The general term (see Fig. 1) in Pascal’s Triangle may be defined as:

(1a)

where

n and r are positive integers, where n≥r.

is equivalent to the binomial coefficient, and is equal to n!/[r!(n-r)!] (1b)

I make here two comments on Chaundy and Bullard’s interesting article with the above title (Math. Gaz. no. 350, vol. XLIV (1960), 253-260).

1. Introduction. If F and G are differentiate functions of one variable, the “function of a function” rule is

where f and g denote the derivatives of F and G respectively.

The flexagon was defined by Mr R. F. Wheeler (M. G. 1958, p.1) as a strip of equilateral triangles which can be folded in various ways into a hexagon to exhibit different colour combinations. Can this idea be generalised? Suppose a plane figure to consist of m regular n-gons, with a common vertex, which can be folded together in pairs.

Important changes are taking place at Westfield College. A new Science Faculty is being created and—although the College is, indeed, no stranger to scientific activity—the scope of the future effort in this field will be on a much larger scale than at any time in the past. I speak to you today as one of the five new professors who have recently joined the College as a result of the expansion; but my position as professor of Mathematics is not quite the same as that of the other four science professors. I am, in a sense, not such a novelty—and this perhaps is why I have been asked to be the first to give my Inaugural Lecture—since my department already exists. In fact mathematics has been taught at the College continuously from its earliest days in the 1880’s, and I should like to begin by paying tribute to the devoted work of my predecessor, Miss G. K. Stanley. Originally a Westfield student, she received her postgraduate mathematical education in Oxford at the feet of the great Hardy, and returned to Westfield College in 1931 to take over the direction of the Mathematics Department from Miss Whitby, who had held office for the preceding 36 years. Not only has Miss Stanley guided the department with a sure hand during a period which has seen profound changes in the teaching of mathematics in London, but she has also given invaluable service direct to the University as secretary of the Board of Studies in Mathematics for the past 16 years. We are very glad indeed to know that she is still with us to give us the benefit of her knowledge and experience in the development that lies ahead.

We discuss the deduction of real euclidean theorems from complex projective theorems, and the reverse process: when is it permissible, in proving a complex projective theorem, to project two given points into the circular points at infinity and use real euclidean methods? We emphasise the difference between the real and the complex, and the consequences of using real numbers to define euclidean geometry, and complex numbers to define projective geometry.

1. The preceding paper considers the most general theorem on change of variable in a Riemann integral: If g(t) is integrable over [a, b] and f (x) is integrable over G ([a, b]), then (G(t))g(t) dt exists and equals (x)dx.

In Note 2949 printed in the May 1961 issue of the Gazette, I gave a method for constructing a rounded closed flexless curve, not necessarily a circle, which is self-inverse with respect to two points.

The difficulty of a good calendar lies in the fact that the earth travels round the sun in 365 days, or to be precise in 365 days 5 hours, 48 mins., 46 secs, which, we should note, is less than 365 days by just over 11 minutes.

In my note in the December 1960 Gazette I proved that if there is an odd prime p and numbers a, b, c prime to p such that

then p necessarily has one of the forms 120k+1, 120k+49.

These designs were prompted by the “pursuit curves” of Mr. I. J. Good The first one consists of his six ‘hyperboloids’ and the second an equal number of ‘spearheads’ The former possesses what may be termed cyclical symmetry and the latter triple axial symmetry

There seems to be a technical improvement in these designs over those of Mr. Good’s insofar as the black areas at the centre of his triangular doodles are conspicuous by their absence over here.

The representation of a twisted cubic as the intersection of a hyperboloid and cylinder with common generator makes an interesting model. This note gives details of its construction.

A square matrix is said to be doubly-stochastic if its elements are non-negative and if all row-sums and column-sums are equal to 1. The study of doubly-stochastic matrices was initiated by I. Schur and was subsequently taken up by Hardy, Littlewood, and Polya, who proved the following fundamental proposition [1, Theorem 46].

Let w be a point and ABCDEF a polygon, convex or not, which we call (P). Projecting w orthogonally on the sides AB, BC, CD, DE, EF, FA of (P), and joining the respective projections, we obtain the polygon A1B1C1D1E1F1, called (P1) [see Figure 1]. We call (P1) the first pedal of w with respect to (P).