The results of this paper have as their background two concepts fundamental in the theory of non-associative algebras (i.e. algebras in which the associative law of multiplication is not assumed). These concepts are:

(1) Isotopy,

(2)The three nuclei λ, μ, ρ of an algebra .

A fundamental result in the theory of measure in the space Ω of real functions x(t) of a real variable t is the following theorem of Kolmogoroff:

Theorem 1. Suppose that functions F(t1, …, tn; b1 …, bn) = F(t; b) are defined for positive integers n and real numbers t1, …, tn, b1, …, bn, and have the following properties:

(1·1) For every fixedt1, …, tn, F(t; b) has non-negative differences

with respect to the variables bl, b2,…, bn, and is continuous on the right with respect to each of them;

if (i1, …, in) is any permutation of (1, 2, …, n). Then a measure P(X) can be defined in a Borel system of subsets of Ω in such a way that the set of functions satisfying

is measurable for any realbi, tiand has measure F(t; b).

The study of differential properties of functions of two real variables in the light of the modern theory of functions of real variables was started by H. Rademacher in 1919. Although much work has since been done on the subject, and a great many general results have been obtained, a number of questions have remained open, some of which will be discussed in the present work.

1. The questions considered in this note are suggested by the elementary topology of the trajectories of systems of non-linear differential equations. Such a system may be assumed in the form

and the values of the dependent variables x1, x2, …, xn at ‘time’ t can be represented by a point P(t) in a ‘phase space’ . As t varies, P(t) describes a curve in , which is a trajectory of (1). Now it often happens that contains a subspace E (usually of lower dimension) with the following properties: (i) by considering the trajectories generated by points P(t) which are, for t = 0, in E, all the trajectories of (1) are obtained; (ii) if P(0) is in E, then P(t) is not in E for 0 < t < c, where c is a constant independent of P(0) in E; (iii) if P(0) is in E, then the trajectory meets E again for some finite t at a point P(T) (T is not necessarily the same for all points of E). By considering P(T) as the image of P(O), a mapping of E into itself is defined which is associated with the system (1), and the topology of the trajectories of (1) can be studied conveniently by discussing this mapping. When the functions fi in (1) satisfy the continuity and Lipschitz conditions of the classical existence-and-uniqueness theorem, the mapping is one-one and continuous. The study of this ‘transformation theory’, initiated by Poincaré, has been developed chiefly by G. D. Birkhoff(l,2). His results have been applied to problems of ‘non-linear mechanics’ by N. Levinson(3).

In connexion with moment problems, S. Verblunsky proved the following two theorems:

Theorem I. (a) If f(x) is integrable in (−∞, +∞) and satisfies 0 ≤f(x) ≤ 1, then there exists a function σ(x), bounded and non-decreasing in (−∞, +∞), such that, for ζ = ξ + iη and η ≠ 0,

The problem of constructing an n-dimensional metric differential geometry based on the idea of a two-dimensional area has given rise to several publications, notably by A. Kawaguchi and S. Hokari (1), E. T. Davies (2), and R. Debever (3). In this geometry the area of a two-dimensional plane element is defined by a fundamental function L(xi, uhk), where the xi are point coordinates and the uhk are the coordinates of the simple bivector representing the plane element. L is supposed to be a positive homogeneous function of the first degree with respect to the variables uij, and to possess continuous partial derivatives up to and including those of the fourth order. With these assumptions the problem of the construction of the metric differential geometry splits into two problems; the first of these is the problem of constructing a metric tensor gij(xr, uhk), and the second is the problem of constructing an affine connexion. We deal with the first problem only in this paper.

