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.

We consider the half linear Sturm-Liouville problem

on the interval [0,1] subject to separated boundary conditions (which may be eigenparameter dependent at x = 1) and use Prüfer techniques to produce an oscillation theory for this problem. Both right definite (r > 0) and left definite (r of both signs) cases are discussed.

The present note is an extension of a previous paper on the same subject. In this paper a concise proof was given of a theorem by Scheffers to the effect that if a linear associative algebra contains the quaternion algebra as a subalgebra, both having the same modulus, then it can be expressed as the direct product of that quaternion algebra and another algebra. It was also shown that this theorem could be generalised to the extent of substituting a matric quadrate algebra for the quaternion algebra. In the present paper the theorem is extended to certain other types of algebras.

Let {Kn} be a sequence of complex numbers, let

and let

Let D be the open unit disc {z: |z| <1}, let be its closure and let .

The primary object of this paper is to prove the two theorems stated below, the first of which generalises a result of Copson (1).

The theorem is:—If a1, a2, a3, …, a22 be the middle points of the sides of any convex polygon A1A2A3…A2n then as regards areas

The following proof depends only on the theorem that the line joining the points of bisection of two sides of a triangle cuts off a triangle equal in area to a quarter of the original triangle.

The functions of Hermite, which are the same as the functions associated with the parabolic cylinder in harmonic analysis, may be defined* by the differential equation which they satisfy, namely,

where n denotes any constant.

Ten papers deal with the theory of relativity, and all are concerned in some way with electromagnetism. Those who know Whittaker through his Modern Analysis and Analytical Dynamics will recognise in these papers the same mastery over complicated situations which enabled him to disdain the support of notational refinements, that same elegance, brevity and persuasive charm which make difficult arguments seem easier than they really are. The new element which emerges is the strongly geometrical approach; but he remains true to the Lagrange tradition and draws no diagrams of space-time, although these must surely have been before his mind's eye and would have helped his readers.

§1. Whittaker has shewn that a general solution of Laplace's Equation

may be put in the form

where f (v, u) denotes an arbitrary function of the two variables u and v; such a representation is valid only in the neighbourhood of a regular point.

Restrictions are given on the dimensions m and k for which there is a map f: ℝ m → ℝk whose Jacobian has rank k in a neighbourhood of a singular point if f is either quadratic or even. The restrictions are shown to be best possible in the quadratic case.

Let P1 and P1′ be images of each other with respect to a circle of radius a. (Fig. 13.) Draw AOB the diameter perpendicular to OP1′P1. If r1,r1′ represent respectively the distances of P1, and P1′ from 0, the centre of the circle, the condition is r1r1′ = a2.

By the results of Rickman [7] and Ralston [6], a finite group G admitting a fixed point free automorphism α of order pq, where p and q are primes, is soluble. If p = q, then |G| is necessarily coprime to |α|, and it follows from Berger [1] that G has Fitting height at most 2, the composition length of <α>. The purpose of this paper is to prove a corresponding result in the case when p≠q.

We study two-parameter nonlinear Sturm-Liouville problems. We shall establish the continuity of the variational eigencurve λ(μ) and asymptotic formulas of λ(μ)as μ → ∞, μ → π2