Attention has recently been directed to the measurement of the relative amplitudes of wireless echoes from the ionosphere, and some interesting methods of automatic registration have been devised (1, 2). For vertical incidence, and for wave-lengths shorter than 200 metres, visual observation shows that a single echo, even when separated from its accompanying oppositely polarized magneto-ionic component, shows considerable rapid fading (3). This fading is evident on the records published by White (1), and it is clear from these that it is not easy to determine the relative amplitudes of two echoes. This paper is an account of an apparatus which automatically integrates the amplitude of an echo over a time long compared with the period of the fading, and at the end of the period registers the integrated value on a recording galvanometer. The process is then repeated so that the final galvanometer trace represents the average value of the intensity throughout successive equal periods of time. It is further described how by using a double thread recording galvanometer, it is possible to record the averaged amplitudes of two different echoes (e.g. ordinary and extraordinary, or E and F) during alternate integrating periods.

1. The object of this paper is to evaluate an infinite integral, involving the product of H-functions, generalized hypergeometric functions and confluent hypergeometric functions by means of finite difference operators E. As the generalized hypergeometric function and H-function are of a very general nature, the integral, on specializing the parameters, leads to a generalization of many results some of which are known and others are believed to be new.

The purpose of this note is to show that Peng's (1) method of approximation is not practicable in the case of quadrilinear interaction (i.e. Fermi's (2) original theory of β-decay), and that it does not remove the infinite self-energy of a nucleon in interaction with a charged scalar meson field. The two examples do not, however, provide a serious argument against Peng's theory, since both refer to field theories which have been abandoned for physical reasons. The principal failure of Peng's method to cope with quadrilinear interactions could even be interpreted in such a way as to mean that interactions of this sort do not exist—a statement which would be equivalent to the assumption that Fermi particles can only interact by means of a Bose field. (2n + 1) linear interaction terms must contain one Bose quantity at least, since Fermi field quantities are not observable and therefore can only enter the interaction term in bilinear form.

Further consideration is given to properties of the Stokes phenomenon associated with a class of nth order differential equations with transition points of order m (a rational fraction) at the origin. The systematic vanishing of the Stokes multipliers leads to the discovery of sets of equations that are transformable from certain specific values of m to particular lower values of m. A series of theorems in the theory of congruences is first proved, from which the main transformation theorem is then deduced. The theory is ifiustrated by various numerical examples.

By a complex function space A we will mean a uniformly closed linear space of continuous complex-valued functions on a compact Hausdorff space X, such that A contains constants and separates the points of X. We denote by S the state-space

endowed with the w*-topology. If A is self-adjoint then it is well known (cf. [1]) that A is naturally isometrically isomorphic to , and re A is naturally isometrically isomorphic to A(S), where (respectively A(S)) denotes the Banach space of all complex-valued (respectively real-valued) continuous affine functions on S with the supremum norm.

In (3), the author defined the notion of a G-space. A G-space is a weaker notion than that of an H-space. The main purpose of this paper is to present various means of constructing G-spaces. As an application of some of the techniques of (3) and of this paper (though not an application of the concept of G-space) we shall prove the following theorem:

Theorem. Let G be a connected compact Lie group and let H be a connected subgroup of maximal rank. Then H3(G/H; Z) = 0. In fact, the Hurewicz homomorphism is trivial for odd dimensions.

In hypergeometry loci defined by parametric equations

have been frequently discussed from various points of view. Among other properties these loci possess two systems of generating spaces obtained by keeping the α's (or β's) fixed while the β's (or α's) vary, the simplest example being the generators of a quadric in S3. Another method of obtaining loci having these properties is by taking the flat spaces through the sets of a linear series of sets of points on an elliptic curve and the flat spaces through the sets of the residual series (the curve being supposed normal. In this way we shall obtain a configuration of ∞PSP's and ∞qSq's, the dual of which gives a locus included among those represented by (1).

It is well known (Watson, A treatise on Bessel functions) that if Jν(z) is a Bessel function of the first kind, and A is a real constant, then all roots of the equation are either real or purely imaginary.

