It was announced in the May issue of the Gazette, that the General Teaching Committee of the Association has appointed a subcommittee to enquire further into the teaching of mathematics to technical students. Many teachers (especially those elementary and secondary teachers who also do part-time evening work) lack the time to explore the vast field of the application of mathematics to other subjects, the knowledge of which would, as they realise, do so much to increase the efficiency and interest of their work.

The solution of the general case of a cable having constant resistance R, inductance L, capacitance C, and leakance G, all expressed per unit length, is usually obtained in the form of an expansion. Using his operational method, Heaviside was the first person to obtain a solution. F. W Carter points out that Heaviside’s result seems to be incorrect, and proceeds to find a solution in the form of a double summation. By aid of complex integration, the integrand being expanded in Bessel functions, H. Jeffreys gives another form of series solution. As this type of problem may arise in lectures on applied mathematics, it is proposed to show how a compact result can be got by contour integration, using a suitable transformation.

I am sorry that Professor Watson (this volume, p. 261) has been so puzzled by the last sentence in a recent note of mine “Query! What is the envelope of the principal axes of conics touching any four lines? ” By “Query ” I did not mean that I was delivering a seventeenth-century challenge to the learned world, but that I should be interested to see a really elementary investigation, provided it was short, though I was disinclined to seek one myself. For from a slightly higher standpoint the question is very simple. Consider the envelope of lines perpendicular to their conjugates with respect to all conics of a pencil of conic envelopes touching four lines.

We propose to study in this paper the properties of a set of points on a proper rectangular hyperbola. We call this set an H-set, which is characterised as pure or mixed according as the points of the set belong to one branch or to both the branches of the hyperbola.

Questions on differentiating from first principles many of the elementary functions, such as sin x, xn, log x, ax, are still often set in examination papers on elementary Differential Calculus, in spite of the difficulties of evaluating rigorously the necessary limits.

1. The reviewer of Berg’s Operational Calculus as applied to Engineering and Physics (2nd ed. 1936) in the Gazette (Oct. 1937, p. 309) remarks that “Heaviside’s methods deserve more attention in this country than they appear to receive”, and a reference is made to articles contributed by Bromwich, Jeffreys and myself to Vol. XIV in 1928.

This Colloquium, held at St. Andrews from July 4 to 15, was in every way as successful as its quadrennial predecessors. A hundred and ten persons attended, including about twenty wives and daughters of mathematicians, including also twenty-one professors of mathematics from the universities of Great Britain, Eire, France, Holland, Denmark, and the United States.

I. There are three ways of trisecting an angle bounded by OA and OB ( = 3αor 3α +2π or 3α +4π), namely:

(i) OX1 and OY1,

(ii) OX2 and OY2,

(iii) OX3 and OY3,

as shown in Fig. 1.

It is well known that if the Euclidean angles of a triangle ABC are trisected as in Fig. 2, then the triangle xyz (commonly quoted as Morley’s Triangle) is equilateral. But angles CAB, ABC, BCA may be trisected each in three ways, so that there are really 27 “Morley” triangles to be considered. The purpose of the present paper is to show that 18 of these 27 triangles are equilateral.

In this address I use the word Euclid not as the name of the man but as a short title for the XIII books of his Elements of Geometry. The “Euclid” of our schooldays was but a small part of the whole, just as the “Newton” of our Cambridge days was a mere fragment of the Principia. We learnt the whole of Books I-IV, about half of VI and about half of XI, of the rest we knew nothing.

If a series of points be pricked at random through a strip of paper, and the strip be held horizontally, certain of the points will appear as peaks, namely those which happen to be higher than either of their immediate neighbours. These we call maximal points. Similarly there are others, minimal points, which are lower than either of their immediate neighbours. Between each maximal point and an adjacent minimal point is a “run”, either a “run up” or a “run down”, the points in a run forming a steadily ascending (or descending) series. These runs will naturally be of various lengths, the length of a run being defined as the number of points in the run, including the first and the last. The shortest runs will thus be of length 2, others may be of length 3, 4, or some greater number, up to m, where m is the total number of points on the strip. Clearly where the points form a run of m there is only one run throughout the whole strip, the greatest possible number of runs is m-1, in which case each run is of length 2.