If P is a point of intersection of two conics which have one common focus, then e1 |PM1| =e2 |PM2|. It follows that if the two directrices cut in O, then OP passes through a second point of intersection, and that if directions of measurement perpendicular to the directrices are assigned, the equations e1 . M1P = ±e2M2P give two of the chords of intersection, we may say that these are the chords associated with the common focus.

E. H. Neville was Librarian of the Mathematical Association for over thirty years, from 1923 to 1954.For more than half of that time Neville was not only Librarian but custodian of the books which were handsomely shelved and displayed in his own home, first in his house on Castle Hill, Reading, and then in his first house in Sonning.

A vivid memory of mine is of being shown that certain 2 × 2 matrices were ‘really’ complex numbers. The correspondence between matrices of the form

and complex numbers

seemed magical and the additional remark that

was an unexpected bonus. The correspondence has been examined in various articles in the Gazette [1, 2]. That it is an isomorphism is seen by comparing

with

in the case of addition, and, in the case of multiplication, by comparing

with

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.