The importance of matric algebras in Function Theory and in Physics (Birtwistle—The new Quantum Mechanics; and Courant and Hilbert—Methoden der mathematischen Physik) has resulted in comprehensive works on finite matrices. Very little progress has, however, been made in the necessary algebras of infinite matrices.

§ 1. The object of this note is to discuss the formula

the integral being supposed convergent for certain ranges of values of x and z. The contour is such that the poles of Γ(– s)lie to its right and the other poles of the integrand to its left. It will be seen that all the Pincherle-Mellin-Barnes integrals are particular cases of this formula.

1. Points, n in number, A, B, C, D, E, … ., are taken at random in a plane, and through each is drawn a line in a random direction. The only condition imposed is that no two of these lines may be parallel.

(i). Two points A, B, define a circle S (AB) which passes through A, B and the intersection of the random lines through A and B. Its centre is denoted by (AB). Each pair of the points gives such a circle and centre.

It is proposed to establish, by elementary methods, a theorem for matrices analogous to Fermat's Theorem in the Theory of Numbers. In Jordan's Traité des Substitutions (Paris, 1870) pp. 127, 128, the order of any given linear substitution or matrix A with reference to any prime number p is determined, but the result given depends on the particular characteristic equation satisfied by the matrix A, and a general result applicable to all matrices of n rows and n columns does not seem to have been published hitherto.

1. The object of this note is to discover a new function which is its own reciprocal in the Hankel Transform of order zero.

I will make use of the following theorem of Hardy and Titchmarsh1:–

A necessary and sufficient condition that a Junction f (x) should be its own Jv transform is that it should be of the form

where 0 < c < 1, and

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 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.

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.