This paper contains two parts:—

I. An endeavour to remove the present confusion in polar diagrams.

II. On the curves derived from a given curve by increasing or diminishing the vectorial angle in a constant ratio.

The transpositions that generate a symmetric group can be represented as real reflections: symmetry operations of a regular simplex. Analogous unitary reflections serve to generate other factor groups of the braid group; they are symmetry operations of regular complex polytopes. Certain relationships among these groups have, as geometric counterparts, unexpected plane sections of the polytopes, beginning with the square sections of the regular tetrahedron. In Section 6, 5-dimensional coordinates will be used to exhibit pentagonal sections of the 4-dimensional regular simplex. The most spectacular instance of such “equatorial” sections occurs in the case of the Witting polytope in complex 4-space, so exquisitely drawn by Peter McMullen for the frontispiece of Regular Complex Polytopes [6]. This has a plane section 3{<5}3 which appears thare as Fig. 4.8B on page 48. Shephard [9, p. 92] called it 3(360)3. Its 120 vertices will be seen to be situated “inside” 120 of the 2160 faces 3{3}3 of the Witting polytope. These “faces” are self-inscribed octagons [7, p. 290].

A. G. D. Watson (1939-41), remarking that there are no Ricci principal directions ata world-point of space-time at which the Einstein equations are satisfied, shows how to define at any world-point a set of principal directions intrinsically related to the Riemann tensor Rijkl itself. These directions are unique except when the space-time has any kind of rotational symmetry about the world-point.

A generalised triangle group has a presentation of the form

where R is a cyclically reduced word involving both x and y. When R = xy, these classical triangle groups have representations as discrete groups of isometrics of S2, R2, H2 depending on

In this paper, for other words R, faithful discrete representations of these groups in Isom +H3 = PSL(2, C) are considered with particular emphasis on the case R = [x, y] and also on the relationship between the Euler characteristic χ and finite covolume representations.

Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.

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.