In the series of papers in the early forties summarised in (1), (2), A. I. Lur'e showed how to utilise Liapunov's second or direct method in the investigation of the stability of linear automatic control systems with a single nonlinear actuator. His approach consists of

1. the transformation of the original system of differential equations via the so-called Lur'e transformation into canonical coordinates in which the construction of the Liapunov function is direct, and

2. the conversion of the differential problem into a purely algebraic problem.

We will be concerned here with the questions of the existence and construction of the Lur's transformation.

If letters a, b, c, … are used to denote points of a nondegenerate plane cubic curve, other than the singular point if any, and if the product ab is defined as the third point of the curve collinear with a and b, we obtain an algebraic system having nonassociative multiplication (ab . c ≠ a . bc in general). It is in fact a totally symmetric entropic quasigroup (these terms are defined below). This idea, which was put forward at a meeting of the Edinburgh Mathematical Society a few years ago, will be exploited in a forthcoming paper. Such quasigroups have many properties which can be interpreted geometrically. Or, conversely, known properties of cubic curves suggest theorems about totally symmetric entropic quasigroups, which if established will involve a gain in generality, since not every such quasigroup can be “placed” on a cubic.

Oldroyd (1) considered non-Newtonian liquids for which the stress tensor sik and the rate of strain tensor eik=½(υk,i+υi,k)are related as follows

Here υi denotes the velocity vector, p an isotropic pressure, gik the metric tensor, and t time; η0 is a constant having the dimensions of viscosity and λ1, λ2, μ0, ν1, ν2, are constants having the dimension of time. Covariant suffixes are written below, contravariant above and the usual summation convention is employed. A suffix i following a comma indicates covariant differentiation with respect to the space variable xi.

This paper concludes a series of papers (1) on a group of axisymmetric boundary value problems in potential and diffraction theory by considering some potential problems for a circular annulus. The Dirichlet problem for an annulus has recently been considered by Gubenko and Mossakovskiǐ (2), who, by a somewhat complicated method, show it to be governed by either one of two Fredholm integral equations of the second kind. The purpose of the present paper is to show how the method developed in previous papers, by which certain integral representations of the potentials in problems for circular disks arid spherical caps are used to reduce such problems to the solutions of either single Abel integral equations or Abel and Fredholm equations, can be applied to both the Dirichlet and Neumann problems for the annulus to give reasonably straightforward derivations of the governing Fredholm equations.

In (1) I obtained † an asymptotic formula for the number of zeros of an arbitrary canonical product II(z) of integral order but not of mean type, all of whose zeros lie on a single radius, from a knowledge of the asymptotic behaviour of (i)log | П(z)| as | z | = r→ ∞ along another radius l, with certain side conditions. After proving the analogous theorem in which log | П(z)| in (i) is replaced by , I show in this note that, at a cost of replacing l by two radii l1 and l2, both of these theorems may be generalised to include a class of canonical products of integral order whose zeros lie along a whole line. In one of the resulting theorems ‡ (Theorem II) I find the asymptotic number of zeros on each half of the line of zeros; another theorem (Theorem III) includes a previous result of mine.§