shown that Green's formulation may be simplified, and has obtained in each case a simple usable solution of the general axially symmetric problem without the need for dual integral equations. As one might expect these simplifications may also be realized in the case of a medium with transverse isotropy. In this paper the author, guided by the work of Green (5) and Shield (3), presents a straightforward method for solving the general axially symmetric crack and punch problems for media with transverse isotropy. This method permits one to write down immediately in an appropriate system of curvilinear coordinates the solution to the general axially symmetric problem, whereas the method given by Shield requires an individual treatment of each problem. In fact it would appear to be extremely difficult if not impossible to obtain the complete solution to the general axially symmetric crack or punch problem by the methods suggested by Shield. Since the formulation given in this paper differs from that given by Shield, his equations for transforming the problem in transverse isotropy to an isotropic problem must be altered slightly. The solutions to the general axially symmetric problems for a medium with transverse isotropy then follow immediately from the solutions to the corresponding isotropic problems in (6). We shall therefore find it necessary to repeat some of the arguments and results of (6).

It is a commonplace observation in experiments on cutaneous sensation that a constant stimulus, applied under supposedly constant conditions, evokes a varial response. In this paper it will be shown that the variability is in part a geometri consequence of the fact that the sense organ is not a continuum but an aggregati of discrete, sensitive ‘spots’. The total extent of the ‘spots’ for one modality, acti at a given time, is only a small fraction of the area of the site tested; this is illustrat in all of the classical maps, and is particularly true of the warmth-spots (Rein (8)). I purposes of theoretical discussion, the sensory spots may therefore be treated points that form an array or pattern. This pattern is unlikely to be regular and mt indeed, be highly irregular. No assumptions about its regularity are necessary the present argument.