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The projective invariants of four mediale
Authors: Turnbull H. W.
Published: January 2009

The theory of four particular linear forms, or matrices of k columns and 2k rows, occurred to me many years ago in an attempt to study the invariants of any number of compound linear forms, or subspaces within a space of n dimensions. In what follows, the invariant theory is given, and its significance for a study of the general matrix of k rows and columns is suggested. The collineation used in §4 was considered by Mr J. H. Grace, who emphasized the importance of the k cross ratios upon transversal lines of four [k−1]'s in [2k−1]. It seemed appropriate to examine these cross ratios which are irrational invariants μi, of the figure of four such spaces, and to work out their relation to the known rational invariants Xi. The main result is given in § 5 (7). In § 5 (10) it is shewn that the harmonic section of a line transversal of the four spaces exists when a linear relation holds between the invariants.
A type of periodicity of certain quartic surfaces
Authors: Edge W. L.
Published: January 2009

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.
An Integral Formula For Qn (cos θ)
Authors: Copson E. T.
Published: January 2009

The function

is, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).
A note on inverse central factorial series
Authors: Aitken A. C.
Published: January 2009

The purpose of this brief note is to draw attention to a type of inverse factorial series which, so far as the writer can judge, has not been intensively studied. The central difference formulae of interpolation and the corresponding infinite series in central factorial polynomials have in the past received much attention, and so also has the ordinary inverse factorial series

which from the time of Stirling (Methodus Differentialis, 1730) has been known to be of value in transforming slowly convergent series into series of more rapid convergence.
The expression of an infinite lower semi-matrix in terms of its idempotent and nilpotent elements
Authors: Gurr C. E. , Turnbull H. W.
Published: January 2009

The importance of matric algebras in Function Theory and in Physics (Birtwistle—The new Quantum Mechanics; and Courant and Hilbert—Methoden der mathematischen Physik) has resulted in comprehensive works on finite matrices. Very little progress has, however, been made in the necessary algebras of infinite matrices.
On the extension of Fermat's theorem to matrices of order n
Authors: Marshall J. B.
Published: January 2009

It is proposed to establish, by elementary methods, a theorem for matrices analogous to Fermat's Theorem in the Theory of Numbers. In Jordan's Traité des Substitutions (Paris, 1870) pp. 127, 128, the order of any given linear substitution or matrix A with reference to any prime number p is determined, but the result given depends on the particular characteristic equation satisfied by the matrix A, and a general result applicable to all matrices of n rows and n columns does not seem to have been published hitherto.
On γ-transformations of series
Authors: Nigam Tapeshwari Prasad
Published: January 2009

The theorems given here deal with the efficiency, mutual consistency, and regularity of γ-transformations.

§1. Theorem 1. The necessary and sufficient conditions that γ-matrices are efficient for all divergent series whose partial sums are bounded, are

where Δgk(a)≡gk(a)−gk+1(a).
On the highest space in which a non-ruled surface of given order can lie
Authors: Timms G.
Published: January 2009

It is well known that a non-ruled (i.e. not consisting of an infinity of lines) surface of order n lies in a space of not more than n dimensions n ≠ 4) and that for n > 9, the maximum dimension actually attained (here denoted by R) is certainly less than n.
On certain expansions involving Whittaker's M-Functions
Authors: Mitra S. C.
Published: January 2009

The object of the present note is to obtain expansions of the square of Whittaker's M-Functions in series of M-Functions and also other expansions involving M-Functions.
Note on certain related conditions in the theory of Cesaro summability
Authors: Hyslop J. M.
Published: January 2009

In a recent paper I discussed, for a given series Σan the relation between the conditions

where 0 < ρ < 1 and p is a positive integer. It was there proved (Theorem 14) that (2) implies (1) but that the converse is not necessarily true. My object in this note is to render more precise the connection between (1) and (2) by showing that (2) is equivalent to (1′), where (1′) is (1) with the additional restriction that Σn−1wn is convergent. The proof of this equivalence relation is based on a technique which was introduced and developed by Andersen and is similar to the proofs in my former paper. The note concludes with a brief discussion of the corresponding results for the case ρ = 1.
The application of γ-matrices to Taylor series
Authors: Vermes P.
Published: January 2009

