The object of the present paper is to establish the equivalence of the well-known theorem of the double-six of lines in projective space of three dimensions and a certain theorem in Euclidean plane geometry. The latter theorem is of considerable interest in itself for two reasons. In the first place, it is a natural extension of Euler's classical theorem connecting the radii of the circumscribed and the inscribed (or the escribed) circles of a triangle with the distance between their centres. Secondly, it gives in a geometrical form the invariant relation between the circle circumscribed to a triangle and a conic inscribed in the triangle. For a statement of the theorem, see § 13 (4).

In some recent work by D. G. Kendall and the author † on the number of points of a lattice which lie in a random circle the mean value of the variance emerged as a constant multiple of the value of the Epstein zeta-function Z(s) associated with the lattice, taken at the point s=. Because of the connexion with the problems of closest packing and covering it seemed likely that the minimum value of Z() would be attained for the hexagonal lattice; it is the purpose of this paper to prove this and to extend the result to other real values of the variable s.

The purpose of this paper is to present a proof of the following theorem of Maclagan-Wedderburn.*

Every finite skew-field† is a field.

The proof depends on group theory and on the properties of Galois fields. As an introduction, §§1–4 are devoted to a systematic and self-contained account of the theory of Galois fields.

§ 1. When a numerical method of obtaining an approximate solution of a linear differential equation is employed, the process involves two distinct types of approximation. The region of integration having been covered with a regular net, the differential equation and the appropriate boundary conditions are replaced by finite difference equations which are linear equations in the values of the dependent variable at the nodes of the net.

The formula

where αp+1 = 1/2(m + n), αp+2 = 1/2(m-n), R(m±n)>0 and x is real and positive, was given by MacRobert (Phil. Mag., Ser. 7, XXXI, p. 258). From it the formula (6) below will be deduced.

The theory of quadratic congruences modulo an integer is dominated by the Quadratic Law of Reciprocity (see § 1), which makes it possible to decide in a very short time whether a quadratic congruence

is solvable or not. The law was first proved by Gauss.* It took him over a year to obtain his first proof, which depends on a tedious lemma in elementary number theory. He subsequently obtained seven further proofs, and today more than fifty proofs are known, most of them based on the ideas of Gauss. The object of the present paper is to present a proof which is a modernised version of Gauss's seventh proof, applying the ideas of that proof to a finite set of objects, the elements of a finite or Galois field.

For a given sequence {am} and p≠0, Schur (2) defined

In particular if p is a prime, a an integer and , then by Fermat's theorem

is integral. Schur proved that if p † a, then all the derivatives

are integral. Zorn (3) using p-adic methods proved Schur's results and also found the residue of Xm (mod pm), where and x = 1 (mod p). The writer (1) proved Zorn's congruences by elementary methods as well as certain additional results of a similar sort.

§ 1. Introductory. The formula to be proved is

where n is zero or a positive integer, R(m)>0, R(αs-l)>0, s = 1, 2, …, p, p≧q.

§1. Introductory. The formula

where w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.

§ 1. Introductory. In § 3 a generalisation of the formula [MacRobert, Phil. Mag., Ser. 7, XXXI, p. 258]

where αp+1 = ½m + ½n, αp+2 = ½m - ½n, R(m ± n) > 0, and x is real and positive, will be established. In the course of the proof Hardy's formula [Mess, of Maths., LVI, (1927), p. 190],

where R(b)>0, will be required. This was originally proved by an application of Mellin's Inversion Formula. An alternative proof is given in § 2, and some related formulae are deduced.

§ 1. Introductory. The integral

where b>0, was given by Hardy (1). It was proved by applying Mellin's inversion formula. An alternative proof, based on the differential equation

satisfied by Kn(x), has been given by the author (2).

An infinite or semi-infinite medium, in which heat is generated or absorbed at a rate proportional to the temperature, is placed at temperature zero in contact with a perfect conductor of finite heat capacity at a higher temperature. Expressions are derived for the subsequent behaviour in linear and spherical cases, and applications suggested.

§ 1. Introductory. The formula to be proved is

where ρq+1 – αp+1 > 0, αr – ρq+1 > 0, r = 1,2, …, p, and p≧q + 1.

Let μ be an isomorphism which maps a subgroup A of the group G onto a second subgroup B (not necessarily distinct from A) of G; then μ is called a partial automorphism of G. If A coincides with G, that is if the isomorphism is defined on the whole of G, we speak of a total automorphism; this is what is usually called an automorphism of G. A partial (or total) automorphism μ,* extends or continues a partial automorphism μ if μ* is defined for, at least, all those elements for which μ is defined, and moreover μ* coincides with μ where μ is defined.

The theory of operational solutions of differential equations in applied mathematics suggests a method of developing the theory of Fourier and allied series that is simpler for ordinary applications than the classical development. It may be useful to those whose interests lie in such applications rather than in the deeper analytical processes associated with this subject.

If the right-hand side of the expansion

is integrated m times from 1 to µ, it becomes

§ 1. Introductory. The formula to be established is

where m is a positive integer,

and the constants are such that the integral converges.

The purpose of the present note is to give some interesting and simple identities connected with basic hypergeometric series of the types 2Φ1 and 3Φ2.

Given the series ,the n-th Casáro sum of order k is defined by the relation

where is the binomial coefficient . Let Then Σan is said to be summable (C; K) to the sum s if, as n → ∞, The series is said to be absolutely summable (C; k), or summable | is convergent. The series is said to be strongly summable (C; k) with index p, or summable [Ck, p], to the sum s if

It is assumed that k and p are positive.

1. Introduction. Given a series we define , by the relations

The series Σan is said to be summable (C, k) to the sum s, if

as n→∞, and strongly summable (C, k), k>0, with index p, to the sum s, or summable [C; k, p] to the sum s, if