Edmund Taylor Whittaker was born at Birkdale, Lancashire, on 24th October 1873 and belonged to a family which had been known for several generations in the district where the Rabble forms the boundary between Lancashire and Yorkshire. He entered Manchester Grammar School in 1885 and spent two years in the Classical side before turning over to the Mathematical side. In December 1891 he won a scholarship to Trinity College, Cambridge, and, after an undergraduate career of unusual distinction, was bracketed with J. H. Grace as Second Wrangler in the Tripos of 1895 ; Bromwich was Senior Wrangler. In 1896 Whittaker was elected a Fellow of Trinity College and in 1897 was first Smith's prizeman, his essay being entitled “On the reduction of the theory of multiform functions to uniform functions.”

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li by

is again in L1, and

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined by

the supremum being over all finite sets of disjoint Borel subsets of G.

In this note a simple application of the Kontorovich-Lebedev transform is described. The problem to which the transform is applied is the one considered in a recent paper (1).

The well-known “basis theorem” of elementary algebra states that in a finite-dimensional vector space, any two bases have the same number of elements; or, equivalently, that a vector space is n-dimensional if it has a basis consisting of n vectors (where the dimension of a vector space is defined to be the least upper bound of the numbers k for which there exist k linearty independent vectors, and a basis is defined to be a maximal set of linearly independent vectors). This theorem has an analogue in the theory of groups : if an Abelian group has a finite maximal set of independent non-cyclic elements, the number of elements in one such set being n, then no set of independent non-cyclic elements can have more than n members.

It is a singular honour to be invited to deliver a lecture commemorating the work of Sir Edmund Whittaker, especially before the Edinburgh Mathematical Society, whose development owes so much to his initiative and co-operation. But when I reflect on the difficulties of the task I can only exclaim in the words of St Jerome's preface to his translation of the New Testament, “Pius labor, sed periculosa praesumptio”.

Let there be given, on an algebraic curve C, of genus p, a linear series and an algebraic series of index v, both without fixed points. The number of groups of r + 1 points which are common to a set of and a set of has been shown by Schubert (1) to be

where d is the number of double points of .

Indices of the free logarithmetic correspond to bifurcating root-trees (cf.(4)), to Evans' non-associative numbers (3) and to Etherington's partitive numbers (2). The free commutative logarithmetic is the homomorph of f determined by the congruence relation P + Q ∼ Q + P. Formulæ for aδ and pα, i.e. the numbers of indices of of a given potency* δ and the number of indices of a given altitude α respectively, were given by Etherington (1), who also gave corresponding formulæ for commutative indices of . Other enumeration formulæ are contained in (5).

In a perfect sequence space α, on which a norm is defined, we can consider three types of convergence, namely projective convergence, strong projective convergence and distance convergence. In the space σ∞, when distance is defined in the usual way, the last two types of convergence coincide and are distinct from projective convergence ((2), p. 316). In the space σ1 all three types of convergence coincide ((2), p. 316). It will be shown in this paper that, if distance convergence and projective convergence coincide, then all three types of convergence coincide. It will not be assumed that the limit under one convergence is also the limit under the other convergence.

Let us denote by α the set of n real numbers α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in αl, …, αn, i.e. ck(α) is the sum of all possible products of k different αi and hk(α) is the sum of all possible products of k αi, where now in any product one or more αi may be repeated any number of times.

Although only three of Whittaker's papers (12, 13, 14) are on the theory of automorphic functions, he retained his interest in it throughout his life. His son, Dr J. M. Whittaker, informs me that his last mathematical conversation with his father was on this subject. The standard English work on automorphic functions by L. R. Ford (4), and its precursor in the Edinburgh Mathematical Tracts series, owe much to Whittaker, and it was he who suggested the term “isometric circle” of which Ford makes such elegant use in his development of the theory. Professor G. N. Watson informs me that Whittaker had expressed his willingness to contributea a chapter on automorphic functions to the revised and expanded edition of Modern Analysis which is in preparation.

It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.

