To find the position of equilibrium of a particle which is acted on by forces of given magnitudes directed towards given points.

This problem was suggested by Mr. P A. Hillhouse, and arose in the first instance from the consideration of forces acting in a crane. The following note does not offer a solution of the problem, but contains a proof by Mr. A. L. Jones that there is at most one position of equilibrium when the forces are all of one sign, i.e. either all tensions or all thrusts. This is followed by an investigation of the total number of solutions, real and imaginary, for all combinations of thrusts and tensions, when the magnitude only of each force, and not its sign, is given. This number is surprisingly large. It would of course be of more practical interest to find the number of real solutions for each particular combination of thrusts and tensions, but I have only succeeded in doing this when the given points are in a straight line, and in one or two other special cases. The results for these are stated without proof at the end of the note.

The Jahresbericht der Deutschen Mathematiker-Vereinigung has lately published an address given by Professor Harzer on “The Exact Sciences in Old Japan.” At first sight it seems difficult to see why the celebration of the birthday of the Emperor of the Germans should have been selected for the delivery of an oration on this topic. The Professor is himself an astronomer, so that it is interesting to note that he brushes aside as of little value anything that Japan has accomplished in the pursuit of his favourite science. Nor does he discover any ground for supposing that anything will be accomplished along this line by those whose sudden prominence in other realms has given the Western world so furiously to think. The history of the science of mathematics in the Island Empire is full of paradoxes. In the days of Pythagoras, it will be remembered, the secrets of the science were entrusted to what was to all intents and purposes a Masonic body, bound by oath not to divulge the mysteries in which they participated. And this body had a further curious and unexpected resemblance to the Masonic order as seen among the Latin races, inasmuch as its intrusion into politics led to more than one popular outbreak. In Japan the parallel almost holds good, save for the utilisation of the society for political purposes. The elementary parts of the science were open to all. He who ran might read. But before the student was permitted to penetrate into the arcana he was compelled to join a brotherhood and to take an oath of secrecy. Things have, of course, considerably changed since those days. The powers that be have become more and more convinced of the paramount necessity of improvement in the direction of mathematical training. And with the courage and directness which is such a marked characteristic of this astonishing race, they refuse to allow entrance to any of the higher courses at the University or higher technical institutions to those who have not the necessary preliminary mathematical equipment. We shall be very much mistaken if the results of these drastic measures do not make themselves very evident in the course of the next half-century.

Having established the expansion of functions without any reference to tests of convergence, we are able to enunciate the following simple test for the convergence of any series that may arise.

(We confine our attention to a series of which all the terms that we need to consider have the same sign.)

To prove that if ƒ represents a function satisfying certain conditions for all values of the variable from a to a + x, then ƒ(a + x) can be expanded in powers of x.

Assume that, as in § I., where ƒ′ is another function which we call the derived function of ƒ.

Several members of the Association expressed their conviction, at the General Meeting and at the subsequent Dinner, that there should be more articles in the Gazette, and more papers read at the Meetings, on the purely pedagogic side of mathematics. It is right and necessary that the general development of the various branches of our Science should not be lost sight of in an organ such as the Gazette, and the reader must be kept in touch by reviews with what is being done in school text-books both at home and elsewhere. But the special aims of the Association (and the object of the Gazette is to secure by publicity the realisation of those aims), is to improve the teaching of our subject in our schools. With this view the Editor is in cordial agreement. He asks the readers of the Gazette to come to his assistance in the matter, to propose subjects for discussion, to offer papers, to ventilate their special difficulties in a “ Question and Answer “ column, in short to do for the Gazette what no Editor can do single-handed. If this proposal is taken up with any vigour by our readers, it should be quite possible to arrange for the regular appearance in the Gazette of a series of Symposia, consisting of (1) articles from the pens of its members and (2) a general summing up of the pros and cons of the points at issue by an authority who combines academic distinction with teaching experience. The latter is easy to find. It is the former who as yet have not come to the front in our Association. The Council has said in the Annual Report that this last year has been a quiet year. We are not sure that this may not be meant for a piece of biting criticism, suggesting that though more remains to be done we have not done it. Be that as it may, we appeal to the readers of the Gazette for their cordial co-operation in providing the material for such discussions as are indicated above. Apart from our differences as to detail in the processes of teaching, there are many other topics which should be of interest to the majority as well as to special groups of our members. Those who are engaged in Preparatory Schools should be glad to expound their views of what is feasible with boys of the age with which they deal, and the Public School Teachers of Mathematics should be glad to indicate the lines along which training should proceed, and any way of improving the preparation of the boys who are ultimately to come under their care. The fairness of the papers set at the various examinations, and their strict adherence to the official syllabus may also prove a fruitful subject of discussion.

By the elementary plane constructions we mean the bisection of angles and segments of lines and the drawing of perpendiculars. The constructions given below are extremely simple, and it is very possible that some or all of them have been given before; but they seem to be of sufficient interest to be reproduced, for the reason that, if they are not new, they are not generally known.