The solution of the general case of a cable having constant resistance R, inductance L, capacitance C, and leakance G, all expressed per unit length, is usually obtained in the form of an expansion. Using his operational method, Heaviside was the first person to obtain a solution. F. W Carter points out that Heaviside’s result seems to be incorrect, and proceeds to find a solution in the form of a double summation. By aid of complex integration, the integrand being expanded in Bessel functions, H. Jeffreys gives another form of series solution. As this type of problem may arise in lectures on applied mathematics, it is proposed to show how a compact result can be got by contour integration, using a suitable transformation.

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities

• (a) matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,

• (b) matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.

Early in the classical period of Greek mathematics there arose a crisis over the meaning of “proportion”, or alternatively of “area”. This crisis, which was resolved by Eudoxus, the pupil of Plato, about the middle of the fourth century B.C., led to the introduction of axioms, or postulates, as “self-evident truths”, necessary for the proof of theorems, equally true but presumably less self-evident. Among these axioms was included the famous Fifth Postulate of Euclid, equivalent to saying (see below for its rather complicated wording) that “through a given point (the point T in Fig. 1) there is at most one (straight) line parallel to a given line (the line MK)”. But this postulate had all the earmarks of a theorem; it looks true yet not altogether self-evident, and its converse, to the effect that there is at least one such line, is an easily provable theorem.

This note describes an example which I have found useful for introducing ideas in linear algebra and group theory. It is simple and yet it is neither trivial nor too abstract.

We consider here three geometrical problems, the solutions of which are not hard, but each involves some rather unobvious point. The reader is invited to attempt them before looking at the solutions. The names of the original proposers are noted in brackets.

A children's television game is played as follows: Four postcards from children who have written in to the programme are selected at random and the children's names are written on a board. The presenters then draw letters of the alphabet one by one from a bingo-style randomising machine and, after each letter is drawn, all occurrences of that letter in any of the children's names are crossed off. The winner of the game is the first child whose name is completely crossed off.

Prodigiously talented with a remarkable flair for anticipating correct results and initiating fruitful areas of research, Stanislaw Ulam (‘Stan’ to his friends) was an unusual mathematician. The considerable breadth of his preoccupations, even in the halcyon days of his youth in Poland, distinguished him from his peers. Sustained by a supreme self-confidence, Ulam preferred pioneering new fields of mathematics to elaborating the ideas of others. Indeed, impatient with detail he tended to leave technicalities to those with whom he collaborated. Uprooted from his native Poland in his twenties, Ulam was to spend two thirds of his working life associated with or employed by the Los Alamos National Laboratory.

The polar equation of the parabola y2=8ax, referred to the origin of coordinates as pole is

r sin2θ = 8a cos θ

or

Inverting with the pole as centre of inversion and 4a as the radius of inversion we find that the equation to the inverse curve is

or r cos θ = 2a sin2θ

or r = 2a sin θ tan θ.