The problem which I enunciate and solve in this paper seems to have originated in the study of properties of polyhedral functions. It is a problem of elementary analytical geometry of three dimensions, and the solution which I give, though somewhat tedious, is both elementary and direct. There are several comments which I have to make about current solutions, but I reserve these until the end of the paper since they will be more easily appreciated when it is possible to compare the current solutions with my solution.

There are two alternative methods of defining the concept of “convergence” of a sequence, one involving explicit mention of the limit, the other (Cauchy's condition) giving a necessary and sufficient condition in terms of the elements of the sequence only. The two definitions are equivalent, because of the property of completeness of the real number system.

Let f be a continuous complex-valued function of a real parameter whose real and imaginary parts are of bounded variation in the range (a, b) of the parameter, so that the range of f is a rectifiable plane curve. The main results connecting the arc-length s with the parametrization are as follows:

Theorem 1 (Tonelli). For any rectifiable curve,

equality holding for all α, β (a ≤ α < β ≤ b) if and only if f is absolutely continuous in (a, b).

A comparison of the rational and classical canonical forms of a square matrix reveals that for a nilpotent matrix the two are identical. In this note I describe how we may utilise this fact in solving the problem of reducing a given matrix to classical canonical form. I believe that the point which I try to make in what follows is one which is not always explicitly remarked upon in the literature, and it has therefore seemed to me to be worth while to stress it here.

The well-known multiple integral

where Rn is the region defined by x1 ≥ 0, x2 ≥ 0, …., xn ≥ 0, x1 + x2 + …. + xn ≤ 1, and where a0, a1, …, an are positive constants, can be evaluated either in the classical way using the Dirichlet transformation or by the use of the Laplace transform. I. J. Good has considered a more general integral and has proved the following result by induction:—

If f1(t), f2(t), …, fn(t) are Lebesgue measurable for 0 ≤ t ≤ 1, m1, m2, …., mn, mn+1 (= 0) are real numbers, Mr = m1 + m2 + … + mr, x1, x2, …, xn are non-negative variables and Xr = x1 + x2 + … + xr, then

It does not seem to be possible to establish this relation by employing the Laplace transform, but we show below that it can be obtained using the Mellin transform.

In the course of mathematical progress new truths are discovered while older ones are sometimes more precisely articulated and often generalised. Because of their elegance and simplicity, however, some classical statements have been left unchanged. As an example, I have in mind the celebrated formula of John Wallis,

which for more than a century has been quoted by writers of textbooks. Usually this formula is written as

In this note it is shown that ¼ < θ < ½. Unquestionably, inequalities similar to this one can be improved indefinitely but at a sacrifice of simplicity, which is why they have survived so long.