This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

In this note we show that if a real square matrix P fulfils certain rather general conditions then there exists a real diagonal matrix D such that the characteristic equation of DP is stable and, furthermore, aperiodic. (A characteristic equation is called stable if the real parts of its roots are all negative. If the roots are all real and simple the equation is said to be aperiodic; see Fuller(3).)

In a previous paper of the same title (1) on

where e(t) is even and has least period 2π, I have shown that the equation has a family of periodic solutions or, equivalently, that a topological transformation T has a family of fixed points. Since much of the detail in (1) is irrelevant here, it will be convenient if the results we must quote about are put into the following form:

Theoerm A. There is a constant integer p such that, whenever k ≥ p, there is at least one solution xk(t) of (1·1) for which

(i) xk(0) > 0, ẋk(0) = 0 and xk(t) has period 2π;

(ii) xk(t) = xk(2π–t);

(iii) xk(t) has, in 0 ≤ t < 2π, k positive maxima, k negative minima and no other stationary points;

(iv)

This except (ii), is a special case of Theorem 2 with the notation x*(t|k, 1, 0) simplified to xk(t) and the constant p introduced to guarantee that (vi) of Theorem 2 implies (iv) above. Finally Lemma 1 of (1), in its original form or as specialized in Theorem B (i) below, gives (ii).

We shall write ak = xk(0) and note, though we do not need to use it here, that, by (v) of Theorem 2, ak = kw + O(k−2) as k→ ∞. If Fk denotes the point (ak, 0) of the (a, b)-plane then Fk is a fixed point of T and we can best state this paper's results in terms of the segment joining a pair of such points.

The distribution of the total (or ‘second’) curvature of a stationary random Gaussian surface is derived on the assumption that the squares of the surface slopes are negligible. The distribution is found to depend on only two parameters, derivable from the fourth moments of the energy spectrum of the surface. Each distribution function satisfies a linear differential equation of the third order, and the distribution is asymmetrical with positive skewness, in general. A special case of zero skewness occurs when the surface consists of two intersecting systems of long-crested waves.

Numerical values of the above integral are given for 3/2λ√3 = 0(0·1)1, to six decimals for |ω| = 0(0·l)2·0(0·2)4·8 and to seven decimals for |ω| = 5(0·5)10(1)15.

The integral satisfies a third-order differential equation; this equation is integrated numerically for ω ≥ 0 and ω ≤ 0, and in each case the method of isolating the solution which satisfies f(ω) → 0 as ω → ∞ is described.

In this paper, we consider the effect of selection in a finite population of haploid individuals in which mutation is occurring in both directions. Unlike Wright's models (2) where the members of each generation all die at the same time to be replaced by a new generation, we suppose that at each change in the state of the system only one individual dies and is replaced by a new individual. In a previous paper (1) we have represented the process by a Markov chain, but it is here more convenient to begin by considering a Markov process with a continuous time parameter. This enables us to introduce selection in a plausible manner.

In this paper we consider a model of a population in which the individuals are diploid and are of two sexes. The individuals have genotypes aa, Aa and AA and we suppose that mutation is occurring in both directions. By supposing that each step in the process is the result of a single death followed by a birth instead of supposing (as in Wright's models (3)) that all the individuals die together at the same instant, to be replaced by a new generation, the mathematical details are much simplified. In this way it is possible to deal exactly with more complicated situations than that of the population of haploid individuals used by Wright in his study of the effects of mutation in a finite population. Moreover, the present model not only involves zygotes and two sexes but is more realistic for those natural populations in which the generations are overlapping.

The general radial integral between oscillator-well wave-functions with a Gauss interaction potential is expressed as a generalized hypergeometric function. A relation between contiguous functions is found and used to obtain a recurrence relation between these integrals.

In recent years much attention has been directed towards the asymptotic solution of diffraction problems. In the present work, consideration is given to the relatively simple problem of the diffraction of an E-polarized plane wave by an infinite slit. The solution takes the form of a series in inverse powers of the ratio of slit-width to wavelength of the incident wave, and is based on the solution by successive substitutions of a pair of integral equations. The current densities induced on both halves of the screen are calculated, from which is deduced the electric field in the slit. The far-field is determined from the aperture distribution, and an asymptotic expression is found for the transmission coefficient as a function of the angle of incidence and the ratio of slit-width to wavelength.

While most previous work has been confined to near-normal incidence, the present theory is uniformly valid for all angles of incidence.

The notions developed in Part I in connexion with the diffraction of an E-polar-ized wave are here applied to the case of H-polarization. Induced current densities, aperture and far fields, and the transmission coefficient are again found in the form of infinite series in inverse powers of the slit-width to wavelength ratio. By means of Babinet's principle, the solution for diffraction by a strip is obtained.

A detailed comparison is made with values of the transmission coefficients for both polarizations found previously and by the present method. It appears that this theory provides accurate information when the slit-width is greater than about a wavelength.

A pair of variational principles are formulated for two-dimensional scattering by obstacles. The first of these is in terms of the obstacle boundary values, and it is shown that a simple ‘optical’ trial function leads to an incorrect frequency dependence for the scattering cross-section. In the second, the obstacle is viewed as the analogue of an aperture coupling two half spaces. The geometric optics part of the cross-section can then be made explicit and is split off to leave a stationary form for the frequency correction. The zero-order calculation for the cross-section of a circle, using corresponding ‘optical’ trial functions, is found to have the correct (Ka)−2/3 frequency dependence.

The scattering of acoustic and electromagnetic radiation by random refractive index fluctuations of the scattering medium (e.g. the troposphere, the ocean) is a problem of considerable contemporary interest. For brevity, we shall refer to these refractive index fluctuations as dielectric noise. The purpose of this paper is to compare two approaches to the problem of the scattering of incident plane waves by a confined region of dielectric noise. In the first approach, which is the conventional one, the scattering dielectric noise is represented as a ‘section’ of a homogeneous (i.e. spatially stationary) random process. In the second approach, which is based on a recent generalization of the notion of stationarity (1), the scattering dielectric noise is represented as a locally homogeneous random process. The advantage of representing the dielectric noise as a locally homogeneous process, rather than as a homogeneous process, is that one can thereby take into account in a natural way the spatial variation of the average noise power, without departing from the physically reasonable requirement that the process be approximately homogeneous in suitably restricted domains.

In this paper the solution is found of certain boundary value problems appropriate to the flow of an invisoid, compressible fluid past a wedge-shaped profile. The most general hodograph equation in the form

is used. Expressions are found for the position coordinates and the drag. The resulting solutions are then discussed for functions k(σ) appropriate to Chaplygin's equation, Tricomi's equation and the equation of Tomotika and Tamada. The case of Helmholtz flow past a wedge with sonic velocity upstream is treated in detail, and the resulting expressions for the drag enable an estimate to be made of the error which results in using approximations to Chaplygin's equation.

The form of Levinson's relation is obtained for the case where the free Hamiltonian contains discrete eigenstates. The relevance of this to the ambiguous solutions of Low equations and dispersion relations is discussed.