In the Report of the Council for 1938 members will have read an account of the work of several sub-committees of the Teaching Committee.

At the meeting of the Teaching Committee in January, 1939, it was agreed that information on work in progress should appear at frequent intervals in the Gazette, for the benefit of members of the Association.

It is hoped that any members who desire to contribute to the work of any of the sub-committees will do so by writing in the first instance to the Secretary of the sub-committee concerned.

I am no artist, but I believe my artistic friends when they tell me that among the first requisites for good art are simplicity and truthfulness. Architecture is also outside my range; but I have architectural friends too, and they tell me of the need for simplicity and truthfulness in Architecture. After taking my degree in Mathematics I went for nearly a year to a Theological College, and I very soon found out for myself what were the qualities which a Theological College most urgently requires.

We have all heard the scheme recommended to Headmasters for their guidance in making the first three reports on a new pupil. The first should be devastating in order to impress the parent with the toughness of the job that the school is up against in remedying the defects in the earlier training of the pupil. The second should be glowing in order to show the progress made when the boy gets to the right school. The third should be very guarded in order to moderate any undue optimism created by the second, and to make it clear that any failure that may lie in the future will be due solely to lack of aptitude or application on the part of the pupil. I fear that some of my colleagues in the universities are prone to adopt a similar line of defence. Certainly one not infrequently hears in college common rooms of the shockingly incompetent teaching given in the schools in this subject or in that. One may sometimes suspect that this is a professional particularisation of the proposition that “all good workmen have bad tools”, though every beginner in logic knows that this may not be simply converted to “all who have bad tools are good workmen” Even if the teachers are as bad as some would have us believe, we professors who have had a hand in the training of so many of them had better keep rather quiet about it.

This note relates to the proof of the formula for the vector triple product (or continued vector product)

a ∧ (b ∧ c) = (a c) b − (a b)c, .....(1)

where the sign ∧ denotes vector multiplication and a dot denotes scalar multiplication.

Of all the reasons urged in favour of the teaching of mathematics and the favoured position of mathematics in the secondary school curriculum, the most widespread is the belief that to learn mathematics is valuable as a training in the art of reasoning. In this paper I want to explore the implications of this belief and its bearing upon the teaching of geometry I shall try to avoid a pronouncement on whether the belief in the value of mathematics as a training in reasoning is justified, by adopting the mathematical method of exploring the consequences of an assumption.

Dr. C. N Srinivasiengar, in his published papers (7, 8, 9 and 10), and Dr. W E. Deming, in a private letter, gave examples of differential equations for which the so-called complete primitives are not really complete, and yet the supplementary solutions are not singular solutions in the envelope sense. In the ensuing correspondence I expanded a remark of E. B. Wilson (11) “It may happen that the arbitrary constant C enters into the expression F(x, y, C) = 0 in such a way that when C becomes positively infinite (or negatively infinite) the curve F(x, y, C) = 0 approaches a definite limiting position which is a solution of the differential equation, such solutions are called infinite solutions.”

A familiar method of proving theorems in elementary projective geometry is to establish by a geometrical construction the existence of a (1, 1) correspondence between the elements of two simply infinite rational systems (such as ranges of points, or pencils of lines) and to conclude that such a correspondence is projective and to use this result to deduce the theorem desired. The argument underlying this reasoning is that the relation between the coordinates x, y of corresponding elements (x), (y) of the two systems is given by an algebraic equation f(x, y) = 0, and that if this relation is such that to each value of x corresponds one value of y and vice versa, then the relation f(x, y) must reduce to a bilinear equation, of the form Axy + Bx + Cy + D = 0, which is easily shown to define a projectivity between the corresponding elements (x), (y).

1. It is well known that the solution of the ordinary linear differential equation with constant coefficients

(Dn + a1Dn−1 + + an−1D+an)y = F(x), .....(1)

where y, Dy, ..., Dn−1y take arbitrary values y0, y1, .., yn−1

when x = 0, ..... (2)

can be obtained by multiplying (1) by e−px, (p > 0) and integrating with regard to x from 0 to ∞

The interchange of salaried appointments between masters and mistresses in British and American schools is arranged in Great Britain by a Committee representative of the English-Speaking Union, the British Federation of University Women, the Incorporated Association of Headmistresses, the Headmasters’ Conference, and the Association of Headmasters, working in conjunction with Committees in the United States.

We know that

log (1 − 2x cos θ + x2)

the series being absolutely and uniformly convergent for all values of θ and for | x | < 1.