A lattice An in n-dimensional Euclidean space En consists of the aggregate of all points with coordinates (xx,…, xn), where

for some real ars (r, s = 1,…, n), subject to the condition ∥ αrs ∥nn ╪ 0. The determinant Δn of Λn, is denned by the relation , the sign being chosen to ensure that Δn > 0.

If A1…, An are the n points of Λn having coordinates (a11, a21…, anl),…, (a1n, a2n,…, ann), respectively, then every point of Λn may be expressed in the form

and Ai,…, An, together with the origin O, are said to generate Λn. This particular set of generating points is not unique; it may be proved that a necessary and sufficient condition that n points of Λn should generate the lattice is that the n × n determinant formed by their x coordinates should be ±Δn, or, equivalently, that the n×n determinant formed by their corresponding u-coordinates should be ±1.

The nature of the eigenvalues of a square quaternion matrix had been considered by Lee [1] and Brenner [2]. In this paper the author gives another elementary proof of the theorems on the eigenvalues of a square quaternion matrix by considering the equation Gy = μȳ, where G is an n x n complex matrix, y is a non-zero vector in Cn, μ is a complex number, and ȳ is the conjugate of y. The author wishes to thank Professor Y. C. Wong for his supervision during the preparation of this paper.

1. Introductory. The following two integrals will be established in § 2.

If m is a positive integer, if p ≧ q + 1 and if R(ar+kt) > 0 (r = 1, 2,…, p, t = 1, 2,…, m),

where co is 1 or e±in according as m is even or odd, the dash denotes that the factor sin (k,–k,)π does not appear and the asterisk that the parameter kt–kt + 1 is omitted. If pp ≧ qthe result holds if the integral is convergent.

In this note I show that

where Jdenotes the Bessel function of the first kind of the orders and arguments indicated, n = 0, 1, 2, 3, … and the real parts of both μand v exceed — 1. This is a generalization of Sonine's first finite integral [1, p. 373] to which it reduces in the special case n = 0.

This paper is paper gives what appears to be a new Rodrigues’ formula for the Associated Legendre Polynomials defined by [5, p. 122]

with the restriction that m is an even positive integer, which helps to evaluate some integrals. Putting m = 2k in (1.1) and replacing Pn(x) by the Gegenbauer Polynomial and using [3, p. 176]

We obtain

Putting a =v–½ in the relation [4, p. 283]

We get

Let C denote Cayley's algebra defined over the field of rational numbers. This paper contains a simple characterization of arithmetics of C in terms of a given basis i0 = 1, i1, i2, …, i7. We deduce that certain of the arithmetics of C are isomorphic. The result that the maximal arithmetics are isomorphic is also contained in the work of van der Blij and Springer [2].

1. Introduction. Recently, I gave an analogue [1] of the MacRobert's E-function [4[ in the form

where the symbol denotes that a similar expression with a and β interchanged is to be added to the expression following it. It has since then been generalized by N. Agarwal [2], who defined and studied the q-analogue of the generalized E-function. In this paper I give some further properties of the Eq-function.

In [2], Tosiro Tsuzzuku gave a proof of the following:

THEOREM. Let G be a doubly transitive permutation group of degree n, let K be any commutative ring with unit element and let p be the natural representation of G by n × n permutation matrices with elements 0, 1 in K. Then ρ is decomposable as a matrix representation over K if and only ifn is an invertible element of K.

For G the symmetric group this result follows from Theorems (2.1) and (4.12) of [1]. The proof given by Tsuzuku is unsatisfactory, although it is perfectly valid when K is a field. The purpose of this note is to give a correct proof of the general case.

1. In this note we consider the formal solution of the dual integral equations

where f(x) and g(x) are given and χ(x) is to be found. The direct solution of these equations has been given by Noble [1] but we shall show that they may be solved more easily if they are first reduced to a form in which g(x) ≡ 0.

Let h (m, n) = αm2 + 2δmn + βn2 be a positive definite quadratic form with determinant αβ–δ2 = 1. It may be put in the shape

with y > 0. We write (for s > 1)

The function Zn(s) may be analytically continued over the whole s-plane. Its only singularity is a simple pole with residue π at s = 1.

In this note a theorem, giving a relation between the Hankel transform of f(x) and Meijer's Bessel function transform of f(x)g(x), is proved. Some corollaries, obtained by specializing the function g(x), are stated as theorems. These theorems are further illustrated by certain suitable examples in which certain integrals involving products of Bessel functions or of Gauss's hypergeometric function and Appell's hypergeometric function are evaluated. Throughout this note we use the following notations:

The integral function

is known as Airy's Integral since, when z is real, it is equal to the integral

which first arose in Airy's researches on optics. It is readily seen that w= Ai(z) satisfies the differential equation d2w/dz2 = zw, an equation which also has solutions Ai(ωz), Ai(ω2z), where ω is the complex cube root of unity, exp 2/3πi. The three solutions are connected by the relation.

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroups

such that L1 ƞA is the kernel of μ and

for ι = 1, 2,…, m–1.

The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

1. We use Cassels's notation and define h (m, n), Q (m, n), Zh (s), Zh (1) – ZQ (1) and G (x, y) as in [1]. Rankin [5] proved that the Epstein zeta-function Zh (s) satisfies, for s ≧ 1·035, the

THEOREM. For s > 0, Zh (s) — ZQ (s) ≧ 0 with equality if and only ifh is equivalent to Q. Rankin then asked whether the theorem is true for all s > 1. Cassels [1] answered this question in the affirmative and proved further that the theorem is true for all s > 0.

This note is concerned with square matrices, denoted by capital letters, whose elements belong to aBoolean algebra with null element 0 and all element 1. Such matrices, which have important applicationsin the theory of electric circuits, can be compounded in the three following ways.

Dr V. O. Ennola has pointed out that there is a mistake in this paper. The inequality for dox at the foot of page 76 does not imply the stated inequality between the second and third lines on page 77. This error vitiates the entire argument, and I have not so far been able to put it right.

Let D be a bounded, closed, simply-connected domain whose boundary C consists of a finite number of analytic Jordan curves. Let γ be any analytic arc of C. Then we shall prove the following theorem.

Theorem 1. Let u(x, y) be harmonic in the interior of D and continuous on γ, and let ϱu(x, y)/ϱn=g(s) when (x, y) is on γ, where g(s) is an analytic function of arc-length s along γ. Then u(x, y) can be harmonically continued across γ.