In a recent paper some general properties of γ-matrices were proved and Dienes' theorem on regular γ-matrices extended to semiregular γ-matrices and the binomial series. In section 2 of this paper the previous results will be extended to certain classes of Taylor series. Section 3 gives some new results on Borel's exponential summation, and section 4 introduces matrices efficient for Taylor series on the circle of convergence and others efficient for Dirichlet series on the line of convergence. A knowledge of the definitions and results of the paper mentioned above is assumed.
Symmetric Determinants and the Cayley and Capelli Operator
Authors: Turnbull H. W.
Published: January 2009

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.
Logarithmetics of Finite Quasigroups (II)
Authors: Popova Helen
Published: January 2009

In the first paper of this series (L.Q.I)1 we have shown that the logarithmetic LQ of a finite quasigroup Q is a quasigroup with respect to addition and that it is a subdirect union of the logarithmetics of the elements of Q.

In this second part we shall discuss further the structure of LQ in its additive aspect, and obtain results concerning the order N of LQ. For plain quasigroups (§3) the structure of LQ(-\-) is studied in more detail and it is shown that N is a power of n, the order of Q.
An Application of Hankel Transforms in Axially Symmetric Potential Flow
Authors: Mackie A. G.
Published: January 2009

In his book on Hydrodynamics, Lamb obtained a solution for the potential flow of an incompressible fluid through a circular hole in a plane wall. More recently Sneddon (Fourier Transforms, New York, 1951) obtained Lamb's solution by an elegant application of Hankel transforms.

Since the streamlines in this solution are symmetric about the wall, it is not of particular physical interest. In this note, Sneddon's method is used to give a solution in which the fluid is infinite in extent on one side of the aperture but issues as a jet of finite diameter on the other side.
On a Certain Series of Abel
Authors: Clunie J.
Published: January 2009

If we express ezλ, qua function of λ, in terms of λeλ by Langrange's theorem we find, for |λ| sufficiently small,

If we now regard λ as a parameter and z as the independent variable then (1.1) can be written

Supposing, therefore, that a function is of the form then
Reflection of a Wave by a Cylindrical Mirror
Authors: Chambers Ll. G.
Published: January 2009

The question of the reflection of a wave by a cylindrical mirror is of interest in a number of fields. It is a problem in which it is difficult to obtain an expression for the reflected or scattered field without recourse to physical assumptions which are sometimes somewhat dubious. An attempt was made by Sommerfeld to solve the problem of a plane wave incident upon such a perfectly conducting mirror by means of what he termed the “Non-Final Determination of Coefficients”. Unfortunately, a close examination of the problem renders it doubtful whether the method can be legitimately employed. It is possible, however, to solve the problem by expressing the scattered field in terms of the currents produced in the mirror, and finding the current generated in the mirror by an arbitrary incident field. The problem which we shall consider is the following two- dimensional one.
Nörlund Methods of Summability Associated with Polynomials
Authors: Borwein D.
Published: January 2009

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
Special Train Algebras Arising in Genetics
Authors: Gonshor H.
Published: January 2009

Ever since Mendel promulgated his famous laws, probability theory and statistics have played an important role in the study of heredity (9). Etherington introduced some concepts of modern algebra when he showed how a nonassociative algebra can be made to correspond to a given genetic system (1, 4). The fact that many of these algebras have common properties has led to their study from a purely abstract point of view (2, 3, 5, 6, 11, 12). Furthermore, the techniques of algebra give new ways of attacking problems in genetics such as that of stability.
On Transient Thermal Stresses in an Infinite Thin Plate
Authors: Derski W.
Published: January 2009

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.
TheJ-Differential and its Integral
Authors: Ingram W. H.
Published: January 2009

We consider the possibility of generalising the statement

to the case
On the Solution of some Axisymmetric Boundary Value Problems by Means of Integral Equations. IV. The Electrostatic Potential due to a Spherical Cap between Two Infinite Conducting Planes
Authors: Collins W. D.
Published: January 2009

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 Solution of Certain Dual Integral Equations
Authors: Williams W. E.
Published: January 2009

A class of dual integral equations involving Bessel functions is solved by formal application of Mellin transforms.
Existence Theorem for Groups
Authors: Chehata C. G.
Published: January 2009

Given a partial automorphism of a group G, i.e. an isomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, it is known (2, Theorem I) that there always exists a group H containing G and an inner automorphism of H which extends µ; i.e. there exists an element t of H, such that the transform by t of any element of A is its image under µ.
A Note on the Kawada-into Theorem
Authors: Pym John S.
Published: January 2009

