In opening this discussion on the influence of the university on the school course in geometry I have two dangers to avoid. Many of the things which it would seem natural to say are true and a few of them may be important, but when I wrote them down and took a second look at them, I saw that they were not concerned in any specific way with geometry; we sometimes forget that there is a Section of the British Association which exists to rescue the rest of us from the generalities and platitudes to which teachers are only too prone. The other danger is of entering so far into particulars as to compose a syllabus round which we might be expected to quarrel instead of talking about principles on which we can hope to agree; here I am fortunate in the subjects chosen by the speakers who are to follow me, for where what I have to say is most in need of illustration, their expositions will provide you with examples worked out at the appropriate length.

I am looking forward with great enthusiasm to the arrival at advanced level of the GCSE student who has experienced a problem solving approach to mathematics through investigations. I hope examination boards are equally enthusiastic. Here for them is the opportunity to broaden their approach to syllabus content and methods of assessment. Can we hope that they are already working with this in mind? Advanced level mathematics courses, particularly in the options of applied mathematics, will need to nurture the good work of the GCSE courses and allow sixth-forms to produce a student capable of modelling real problems, in terms of sound and necessary mathematical techniques.

1. Introduction. While the operational calculus for the solution o initial value problems for linear differential equations with constam coefficients is now commonly introduced in a thoroughly satisfactory manner—either by means of the Laplace transform (e.g. ref. 1) of by the method of J- Mikusinski (ref. 2)—the same cannot be said of the simpler method for the general integration of such equations which is known as the D calculus. In fact, I cannot recall any text which gives a consistent definition of the meaning of expressions of the type [F(D)/G(D)]y, where F and G are polynomials. The reason for this state of affairs seems to be that the scope of the D calculus is rather limited so that the validity of the result can be verified in each case (e.g. ref. 3). In particular, this applies to the decomposition of an operator F-1(D) into partial fractions, which is the central step: in the solution of an equation with nonvanishing right hand side of general form. Nevertheless a rational approach to the entire problem is perhaps not out of place. This is attempted in the present paper.

Apart from the conic sections, many well known curves have physical and geometrical applications which are readily understood by sixth formers and undergraduates. The catenary, cycloid, spirals, involutes, evolutes and many others appear in courses in civil engineering and control mechanisms, and frequently in A level calculus questions.

The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative. We shall often ignore the fact that A3, A5, .. vanish, thereby refusing to complicate formulae unnecessarily.