Denoting the sum of the products of the first n natural numbers taken r at a time by the symbol G(n, r), I have shown that

with the initial values and

In general it was shown that

Defining G (x, r) by the fundamental relation:

for all values of x, we get

The following rational method of dealing with the reduction of a singular matrix pencil to canonical form has certain advantages. It is based on the principle of vector chains, the length of the chain determining a minimal index. This treatment is analogous to that employed by Dr A. C. Aitken and the author in Canonical Matrices (1932) 45–57, for the nonsingular case. In Theorems 1 and 2 tests are explicitly given for determining the minimal indices. Theorem 2 gives a method of discovering the lowest row (or column) minimal index. Theoretically it should be possible to state a corresponding theorem for each of these indices, not necessarily the lowest, and prior to any reduction of the pencil. This extension still awaits solution.

Theorem 1 of a recent paper “On the asymptotic periods of integral functions” can be replaced by the following more precise result.

If f(z) is an integral function of order ρ there is an integral function g(z), of order ρ, such that

The improvement consists in showing that, if ρ < 1, g (z) can be chosen to be of order ρ, not merely of order less than or equal to one.

A generalised lemma used in the second of two papers enables us, as was suggested there, to extend results of the first. Thus, among others, we easily get the following:

Let tn be determined by the relation

In a previous paper in these Proceedings the author discussed conditions for a maximum or minimum of functions of integrals of the type

using the methods of the Calculus of Variations. In the effort to establish a third necessary condition for a minimum—the analogue of Jacobi's condition in the ordinary variational problem—it was found that the analogue of Jacobi's Equation was an integrodifferential equation of the form

The use of geodesic polar coordinates in the intrinsic geometry of a surface leads to the concept of a geodesic circle, i.e. the locus of points at a constant distance from the pole 0 along the geodesics through 0. A geodesic hypersphere is the obvious generalisation of this for a Riemannian Vn. We propose to consider more general central quadric hypersurfaces of Vn, which we define as follows. Let xi (i = 1, 2, …, n) be a system of coordinates in Vn, whose metric is gijdxidxj, and let aij be the components in the x's of a symmetric covariant tensor of the second order, evaluated at the point 0, which is taken as pole.

Let ρA + σB =[ραμν + σbμν] be a pencil of type m × m′, i.e. with m rows and m′ columns, where A and B are matrices with constant elements which are not mere scalar multiples of each other; and ρ and σ are homogeneous parameters.

The problem of statistical “selection” is concerned with the alteration induced in a frequency distribution in several variables by an alteration of the parameters in a subsection of the distribution. It may be illustrated by a simple trivariate case, as follows:

From a population characterised by variables x, y, z, correlated and normally distributed, with means 0, 0, 0, variances and product variances r12σ1σ2, r13σ1σ3, r23σ2σ3, a sub-population is extracted by selection in x alone, in such a way that after selection x is still normally distributed, but with mean h and variance s2 . It is required to determine the new values, in the selected population, of the means and variances of y and z, and of the product variances.