An elementary proposition states that an absolutely convergent series is convergent, i.e. that if

then

We show that there is a pair of handlebodies H1 and H2 with common boundary F with the properties:

(a) There is no essential simple closed curve in F bounding a disc in both H1 and H2.

(b) Given any positive number, there are essential simple curves Cii = 1, 2 on F, bounding discs in Hi whose distance apart in the Hausdorff topology on F is less than this positive number.

Such an example has consequences for Heegaard splittings and recognising the 3-sphere.

If we join the angular points ABC of a triangle to any point O the locus of the centres of conies passing through A, B, C, O is a conic bisecting the six joins of the four points and passing through the intersections of OA, BC : OB, AC; and OC, AB. This conic is analogous to the nine-point circle, and at last meeting of the Society Mr Pinkerton showed that its centre lies on the line joining O to the centroid. In what follows an attempt is made still further to generalise this conception.

The infinite integrals

in which λ is positive and – π < θ < π were encountered by Kottler in a problem in the theory of diffraction. They have more recently been studied by Copson and Ferrar, who obtained the remarkably simple Fourier series

in which denotes a “cut Bessel function” of the third kind; this expansion is valid when the term has to be added to the expansion on the right.

The application of the geometrical properties of the Brocard and Tucker circles of a triangle to a quadrilateral appears never to have been adequately worked out, as far as the author can discover. Hence, the object of this paper.

Some of the problems involved have been published, under the author's name, as independent questions for solution, and where, in the author's opinion, solutions other than his own have seemed more satisfactory for the logical treatment of the subject, these solutions have been employed, with due acknowledgments to their authors.

The chain is supposed to be rotating bodily about a vertical axis with constant angular velocity, and to have taken up a shape of relative equilibrium; the links are like separate pendulums jointed together, and the condition is investigated where the chain is composed of one, two, three, or any number of such pendulum links; finally the passage is considered to the ultimate case of a chain of small links, which may be considered as a continuous flexible cord.

Groups called amalgamated sums that arise as inductive limits of systems of groups and injective homomorphisms are studied. The problem is to find conditions under which the groups in the system do not collapse in the limit. Such a condition is given by J. Tits when certain subsystems are associated to buildings. This condition can be phrased to apply to certain systems of abstract groups and injective homomorphisms. It is shown to imply that no collapse occurs in the limit in a strong sense; namely the natural homomorphism of the amalgamated sum of any subsystem into the amalgamated sum of the full system is injective. This answers a question of S. J. Pride.

In connection with their proof of the fundamental relation

Whittaker and Watson state that “several proofs of this important proposition have been given, but none are simple.”

In this paper we generalize techniques used by Klyachko and the authors to prove some tessellation results about S2. These results are applied to prove the solvability of certain equations with torsion-free coefficients.

The six quaternary quartic forms

were first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

The problem discussed is that of dissecting two given triangles into triangular parts which shall consist of mutually similar pairs of triangles, so that the first given triangle A being dissected into the triangles a1, a2, a3 …, and the second given triangle B being dissected into the triangles b1, b2, b3 …, we shall have a1 similar to b1, a2 to b2, and so on.

The integral equations we consider are

and

where 0 < α < 1. The asymptotic behaviour of the eigen-values of the latter equation is already known (see (1) and (4)). The former equation has been studied by many authors but as yet no explicit statement seems to have been made about the behaviour of its eigen-values.