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),

defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).
Approximate Solutions of Boundary Value Problems in Potential Theory with Applications to Electrostatics and Elastostatics
Authors: Williams W. E.
Published: January 2009

A simple approximate formula is obtained for the capacity of the condenser formed by a “small” conductor placed inside a much larger one. The formula involves a constant whose choice is, to a certain extent, arbitrary and it is shown that, for problems involving spheroids inside cylinders and between parallel plates, the constant may be found in a simple fashion so as to give very accurate results. A similar formula is obtained for the loss in potential energy due to a crack or cavity in a circular beam or a thick plate. For the particular cases of the boundary value problems considered which have been treated by other means very close numerical agreement is obtained between those results and ones deduced in the present paper.
The Temperature Distribution in a Semi-Infinite Solid whose Surface is Maintained at an Arbitrary Temperature
Authors: Pugh H. LL. D.
Published: January 2009

1. The problem discussed is the temperature distribution in a semi-infinite solid, initially at zero, when the surface is suddenly raised to and maintained at a temperature which is an arbitrary function of the distance from some line which lies in that surface.
The weak-star closure of the unit ball in a hyperplane
Authors: Jameson G. J. O.
Published: January 2009

Let X be a normed linear space. We regard X as a subspace of its bidual X**. Polars will always be evaluated in the pair (X**, X*). We denote the closed unit ball in X by U, so that U0, U00 are the closed unit balls in X*, X** respectively. The weak topology induced by X on X* (the “weak-star” topology) will be denoted by σ(X), and cl() will denote σ(X)-closure.
Sequences in genetic algebras for overlapping generations
Authors: Heuch I.
Published: January 2009

When Etherington (2) introduced linear commutative non-associative algebras in connection with problems in theoretical genetics, he pointed out that various sequences of elements in these algebras represented different mating systems. In all such systems it was however assumed that the generations did not overlap, and this restriction has been kept in later work in this field. In this paper we treat sequences which make it possible to find the probability distribution in successive generations in a discrete time model where the generations may be overlapping. We also consider idempotents in genetic algebras and outline how the method used on the overlapping generation sequence may be applied to other sequences.
Some inequalities for a hypergeometric integral
Authors: Flett T. M.
Published: January 2009

In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric function

in the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.
Self-adjoint square roots of positive self-adjoint bounded linear operators
Authors: Stuart C. A.
Published: January 2009

A corollary of the main theorem presented in this note is a generalisation of the well-known result that a self-adjoint square root of a positive self-adjoint compact linear map in a Hilbert space is itself a compact linear map. The method used here exploits the techniques developed recently in the study of k-set contractions ((1), (2)).
Divisibility of ordered groups
Authors: Wright I. W.
Published: January 2009

In this paper it is shown that divisibility of a complete lattice ordered (abelian) group is closely related to the existence of a sufficient number of small elements in the positive cone.

We shall denote the set of all real numbers by R which symbol will be reserved for this purpose. All terms used are as defined in Birkhoff(1). For the reader's convenience we now define the two terms most used in the sequel.
Completely reducible near-rings
Authors: Oswald A.
Published: January 2009

To establish our notation N will always denote a (left) near-ring without any type of multiplicative identity (unless the contrary is stated) satisfying On = 0 for each n ∈ N where 0 is the additive identity of N. A group M, written additively, which admits N as a set of right multipliers is a (right) N-module if a ∈ M, n1, n2 ∈ N implies a(n1 + n2) = an1 + an2 and a(n1n2) = (an1)n2. When N has a two-sided identity, 1, we suppose that a 1 = a for each a ∈ M. A subgroup X of M is an N-subgroup of M if it is an N-module; X is a submodule of M if it is a normal subgroup of M and a ∈ M, x ∈ X. n ∈ N implies (a + x)n – an ∈ X. We denote by SL(M) the set of N-subgroups and by L(M) the set of submodules of M. Since N may be regarded as an N-module we can talk about N-subgroups and submodules of N although we usually call the submodules of N right ideals of N. Other definitions can be found in (6).
New sets of equi-isoclinic n-planes from old
Authors: Hoggar S. G.
Published: January 2009

Two n-planes Γ and Δ in real Euclidean r-space Rr are called isoclinic with parameter λ if the angle θ between any x in Γ and its orthogonal projection Px on Δ is unique, with cos2 θ = λ. Let vλ(n, r) denote the maximum number of equi-isoclinic (i.e. pairwise isoclinic) n-planes in Rr with parameter λ.
Linear operators whose domain is locally convex
Authors: Kalton N. J.
Published: January 2009

