In the paper by H. S. Ruse (this volume 144-152) in equation (1.5) read “Rαβ” instead of “Rαβ”; in equation (1.10) read “€αβ” and “€βα” respectively instead of “€αβ” and “€βα”.

In the paper by C. T. Rajagopal (this volume 162-167) on page 165 line 2 read “dt1” instead of “dt” on page 166 line 2 read “dt1” instead of dt” and in line 2 of the footnote on page 166 read “(16)” instead of “(15)”.

The abstract theory of positive compact operators (acting in a partially ordered Banach space) has proved to be particularly useful in the theory of integral equations. In a recent paper (2) it was shown that many of the now classical theorems for positive compact operators can be extended to certain classes of non-compact operators. One result, proved in (2, Theorem 5), was a fixed point theorem for compressive k-set contractions (k<l). The main result of this paper (Theorem 3.3) shows that some of the hypotheses of (2, Theorem 5) are unnecessary. We use techniques based on those used by M. A. Krasnoselskii in the proof of Theorem 4.12 in (4), which is the classical fixed point theorem for compressive compact operators, to obtain a complete generalisation of this classical result to the k-set contractions (k < 1). It should be remarked that J. D. Hamilton has extended the same result to A-proper mappings (3, Theorem 1). However apparently it is not known, even in the case when we are dealing with a Π1-space, whether k-set contractions are A-proper or not.

Obscurity in the direct discussion of The Envelope, as given in works on Differential Equations, has led writers on The Calculus to define the envelope of a family by a property which all know that it shares with any locus of multiple-points belonging to the family. The following presentation is an attempt by use of systematic notation to make clear the details of the direct process:—

Starting from the definition that

A curve is an envelope of a given family, if at each of its points it touches a member of the family:

let us suppose that a family is specified by the equation

in which ψ is a continuous function of the three variables x, y, u; continuous variation of u corresponds to continuous motion and deformation of a variable curve in the xy-plane, which takes in succession the curves of the family as positions.

The length of an arc of a flexible rope or chain suspended in the catenary y = c cosh x/c is s = c sinh x/c when measured from the vertex, but the practical determination of s is troublesome, owing to the difficulty in finding the parameter c from the transcendental equation when the coordinates of the point of suspension are given. The importance of a formula such as Huygens' approximation to the length of a circular arc s = 2B+⅓(2B−A), where A is the chord of the arc and B that of half the arc, is due to the fact that one can scale directly these lengths by rectilinear measurements without requiring to find the central angle or to make any subsidiary calculations. Formulae of this nature applicable to the parabola, or to curves whose arcs might be replaced by parabolic arcs, would be useful in the design of structural works dealing with ropes or chains. In such cases, as the dip is frequently less than one-eighth of the space, the catenary may be replaced by a parabola.

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

The object of this note is to generalize the notion of quasi-monotony for sequences of real numbers and to prove corresponding generalizations of certain known theorems. First, we recall the definition of quasi-monotony.

Precise asymptotic estimates for the eigenvalues of a uniformly right definite two parameter system of Sturm–Liouville problems are developed. The work extends recent results of B. P. Rynne.

In their paper N. Divinsky and A. Sulinski [6] have introduced the notion of mutagenic radical property—that is, a radical property which is far removed from hereditariness—and constructed two such examples. The first is the lower radical property determined by a ring Swo (N. Divinsky [5]) and is an almost subidempotent radical property in the sense of F. Szász [9], and the second is a weakly supernilpotent radical property, that is the lower radical property determined by Swo and all nilpotent rings.

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

While engaged in a study of the Methodus Differentialis of Jas. Stirling (1730) I have been struck by the fact that Nicole's Papers on the same subject, printed in the Memoires de l'Academie Royale des Sciences (Paris), appear to form a fitting prelude to the work published by Stirling. The dates of Nicole's Papers are 1717, 1723, 1724, 1727, and it is almost certain that Stirling was well acquainted with their contents, for he remarks on page 24 of the Methodus Differentialis:—“Hac de re primus quod sciarn egit D. Taylor in Methodo Incrementorum. Eadem etiam fusius et elegantissime traditur a D. Nicol in Actis Academiae Regiae Parisiensis.”

The closed wedges in C(X) (the space of real continuous functions on a compact Hausdorff space X) which are also inf-lattices have been characterized by Choquet and Deny (2); see also (5). The present note extends their result to certain wedges of affine continuous functions on a Choquet simplex, the generalization being in the same spirit as the generalization of the Kakutani- Stone theorem obtained by Edwards in (4).

I should like to thank my supervisor, Dr D. A. Edwards, for suggesting this problem and for his subsequent help. I am also grateful to the referee for correcting several slips.

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.

Let E be a real Hausdorff locally convex space with topological dual E′, topologised by the strong topology. Let (x, x′) denote the bilinear mapping defining the duality between E and E′ (x∈E, x′∈E′). By a unitary representation of E′ we mean an operator valued function U(x′) = Ux′. defined on E′, whose values are unitary operators in a separable Hilbert space H such that