§ 1. The theory of symmetric polynomials abounds in dual identities and symmetries of various kinds. It has been investigated from the determinantal standpoint largely by means of quotients of alternants, such as,

the denominator being the difference product of a, b, c, …, a simple alternant.

Let V be a group, written additively but not necessarily abelian, and let S be a semigroup of endomorphisms of V. The set C( S; V)={ f: V→ V| fσ=σ f for all σ∈ S and f(0)=0} forms a zero-symmetric near-ring with identity under the operations of function addition and composition, called the centraliser near-ring determined by S and V. Centraliser near-rings are very general, for if N is any zero-symmetric near-ring with 1 then there exists a group V and a semigroup S of endomorphisms of V such that N≃ C( S; V).

In his Theory of Heat, Clerk-Maxwell gives a very elegant and simple geometrical proof of the four thennodynamical relations, and points out that his construction shows that the truth of any one is a necessary consequence of the truth of any other. The usual analytical proof of these relations can be made as simple as Maxwell's geometrical proof, and the fact that any one is a necessary consequence of any other becomes evident when it is considered that they are all deduced by a common process from identical transformations of one equation. The following method of proof seems to me to bring out more directly their necessary interdependence.

Longitudinal elastic waves in solid circular cylinders were first investigated by Pochhammer (1) and Chree (2) but they did not take into account the effects that thermal properties have on the propagation of these waves. In this paper we shall consider waves in solid and hollow cylinders as well as in the infinite medium with a cylindrical cavity, and in each case we shall take account of the thermoelastic effects.

In his article in No. 15 of Mathematical Notes, Dr Dougall has shown how by introducing series with n terms and a remainder, instead of infinite series, some well-known proofs of Newton's Theorems and others can be rendered more elementary in character. Once the idea has been suggested, it is of course easy to find further applications for it.

Several structure theorems are proved for groups G having the following property. There is a prime p and a collection of subgroups of G such that the elements of G which lie in the complement of every subgroup of the collection all have order p.

The property is that if any quadrilateral, ABCD, skew or otherwise, have its sides AB, DC divided in E, F so that

AE : EB = DF : FC = AD : BO

then the direction of the line EF shall bisect the angle between the directions of BC, AD.

This extension of Euclid's VI. 3 follows immediately from the proposition that if AE : EB = DF : FC, then all such lines as EF are parallel to one plane, namely, the plane parallel to BC, AD; and that they each cut similar lines drawn with reference to BC, AD.

Let be a category with finite products and a final object and let X be any group object in . The set of -morphisms, (X, X) is, in a natural way, a near-ring which we call the endomorphism near-ring of X in Such nearrings have previously been studied in the case where is the category of pointed sets and mappings, (6). Generally speaking, if Γ is an additive group and S is a semigroup of endomorphisms of Γ then a near-ring can be generated naturally by taking all zero preserving mappings of Γ into itself which commute with S (see 1). This type of near-ring is again an endomorphism near-ring, only the category is the category of S-acts and S-morphisms (see (4) for definition of S-act, etc.).

Being given a fixed line (the directrix) and two fixed points S, s (Fig. 15), then, if z and Z are two points on the directrix, the lines zp and ZP are said to correspond if zp be parallel to SZ and 8zsz be parallel to ZP.

Theorem 1. If pairs of corresponding lines meet in p and P respectively, then sp is parallel to SP ; and, if any line goes through p, the corresponding line goes through P.

The problem was proposed by Steiner of finding certain loci and envelopes connected with a system of similar conics through three points. In previous papers I have found the locus of centres and the envelope of asymptotes of such a system of conics, also the envelope of axes, and the locus of foci. I now propose to discuss the envelope of the directrices.