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on S separates the points of S. If X is a Banach space and T : X → F is a continuous linear operator, then T is quasi-convex if is quasi-convex, where U is the unit ball of X.
General hereditary for radical theory
Authors: Rossa R. F. , Tangeman R. L.
Published: January 2009

LetWbe a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong toW. The notationI⊲Rwill meanIis an ideal ofR. A relation σ onWwill be called anH-relation if σ satisfies the properties:

(1)IσRimpliesIis a subring ofR.

(2) IfIσRand ø is a homomorphism ofR, thenIØ σRø.

(3) IfIσRandJis an ideal of R, thenI∩JσJ.
Strongly E-reflexive inverse semigroups
Authors: O'Carroll L.
Published: January 2009

LetSbe an inverse semigroup with semilattice of idempotentsE. We denote by σ the minimum group congruence onS(6), and by τ the maximum idempotent-determined congruence onS(2). (Recall that the congruence η onSis calledidempotent-determinedif (e,x)∈ η ande∈Eimply thatx∈E.) In general τ ⊆ σ.
Standard subgroups of GL2(A)
Authors: Mason A. W.
Published: January 2009

Let R be a commutative ring and let q be an ideal in R. Let En(R) be the subgroup of GLn(R) generated by the elementary matrices and let En(q) be the normal subgroup of En(R) generated by the q-elenientary matrices. The order of a subgroup S of GLn(R) is the ideal q0 in R generated by xij, xii−xjj, where (xij)∈S, with 1≦i, j≦n and i≠j. The subgroup S is called a standard subgroup if En(q0)≦S. An almost-normal subgroup of GLn(R) is a non-normal subgroup which is normalized by En(R).
Extensions, crossed modules and internal categories in categories of groups with operations
Authors: Porter T.
Published: January 2009

In [6] Brown and Spencer noted that internal categories within the category of groups are equivalent to crossed modules. As they remarked, this result was known to various others before them, but it had not until then appeared in print. That paper led me to investigate the question of which algebraic categories, C, were such that a similar result held i.e. internal categories in C are equivalent to crossed modules of the appropriate type. The resulting work was written up in 1980 but was not submitted for publication.
Distributive elements in centralizer near-rings
Authors: Maxson C. J. , Meldrum J. D. P.
Published: January 2009

Let  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.
Principal congruences in de Morgan algebras
Authors: Adams M. E.
Published: January 2009

A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.
Locally nilpotent skew linear groups II
Authors: Wehrfritz B. A. F.
Published: January 2009

Our paper [6] studied in some depth certain locally nilpotent skew linear groups, but our conclusions there left some obvious gaps. By means of a trick, which now seems obvious, but then did not, we are able to tidy up the situation very satisfactorily. This present paper should be viewed as a follow up to [6]. In particular we do not repeat the motivation, basic definitions and references to related work given here.

The following was conjectured in [6], where substantial steps were taken towards its solution.
The osculating of a certain curve in [4]
Authors: Edge W. L.
Published: January 2009

The equation of the osculating plane at a point on the complete irreducible curve of intersection of two algebraic surfaces in [3] was found by Hesse (5, p. 283); the plane, having to contain the tangent of the curve, belongs to the pencil spanned by the tangent planes of the two surfaces, and it is a question of determining which plane of the pencil to choose. The equation also appears in the books of Salmon (6, p. 378) and Baker (1, p. 206). The analogous problem for the osculating solid at a point on the complete irreducible curve of intersection of three algebraic primals, or threefolds, in [4] does not appear to have been considered. The simplest instance is the octavic curve C of intersection of three quadrics, and this has the special interest of being a canonical curve; moreover the quadrics are of the same order, and so can be replaced by any three linearly independent members of the net which they determine, a replacement of which it may be prudent to take advantage with a view to simplifying the algebra. It is a question of determining which solid to choose among the tangent solids to the quadrics of the net at a point on C, but while Hesse's methods serve to carry one a certain distance there seems no obvious way of pushing them to a conclusion. It is then natural, with a view to reaching a conclusion, to choose a net of quadrics that, through having some particular property, is more amenable.
Duplication of linear algebras
Authors: Etherington I. M. H.
Published: January 2009

