This paper contains

(a) an explicit solution of the problem of finding a function which is harmonic within a given sphere and takes at the surface the same value as a given rational integral homogeneous function of the rectangular coordinates of a point referred to the centre of the sphere as origin;

(b) a concise symbolical expression for the integral, over the surface of the sphere, of the product of any three rational integral spherical harmonics.

Recently Widder (3) has obtained inversion integrals for a convolution transform whose kernel is a Laguerre polynomial. Buschman (1) has considered convolution equations withgeneralised Laguerre polynomial kernel. His result includes that of Widder. However no attempt has so far been made towards the study of singular integral equations involving Laguerre polynomials as kernel. Here the author obtains an inversion integral for such an equation.

It is a very convenient method to begin the study of Trigonometry with one or two lessons on Coordinates and the Coordinate Diagram. This leads to a general definition of the Sine and Cosine of any angle which the beginner has little difficulty in comprehending. With the help of the Coordinate Diagram the theorems on projection which are of so much importance can be proved with great precision. It is in such a course that the present proof of the Addition Theorems might find a place.

The problem to be considered may be stated as follows:—How many necklaces may be formed with p pearls, r rubies, and d diamonds?

Let <G,+> be a group with identity 0 and let S be a semigroup of endomorphisms of G. The set Ms(G)={f:G→G; f(0)=0, fσ=σf, for all σ∈S} with the operations of unction addition and composition is a zero-symmetric near-ring with identity called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings have been studied for many classes of semigroups of endomorphisms. (See [8] and the references given there.) In this paper we continue these investigations into the structure of centralizer near-rings via our study of the relationship between distributive elements in Ms(G) and endomorphisms in Ms(G). More specifically, let N = Ms(G) and let Nd={f∈N; f(g1+g2)=fg1+fg2}, the set of distributive elements in N. Under the operation of function composition, Nd is a semigroup containing the identity map, id. Moreover, Nd contains as a submonoid = {α ∈ End G; ασ=σα for all σ∈S}. Here we determine for certain semigroups S, whether or not = Nd.

§1. The following three illustrations of cyclant substitutions are supplementary to those given in § 5 of a previous paper.

In the cyclant substitution in three variables with equimodular multipliers

in which the coefficients are real and satisfy the relations

the following expressions, as may at once be verified, are absolute invariants:

We calculate K0 of the rational group algebra of a certain crystallographic group, showing that it contains an element of order 2. We show that this element is the Euler class, and use our calculation to produce a whole family of groups with Euler class of order 2.

The number of self-complementary (s.c.) graphs and digraphs with a given number of vertices was found by R. C. Read in [1]. That paper used a special case of De Bruijn's generalisation of Polya's theorem that involved the cycle-index of Gn, the group of permutations of pairs of vertices induced by permutations of the vertices. We obtain Read's formulae by using only well-known elementary facts about s.c. graphs and their complementing permutations.

In [1], Calderón proved that, if u is a harmonic function on Rn × ]0, ∞[, and at each point ξ of a subset E of Rn, u is bounded in some cone with vertex (ξ, 0), then u has a nontangential limit at almost every point of E × {0}. The main result of this note is a stronger version of this theorem, in which the hypotheses remain unchanged but the nontangential limits in the conclusion are replaced by limits through the more general approach regions first considered by Nagel and Stein in [7].

The first part of the following investigation was begun before the discovery that Mr E. Kasner had already touched upon the apolarity theory of double binary forms in an important work on the Inversion Group (Transactions of the American Mathematical Society, Vol. I (1900), pp. 471–473). The theory is carried further in what follows, with special reference to the (2, 2) form. The second part answers questions raised by Professor A. R. Forsyth in the Quarterly Journal, 1910, p. 113. It appears that the general (2, 2) form admits of three independent automorphic transformations, but the general (n, n) form admits of none, if n exceeds two.

Using the Eichler-Shimura isomorphism and the action of the Hecke operator T2 on period polynomials, we shall give a simple and new proof of the following result (implicitly contained in the literature): let f be a normalized Hecke eigenform of weight k with respect to the full modular group with eigenvalues λp under the usual Hecke operators Tp (p a prime). Let Kf be the field generated over Q by the λp for all p. Let p be a prime of Kf lying above 5. Then

The differential equation of Mathieu, or “equation of the elliptic cylinder functions,”

occurs in many physical and astronomical problems. From the general theory of linear differential equations, we learn that its solution is of the type

where A and B denote arbitrary constants, μ is a constant depending on the constants a and q of the differential equation, and φ(z) and ψ(z) are periodic functions of z. For certain values of a and q the constant μ vanishes, and the solution y is then a purely periodic function of z; but in general μ is different from zero.

In his thesis, A. A. Hussein Omar, motivated by the study of possible shapes of generic Dirichlet regions for a surface group, made a detailed study for g = 2,3 of the groups generated by pairs (μ, τ) of regular (i.e. fixed-point-free) permutations of order 2,3 respectively and of degree n = 6(2g − 1), such that μ ْ τ is an n-cycle. He observed that, for g = 2,3, precisely one pair generates what he calls a superimprimitive group, and raised the question whether such pairs exist for all g, and, if so, whether they areunique. Our main result is that they do always exist, but that, for large values of g, theyare far from unique. (For details and some motivation for the notation, see [4, 5].)

Take first the case of a surface S rolling on a plane, the instantaneous axis of rotation being a line in the tangent plane at the point of contact. Take that line as x-axis, and the normal as z-axis, and let the equation to the surface be

Then if p is small, the equation of the section of the surface S by the plane z = p is

Now for rolling of this sort it is clear that the successive axes of rotation during a short time will be generators parallel to OX of a cylindrical surface which will touch S in the neighbourhood of O.

Given a poset (X, ≦), the covering poset (C(X), ≦) consists of the set C(X) of covering pairs, that is, pairs (a, b)∈X2 with a<b such that there is no c∈X with a<c<b, partially ordered by (a, b)≦(a′, b′) if and only if (a, b) = (a′, b′) or b≦a′. There is a natural homomorphism v from the automorphism group of (X, ≦) into the automorphism group of (C(X), ≦). It is shown that given groups G, H and a homomorphism α from G into H there exists a poset (X, ≦) and isomorphisms φψ from G onto Aut(X, ≦), respectively from H onto Aut(C(X), ≦) such that φv = αψ. It is also shown that every group is isomorphic to the automorphism group of a poset all of whose maximal chains are isomorphic to the nationals.