In the following pages we give a complete system of projective concomitants for any number of linear complexes and one quadric, in a quaternary field. The investigation follows the earlier1 paper on mixed quaternary forms, in which the corresponding system, omitting the quadric, was given. Such a system, for one quadric and one linear complex was given by Weitzenböck who used complex symbols. A detailed investigation of the case of two linear complexes and a quadric, together with their geometry, has been given elsewhere.

It is well known that the polynomial in x,

has the following properties:—

(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;

(B) it satisfies the three-term recurrence relation

(C) it is the solution of the second order differential equation

(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),

i.e. when

Several other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

It is well known that any map of n regions on a sphere may be coloured in five or fewer colours. The purpose of the present note is to prove the following

Theorem. If Pn(λ)denotes the number of ways of colouring any ma: of n regions on the sphere in λ (or fewer) colours, then

(1)

This inequality obviously holds for λ = 1, 2, 3 so that we may confine attention to the case λ > 4. Furthermore it holds for n = 3, 4 since the first region may be coloured in λ ways, the second in at least λ — 1 ways, the third in at least λ — 2 ways, and the fourth, if there be one, in at least λ — 3 ways.

The earliest proof that every rational number (R) can be expressed as a sum of cubes of three rational numbers (x, y, z), not necessarily positive, was published in 1825 by S. Ryley, a schoolmaster of Leeds: formulae were given for x, y, z in terms of a parameter, such that every value of the parameter led to a system of values of x, y, z satisfying the above relation, and every rational value of the parameter led to a system of rational values of x, y, z. The later solutions referred to by Dickson are found to give the same results as Ryley's formula, as does another method, quoted in a modified form by Landau from a paper by the present writer. Thus it might almost be believed that Ryley's century-old result embodies all that is known with regard to the resolution of a number into three cubes, and that his formula is unique. I propose to examine the rationale of his method and the causes of its success; it will then appear that an infinity of similar formulae exist, and that one of them is at least as simple as his. It is convenient to state Ryley's formula, and the modification made by Landau, in section 2; and to generalize the method in section 3.

An algebraic equation

determines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.

§ 1. It is well known that, if , the convergence of sn to a limit implies the convergence of tn to the same limit. The converse theorem, that the convergence of tn implies the convergence of sn, is false. Mercer1 proved, however, that if , then both sn and tn tend to l. This theorem has recently been extended in various directions.2 In the present note the case of Abel limits is considered.

The term “flat” is used to indicate that the minimum modulus of a function in a region is (in some sense) of the same order as the maximum modulus. Some properties concerned with this notion are described below. They came to light during an attempt to answer a question put to me by Professor Littlewood.