The process of duplication of a linear algebra was defined in an earlier paper, where its occurrence in the symbolism of genetics was pointed out. The definition will now be repeated with an amplification. Although for purpose of illustration it is applied to the algebra of complex numbers, duplication will seem of no special significance if attention is fixed on algebras with associative multiplication and unique division; for duplication generally destroys these properties. The results to be proved, however, show that it is significant in connection with various other conceptions which appeared in the discussion of genetic algebras; namely baric algebras and train algebras (defined in G.A.), also nilpotent algebras, linear transformation and direct multiplication of algebras.
Bessel-integral functions
Authors: Humbert Pierre
Published: January 2009

In a very remarkable work on the operational Calculus, Dr Balth. van der Pol has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine- or sine-integral with respect to the ordinary cosine or sine. He showed that this function—which he called Bessel-integral junction—can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order.
The Boltzmann Equations in General Relativity
Authors: Walker A. G.
Published: January 2009

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.
A Geometrical Interpretation of the Symmetrical Invariant of Three Ternary Quadratics
Authors: Scott T.
Published: January 2009

In the paper, “Sul sistema di tre forme ternarie quadratiche,” Ciamberlini has derived the complete irreducible system of concomitants for three ternary quadratics and has given a short treatment of their geometrical interpretations. Among the concomitants is the invariant (abc)2 which is symmetrical and linear in the coefficients of each quadratic. The purpose of this note is to give a geometrical interpretation of the invariant, and to extend the result for symmetrical invariants of forms in higher dimensions.
On the Riesz-Riemann-Liouville Integral
Authors: Copson E. T.
Published: January 2009

In a lecture at the Oslo Congress in 1936, Marcel Riesz introduced an important generalisation of the Riemann-Liouville integral of fractional order. Riesz's integral Iaf of order α is a multiple integral in m variables which converges uniformly when the real part of αexceeds m —2 and so represents an analytic function of the complex variable α. This integral is important in the theory of the generalised wave equation, for it provides a direct method of solving Cauchy's initial-value problem. The most recent developments show that it is likely to be also of great importance in quantum electrodynamics.
On Keller's conjecture for certain cyclic groups
Authors: Sands A. D.
Published: January 2009

Keller (6) considered a generalisation of a problem of Minkowski (7) concerning the filling ofRnby congruent cubes. Hajós (4) reduced Minkowski's conjecture to a problem concerning the factorization of finite abelian groups and then solved this problem. In a similar manner Hajós (5) reduced Keller's conjecture to a problem in the factorization of finite abelian groups, but this problem remains unsolved, in general. It occurs also as Problem 80 in Fuchs (3). Seitz (10) has obtained a solution for cyclic groups of prime power order. In this paper we present a solution for cyclic groups whose order is the product of two prime powers.
Evaluation of Permanents
Authors: Minc Henryk
Published: January 2009

The permanent of an m x n matrix A = (aij),m≤n, is defined by

where the summation is over all one-to-one functions σ from {1, … ,m} to { 1, …,n}. In other words, the permanent of A is the sum of all thediagonal productsofA, that is, all the products ofmentries ofAno two of which lie in the same row or in the same column. Thus the permanent ofAmay be evaluated by first multiplying all the row sums ofAand then subtracting from the product all terms that contain as factors two or more entries from the same column ofA. This is the idea behind the formulas of Binet and of Ryser.
On quasinormal subgroups of certain finitely generated groups
Authors: Lennox John C.
Published: January 2009

A subgroup Q of a group G is called quasinormal in G if Q permutes with every subgroup of G. Of course a quasinormal subgroup Q of a group G may be very far from normal. In fact, examples of Iwasawa show (for a convenient reference see [8]) that we may have Q core-free and the normal closure QG of Q in G equal to G so that Q is not even subnormal in G. We note also that the core of Q in G, QG, is of infinite index in QG in this example. If G is finitely generated then any quasinormal subgroup Q is subnormal in G [8] and although Q is not necessarily normal in G we have that |QG:Q| is finite and |QG:Q| is a nilpotent group of finite exponent [5].
Automorphism groups of laminated near-rings determined by complex polynomials
Authors: Magill K. D. , Misra P. R. , Tewari U. B.
Published: January 2009