1. It often happens that the greater parts of two variables x, y may be connected linearly but there may be other disturbances of both. A treatment given by one of us ((1), section 4·42, pp. 216–17) assumes that the other disturbances are independent, as for observational errors. This is not altogether satisfactory because unless the variations of xr, yr are much greater than the standard errors there is an asymmetry in the treatment corresponding to the well-known distinction between the two regression lines.

An addition theorem, expressing the spherical harmonics in terms of the spherical harmonics and , is derived.

We shall prove the following Theorem. There are exactlyisomorphism classes of residually finite groups with every proper quotient finite.

We shall be concerned with the differentiation of an arbitrary set function ψ from certain classes of set functions with respect to another fixed set function φ. In (1), § 2·5, a halo evanescence condition is shown to be necessary and sufficient for the differentiation of a fixed measure ψ with respect to φ. Here we formulate a type of halo condition which ensures the differentiability of entire classes of set functions. Although the necessity of this condition is not shown, we demonstrate by means of an example that the results cannot be improved upon in certain respects.

1. Elsewhere I have expressed the opinion that a linearly measurable plane set of points, of upper density ½, has zero projection (i.e. projection of measure zero) on almost all directions. Though unable to prove this I have there managed to obtain a partial result. I would not attempt to guess to what extent this is true of a general irregular linearly measurable plane set; these sets are not amenable to the methods used in the case of those of upper density ½, the latter having a remarkable and simple structure. It is a quite trivial matter to construct an irregular set, of upper density 1, with positive projection on some one direction; and from it we can deduce another such set with positive projections on a denumerable every-where-dense set of directions. But, working on a suggestion due to Mr Besicovitch, I have obtained an irregular linearly measurable plane set, of upper density 1, with positive projection on a set P of directions, where the members of P in any sector have cardinal t, though P has measure zero. It may be that the set of points has positive projection on directions other than those of P, but there are no obvious indications of any such directions.

Let R denote a commutative Noetherian ring with an identity element. Hilbert's basis theorem states that the polynomial ring in a finite number of indeterminates over R is also Noetherian. (See Northcott ], theorem 8, p. 26; Zariski and Samuel [4], theorem 1, p. 201). Hilbert's original theorem in [2] is stated for the case when R is a field or the ring of integers. The standard proofs of this fundamental theorem are essentially of a direct type. The analogue of Hilbert's basis theorem in the ring of formal power series in a finite number of indeterminates over R is also true (Chevalley [1]; see also Northcott [3], theorem 3, p. 89; Zariski and Samuel [5], theorem 4, p. 138). In the present note we bring together concise proofs of Hilbert's and Chevalley's theorems by using a single indirect (i.e. reductio ad absurdum) argument. By induction it is sufficient to consider the case of a polynomial ring S and a formal power series ring T in a single indeterminate X over R.

An incompressible magnetohydrodynamic flow, bounded below by a semi-infinite solid wall insulator with a plane surface, is weakly perturbed by two-dimensional (i.e. cylindrical) sources immersed within the fluid. Certain results of Chee-Seng(1) are employed to establish an exact analytic solution for arbitrary source functions. Both direct emission and reflected disturbance fields are hyperliptic, i.e. partly hyperbolic and partly elliptic. The reflected hyperbolic mode is propagated as a Riemann invariant of a modified incident field and its Hilbert transform. This propagation occurs along reflected characteristics whose (acute) inclination with the interface is generally unequal to that of the incident characteristics, but which are, instead, parallel to those non-incident characteristic rays emanating away from the interface.

Within the limitations of the classical small deflexion theory of thin plates and using complex variable methods, exact expressions are obtained in series form for the deflexion at any point of a thin isotropic circular plate simply supported along a concentric circle and subject to loading symmetrically distributed over an eccentric circular patch which lies inside the circle of support. In special and limiting cases the solutions reduce to those obtained before.

The modified field equations proposed by the author can be derived from the energy density

(where is the electric displacement, the magnetic induction, the Poynting vector) with the help of the commutation laws for these vectors.