1. There has been considerable debate on the conditions under which a given function f(r) can be expanded as a series of the orthogonal functions where υ1, υ2, … are the zeros of the Hankel function regarded as a function of its order. Here k, a are positive constants and 0 < a ≤ r < ∞. The formal expansion can be written as

where

A space time is stationary if and only if it admits a congruence of time-like curves such that the distance between any two curves is time independent. The distance between two curves A and B, which are not necessarily infinitesimally close to each other, is defined as half the trip-time of a photon, which travels from A to B and back to A.

A new source potential is constructed in the linearized theory of water waves. It is shown how this source potential may be used to reduce the problem of the reflexion and transmission of waves by a shelf of arbitrary profile to an integral equation. It is further shown that by suitably restricting the shelf profile, the Fred-holm theory is applicable to the integral equation so that in general a solution of the equation and hence of the problem, exists and is unique except possibly for certain discrete values of the parameters of the problem corresponding to trapping modes over the shelf.

The two-dimensional configuration is considered of a fixed, semi-infinite, vertical barrier extending downwards from a fluid surface and having, at some depth, a gap of arbitrary width. A train of surface waves, incident on the barrier, is partly transmitted and partly reflected. The velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation. It is shown that the two methods lead to the same Riemann–Hilbert problem. Transmission and reflexion coefficients are calculated for several values of the ratio gap width/mean gap depth.

The problem considered is that of transforming the integral equations obtained by Jones (2, 3), for diffraction by a circular disc so that they can be solved by iteration for any general incident field. A method of achieving this transformation is presented for the case of axi-symmetric incident fields. The generalization of the method to non-axi-symmetric problems is also examined.

J. W. S. Cassels, A note on the division values of ℘(u).

Monsieur Y. Hellegouarch has pointed out to me that the inequality (131) of my paper “A note on the division values of ℘ (u)” (Proc. Cambridge Phil. Soc.45 (1949), 169–172) does not follow from the argument hinted there and is almost certainly incorrect. All the argument shows is that the rather weaker estimate in (132) holds for m = pk as well as for m = 2pk.

The erroneous claim is reproduced in my report “Diophantine equations with special reference to elliptic curves” (J. London Math. Soc.41 (1966), 193–291) in expression (17.14) on page 247.