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

Let n be a positive integer. Then we know that, if m>– 1,

Consider the integral

which is equal to

The Cubic Surface, as is well known, can be formulated as the locus of a point which, when joined to six given lines—one of which cuts the other five—forms planes enveloping a quadric cone.1 It might be of some interest to show how such a definition leads to one or two of the better known forms of the equation to the surface. The method of approach is by means of the Clebsch Transformation Principle in Geometry and the use of general coordinates. In particular, compound symbols and bracket factors, as developed by H. W. Turnbull2 in his works on Geometry and Invariant Algebra, have been largely used.

It is well-known that a general net of quadric surfaces cannot be obtained as the net of polar quadrics of the points of a plane in regard to a cubic surface; in order that it may be so obtained it must have various properties that a general net of quadrics does not have. The locus of the vertices of the cones which belong to the net of quadrics is a curve ϑ –the Jacobian curve of the net of quadrics, and the trisecants of ϑ generate a scroll. Any plane which contains two trisecants of ϑ is a bitangent plane of the scroll and, for a general net of quadrics, there are eighteen of these bitangent planes passing through an arbitrary point. When however the net of quadrics is a net of polar quadrics it is found that any plane which contains two trisecants of ϑ contains two other trisecants also; it thus contains four trisecants in all and counts six times as a bitangent plane of the scroll. The bitangent developable of the scroll, which is, for a general net of quadrics, of class eighteen, degenerates, in the special case when the net of quadrics is a net of polar quadrics, into a developable of class three counted six times; the planes of the developable are therefore the osculating planes of a twisted cubic γ. The plane which, together with a cubic surface, gives rise to the net of polar quadrics must be one of the osculating planes of γ. It is also found, further, that the osculating planes of γ are grouped into pentahedra, the vertices of all these pentahedra lying on ϑ.

In this note we wish to study some properties of a square matrix of order n and of rank r,

whose coaxial minors |M| of certain orders, or some of them, are known to be zero. Our main results will lie in two directions, according as the vanishing minors are, within certain limits, of an arbitrary order (Part II) or of two consecutive orders only (Part III).

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.

Sufficient conditions for the absolute summability (A)1 of a Fourier series have been given by J. M. Whittaker and B. N. Prasad. They obtained theorem 1 below in the cases α = 0 and α = 1 respectively. Theorem 1 is contained in theorem 2, which was given by Prasad in the case α = 0.

Attention has recently been drawn to the obscurity of the usual presentations of Mayer's method of solution of the total differential equation

This method has the practical advantage that only a single integration is required, but its theoretical discussion is usually based on the validity of some other method of solution. Mayer's method gives a result even when the equation (1) is not integrable, but this cannot of course be a solution. An examination of the conditions under which the result is actually an integral of equation (1) leads to a proof of the existence theorem for (1) which is related to Mayer's method of solution in a natural way, and which moreover appears to be novel and of value in the presentation of the subject.

Professor Hemraj has given a proof of a part of a theorem of Gauss without using the theory of quadratic residues. Proceeding on similar lines, I have obtained a complete proof which is rather simpler and certainly more concise.

The method of summability with which I shall be concerned here is denoted by (V, α ) and is defined as follows:—The series Σαn is said to be summable (V, α ) to the sum s if

This is a particular case of a method due to Valiron in which μ–2α is replaced by a function of μ.

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.

In this study, the use of factorial moments and factorial moment generating functions as applied (2) to the Poisson frequency function and Charlier's Type B function is further extended towards developing a theory of these distributions for the case of two or more correlated variables.

Consider a plane curve C of order n and class X; it is to be supposed throughout that C has only ordinary Plücker singularities, i.e. nodes, cusps, inflections and bitangents. Through any point P1 of C there pass, apart from the tangent at P1 itself, X – 2 lines which touch C; let T12 be the point of contact of any one of these tangents and P2 any one of the n – 3 further intersections of P1T12 with C. Through P2 there pass, apart from the tangent at P2 itself and the line P2P1, X — 3 lines which touch C; let T23 be the point of contact of any one of these with C and P3 any one of its n — 3 further intersections with C.

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 recent paper Turnbull, discussing a rational method for the reduction of a singular matrix pencil to canonical form, has shown how the lowest row, or column, minimal index may be determined directly without reducing the pencil to canonical form. It is the purpose of this note to show how all such indices may be determined, and at the same time to give conditions, somewhat simpler than the usual ones, for the equivalence of two matrix pencils.

The parabolic cylinder functions Dn(x) and D−(n+1) (± ix) are defined by

for all values of n and x.

If x1, x2, …., xk, …. are independent random variables each of which is subjected to a distribution law σ = σ(x) independent of k and having a finite positive dispersion, then x1 + x2 + …. + xn is known to obey the Gauss law as n→ + ∞, no matter how σ (x) be chosen. There arises, however, the question whether it is nevertheless possible to determine the elementary law σ (x) from the asymptotic behaviour of the distribution law of x1 + x2 + …. + xn for very large but finite values of n. It will be shown that the answer is affirmative under very general conditions.

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

1. A given function f(x) is said to be represented operationally by another function φ(p), if

provided that the integral converges.

In this paper I give some operational representations, according to the Carson-Heaviside Calculus, which, as far as I know, are new; and I deduce some properties of the functions thus introduced.