In 1914 Mitchell (13) underook the determination of the finite primitive collineation groups in more than three dimensions which are generated by homologies. Among these groups are five which are remarkable as not being members of an infinitely extended series, and these five isolated groups fall naturally into two sets. One set comprises three groups of orders 27.34:5, 29.34.5.7 and 213.35.52.7, belonging respectively to space of five, six and seven dimensions, the centres of the generating homologies of the smaller groups being among those of the larger ones; these groups had been considered earlier by Burnside (3) as groups of linear substitutions with rational coefficients, and both the groups and the configurations formed by the centres of the generating homologies are closely related to polytopes in Euclidean space of six, seven and eight dimensions first described by Gosset (9) and later studied in detail by Coxeter (6, 7). The two remaining groups, of orders 26.34.5 and 28.36.5.7, belong respectively to four and five dimensions, and the centres of the forty-five generating homologies of the former appear among the centres of the 126 generating homologies of the latter. The smaller group is the well-known simple group of order 25920, and is probably the most extensively studied of all special groups of finite order. Its representation as a collineation group in four dimensions was first obtained by Burkhardt (2), who determined the invariants of the group; the simplest of these is a quartic primal with nodes at the 45 centres of homologies, which was later studied by Coble (5) and myself (14). The geometry of the configuration formed by these 45 nodes has been investigated in detail by Baker (1), whose results I have used elsewhere (15) to classify the operations of the group and obtain their distribution into conjugate sets.

In a recent tract (Baker (1)) there is described in considerable detail a configuration of forty-five points which are nodes of a quartic primal in four dimensions. The geometry of this primal is very fascinating; among its interesting properties is the fact that a number of well-known geometrical configurations, which usually arise as unrelated phenomena, here all appear in connexion with the one figure. The interest of the primal, of course, lies chiefly in the large number of collineations which leave it invariant. The group G* of these collineations is considered in a paper by Burkhardt (2), in which are given explicit expressions for five algebraically independent functions of the five variables, which are left invariant by the operations of the group. The simplest of these invariants is of the fourth order, and when equated to zero represents the quartic primal which is the subject of Baker's tract.

1. Any mention of the word ‘grouping’ immediately brings to a statistician's mind the Sheppard corrections. These are usually used to make inferences about the underlying ungrouped population from observations made on the grouped population, but it is important to realize that, as stated and proved, they have nothing to do with sampling or inference and are merely expressions for the moments of one population in terms of the moments of another population derived from it. They can only be used for the inference problem when allied to the method of moments. This method, as formulated by K. Pearson, consists in taking for θ* the estimate of the population parameter θ, the same function of the sample moments mi that θ is of the population moments μi, each mi being an estimate of the corresponding μi. If the population is grouped the mi are estimates of the , the grouped population moments, so we require θ as a function of the to apply Pearson's method. This can be done since θ is known as a function of the µi and the µi are known as functions of the by the corrections. Use of the Sheppard corrections with any other inference method, even when this method, when applied to the continuous population, yields an estimate which is a sample moment, so far as I am aware, has not been examined except for the normal curve.

Recent investigations by F. Yates (1) in agricultural statistics suggest a mathematical problem which may be formulated as follows. A function f(x) is known to be of bounded variation and Lebesgue integrable on the range −∞ < x < ∞, and its integral over this range is to be determined. In default of any knowledge of the position of the non-negligible values of the function the best that can be done is to calculate the infinite sum

for some suitable δ and an arbitrary origin t, where s ranges over all possible positive and negative integers including zero. S is evidently of period δ in t and ranges over all its values as t varies from 0 to δ. Previous writers (Aitken (2), p. 45, and Kendall (3)) have examined the resulting errors for fixed t. (They considered only symmetrical functions, and supposed one of the lattice points to be located at the centre.) Here we do not restrict ourselves to symmetrical functions and consider the likely departure of S(t) from J (the required integral) when t is a random variable uniformly distributed in (0, δ). It will be shown that S(t) is distributed about J as mean value, with a variance which will be evaluated as a function of δ, the scale of subdivision.

1. The following sentence occurs in a recent paper ((1), p. 54): we do not understand precisely what it means, but we shall show that the idea behind it can be put on a reasonably quantitative basis: ‘In the vocabulary of solvers of simultaneous equations, the operator (1, −2, 1) tends to lead to ill-conditioning when the number of equations is large.’

