We have recently discussed in (1) the general solution of a certain functional equation arising in statistical thermodynamics, and we propose in this note to deal with another functional equation arising from the same source(2).

The problem is to obtain the most general function f(x) which, for all positive integral values of m, n, satifies the functional equation

where

where A is an arbitrary constant. It is the object of this note to prove that this is the only continuous solution.

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.

Various improvements in the formula

which was discovered by Wallis in 1669, were studied by D. K. Kazarinoff in No. 40 of these Notes (December 1956).

The problems considered here are essentially algebraic; but it is convenient to begin with a picturesque formulation.

§ 1. The preceding Note has shown the connection between the partition of a convex polygon by non-crossing diagonals and the insertion of brackets in a product, the latter being more commonly represented by the construction of a tree. It was shown that the enumeration of these entities leads to a generating function y = f(x) which satisfies an algebraic equation of the type

In simple cases, the solution of the equation was found as a power series in x, the coefficient An of xn giving the required number of partitions of an (n + 1)-gon.

A considerable amount is known about the latent roots of matrices of the form

in the case when each cross-product of non-diagonal elements, aici-1, is positive. One forms the sequence of polynomials fr(λ) = |Lr−λI| for r = 1, 2, … n, and observes that

then it is easy to deduce that (i) the zeros of fn(λ) and fn_1(λ) interlace—that is, between two consecutive zeros of either polynomial lies precisely one zero of the other (ii) at the zeros of fn(λ) the values of fn-x(λ) are alternately positive and negative, (iii) all the zeros of fn(λ)— i.e. all the latent roots of Ln—are real and different.

The object of this note is to give simpler proof* of two formulae involving the function ψ (z) which I have proved elsewhere by more complicated methods.

1. If there are n individuals A1, A2,.…, An, in how many ways can they be put into groups? For example, if there are three individuals A, B, C, they may be grouped as

that is, in 5 ways, the respective subgroups, 1, 3, 1 in number, corresponding to the partitions 1 + 1 + 1, 1 + 2, 3 of the integer 3. Hence P (3), say, is 5.

This paper contains (i) a short history of the geometrical theorem proposed in 1840 by Prof. Lehmus of Berlin to Jacob Steiner—“If BJY, CJZ are equal bisectors of the base angles of a triangle ABC, then AB equals AC,” (ii) a selection of some half-dozen solutions from the 50 or 60 that have been given, (iii) some discussion of the logical points raised, and (iv) a list of references to the extensive literature of the subject.

Let 0 < r < 1 and let f(z) be regular for |z| ≤ 1.

Then from Cauchy's integral

we have the inequality

For some questions in arithmetic it is convenient to be able to decimalise shillings and pence mentally to any required number of decimal places.

The pencil of quartic curves

where x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.

Since (1) may also be written

the four lines

are, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

A lamina of mass m can be replaced by an equimomental system of three equal particles m placed at the vertices of a maximum inscribed triangle of a momental ellipse for the centre of mass. Similarly any rigid body can be replaced by an equimomental system of four equal particles (cf. Routh, Elementary Rigid Dynamics, 7th ed. (1905), Art. 44 and Note, p. 423).

By defining a logarithm as

we may visualise the function as the area under the curve

, measured to the right from the zero value at the ordinate AB. The fundamental properties follow at once from (1): for if u = av, then

1. In elementary geometry, the tangent to a curve C at a point P is defined as the limiting position of the chord PQ as Q tends to P along the curve. Further, C is said to have a continuous tangent at P if it has a tangent at every point Q in the neighbourhood of P, and if the tangent at Q tends to the tangent at P as Q tends to P along C.1

In this note an expansion is found for the expression xn + yn in terms of x + y and of xy. Two illustrative applications are appended.

Consider the nth order differential equation

where the coefficients cv are real constants and f is a real function continuous in the interval a≦ x ≦ b. The following theorem will be proved in §4:

If the characteristic equation of (I) has no purely imaginary roots, then a particular integral η (x) can always be found which satisfies the inequality

where C is a certain function of the cv only and M is the maximum of |f|. In particular we may take C = 1 if all roots of the characteristic equation are real.

In many cases, when a colony of micro-organisms such as moulds, yeasts or bacteria grows on the plane surface of a solid medium (e.g. agar), starting from a single cell, the colony tends to grow as an ever expanding circle. The reason for this is that every cell, if free from competition, can multiply at roughly a constant rate in all directions in a plane, limited by the fact that territory occupied by one cell cannot be occupied by another. For the purposes of the present discussion, we can assume, as a first approximation, that the whole process is two-dimensional.