The subject of this note is that dealt with in Mr Tweedie's paper in the Proceedings, vol. XVII., 33–37, and my only reason for bringing it before the Society is to call attention to a slightly different method of presenting the same order of ideas. The method is that adopted by Peano, Lezioni di Analisi Infinitesimale, vol. I., §23, but as the book is not readily accessible to teachers, there may be some interest in having the method reproduced in our Proceedings. I add one or two remarks.

A representation of degree 3 is used in § 4 to establish the necessary and sufficient condition

for the finiteness of the group (3, p | q, r) defined by

The questions involved in the consideration of three-bar motion have attracted a good deal of attention (Proceedings of Mathematical Society of London passim, and elsewhere); but I am not aware of any complete account of the figures that can be derived from such a motion. The present paper gives a complete list of all the different kinds of curve that are obtained by a tracing point at the middle of the middle bar, the two outer bars being equal.

1. Let

where D and E are integers, and C a positive integer not a perfect square, k being the number of partial quotients in the non-recurring part of the continued fraction, and c the number in the cycle,

and let

This paper is a continuation of my paper, “On a Method of Studying Displacement,” in last year's Proceedings. In that paper I showed how the chief theorems as to the displacements of rigid bodies could be simply demonstrated by the use of what I called a Displacement-chain or Displacement-sequence

For a measurable function f on the unit ball B in ℂn we define (M1f)(w), |w|<1, to be the mean modulus of f over a hyperbolic ball with center at w and of a fixed radius. The space , 0<p<∞, is defined by the requirement that M1f belongs to the Lebesgue space Lp. It is shown that the subspace of Lp spanned by holomorphic functions coincides with the corresponding subspace of . It is proved that if s>(n+1)(p−1−1), 0<p<1, then this subspace is complemented in by the projection whose reproducing kernel is . As corollaries we get an extension of the Forelli–Rudin projection theorem and we show that a holomorphic function f is Lp-integrable, 0<p<∞, over the unit ball B iff u = Ref is Lp-integrable over B. Finally, we sketch an alternative proof of the main result of this paper in the case 0<p<1.

Given a finite irredundant system of equations to be solved over the free group, one has four non-negative integers: the number of equations, the number of indeterminants, the rank of the system and the Abelian rank of the system. We show which four-tuples can actually occur.

In this paper we shall employ the nonlinear alternative of Leray–Schauder and known sign properties of a related Green's function to establish the existence results for the nth-order discrete focal boundary-value problem. Both the singular and non-singular cases will be discussed.

The following changes should be made in the statement of Theorem 3.1 and its proof.

A set function is said to be λ-convex if whenever A and B are in Σ with A ∩ B = φ, then

Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

The classical Dvoretzky-Rogers theorem states that if E is a normed space for which l1(E)=l1{E} (or equivalently , then E is finite dimensional (see [12] p. 67). This property still holds for any lp (l<p<∞) in place of l1 (see [7]p. 104 and [2] Corollary 5.5). Recently it has been shown that this result remains true when one replaces l1 by any non nuclear perfect sequence space having the normal topology (see [14]). In this context, De Grande-De Kimpe [4] gives an extension of the Devoretzky-Rogers theorem for perfect Banach sequence spaces.

We show that if X is a Banach space of infinite dimension and μh is a Hausdorff measure, where h is continuous, then there exists a measurable set K ⊂ X such that 0<μh(K)< + ∞. We also characterize the normed spaces in which the unit ball can be covered by a sequence of balls whose radii rn < 1 converge to zero as n → ∞.

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.