Recently I have given a generalisation of the Laplace integral

in the form

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

An extensive investigation into the geometry associated with the binary cubic form as representing a triad of points on a twisted cubic curve has been carried out by Sturm. The object of the present note is to derive two further geometrical properties, one associated with the twisted cubic, and the other with the rational plane cubic curve.

Among the many solutions of the wave equation investigated by the late Professor Bateman there is one type which has so far received little attention.

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.

Einstein's fundamental equations of the gravitational field are

where Tμν are the components of the energy tensor and λ is the cosmical constant. In empty space these equations become

which may be reduced to

since G = 4λ, by contraction of (2).

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].