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.

It is not perhaps generally realised that the special bilinear substitution , used to reduce the integral

1. The invariants and covariants of a system of two conics have been much studied2 but little has been said about those of three conies. Three conics have a symmetrical invariant Ω123, or in symbolical notation (a b c)2. According to Ciamberlini3 the vanishing of this invariant signifies that the Φ conic of any two of f1, f2, f3 is inpolar with respect to the third; and in a previous paper4 I have derived by symbolical methods a more symmetrical result, viz., if Ω123 vanishes, then u being any line in the plane, u1, u2, u3 are concurrent, where ui is the polar with respect to fi of the pole of u with respect to Φjk.

Cardboard or wire models of ellipsoids and hyperboloids exist which consist of two sets of circular sections. They cover the quadric surface with curvilinear quadrilaterals, whose sides remain constant in length when the model alters in shape. In fact the models admit of one degree of freedom—they are collapsible—and the angle between the two sets of circular sections can be varied.

The determinant to be described made its appearance in the course of a search for the frequency distribution of Spearman's rank correlation coefficient (2). This name is given to the form, , taken by the product moment correlation coefficient in the special case where the two variables are separate arrangements of the first n natural numbers.

In my experience answers to questions about maxima and minima are as a rule hazy and unsatisfactory. I suggest that the principle “according as a function is increasing or decreasing, its derivative is positive or negative” might be applied more fully. All functions are here assumed continuous.

An interesting determinant occurs in the fifth volume of Muir's History1. It is

where n = ½ (p – 1), and ars is the smallest positive integer such that

rars = s (mod p), (1) p being any odd prime number. It is evident that each element ars is unique and non zero. For p = 5, 7, 11 the determinants are

(2) respectively, and their values are

– 5 , 72, 114.

Let f(x) ≡ (1 – x)b + b ab-1 x

ø (x) ≡ xc – c Βc-1 x

where b ≧ 1, c ≧ 1, 0 ≦ α ≦ 1, 0 ≦ β ≦ 1, and x is assumed to lie in the range (0, 1). By differentiation, or otherwise, it is easily shewn that f(x) and ø (x) have minima when x = 1 – α and when x = β, respectively. Hence

(1 – x)b + ab-1 x ≧ b ab-1 + (1 – b) ab

xc βc-1 x ≧ (1 – c)βc.