In Fig. 1, AA1A2A3 … is a rectangular hyperbola, whose asymptotes are OT, OT', and whose vertex is A. We make the following construction

At A1, a point on the hyperbola, a tangent is drawn, cutting OA at P The tangent at A is drawn, cutting OA1 at P1. Let the other tangent from P1 touch the hyperbola at A2. Produce PA1 to cut OA2 at P2. Let the other tangent from P2 touch the hyperbola at A3, and let P1A2 produced cut OA3 at P3.

Consider this process to be continued indefinitely and to be repeated on the other side of the axis OA, A′1 being the point of contact of the other tangent from P We obtain a pencil of rays OAn, OA′n which gradually lie closer together, becoming infinite in number as the asymptotes, which are the ultimate rays of the pencil, are approached. The system is obviously symmetrical about OA, and I shall show that this pencil has further remarkable properties, which I have named “pansymmetry” These properties can be stated thus

If a tangent to the hyperbola at the point An be produced both ways, it will be symmetrically divided by all the rays of the pencil, the p th section on the one side of An being equal to the p th section on the other side.

The study of the turbulent motion of fluids is of great importance in aerodynamics and engineering, besides being a fundamental problem of hydrodynamics. Several attempts have been made to develop an adequate theory of turbulent motion, but, except for a few good empirical formulae, little advance has been made. L. Prandtl has given a clear and concise account of his own theory and that of Th. von Kármán in a recent paper (1), which contains full references to the literature of the subject. H. Bateman has also contributed an excellent account of the present state of the theory in the Report on Hydrodynamics of the National Research Council of America. In neither of these accounts, however, has reference been made to the theory put forward by J. M. Burgers of Delft, in a series of papers (2) published in 1929, and further developed in 1933. In these papers Burgers has attempted to apply the methods of classical statistical mechanics to the theory of turbulent fluid motion. In this article a simplified account of his methods will be given.

The title of this paper is “The Eternal Triangle” ; but titles, like politicians, are sometimes misleading. Only last year, for example, you, Mr. President, skilfully cloaked an address on elliptic arcs under the beguiling title of “The Marquis and the Land Agent : a Tale of the Eighteenth Century”. And so now, Ladies and Gentlemen, any hopes you may have entertained of hearing a new solution of an age-old problem must be abandoned. I cannot even say with George Canning.

1. Let ABC be a triangle, H its orthocentre, P a point on the circumcircle, and D, E, F the feet of the perpendiculars from P on BC, CA, AB respectively. Then it is known that D, E, F are on a straight line which bisects PH. Through P draw lines making the angle ½π–α in the same sense with PD, PE, PF, PH to meet the sides in D′, E′, F′ and a line through H perpendicular to PH in H′. Then D′, E′, F′ are also on a line which bisects PH′. Call this line the Simson line (α) of P with respect to ABC

One service which Euclid rendered to the school-teacher in the days when his supremacy was unquestioned was, that he set a logical standard.

Euclid’s logic has been attacked by many writers, and by none more severely than by a philosopher whom I have worshipped this side idolatry as fervently as any But I doubt whether even Russell, in his less controversial moods, would deny that broadly speaking Euclid pitches the standard reasonably When he is not proving the obvious, what he takes for granted is usually what we should take for granted. This is the more interesting because it is to some extent accidental. Since Euclid wrote down such axioms as “ Things which are equal to the same thing are equal to one another ” and “ Magnitudes which coincide are equal ” he seems to have thought that he had made explicit every assumption involred in his work. We know that this opinion was mistaken, but we know also that the ideal of exposing every assumption is not appropriate to school geometry—I would go so far as to say not appropriate to any subject except that which it dominates, namely, formal logic as expounded by Peano and in Principia Mathematica.

It is known that if S be one of a range of conics touching four given lines, and if XP be a variable line through a given point X, then (a) there exists a line p′ which is conjugate to XP with respect to every S, and (b) p′ touches a fixed conic σ, which is the eleven-line conic of X with respect to the range, and which is also the envelope of the polar of X with respect to the range. Let us call this conic the σ of X.

If two segments AB, XX1 of a line have their mid-points coincident, X, X1 are said to be isotomically conjugate with regard to AB, and vice versa. For brevity in this paper A, B and X, X1 will be referred to as “ isotoms ”

If pairs of isotoms XX1, YY1, ZZ1 are taken on the sides BC, CA, AB of a triangle, and if X, Y, Z are collinear, then by Menelaus' theorem X1Y1, Z1 are also collinear, and the two lines are said to be isotomically conjugate with regard to the triangle. For brevity pairs of lines related in this way will be termed “ isotomic lines”. Conversely if two transversals make XX1, YY1 isotoms with regard to two sides of a triangle then ZZ1 will be isotoms on the third side. These simple results, the source of which I have been unable to trace, are the only results on isotomically related figures made use of in the following investigation.

It has been pointed out that Mayer's method of solving the differential equation

is quite general and only requires one integration, whereas the other general methods require two, or even thee, integrations. The reason for this is rather obscure in most text-books ; and the proof of the validity of the method rests on an existence theorem which leads to a solution of an apparently different form. I shall prove here that the two forms are identical.

Arithmetic has suffered in the past, and still suffers, from being regarded as merely a tool subject, without interest or content of its own. The three R's have a very definite meaning for many an English business man of to-day, who himself had little or no schooling after the age of 14 : he requires of his would-be employees that they shall be able to read print, to write a legible hand, to add up a column of figures and make certain specialised mental calculations, chiefly concerned with our complicated monetary system.

The University Honours graduate who teaches in our Secondary Schools tacitly admits that these rudiments are necessary, but is in a hurry to turn the reading lesson into literary appreciation, writing into self-expressive composition, and arithmetic into the various branches of mathematics, or the hand-maid of science.