In previous papers (Bartlett 1951, Williams 1952a, b, c; 1953, 1955) certain exact tests for discriminant functions have been discussed. These papers have also indicated how the tests may usefully be applied to a wide range of problems: the significance of discriminant function coefficients, the interpretation of interactions in factorial experiments, the concurrence of regression lines with the same independent variables and the proportionality of regression lines with different independent variables, the comparison of sets of ratios, and the testing of arbitrary scores applied to frequency data.

The theory of relativity shows that the times measured by two observers will in general be different if they are in relative motion, so that their respective times between any two coincidences will differ. Bergmann [1] has investigated the problem of a particle moving in a small simple harmonic motion in a static gravitational field, and has found that the time difference for this particle and an observer at rest becomes zero whenever the particle passes through the centre and limits of its swing. This problem will now be dealt with in a different manner, using Schwarzschild's interior solution of the gravitational equations. The exterior solution for a point mass is not suitable in the present case, due to the singularity of the field at a point in the path of the particle.

Determinants [1] of 5th and 7th orders have already been discussed in connection with the Theorems 4 and 5 of Atkin and Swinnerton-Dyer [2]. The determinant under consideration occurs in the investigation of Dyson's rank function for q = 11 given by Atkin and Hussain [3]. As regards the notation it may be mentioned that the author has adopted the same as that of Atkin and Hussain [3].

A number of authors have studied busy period problems for particular cases of the general single-server queueing system. For example, using the now standard notation of Kendall [7], GI/M/1 was studied by Conolly [3] and Takacs [18]. Earlier work on M/M/1 includes that of Ledermann and Reuter [8] and Bailey [1]. Kendall [6], Takacs [17], and Prabhu [11] have considered M/G/1.

The order cycle system of stock control can be formulated as follows: demands for stock occur according to some pattern which we call the demand process and specify in detail later in this section. At fixed intervals of time orders are placed to replenish stock. Let orders for stock be placed at the instants jN, j = 1,2, …; the interval [(j — 1)(N, jN) which we suppose closed at its lower end-point and open at its upper end-point, is called the j-th order cycle and N the length of this interval is called the order cycle period. We suppose that an order placed at time jN is delivered into stock at time jN + li where {li} is a sequence of non-negative random variables independent of the demand process and the order cycle period, we suppose also that the li are mutually independent and identically distributed with common distribution function L(x) with L(0 +) = 0 and finite expectation . The quantity li is called the lead time of the j-th order, that is of the order placed at jN and is supposed independent of the amount ordered. That portion of a demand, if any, which cannot be satisfied immediately is satisfied from future deliveries, thus every demand is satisfied eventually and a negative inventory or back orders can be held. We shall suppose that all order cycle periods under consideration are multiples of some fixed time interval of length τ which we shall take as our unit of time. For example, the interval of length τ could be one day and we would consider order cycle periods which were integral multiples of days. For convenience we take τ = 1 and suppose that the order cycle peiod N is an integer. In this paper we shall consider the following two demand processes.

The great theorem on convergence of integrals is due in its usual form to Lebesgue [2] though its origins go back to Arzela [1]. It says that the integral of the limit of a sequence of functions is the limit of the integrals if the sequence is dominated by an integrable function. This paper investigates the converse problem — if we know that we may take limits under the integral sign, then what can we say about the convergence? The answer is found for functions of a real variable, but it is easily extended to any space with a countably additive measure. Finally the result is illustrated by an application to Fourier series.

The familiar variation-iteration method for solving the eigenvalue equation Cψ = λBψ (C and B are Hermitean operators), is applied to a case in which the operator C, and hence also the eigenvalues λ, depend on a continuous parameter a. It is shown that certain exact properties of the functions λ = λ(a) can be deduced from low-order results in the variation-iteration scheme.

In a recent paper [1] on scales of functions (which will be called SF for short) P. Erdös C. A. Rogers and S. J. Taylor developed the theory of scales of functions and used the Continuum Hypothesis to establish the existence of scales of functions having certain desirable properties. One of these properties was that of being dense in a certain sense. In the course of some joint work Taylor and I have felt the need for scales which are dense in a rather stronger sense. The object of this note is to indicate how the methods of SF can be used to show that the Continuum Hypothesis implies the existence of scales with the required properties.

A topological space is paracompact if and only if each open cover of the space has an open locally finite refinement. It is well-known that an unusual normality condition is satisfied by each paracompact regular space X [p. 158, 5]: Let α be a locally finite (discrete) family of subsets of X, then there is a neighborhood V of the diagonal Δ(X) (in X × X), such that V[x] intersects at most a finite number of members (respectively at most one member) of {V[A]: A ∈ α} for each x ∈ X. In this not we will show that a variant of this condition actually characterizes paracompactness. Among other results, an improvement to a recent result of H. H.Corson [2] is given so as to accord with a conjecture of J. L. Kelley [p. 208, 5] more prettily, and we connect paracompactness to metacompactness [1]

The classical theory of canonical correlation is concerned with a standard description of the relationship between any linear combination of ρ random variables xs, and any linear combination of q random variables yt insofar as this relation can be described in terms of correlation. Lancaster [1] has extended this theory, for p = q = 1, to include a description of the correlation of any function of a random variable x and any function of a random variable y (both functions having finite variance) for a class of joint distributions of x and y which is very general. It is the purpose of this paper to derive Lancaster's results from general theorems concerning the spectral decomposition of operators on a Hilbert space. These theorems lend themselves easily to the generalisation of the theory to situations where p and q are not finite. In the case of Gaussian, stationary, processes this generalisation is equivalent to the classical spectral theory and corresponds to a canonical reduction of a (finite) sample of data which is basic. The theory also then extends to any number of processes. In the Gaussian case, also, the present discussion-is connected with the results of Gelfand and Yaglom [2] relating to the amount of information in one random process about another.

