Ladies and gentlemen—It has become customary for the President of the Association, when a university teacher, to speak of some aspect of mathematics in which he or she is an expert. When the Council of the Association did me the very great honour of inviting me to preside over this conference, the first, I believe, ever to be held in Oxford or Cambridge, I decided to break away from tradition. This decision was strengthened when I knew that I was to have the privilege and responsibility of addressing you in the Shel-donian, which is furnished with no apparent visual aids.

This investigation grew out of the need to design a piece of electronic test-equipment. The purpose of the equipment was to study the effect of varying capacity upon the behaviour of a number of electric circuits, and it consisted basically of an automatically-stepping switch which presented at its output terminals a controlled sequence of values of capacity. The nature of the test demanded that the set of capacity values should as nearly as possible cover a given range in a logarithmic scale, and to economise in components the equipment was to generate the set of values from a basic set of condensers, by creating all possible parallel-connected combinations. The problem was, how to choose the basic set? The analysis has a number of points of combinatorial interest, and illustrates fairly well the ad hoc approach of the industrial mathematician.

In a previous article [2] I gave a synthetic geometric proof of theorem which had been proved algebraically by T. G. Room [1] In the second part of my proof I used Pascal’s theorem, but Professor Room has since told me that he would have liked a proof which does not involve the theory of conies. I give such a proof here having first restated the theorem in two parts, as in [2],

S. Zylbertrest in his interesting generalisation of the Wallace-line has opened the geometric gate to a new field of research. D. G. Tahta then followed with an algebraic approach and derived a few theorems from his equation of the Zylbertrest-line.

It is a standard textbook exercise to show that by considering the coefficients of xn first in (1 + x)n. (1 + x)n and then in (1 + x)2n It is the purpose of this note to show that formulas for a wide class of sums of products of binomial coefficients can be obtained by using Leibniz’ rule for the differential coefficient of a product; all expressions are derived for three indices, but generalization to any other number is obvious.

In Note 2968 a definition of dual numbers a1 + ∊a2, where a1 and a2 are real numbers and ∊2 is put equal to zero whenever it occurs, is given. Similarly, dual vectors are expressions q1 + ∊q2, where q1 and q2 are ordinary three-component vectors (see [1]). The dual vector q1 + ∊q2 is said to be proper if |q1| 0, and normal if |q1| = 1, q1 . q2 ≠ 0.

Let Gn be a complete directed graph of order n. That is, Gn has n nodes and every pair of nodes is connected by exactly one directed edge. We say, Gn has the property Sk (after Schütte who posed the problem) if for every set of k nodes there is at least one node in Gn from which the edges go out to the given k nodes.

It appears to have been customary, from the most ancient boo] on the subject to the most modern, to analyse the relationsh between distributed load function q, shearing force V, and bendii moment M, acting on a beam (Fig. 1) by considering the translation and rotational equilibrium of an element of the beam (Fig. 2).