This note is a sequel to the article by Mine (2) on the same problem.

I described in (1) a notation for indices of powers in non-associative algebra, defined the degree † and altitude of a power or index, and observed that powers can be represented by bifurcating root-trees. For example, the power xx.x is denoted x2 + 1, with index 2 + 1, and is represented by the tree ; the degree (the number of factors, or free knots in the tree) is 3, and the altitude (the height of the tree) is 2. Multiplication being non-commutative or commutative, one maintains or ignores the distinction between left and right in the tree.

We consider the following problem: A potential function φ satisfies Laplace's equation ∇2φ = φxx + φyy = 0 in a region R bounded by a closed curve C on which mixed boundary conditions are specified, i.e. φ = f(s) on a part A of the boundary and ∂φ/∂n = g(s) on a part B, where C = A + B and distance along C is denoted by s. Electrostatic problems of this type have been solved approximately in (1) and (2) by formulating them in terms of integral equations and then applying variational principles to the integral equations. In that approach, attention is concentrated on integrals over the boundary of the region R. The most common type of variational principle for potential problems involves integrals over the region R rather than integrals over the boundary of R. An example is given by the Rayleigh-Ritz method which depends on the stationary character of Dirichlet's integral

In this paper we show that the variational principles used in (1), (2), are closely connected with the more usual type of variational principles, by deriving the principles used in (1), (2) from inequalities deduced by considering integrals of type (1) over the region R.

It is known that the behaviour of an electromagnetic field is consistent with it possessing a linear momentum density (1, p. 103)

(Girogi Units are used and the system of notation is the usual one).

The presence of a non-steady state of temperature in an elastic solid gives rise to an additional term in the generalised Hooke's Law connecting the stress and strain tensors and terms involving the time rate of change of the dilatation. This time-dependent dilatation may produce so-called thermoelastic stress waves. The present note is concerned with the effects produced by these additional terms in a simple situation, in which the elastic solid is regarded as a thin plate of infinite extent. The distribution of temperature in the plate is produced by a point heat source of Dirac type.

In (1), § 6.2, a multiplying factor method has been used to solve certain dual integral equations. The results are then used to solve a single integral equation of the Wiener-Hopf type. In this note we indicate how a related technique can be used to solve Wiener-Hopf integral equations directly. Consider

where

Define

where α = σ+iτ, and F+(α) is regular for τ>q; K(α) is regular and non-zero in −p < τ < p. For simplicity we restrict ourselves to the case where

In this paper we consider a function f(x) defined by

All quantities are taken to be real, it is assumed that R is a function of the variable x, b is a constant, N and G are functions of the variable t and all the functions are such that the integral (1) exists when x is large enough. We wish to find an asymptotic representation of f(x) as x → + ∞, assuming that we are given certain information about the limiting behaviours of the functions R, N and G.

Let s, sn(n = 0, 1, …) be arbitrary complex numbers, and let

be a polynomial, with complex coefficients, which satisfies the normalizing condition

Associated with such a polynomial is a Nörlund method of summability Np: the sequence {sn} is said to be Np-convergent to s, and we write sn →s (Np), if

When an elastic solid is subjected to a dynamical system of surface or body forces, not all of the work done by these forces is employed in deforming the material. The remainder is converted into heat energy producing a distribution of temperature throughout the body. Similarly the application of a surface temperature distribution, or the introduction of heat sources within the body, produces elastic as well as thermal effects. Thus we see that in the dynamical case there is a link between these two types of condition—thermal and elastic.

It is known that there is a connexion between the functional equation of the theta-function

and the reciprocity formula for Gauss sums. The latter formula is usually written as

where p and q are positive integers, mutually prime, and not both odd.

A complete system of combinantal forms (generalised and ordinary) of a pencil of quadrics fλ ≡ λ1f1 + λ2f2 can be chosen such that the coefficients of the various power products of λ1, λ2 in the former give a complete irreducible system of concomitants of the two quadrics f1, f2, and conversely. This result was proved by Todd (1), who used it in conjunction with Schur function analysis (2) (3) to derive the complete irreducible system of concomitants of two quaternary quadratics (4).

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.

We consider the possibility of generalising the statement

to the case

The differential operation known as Lie derivation was introduced by W. Slebodzinski in 1931, and since then it has been used by numerous investigators in applications in pure and applied mathematics and also in physics. A recent monograph by Kentaro Yano (2) devoted to the theory and application of Lie derivatives gives some idea of the wide range of its uses. However, in this monograph, as indeed in other treatments of the subject, the Lie derivative of a tensor field is defined by means of a formula involving partial derivatives of the given tensor field. It is then proved that the Lie derivative is a differential invariant, i.e. it is independent of a transformation from one allowable coordinate system to another. Sometimes some geometrical motivation is given in explanation of the formula, but this is seldom very satisfying.

1. Guinand (2) has obtained finite identities of the type

where m, n, N are positive integers and either

or

where γ is Euler's constant and the notation ∑′ indicates that when x is integral the term r = x is multiplied by ½. Clearly there is no loss of generality in taking N = 1 in (1.1).

This paper is a sequel to previous papers (1, 2, 3) on the solution of axisymmetric potential problems for circular disks and spherical caps by means of integral equations and applies the methods developed in these papers to the electrostatic potential problem for a perfectly conducting thin spherical cap or circular disk between two infinite earthed conducting planes.

The present account is an application of the principles of combinantal forms and Schur function analysis given in a previous paper (A), the references therein being henceforward denoted by A1 to A9, and the complete irreducible system of invariants of three quaternary quadrics will now be obtained from the complete system (not necessarily itself irreducible) derived by Turnbull (A5, p. 483). This latter system comprises 47 invariants, viz. 15, 1, 6, 6, 1, 15 and 3 members of total degrees 4, 6, 8, 10, 12, 14 and 18 respectively in the coefficients of the quadrics. It will be proved that all of these are irreducible except for the one of degree 12 and the three of degree 18, the former being of especial interest as it is a real combinant and moreover, involves unusual features in the proof of its reduction and also in the derivation of the form expressing it in terms of irreducible invariants.

For any positive integer n and positive real variables x1, x2, …, xn write

where xr is defined for all integers r by the relations

If

and

it is known that (1, 3, 4)

and

By a theorem of Perron, a non-negative irreducible (n × n) matrix A = (aμν) has a positive fundamental root σ, the “ maximal root of A ”, such that the moduli of all other eigenvalues of A do not exceed σ. If we put

σ lies between R and r. Since σ is not changed if A is transformed by a positive diagonal matrix D(p1,…pn, σ lies also between the expressions

By a theorem of Frobenius, to σ as an eigenvalue of A belongs a positive eigenvector ξ, = (x1 …, xn), satisfying

A multi-partite number of orders j is a j dimensional vector, the components of which are non-negative rational integers. A partition of (n1, n2, …, nj) is a solution of the vector equation

in multi-partite numbers other than (0, 0, …, 0). Two partitions, which differ only in the order of the multi-partite numbers on the left-hand side of (1), are regarded as identical. We denote by P1(n1, …, nj) the number of different partitions of (n1 …, nj) and by p2(n1, …, nj) the number of those partitions in which no part has a zero component. Also, we write P3(n1, …, nj) for the number of partitions of (n1; …, nj) into different parts and p4(n1, …, nj) for the number of partitions into different parts none of which has a zero component.

In the case of a wave function with spherical symmetry, the wave equation can be separated using spherical polar coordinates, and the equation for the radial component becomes

where λ is a constant parameter, proportional to the energy of the particle under consideration, q(r) is proportional to the potential energy, and l is a positive integer or zero.