The Junior Mathematical Association of London, of which I expect very few of you have even heard, is an association in a sense of the word rather different from that in which it is used in the title of the Association I am now addressing. It is the name adopted by some eight large schools (of the Public School type) in and around London for a joint Mathematical Society which has been running amongst the Sixth Form mathematical specialists in those schools for nearly five years. It is essentially an association of institutions rather than of persons, and this type of organisation will at once be seen to be necessary by any one who considers the effect of the changing personnel of Sixth Forms. It possesses therefore no strict membership roll, no subscription, and, incidentally, no harassed treasurer. All the same, it is alive, as I hope to be able to show you by the end of this short paper. It might also be recorded at this point that a somewhat similar association exists amongst some of the Classical Sixths.

As the President has told you, the subject of this lecture was not my own choice. When first asked to speak on it my impression was that, as the phrase goes, the answer was a lemon; but, on thinking it over further, it seemed that quantum mechanics might have some bearing on school work, although that influence is in a sense only accidental, as I will explain. As far as I can see, however, statistical mechanics, the other half of the subject of the paper, has no bearing on school work, and it only occurs in the title because that title was given me!

The article on “Differentials”, in the May Gazette—based on a discussion at the last Annual Meeting of the Association, and referring back to articles of July 1931 and February 1932 —has so clarified certain issues for the writer as to confirm a long-standing conviction that the standard definition of “differentials” is quite unsound, and that the view set out below (which he had developed in his own teaching ) is much truer to all the facts of the situation.

I. In a note in the Mathematical Gazette, February 1934, pp. 43–46, I gave a method for determining the foci, etc., of a conic in general and showed that the method could be applied to the parabola. A direct solution for the parabola can, however, be put down more quickly.

Voici quelques compléments au remarquable article de M. N. M. Gibbins sur la figure constituée par un triangle bordé de triangles équilatéraux.

1. Considérons un triangle BAG, et les triangles équilatéraux BCA1BC1A, AB1C et BA3C, BAC3 , ACB3 construits extérieurement puis intérieurement sur les côtés CB, BA, AC.

Consider x : y : 1 = t2 - 2t : t : t2 - t - 1. ........................ ( i)

The curve meets the line lx +my + n = 0 in points given by

l(t2 -2t) +mt +n(t2 -t-1) =0.

For any given Z, m, n there are two values of t. Hence the curve meets any line in two points and is therefore a conic. For a given l, m, n the values of t are the roots of the equation

t2 (l +n) + t(m -2l -n) -n =0.

The following notes are the outcome of a course of lectures to pupils on invariants. The methods are new to me, but I publish them with hesitation and apologise beforehand should they prove well known.

By a simplex is meant the figure in general flat space of which the triangle and tetrahedron are examples. In space of n dimensions the simplex has (n + 1) vertices and (n + 1) faces, each face consisting of a prime, i.e. a space of (n - 1) dimensions, containing n of the vertices and hence itself a simplex in (n - 1)-dimensional space. There is a general likeness between the triangle and simplexes of higher order, but this is marred by the fact that, in general, a simplex does not possess an orthocentre.

The idea of attaching signs to certain geometrical magnitudes such as lengths, angles and areas is fundamental in analytical geometry. It has its uses, too, in pure geometry in simplifying certain theorems and constructions. Many theorems, for instance, such as “ the area of a triangle is half that of a rectangle with the same base and height ”, or “ the angle subtended at the centre of a circle is double that at the circumference”, have to be proved differently for two or more figures, the word “ add ” being in some cases changed to “ subtract ”. In such theorems the one proof will suffice for any figure provided the proper interpretation is made of the signs of the magnitudes.