Let ф(t) be an even function integrable in the Lebesgue sense and periodic with period 2п. Let Write By an indirect method based on the method of Riesz summability for the Fourier series, the author has established the following convergence test for the oscifiating series Σan. The theorem is as follows:

THEOREM A [1]. If, for some Δ > 0 as t → +0 andthen фan converges to the sums = 0. Here K is an absolute constant independent of n.

If the finite group G has a 2-Sylow subgroup S of order 2a+1, containing a cyclic subgroup of index 2, then in general S may be one of the following six types [8]:

(i) cyclic; (ii) Abelian of type (a, 1), a > 1; (iii) dihedral1; (iv) generalized quaternion; (v) {α, β}, α2a = β2, α2a−1+1, a ≧ 3;

(vi) {α, β}, α2a = β2, α2a−1+1, a ≧ 3.

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

An attempt is made to develop the statistical mechanics of the liquid state based not on the usual concept of a “radial distribution function” but on that of a “next neighbour distribution function” which is closely linked up with Bernal's ideas on the characteristic features of liquid structure. Making certain simplifying assumptions it is indeed possible to construct a partition function for an atomic liquid in this way and from this to derive the thermodynamic properties of the system according to the principles of classical statistical mechanics. It is shown that the free energy, the equation of state, the specific heat and entropy as obtained from the theory are consistent with the expected behaviour of such liquids. It is further shown that the computed next neighbour distribution function for close packing is in good agreement with the one derived empirically from a model by Bernal and Mason.

1. Dual integral equations of the form

where f(x) and g(x) are given and Ψ(x) is the unknown, have been increasingly studied in recent years; the first solutions were given for the case g(x) ≡ 0 by Titchmarsh [1] (for 0 < α < 2) and Busbridge [2] (for — 2 < α < 0). An interesting and much simpler method of solving the equations in the same case, g ≡ 0, was given by Gordon [3]. He also showed that the problem of solving the general equations (1) and (2) can be reduced to a problem in which g ≡ 0. He did not pursue this idea as far as finding and simplifying the solution of (1) and (2) but this has been done recently (see [4]) and Noble [5] used a similar idea in treating the case f ≡ 0.

A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), where

for some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.

If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the form

and Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.

It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.

For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

The synthesis of 3-bromofluorenone is described and attention drawn to erroneous statements by previous workers in the field. The reduction of the ketone to 3-bromofluorene has been investigated and shown to be readily effected by heating with hydrazine hydrate in diethylene glycol. This appears to be the method of choice for reducing fluorenones to fluorenes.

In a previous paper in this journal [1], I gave formulae for determining the coefficients in certain dual trigonometrical series. The derivation of these formulae involved rather sophisticated assumptions and some intricate manipulation of the hypergeometric function and relied heavily on the solution of Schlömilch's integral equation. I have now found a much simpler formal solution by using Mehler's integral representation of the Legendre polynomial and the final formulae for the coefficients can be given in a more attractive form. As the results of my previous work have had several recent applications to physical problems, it seems worth while to give some details of this improved solution.

There exist several different approaches to the problem of solving dual integral equations involving Bessel Functions [1, 2, 3, 4, 5, 6,7], and Erdelyi and Sneddon in a recent paper [8] have shown that the introduction of certain operators occurring in the theory of fractional integration enables the relationships between the various methods to be clearly demonstrated. For dual integral equations other than those involving Bessel Functions the operators introduced by Erdélyi and Sneddon are not always the appropriate ones to use and it seems to be of interest to consider this more general type of situation.

The methods employed in papers I–IV of this series are modified to provide the solution of certain dual equations involving trigonometric series. It is necessary to introduce a modified form of the conventional operators of fractional integration and to discuss their relation with generalized Schlömilch series expansions of an arbitrary function. These general methods are illustrated by detailed reference to a particular special case.

This paper is paper gives what appears to be a new Rodrigues’ formula for the Associated Legendre Polynomials defined by [5, p. 122]

with the restriction that m is an even positive integer, which helps to evaluate some integrals. Putting m = 2k in (1.1) and replacing Pn(x) by the Gegenbauer Polynomial and using [3, p. 176]

We obtain

Putting a =v–½ in the relation [4, p. 283]

We get

1. Introductory. In this paper certain infinite integrals involving products of four Bessel functions of different arguments are evaluated in terms of Appell's function F4 by the methods of the operational calculus. The results obtained are believed to be new.

As usual, the conventional notation will be used to denote the classical Laplace integral relation

In the proofs of the formulae the following results will be required [1, pp. 281, 284], [3, pp. 78, 79].

A fundamental problem in the theory of ordinals is the assignation of principal sequences to limit numbers of the second number class.

It is our main object here to show that a certain class of methods, which are a natural generalisation of those used in the solution of the corresponding problem for the real numbers (the description of which we omit), must fail to solve the problem. The methods are those which rest on the following assumption: the principal sequence assigned to any limit number of the second number class is determined once the first i terms of that sequence are known.

A new proof of the Cayley-Hamilton theorem avoiding the use of determinants makes it possible to apply this theorem to matrices over a commutative semi-ring. The relationship of this theorem to a theorem by Lunts concerning switching matrices is investigated.