The first part of the present paper reproduces an article contributed by me to the July issue of Mathesis, 1900, entitled “Sur les triangles trihomologiques”; the second part contains fresh developments.

Let A be an abelian surface and let |D| be a polarization of type (1,3) on A. If (A,|D|) is not a product of elliptic curves, such a polarization induces a finite morphism Q: A →p2C of degree 6. In this paper we describe the branch locus of Q when A is bielliptic in thesense of K. Hulek and S. H. Weintraub (see [13]), generalizing the results proved by Ch. Birkenhake and H. Lange in [4].

The property referred to comes to light on consideration of the problem, “To inscribe in a given n-gon the n-gon of minimum perimeter.”

A number of papers have been written concerning the properties of triangles which circumscribe convex sets, see for example (1), (2), (3).

In this note we shall characterise two set functionals, one of which has already been introduced in (6). This will enable us to produce some new results in the spirit of those obtained in the papers mentioned above. These two functionals are interesting in that although they are defined in a rather abstruse manner, they turn out to have an intuitive geometric meaning, namely that they are the minimal widths of certain circumscribing triangles.

Mohanty (1) and (3) considered the absolute summability of conjugate series and Fourier series by Borel's integral method by proving the following theorems.

Let A and B be function algebras. The well-known Nagasawa theorem [5] states that A and B are isometric if and only if they are isomorphic in the category of Banach algebras. In [2] it was shown that this theorem is stable in the sense that if the Banach–Mazur distance between the underlying Banach spaces of A and B is close to one then these algebras are almost isomorphic, that is there exists a linear map T from A onto B such that . On the other hand one can get from Theorems 1 and 3 of [3] that the Nagasawa theorem can be extended to some operator algebras as follows:

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the condition

is both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

§ 1. The famous theorem of the pedal line of a triangle in ordinary geometry can be stated as follows:—“Given a triangle ABC and a point P such that the feet of the perpendiculars X, Y, Z, dropped from P on the sides of the triangle, are collinear, then the locus of P is the circumcircle.” In noneuclidean geometry this locus is not a circle or even a curve of the second degree, but a cubic; and in both cases the envelope of the line XYZ is a curve of the third class. The explanation of the inconsistency in ordinary geometry is that the complete locus consists of the circumcircle together with the straight line at infinity.

§ 1. The inequality of the Arithmetic and Geometric Means of n positive quantities has been proved by many different methods; of which a classified summary has been given in the Mathematical Gazette (Vol. II., p. 283). The present article may be looked on as supplementary to that summary. It deals with proofs that belong to a general type, of which the proof given in the Tutorial Algebra, §205, and that given by Mr G. E. Crawford in our Proceedings, Vol. XVIII., p. 2, are very special limiting cases. Proofs of the type in question consist of a finite number of steps, by which, starting from the n given quantities, and changing two at a time according to some law, we reach a new set of quantities whose arithmetic mean is not greater, and whose geometric mean is not less than the corresponding means of the given quantities.

In problems in the mathematical theory of elasticity related to the symmetric deformation of an infinite elastic solid with an external crack we encounter the problem of determining an axisymmetric function φ(ρ, z) which is harmonic in the half-space z>0 and satisfies the mixed boundary conditions

on the plane boundary z = 0, where it is assumed that f(ρ) is continuously differentiable in [1, ∞). Further φ→0 as √(ρ2+z2)→∞.

We consider some new types of realization problem for obstructions in the Browder-Livesay groups by homotopy equivalences of closed manifold pairs. We give several examples of calculations. We also consider relations with classical surgery problems.

In text books of Plane Coordinate Geometry, two methods are usually given for investigating the condition that the general equation of the second degree:

may represent a pair of real or imaginary straight lines.

Let AOB, DOE be any two intersecting chords of a circle. Required to inscribe a circle in one of the compartments, as BOE.

Draw OF bisecting the angle BOE and let it cut the circle in F. Draw FM perpendicular to AB.