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.

The formula to be proved is

The formulae required in the proof are the Barnes Integral

and Whipple's Formula (1)

§ 2. Proof of the Formula. From (2) the E-function on the left of (1) is equal to

In § 2 the following two formulae will be established.

If, when p≧q + l, R(ar + k)>0, r = 1, 2,…, p and | ampz |<π,

If p≧q, the result holds if the integral is convergent.

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.

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

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.

The formula to be established is

where l, m, n. are any numbers real or complex and R(b)>0. A similar result, involving Bessel Functions of the First Kind, was obtained by Hanumanta Rao [Mess, of Maths., XLVII. (1918), pp. 134–137].

This paper describes a simple method of evaluating integrals of a type some cases of which have been discussed by Dr. Ragab.

The following formulae are required in the proofs

If p≧q + 1, z ≠ 0,

If. p = q + 1, | z |>1 in this formula.

In Muir's Theory of Determinants, Vol. III, pp. 232–237, there will be found accounts of papers by H. Nägelsbach, J. Hammond and J. W. L. Glaisher, in which expressions for the Bernoulli numbers are obtained in terms of determinants. In the present paper an expression for Bn will be derived which appears to be new, but which is very like some of those mentioned by Muir.

§ 1. Introductory. The formula to be established is

where m is a positive integer,

and the constants are such that the integral converges.

1. There are exceptional integrals of the total differential equation

in the case when it is not completely integrable, and so when the invariant

is not identically zero, which do not seem to be mentioned by any standard authorities such as Cartan, Goursat, de la Vallée Poussin, and Schouten and Kulk. These are integrals of (1) which do not reduce I to zero. They arise only when the first partial derivates of P, Q, R are not all continuous. A simple example is z = 0 as an integral of

§ 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. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have

The earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

§ 1. Introduction. A problem of some interest in mathematical statistics is that of determining conditions under which the randomisation distribution of a statistic and its normal theory distribution are asymptotically equivalent, as these two distributions are used in alternative approaches to the same inference problem.

§ 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.

§ 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.