This paper deals with the multivariate normal distribution in the symmetrical case when the frequency function may be reduced to

This is the usual distribution occurring for a series of similar independent observations or for the firing of shots at a target.

I propose to prove the following theorem.

With n > 2, the (n + 2) equations derived from the matrix

where

by equating to zero all (n + 1)-rowed determinants from the matrix ‖Δ‖ are equivalent to only two, one of which is linear in li (i = 1, 2, …, n) and the other is homogeneous and quadratic in a certain n – 1 of li (i = 1, 2, …, n); the elements of the matrix are real; (r, s = 1, 2, …, n) and d is arbitrary.

Let P be an n-rowed skew-symmetric matrix of rank 2r with elements out of an infinite field F. Denoting by x, y columns of n variables (indeterminates over F) xv, yv (v = 1, …., n), and by x′, y′ the corresponding row matrices, we consider the skew-symmetric bilinear form y′Px. It is well known that for every P a regular homogeneous substitution over F can be found so that

The present paper is concerned with the “logarithmetic”, or arithmetic of shapes of non-associative combinations as defined by Etherington in ref. (1). The shape of a non-associative product is defined as “the manner of association of its factors without regard to their identity”. Thus, for a binary non-communicative operation, the products ((AB)C)D and ((BA)C)D and ((AA)A)A all have the same shape, while D((AB)C) has a different shape. The sum of the two shapes a and b is defined as the shape of the product of two expressions, of shapes a and b respectively, in the original system of non-associative combination. The product of two shapes a and b is defined as the shape of any expression obtained by replacing every factor in an expression of shape b by an expression of shape a. It is readily shown that these definitions are unambiguous.

In this paper, I introduce a class of arrangements called arrays of strength d and discuss methods of constructing them with the help of finite geometrical configurations and algebraic groups involving elements of a Galois field. The definitions of arrays of strength d and other configurations that are used are given below.

Recently I have given a generalisation of the Laplace integral

in the form

where Wk, m (x) stands for Whittaker Functions.

Let

be a series for which the Abelian generator,

converges whenever s > 0. Then the same is true of the Lambertian generator,

and vice versa.

The Kronecker product of two irreducible matrix representations D(λ), D(μ) of the symmetric group on n letters, furnishes a representation of that group, which is, in general reducible. The question of what irreducible representations will appear in the analysis of such products has been dealt with by Prof. F. D. Murnaghan. Indeed he has obtained the analysis of D(n − p, λ2, …) × D(n − q, μ2, …), for the particular values, p = 1, q = 1, 2, 3, 4, 5; p = 2, q = 2, 3, 4; p = 3, q = 3, 4, applying a method which is a recurrence one, in the sense that to obtain such an analysis we have to look at some other analyses which come first in order.

In this note all small latin letters denote rational integers. We write k ≧ 1, s ≧ 1 and consider the simultaneous equations

A solution of these equations is said to be non-trivial if no set {xiu} is a permutation of another set {xiv}. In 1851 Prouhet constructed a non-trivial solution of these equations with j = sk and Lehmer has recently found a parametric solution for the same j. Here I give two alternative elementary proofs of Lehmer's result. Lehmer's own proof depends on the ideas of generating functions, exponentials, differentiation, matrices, and complex roots of unity, though all at a fairly simple level. One of my proofs requires only the factor theorem for a polynomial and the other only the multinomial theorem for a positive integral index.

I prove in this note some theorems on Rieszian and Dirichlet summabilities involving a Tauberian hypothesis with gaps. One of the theorems (§ 2, Theorem A) has been proved by Ricci [4, § 6] in a slightly less general form. Another theorem (§ 3) contains a Riesz version of a (C, k)-summability problem studied by Meyer-König [1, Satz 1].