If AC, BD, EF are the diagonals of the quadrilateral and PQR is the triangle formed by the diagonals, the circles on the diagonals as diameters form a co-axal system and the circle PQR is a circle of the complementary system. If the circles on the diagonals as diameters intersect in real points S1, S2, then S1, S2 are the limiting points of the system to which the circle PQR belongs and if S1, S2 are not real points and if the straight line joining the mid-points of the diagonals cuts the circle PQR in L1, L2 then all the circles of the system to which the circle PQR belongs pass through L1, L2 and L1, L2 are the limiting points of the system to which the circles on the diagonals as diameters belong.

In the textbooks there are many examples concerning the leading features of the triangle whose sides are given by the equations ax2 + 2hxy + by2 = 0 and L ≡ lx + my + n=0. These are usually easy because of the use that can be made of the origin. But when we replace the first equation by the general equation

During the year 1937, 153 new members have joined the Association, of whom 40 are junior members. The number of members now on the roll is 1648, of whom 9 are honorary members, 108 are life members, 1463 are ordinary members and 68 are junior members.

The Council reports with the deepest regret the untimely death of the President, Professor L. N. G. Filon, and wishes to record its appreciation of the services rendered by him to the Association.

The principles underlying my proposals with regard to this matter were published in an article in the Mathematical Gazette (May 1922). They may be briefly restated as follows.

The first principle is that school geometry should be organised not as a purely logical system but rather on the lines of the natural sciences. This means that it should be based explicitly upon the pupils’ spatial experiences, which it should seek to extend, to make precise, to analyse and to co-ordinate. Under this description one includes the important field in which geometrical observations and conclusions have practical utility.

It is only after considerable hesitation that I have allowed myself to be persuaded to publish this paper, for it contains nothing whatever that cannot be obtained from any of the standard text-books on the Theory of Numbers. I have, however, been influenced by the fact that to many school-masters such text-books are not easy of access, and that this is likely to be so more particularly in the case of those who were unable to attend the January meeting. Thus, although I feel that the pages of the Mathematical Gazette are hardly the place for the reproduction of selected passages from standard mathematical works, I have in this case allowed my scruples to be overcome, in the hope that the concession may prove useful to those who have neither the leisure nor the opportunity to become acquainted with the subject from the pages of Mathews or Dickson.

The general solution in positive integers of the equation

(1) 2a(x2−y2) + l =z2

where a is a prime number (not 2).

In recent years there has been a great development in the design and use of machines for carrying out calculations of various kinds, and this is a technical advance of some importance, not only because of the saving of time and labour in calculations which would be carried out in any case, but still more because it makes it possible to carry out extensive calculations which would be too laborious and time-consuming to undertake without such mechanical or electrical aids.

Recently, in the course of conversation with a friend, I was asked in what ways was it possible to liven up the mathematical teaching of a school. I made a few vague suggestions at the time, but it occurred to me later that I might consider the question, and then apply the results of my deliberations not only to my own work, but to put them down on paper in case others might believe them worth while.

The considerations which I propose to put before you to-day arose from an attempt, when teaching students, to build up the classical laws of Newtonian Mechanics inductively from laboratory experiments.

In this task we are immediately confronted with two fundamental notions, those of mass and force.

It will probably be admitted that no definition of a physical quantity can be accepted as adequate unless it implies a criterion leading to an unambiguous measure of the thing defined.

The following notes are the result of an attempt to gain some insight into the meaning of the Lorentz formulae, avoiding the use of technical terms such as frames of reference, coordinates, and point-events.

The hypothesis of Einstein that the velocity of light is a constant, independent of the relative velocity of the source and observer, is assumed throughout.

Operational methods in pure mathematics, e.g. the work of Boole, Graves and Murphy, existed many years before Oliver Heaviside introduced new ideas into mathematical physics. His first paper using such ideas was published in the Proceedings of the Royal Society in 1893. Remarks therein indicate his unfamiliarity with some of the literature of the subject, owing to lack of reference facilities, for example, Liouville’s work on fractional differentiation.

I Suppose that most of us have some interest in the teaching of mathematics in other countries. I have recently returned from a year spent in the United States as visiting instructor in mathematics in a large preparatory school, a boarding school of seven hundred boys with a staff of about eighty-five men, twelve of whom form the mathematical department. (The term preparatory school as applied to an American school is roughly equivalent to our public school, the word preparatory being used in the sense that the school prepares its pupils for college.)

On connaît le problème classique ou le problème simple de Dirichlet. On donne une surface fermée S qui partage l’espace en deux domaines, l’un intérieur, l’autre extérieur. On suppose que la surface satisfait à certaines conditions de régularité, par exemple, être douée d’un plan tangent. On assigne une fonction continue U(M) d’un point M sur la surface S, et l’on demande de déterminer la fonction U(P), harmonique et bornée dans l’un des deux domaines précédents, qui tend vers U(M) quand le point P tend vers le point M de la frontière. Le problème est intérieur ou extérieur selon le domaine considéré.

If the bisectors of the base angles of a triangle are equal, the triangle is isosceles.

Several solutions of the problem are given in the Gazette for July 1932, and again in the issue of May 1933. It is stated in the former that the problem was mentioned in the Lady’s and Gentleman’s Diary of 1859 and of 1860. The history of the problem is interesting. It appears that it was first proposed in 1840 by Professor Lehmus of Berlin to Jacob Steiner, who had been Professor of Geometry at Berlin since 1834 and held that post until his death in 1863.

The point of the question, “What is the envelope of the principal axes of conics touching any four lines?” posed by Prof. Forder on p. 83 of the current volume of the Gazette has puzzled me not a little; because (i) though my ignorance of the literature of coniCs is profound, yet that literature is so extensive that it seems unlikely that no solution of the problem has ever been published; and (ii) it seems even more unlikely that Prof. Forder is not in possession of a solution. However that may be, here is a solution which might have been submitted by a well-prepared candidate, had the question been set in a Cambridge Entrance Scholarship Examination, whether in December 1937 or in December 1906—or even in April 1877.

Of all subjects which form part of a mathematical course, I believe that the average student feels most at home in his study of trigonometry There are, however, certain applications which escape him and which are of interest. Amongst these are a well-defined group of problems in heights and distances.