1. Whittaker (7), generalizing a result of Ferrar(3), showed that the cardinal series based on the positive and negative integers is consistent in the sense that, if

and an integral function f (x) is denned by

then provided 0 < λ > 1

In this note I show that results of Paley- Wiener, Levinson and others on biorthogonal series can be made to yield a consistency theorem for cardinal series based on sequences λn, where

that is, series

where

We use Fourier transform technique and need hypotheses, a little more restrictive than Whittaker's, which would reduce in the case .

In a recent paper (3) the Lindeberg-Lévy theorem (2) was extended for certain types of stationary dependent variables. In the present paper mainly the same basic ideas as were used in (3) are employed to give central limit theorems for m-dependent scalar variables (a) stationary to second order and (b) asymptotically stationary to second order, the sufficient condition in each case being akin to the Linde-berg condition ((1), p. 57) for independent variables. The analogue of the main theorem for vector variables is given. An extension of the Lindeberg-Lévy theorem is derived.

It is shown that Wild's formal solution of the Boltzmann integro-differential equation can be used to obtain Maxwell's classical relaxation behaviour of the second velocity moments of the molecular distribution function.

In an earlier note (3) we have shown by examples that it is possible to have an irreducible non-degenerate point-point correspondence on an algebraic surface such that the correspondence and its inverse have arbitrary unequal negative valencies. We have also remarked that, by multiplying such a correspondence by one with equal positive valencies, we can find correspondences with unequal positive valencies. In (4) we have given reasons, falling short of complete proof, which suggest that, on any surface whose intersection group is cyclic, correspondences with unequal Albanese valencies exist. But the only proofs of the existence of such correspondences are the examples due to the author, Severi and Todd in (3) which all refer to ruled surfaces. Severi has rightly pointed out that these examples are not entirely satisfactory, since, as ruled surfaces have zero geometric genus, there is a confusion between valency correspondences in Severi's and in Albanese's senses, whereas the theoretical arguments of (4) apply only to Albanese valency correspondences. In this note we remark that on hyperelliptic surfaces there exist correspondences with valencies γ and γ3 in the two directions for any integer γ, and that unless |γ| = 1, in which case the two valencies are the same, these correspondences cannot have a Severi valency in either direction. This result is an immediate consequence of some classical results on Abelian varieties (c.f.(2)), but the application seems of sufficient interest to note explicitly. We remark that we are still not able to find examples of correspondences which are irreducible and non-degenerate and whose valencies in the two directions are of opposite sign; it seems a plausible conjecture that such correspondences do not exist.

The object of this note is to give an example of an infinite locally finite p-group which has no proper characteristic subgroup except the unit group. (A group G is a locally finite p-group if every finite set of elements of G generates a subgroup of finite order equal to a power of the prime p.) It is known that an infinite locally finite p-group cannot be simple, for if it were it would satisfy the minimal condition for normal subgroups, and so have a non-trivial centre (see(1)). However our example shows that it can be characteristically-simple. Examples are known of locally finite p-groups with trivial centre ((2), (4)), and of locally finite p-groups coinciding with their commutator groups ((1), (5)). Since the centre and commutator subgroup of a group are characteristic subgroups our example will have both of these properties. We may remark that the direct product of a simple, or even of a characteristically-simple group with itself any number of times is also characteristically-simple, but by Corollary 2.1 our group cannot be so decomposed.

The Schroedinger equation of the helium atom is separable between the radial and angular variables. The separated equations in the radial variables are found by using a new technique of calculation with symmetric traceless tensors. Account is taken of reflexion and permutation characters.

The notation which we use is that of a recent paper by one of us, and we quote results from that paper as they are required. It is known (see R, Theorem 1, for example) that the vector space k of all cusp-forms f(z) of even negative dimension – k (k ≥ 12), belonging to the full modular group Γ(1), possesses a finite basis of forms

where k is defined by (2·10) of R and the coefficients possess the following properties:

for a prime p, where p is a positive integer.

The diffraction of a plane harmonic electromagnetic wave by two parallel, perfectly conducting planes of infinite width and finite length is considered. The wave is assumed to be polarized perpendicular to the edges of the plane, and thus the problem can also be regarded as that of diffraction of a plane sound wave. Two integral equations are obtained, and the equations are solved approximately when the length of the planes is large compared with the wave-length. The resonance of the system is also considered and an equation obtained for the resonant lengths. This equation is solved by an approximation depending on the assumption that twice the product of the wave number (k) and the resonant length (l) is always large.

It is shown that the equations governing oblique propagation in a horizontally stratified ionosphere with an oblique magnetic field can be cast into a form suitable for solution by successive approximations. In the general case this ‘coupled’ form consists of four first order differential equations, each being associated with one characteristic wave. In the special cases (a) horizontal magnetic field, (b) plane of incidence perpendicular to the magnetic meridian, two second order coupled equations of a particular type can be derived, each of which is associated with a pair of corresponding upgoing and downgoing characteristic waves. These latter equations are similar to, and include, those already given by Försterling(7) for vertical incidence.

The cases in which there is no coupling are briefly considered from the point of view of the first order equations.

The general formulation provides a basis for assessing the validity of the standard ‘ray’ approximation, alternative to that developed by Booker (2), and brings out the nature of its breakdown in coupling and reflexion regions.

Specific applications and extensions of the theory are left for later consideration.