Models which treat molecules as graphs provide the first approximation to practically all chemical and physical theories of polymers. This paper, worded so as to aim at intelligibility by both chemists and mathematicians, deals with statistical effects of correlations on distributions of molecules in their graph-like states. The correlations are formally equivalent to fertility correlations in family trees, and the theory covers the range of correlation to the point where a man's fertility expectations depend on how many brothers he has. Chemically, this describes the ‘second shell substitution effect’ which includes certain steric hindrance situations; mathematically, the problem is solved by restoring the Markovian nature of a conventional Galton-Watson process. Long-range correlations are an inevitable and troublesome aspect of the structure of gels and this work represents a step towards improving models for real gels.

§ 1. Let L2(0, ∞) denote the Hilbert space of Lebesgue measurable, integrable-square functions on the half-line [0, ∞).

Integral operators of the form

acting on the space L2 (0, ∞) occur in the theory of ordinary differential equations; see for example the book by E. C. Titchmarsh [4; § 2.6]. It is important to establish when operators of this kind are bounded; see the book by A. E. Taylor [3; § 4.1 and §§4.11, 4.12 and § 4.13].

The effect of the presence of singularities on the method of stationary phase for a certain integral of two variables is examined. The singularities lie on two straight lines whose positions depend upon parameters. For certain values of the parameters the straight lines can pass through the point of stationary phase.

The asymptotic development, which is uniform with respect to the parameters, is determined. It is found that it can be expressed in terms of a function which, in certain circumstances, reduces to a Fresnel integral. The main properties of this function are derived in an appendix.

Explicit formulae are given for the dominant asymptotic terms in the cases when the point of stationary phase is approached by (i) neither line, (ii) only one of the lines, (iii) both lines.

This paper is concerned with high-frequency scattering in a medium, the square of whose refractive index varies linearly with height from a plane boundary. Two asymptotic methods are examined, namely the method of stationary phase and evaluation by residue series. The first of these corresponds to geometric optics and gives the high-frequency field in the illuminated region, while the second complements the first in the sense that if thsre is no point of stationary phase, the residue series is an asymptotic expansion of the field. The Airy functions in the residue series can be replaced by their asymptotic developments in terms of exponentials, and when this is done only the first term or first creeping wave is of genuine significance.

p-Toluenesulphinic acid, with formaldehyde or an aromatic aldehyde, attack the nucleus of a secondary aromatic amine, giving a 4-alkylamino-benzyl (or -benzhydryl) p-tolyl sulphone.

On treatment with t-amyl-alcoholic sodium t-amyloxide, sulphones of the type TsCHAr.C6H4.NPhCH2R lose the elements of toluenesulphinic acid giving the Schiff base ArCH2.C6H4.N = CHR, which can be hydrolysed to ArCH2.C6H4.NH2.

Closed form solutions of some triple equations involving trigonometric series have been obtained, in each case, by reducing them to a single integral equation. The results have been applied to determine the distribution of stress in the interior of an infinitely long elastic strip containing a pair of Griffith cracks situated on a line perpendicular to the bounding lines of the strip.