It is well known* that certain types of partial differential equation may be solved using integral transforms with suitable kernels. In general, these equations may be solved by the classical method of separating variables, but the use of an integral transform yields the solution in a more direct way in the sense that the boundary values are contained in the solution.

It is the purpose of this note to apply this technique to obtain the solution of the differential equation associated with the transverse motion of an elastic beam for a wide class of boundary conditions.

In many branches of applied mathematics there exists a class of problems which depend for their solution upon the integration of a set of simultaneous linear partial differential equations subject to certain boundary conditions. In all but the simplest cases it is not practicable to deal with these equations by standard methods. For problems involving infinite regions, solutions can often be found by the use of integral transforms. However, in many problems we are concerned with media of finite extent, so that if we are to make a direct application of this method, we shall have to use finite transforms, and under certain conditions these are much more difficult to apply than transforms over an infinite range.

Let T(α1, α2, …, αn−1) be the n-ary transformation which takes the sequence {si}, (i = 0, 1, …), into the sequence {s'i} where

with sm = 0 when m is a negative integer, and where α1, …, αn an are real numbers with sum unity. In a previous note (1) conditions were found on α and β for the ternary transformation T(α, β) to be equivalent to convergence. A method is given here for treating the similar problem for the general n-ary transformation.

Whittaker's contributions to algebra are not numerous and are confined to special problems ; in one or two cases they are rediscoveries, of the kind that add a certain illumination to the original. His outlook on algebra was that of his time ; one might characterise it by referring to the spirit and content of Perron's Algebra, as contrasted with the Moderne Algebra of van der Waerden. Bred in the Cayley-Sylvester tradition of matrix algebra and the invariant theory of forms, he was expert also in the manipulation of continued fractions and determinants. His lectures to honours classes in Edinburgh always included a course on matrices, vector analysis and invariants, historical reference being made to Grassmann, Cayley and Sylvester; it is to such a course that the writer owes his first acquaintance with matrix algebra. In later years, from about 1925, Whittaker acquired an adequate knowledge, though never a marked taste, for those parts of modern abstract algebra, in particular the representation of groups and algebras, that serve a purpose in mathematical physics; his courses of research lectures, which he kept up until a late period in his tenure of the chair, bore witness of this.

Longitudinal elastic waves in solid circular cylinders were first investigated by Pochhammer (1) and Chree (2) but they did not take into account the effects that thermal properties have on the propagation of these waves. In this paper we shall consider waves in solid and hollow cylinders as well as in the infinite medium with a cylindrical cavity, and in each case we shall take account of the thermoelastic effects.

Ten papers deal with the theory of relativity, and all are concerned in some way with electromagnetism. Those who know Whittaker through his Modern Analysis and Analytical Dynamics will recognise in these papers the same mastery over complicated situations which enabled him to disdain the support of notational refinements, that same elegance, brevity and persuasive charm which make difficult arguments seem easier than they really are. The new element which emerges is the strongly geometrical approach; but he remains true to the Lagrange tradition and draws no diagrams of space-time, although these must surely have been before his mind's eye and would have helped his readers.

Much of the work on the theory of diffraction by an infinite wedge has been for cases of harmonic time-dependence. Oberhettinger (1) obtained an expression for the Green's function of the wave equation in the two dimensional case of a line source of oscillating current parallel to the edge of a wedge with perfectly conducting walls. Solutions of the time-dependent wave equation have been obtained by Keller and Blank (2), Kay (3) and more recently by Turner (4) who considered the diffraction of a cylindrical pulse by a half plane.

A groupoid is a set closed with respect to a binary operation. It is commutative and entropic if xy = yx and xy.zw = xz.yw hold for all its elements. It is cyclic if it is generated by one element. Let x be the generator of the free commutative entropic cyclic groupoid . Then any element of can be written in the form xP where x1 = x and xQ+R = xQxR. Two indices P, Q are equal (called “concordant” in (3)) if and only if xP = xQ. The groupoid of these indices, the free additive commutative entropic logarithmetic (cf. (3)), is clearly isomorphic to .