1. It is frequently required to find the numerical value of the definite integral It is, however, often found that even if the analytical expression of f(x) is given, it cannot be integrated in terms of known elementary functions. The elliptic integrals are perhaps the best known examples of functions of this type; and more common are cases where f(x) is not even defined by an analytical expression but merely specified by a table of numerical values. In the two cases mentioned one has to resort to a numerical integration procedure in which the integral is evaluated in terms of the values of f(x) at a finite number of the arguments x1, x2, …, xn. There are essentially two different types of quadrature formulae available, those which depend on the use of finite differences and the tabulation of f(x) at equidistant points, and those which depend on the values of f(x) at suitably selected points xi. These latter, so called mean-value methods, will be discussed in the present paper. For a recent treatment of such methods see, for example, Beard (5).

The problem considered in this paper is the fluid motion arising from a thin semi-infinite plate started to move impulsively from rest (in viscous incompressible fluid at rest) with a velocity, subsequently maintained uniform, parallel to the edge. Two solutions are given, one obtained in polar coordinates is in the form of an infinite series, whilst the other, derived operationally in parabolic coordinates, leads to a single integral for the velocity distribution. The former is convenient for computation in the vicinity of the edge, the latter being more convenient elsewhere.

A hypothesis originally introduced by Rayleigh after he had discussed the corresponding flow for an infinite plane has been used here to draw deductions about the steady flow past a quarter-plane whose leading edge is normal to the direction of flow and also to obtain approximate expressions for the effect of the edges on the skin friction of a sufficiently broad rectangle whose length is parallel to the incident stream.

A method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.

In the first part of this paper a method was given for constructing a wave potential when the normal velocity is a prescribed function of the angular variable on a submerged circular cylinder. It was shown that the method breaks down for values of the parameters Ka and Kf for which a certain infinite determinant vanishes. The vanishing of this determinant implies the existence of a non-trivial velocity potential, such that the normal velocity vanishes on the cylinder and both velocity components vanish at infinity. In this part of the paper it is shown that there can be no non-trivial solution of this kind; in other words the infinite determinant does not vanish. In the absence of a general uniqueness theory for surface waves it seems worth while to establish this particular result.

This paper is mainly concerned with the distribution of stress near a flat elliptical crack in a body of infinite extent under a uniform tension at infinity perpendicular to the plane of the crack. After an analytical solution of this problem was found the authors received a copy of a paper by Sadowsky and Sternberg (3) in which they solved the more general problem of the stress concentration around a tri-axial ellipsoidal cavity in an elastic body of infinite extent, the body at infinity being in a uniform state of stress whose principal axes are parallel to the axes of the cavity. The method of solution adopted by the present writers for the special case of the elliptical crack is somewhat different from the more general work of Sadowsky and Sternberg, and is, moreover, surprisingly simple, so it seems to be of value to present this solution here.

In this paper, Fourier integrals are used to solve some elastic problems of generalized plane stress and small transverse displacements in infinitely long, rectangular, isotropic plates stressed only at their edges. The Airy stress function and the transverse displacement satisfy the two-dimensional bi-harmonic equation, and the basic mathematical problem is to solve this equation subject to different sets of boundary conditions. Little attention has been given hitherto to problems in which some of the boundary conditions depend directly upon displacements. Here the general problem is solved when one long edge is fixed, and stresses or displacements are arbitrarily prescribed at the other, with no stresses and displacements at infinity. The problem of a concentrated edge force is discussed in detail and numerical values of the stresses at the fixed edge are given.

It is shown that all the classical thermodynamical properties and the equation of state of a rectilinear assembly of spherical molecules with short-range attractive forces acting only between neighbours can be derived from a single partition function which is an analytic function of the pressure and the temperature. The thermodynamical functions are simply related to the Laplace transform of the function e−E(x)/(kT), where E(x) is the interaction energy between two neighbouring molecules. Although this linear model behaves like a perfect gas at high temperatures and like a crystal near the absolute zero, there are no critical phenomena, owing to the analytic character of the partition function. It is, however, pointed out that at very low temperatures the slope of the isothermal curves can suffer extremely sharp changes which may be interpreted as changes of phase, and this point is illustrated by an example.

Finally, I should like to express my sincere gratitude to Prof. H. Jones for his guidance of this work and for his invaluable advice throughout. My thanks are also due to the Turkish Ministry of Education for the award of a scholarship.