In (l) Khuri has proved the validity of a dispersion relation for non-relativistic potential scattering. More precisely, he has shown that if the potential V(r) is central and satisfies: then the scattering amplitude f(k, τ) = M(E, τ) (where E = k2 is the energy) satisfies the following dispersion relation for fixed momentum transfer τ ≤ 2α: In (2), Rj(τ) is the (real) residue of the scattering amplitude at the bound state Ej and is the Fourier transform of the potential .

Atomic wave functions calculated using the Hartree—Fock equation and a polarization potential are used to obtain mean square radii of atomic orbitals for the ions Si8+ and Ca+, and transition integrals for the Si IV and Ca II spectra. A table of radial wave functions for Ca+ is given. Our results are compared with those obtained from other wave functions, and in one case with an experimental value, and generally good agreement is obtained.

The propagation of a scalar wave is studied, after it has suffered changes of phase in its passage through an irregular refracting medium, such as the ionosphere. The rms phase fluctuations may be large; i.e. greater than one radian. The paper is in three parts. In the first, the receiver is at a great distance from the screen (Fraunhofer diffraction), and a formula is found for all even order moments of the amplitude, and for the correlation function of the even powers of the amplitude. The probability distribution of the amplitude is proved to be a Rice distribution. The multiple integrals expressing the moments of the amplitude reduce to sums of correlations of the original phase fluctuations amongst points over the screen, which are evaluated using combinatorial arguments.

In the second part the variance of the square of the amplitude is calculated numerically for points near the screen (Fresnel diffraction). When the rms phase fluctuation is greater than one the variance is maximum near the point where the irregularities in the screen would bring the rays to a focus.

In the third part radio star scintillation and scatter propagation are discussed in the light of the theory.

The possible existence of a magnetic pole has been under discussion for many years. Dirac worked out in detail an electro-dynamics in which a single magnetic pole is assumed to exist (1,2). He has shown that the existence of such a pole leads naturally to the quantization of charge, the magnetic pole strength g and the electric charge e being related by the equation where p is an integer, h is Planck's constant divided by 2π, and c is the velocity of light.

Two methods of solution are given for the flow of a viscous liquid in a circular pipe whose radius has a slight sinusoidal variation. Numerical examples are given, and some related problems are briefly discussed.

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.

If five spheres in 3-space are such that each pair is inclined at the same non-zero angle θ, then where b1, …, b5 (the ‘bends’ (1) of the spheres) are the reciprocals of their radii. To prove this result, establish a system of rectangular cartesian coordinates (x, y, z) and let the spheres have centres (xi, yi, zi) and radii , where i = 1,…, 5. Then for x5, y5, z5, r5 we have the equations which, on subtraction, yield three linear equations and one quadratic equation. Solving the three linear equations for x5, y5, z5 and substituting, we see that the required relation is algebraic (indeed quadratic) in r5 and hence in b5. Since it is also symmetric in b1,…, b5, it follows that it can be expressed as a polynomial relation in the elementary symmetric functions p1, …, p5 in b5, …, b5.

We refer for general background to N. Bourbaki, Intégration, chapters iv and v. We consider a locally compact Hausdorff space R and denote the set of continuous functions with compact support by The Riesz-Markov theorem shows that there is a 1−1 correspondence between the set of regular Borel measures on R and the set of positive elements of the topological dual of . Let {μn}, μ be regular Borel probability measures on R. The sequence of measures {μn} is said to converge vaguely to μ if, for all . Thus the vague topology is that of simple convergence in . We shall call a μ-measurable set μ-quarrable if its boundary is a μ-null set.

The notation of the introduction of the first paper (1) will be used without further explanation. All page references, and also references to Theorems 1, 2, 3, 4 or to numbered equations, will be understood to refer to the pages, theorems or equations* so numbered in (1). The theorems of the present paper are denoted by primed numbers.

In a recent paper (2) Rankin has proved that for all positive integral n where As stated by Rankin, his proof is not simple although it is based on elementary ideas. Also, as noted by him, no other proof is available for any inequality of the form with λ a positive constant independent of n. It is the main purpose of this note to give a simple proof for the following result which implies the truth of (3) with λ = ¼, though not the truth of (1).

The expectation, variance and covariance for different states of a k-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve the k latent roots of the determinantal equation. As it is not easy to determine the latent roots when k > 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).

Some years ago, in connexion with some unpublished work in the theory of queues, the question arose as to whether the characteristic function of a non-negative random variable could vanish identically in an interval. The purpose of this note is to show that such a thing is impossible.