For a theory of the flow about an aerofoil equipped with a split flap to be satisfactory, account must be taken of the wake which is formed downstream of the combination. In this paper the wing and the flap are each represented by a flat plate and it is assumed that the pressure in the wake is constant and equal to the free stream value, i.e. (it is assumed that the wake may be represented by the region between two free stream-line which extend downstream from the trailing edges of the wi ng and the flap1) (see Fig. 1).

By a theorem of Hurwitz [3], an algebraic curve of genus g ≧ 2 cannot have more than 84(g − l) birational self-transformations, or, as we shall call them, automorphisms. The bound is attained for Klein's quartic

of genus 3 [4]. In studying the problem whether there are any other curves for which the bound is attained, I was led to consider the universal covering space of the Riemann surface, which, as Siegel observed, relates Hurwitz's theorem to Siegel's own result [7] on the measure of the fundamental region of Fuchsian groups. Any curve with 84(g − 1) automorphisms must be uniformized by a normal subgroup of the triangle group (2, 3, 7), and, by a closer analysis of possible finite factor groups of (2, 3, 7), purely algebraic methods yield an infinite family of curves with the maximum number of automorphisms. This will be shown in a later paper.

Let

It is familiar that

(1) In the addition theorem [3, p. 363]

where

take θ = π and replace v by v−½. Since

we get

The object of this note is to study the regular coverings of the closed orientable surface of genus 2.

Let the closed orientable surface Fh of genus h be a covering of F2 and let and f be the fundamental groups respectively. Then is a subgroup of f of index n = h − 1. A covering is called regular if is normal in f.

Conversely, let be a normal subgroup of f of finite index. Then there is a uniquely determined regular covering Fh such that is the fundamental group of Fh. The covering Fh is an orientable surface. Since the index n of in f is supposed to be finite, Fh is closed, and its genus is given by n = h − 1.

The fundamental group f can be defined by

.

Let Rn denote real Euclidean space of n dimensions. If

define , and (as usual) so that, by the inequality of arithmetic and geometric means,

Let Λ0 be the integer lattice, consisting of those points in Rn whose co-ordinates are integers. A non-singular n × n matrix M will be called a Minkowski matrix, if, for any point a ∈ Rn, there exists a point x ∈ Λ0 such that

It was shown by Minkowski that, when n = 2, every non-singular matrix is a Minkowski matrix, and that, for general n, every rational non-singular matrix is a Minkowski matrix. Minkowski is also said to have conjectured that every non-singular matrix is a Minkowski matrix, whatever the value of n. For n = 3, this was proved by Remak [5], and a much simpler proof was given later by Davenport [2]. For n = 4, it was proved by Dyson [3], who used a method similar to that of Remak and Davenport, but required the methods of algebraic topology to deal with some of the complications which arise in the higher dimension. Since this method depends also on the reduction of quadratic forms, it is quite likely that it might fail for higher values of n even if the topological difficulties could be overcome. Therefore, an alternative proof for n = 3, due to Birch and Swinnerton-Dyer [1], is of some interest, though in this dimension it is more complicated than Davenport's proof. A good deal of their analysis applies to general n, and they showed that for all n there is a neighbourhood of the unit matrix I that consists entirely of Minkowski matrices.

In this note we consider measures on a left coset space G/H, where G is a locally compact group and H is a closed subgroup. We assume the natural topology in G/H and we denote the generic element of this space by xH (x∈G). Every element t∈G defines a homeomorphism of G/H given by t(xH) = (tx)H. A. Weil showed that a Baire measure on G/H invariant under all these homeomorphisms can exist only if

Δ(ξ) = δ(ξ) for each ξ ∈ H,

where Δ(x), δ(ξ) denote the modular functions in G, H [6, pp. 42–45]. We shall devote our investigations to inherited measures on G/H (cf. [3] and the definition below) invariant under homeomorphisms belonging to a normal and closed subgroup T ⊂ G.

Let σ1(n) denote the sum of the tth powers of the divisors of n, σ(n) = σ1(n). Also place

where γ is Euler's constant, ζ(s) is the Riemann ζ-function and x ≧ 2. The function Δ(x) is the remainder term arising in the divisor problem for σ((m, n)). Cesàro proved originally [1], [6, p. 328] that Δ(x) = o(x2 log x). More recently in I [2, (3.14)] it was shown by elementary methods that . This estimate was later improved to in II [3, (3.7)]. In the present paper (§ 3) we obtain a much more substantial reduction in the order of Δ(x), by showing that Δ(x) can be expressed in terms of the remainder term in the classical Dirichlet divisor problem. On the basis of well known results for this problem, it follows easily that . The precise statement of the result for σ((m, n)) is contained in (3.2).

The purpose of this paper is to call attention to a very simple example of intersections on a singular variety, and to its effect on intersection theories relating to ambient varieties with singularities; in particular it will be shown how the example—to which we refer throughout as Example A—invalidates an aspect of the theory put forward by P. Samuel ([1], Ch. VI).