Adopting the notation of Barnes1 and Fox, let us write

If q > p — l, the series on the right of (1) represents an integral function, while if q = p — 1, the series converges only inside or on the circle | x |=1.

1·1. The two functions F (λ, θ) and G (λ, θ), defined by the infinite integrals (1) and (2) respectively, below, occur in Kottler's theoretical discussion of the diffraction of a monochromatic plane wave by a perfectly black half plane. Some properties of these functions have been investigated by several recent writers.

The object of this paper is to show how some formulae in Analytic Number Theory, in particular, the formula for N(T), the number of zeros of the Zeta-function between t = 0 and t = T, are easy deductions from the Generalised Poisson-Jensen formula. A similar method, using Green's function instead of the general function g(s) of § 2, has been published by F. and R. Nevanlinna (Math. Zeitschrift, 20 (1924), and 23 (1925), but the result contained in (vi) below appears to be new, although the writer has not been able, as yet, to make any effective use of it. It is clear that other applications could be made, but it seems sufficient to give here an indication of the method. The notation throughout is the usual one, and the references are to the Cambridge Tract by E. C. Titchmarsh on “The Zeta-function of Riemann”. Finally, I am indebted to the referee for the reference to the papers of F. and R. Nevanlinna.

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered was

where is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-element

was that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.

The object of the present note is to obtain expansions of the square of Whittaker's M-Functions in series of M-Functions and also other expansions involving M-Functions.

The problem which we discuss in this paper can be easily settled for a closed plane set; after a brief introduction giving the definitions and theorems used later we indicate how this may be done; we then proceed to the main problem.

Introduction. Nous nous proposons ici de pousser un peu plus loin les conséquences que Broderick a tirées d'un théorème dont il nous a fait connaître l'existence au dernier Colloque de St Andrews.

L'un des buts de Broderick était de prouver que certaines formules de King restent valables quand les événements considéréd ne sont plus indépendants, démonstration dont il n'avait pas trouvé trace dans la littérature, sauf relativement à des cas particuliers.

It is well known that if f(t) is (a) integrable in the Lebesgue sense, or more generally (b) integrable in the Perron sense, over every interval (α, β) interior to (a, b), and if

exists, then f(t) is integrable in the Perron sense over (a, b) to the value (1·1).

The elements of the abstract number systems termed groups, rings, ideals, modules and algebras are mere symbols arranged in systems by means of consistent and independent postulates which isolate these systems from the complete realm of abstract mathematics. The postulates are usually chosen so as to generalise the special number systems which have been noticed in traditional mathematics and their independence and consistenc}" are usually proved by means of numerical examples. It is suggested in this note that the extents of the consistency and independence of a set of postulates should also be studied

denote the n-th Cesàro mean of order k for the series aΣan, that is,

where

and let

In a recent paper I discussed, for a given series Σan the relation between the conditions

where 0 < ρ < 1 and p is a positive integer. It was there proved (Theorem 14) that (2) implies (1) but that the converse is not necessarily true. My object in this note is to render more precise the connection between (1) and (2) by showing that (2) is equivalent to (1′), where (1′) is (1) with the additional restriction that Σn−1wn is convergent. The proof of this equivalence relation is based on a technique which was introduced and developed by Andersen and is similar to the proofs in my former paper. The note concludes with a brief discussion of the corresponding results for the case ρ = 1.

1. It is convenient to begin with a brief statement of the notation which will be used throughout this paper.

Let k be any positive number and let

where is the coefficient of xn in the formal expansion of (1 – x )–k–1, and let

Then the series Σαn is said to be summable(C, k) if is convergent, that is, if tends to a limit, and absolutely summable (C, k), or summable |C, k|, if is absolutely convergent.

The object of this note is to deduce from first principles the probability differential of Pearson's x2, and to account for “degrees of freedom.”

Let υλ be a vector (i.e. a vector field) in an affinely connected space Ln, ∇k the symbol of covariant differentiation, and r the rank of the matrix ‖∇k υλ‖ then there exist two sets of n − r independent vectors and which satisfy respectively the equations