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.

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.

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.

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

for all values of n and x.

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

provided that the integral converges.

In a recent paper J. M. Whittaker considered the problem of resolving a linear differential system into a product of two or more systems of lower order. This note is a contribution to this problem and furnishes the necessary and sufficient conditions for the resolution of the system into two equivalent systems of lower order of the type considered by Whittaker.

A few definite integrals involving more than two Legendre functions in the integrand have been considered by Ferrers, Adams, Dougall, Nicholson and Bailey. We take for example the following integrals.

§ 1. The theory of symmetric polynomials abounds in dual identities and symmetries of various kinds. It has been investigated from the determinantal standpoint largely by means of quotients of alternants, such as,

the denominator being the difference product of a, b, c, …, a simple alternant.

1. The question, “How, from a given function which is self-reciprocal for a transform of a particular order, can we construct other functions which are self-reciprocal for transforms of different orders?” was first raised by Hardy and Titchmarsh who gave some rules for constructing such functions. Following their method, I have shown, in a recent paper, that there are certain general theorems of the following type:—

If f (x) is its own Jμtransform, g (x) is its own Jv transform. In this note I add a few more such theorems, the interest lying mainly in the results themselves and not in a rigorous proof thereof; and hence only the formal procedure is given here.