On reading a recent paper by R. S. Varma (Varma 1949) I recalled that in May 1942 I investigated an integral transformation which is very similar to Varma's. Varma has

and points out that this reduces to a Laplace integral for k = ¼, m = ± ¼. Instead of (1), one could consider the integral

which was introduced by C. S. Meijer (Meijer 1940 b); this integral reduces to a Laplace integral whenever k = m + ½. Now, apart from comparatively unimportant factors, the nucleus of (2) is a fractional derivative or integral, as the case may be, of e−st, and on carrying out a fractional integration by parts, it appears that (2) is essentially the Laplace transform of a fractional integral or derivative of f. Thus, the whole theory of the transformation (2), including inversion formulae, representation theorems, etc., can be deduced from the well-known theory of the Laplace transformation. It is not quite clear that a similar reduction is possible for (1), although it is certainly possible when k = 0.

Many-valued or non-Aristotelian calculi of propositions (logics) were originally introduced by generalisation of the truth-table method. It was known by the end of the nineteenth century that ordinary “binary” formulae of the calculus of propositions, such as

could be verified directly by means of the truth-table:

although the terminology and symbolism used were different.

1. Let

be an integral function, λn being a strictly increasing sequence of nonnegative integers. We shall use the notations

describing M (r) as the maximum modulus, m (r) as the minimum modulus and μ(r) as the maximum term of f (z).

Integral equations are obtained with nuclei (l — zt/a)2n and (z–t)2n which are satisfied by characteristic solutions of the transformed Lamé-Wangerin equation of order n, and each of the two characteristic solutions is expressed in terms of the other by a contour integral.

Among the many papers on the subject of lattices I have not seen any simple discussion of the congruences on a distributive lattice. It is the purpose of this note to give such a discussion for lattices with a certain finiteness. Any distributive lattice is isomorphic with a ring of sets (G. Birkhoff, Lattice Theory, revised edition, 1948, p. 140, corollary to Theorem 6); I take the case where the sets are finite. All finite distributive lattices are covered by this case.

In a previous paper (Shenton, 1953) we have given an expansion for integrals of the form This expansion may be expressed as a determinantal quotient or Schweinsian series. In the present paper we state more general terms under which the expansion holds and consider the case when the limits of integration are infinite and the weight function of the form In particular we giye expansions for the Psi function, and where C (x) is a positive polynomial.

The diffraction of a simple harmonic wave train by a straightedged semi-infinite screen was originally discussed by Sommerfeld in 1895. The analysis is of a recondite character, involving the use of multivalued functions and Riemann surfaces (1). An alternative formulation of the problem is as an inhomogeneous Wiener-Hop integral equation, the solution of which also involves considerable difficulties (2). It is the purpose of this note to show that following Friedlander (3) it is possible by the use of parabolic co-ordinates to solve the problem by elementary methods. The method can be applied either to the case of sound or that of electromagnetism, the results being formally identical.

This note gives a proof of the result:

A necessary and sufficient condition that a trigonometrical series T (x) be the Fourier series of a function is that σn – σm = O(n-n) uniformly in [0, 2π] for all m≤n, where σn is the nth (C, l) mean of T (x).