The concept of a laminated near-ring was introduced in [2]. We recall briefly what it is. Let N be a near-ring and let a∈N. Define a new multiplication on N by x * y = xay for all x,y∈N. With this new multiplication and the same addition as before we have another near-ring which we denote by Na. The near-ring Na is referred to as a laminated near-ring, the original near-ring N is the base near-ring and a is the laminator or laminating element.
Lattices whose ideal lattice is Stone
Authors: Beazer R.
Published: January 2009

An elementary fact about ideal lattices of bounded distributive lattices is that they belong to the equational class ℬω of all distributive p-algebras (distributive lattices with pseudocomplementation). The lattice of equational subclasses of ℬω is known to be a chain

of type (ω+l, where ℬ0 is the class of Boolean algebras and ℬ1 is the class of Stone algebras. G. Grätzer in his book [7] asks after a characterisation of those bounded distributive lattices whose ideal lattice belongs to ℬ (n≧1). The answer to the problem for the case n = 0 is well known: the ideal lattice of a bounded lattice L is Boolean if and only if L is a finite Boolean algebra. D. Thomas [10] recently solved the problem for the case n = 1 utilising the order-topological duality theory for bounded distributive lattices and in [5] W. Bowen obtained another proof of Thomas's result via a construction of the dual space of the ideal lattice of a bounded distributive lattice from its dual space. In this paper we give a short, purely algebraic proof of Thomas's result and deduce from it necessary and sufficient conditions for the ideal lattice of a bounded distributive lattice to be a relative Stone algebra.
The natural partial order on an abundant semigroup
Authors: Lawson Mark V.
Published: January 2009

In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].
Divisible S-systems and R-modules
Authors: Gould Victoria
Published: January 2009

Throughout this paper S will denote a given monoid and R a given ring with unity. A set A is a right S-system if there is a map φ:A × S→A satisfying

and

for any element a of A and any elements s, t of S. For φ(a, s) we write as and we refer to right S-systems simply as S-systems. One has the obvious definitions of an S-subsystem, an S-homomorphism and a congruence on an S-system. The reader ispresumed to be familiar with the basic definitions concerning right R-modules over R. As with S-systems we will refer to right R-modules just as R-modules.
Explicit expressions for Sturm-Liouville operator problems
Authors: Jodar Lucas
Published: January 2009

Throughout this paper H will denote a complex separable Hilbert space and L(H) denotes the algebra of all bounded linear operators on H. If T lies in L(H), its spectrum σ(T) is the set of all complex numbers z such zI–T is not invertible in L(H) and its compression spectrum σcomp(T) is the set of all complex numbers z such that the range (zI-T)(H) is not dense in H ([3, p. 240]). This paper is concerned with the Sturm–Liouville operator problem

where λ is a complex parameter and X(t), Q, Ei, Fi for i = l,2, and t∈[0,a], are bounded operators in L(H). For the scalar case, the classical Sturm-Liouville theory yields a complete solution of the problem, see [4], and [7]. For the finite-dimensional case, second order operator differential equations are important in the theory of damped oscillatory systems and vibrational systems ([2, 6]). Infinite-dimensional differential equations occur frequently in the theory of stochastic processes, the degradation of polymers, infinite ladder network theory in engineering [1, 17], denumerable Markov chains, and moment problems [10, 20]. Sturm-Liouville operator problems have been studied by several authors and with several techniques ([12, 13, 14, 15, 16]).
Real Linear Substitutions with Equimodular Multipliers, and their expression in terms of their Invariants
Authors: Taylor D. G.
Published: January 2009

§1. Let

denote a linear substitution of non-vanishing determinant; and let the roots k∈ of its characteristic equation

be for the present assumed distinct. Then with each root k∈ is associated an invariant point or poleP∈, and a linear invariant, or invariant (n−2)-plane ξ∈. If the n points P∈ do not lie on an (n−2)-plane, the determinant of their coordinates,
Note on the Peano-Baker Method of solving Linear Differential Equations
Authors: Milne Arch
Published: January 2009

In Mathematische Annalen, Vol. 32 (1888) Peano discusses the solution of a system of homogeneous linear differential equations

where rij denotes a real function of the variable t, and shows how, by a series of repeated substitutions, this system of equations may be replaced by the equivalent equation

where X denotes the complex [x1, x2, … xn] and R the matrix

of which equation the solution X can be represented as a sum of integrals.
On a Group of Parabolas associated with the Triangle
Authors: Philp George
Published: January 2009

§1. The general theorem underlying the subject of this paper is as follows:—

If through c1, c2, c3, … points on AB, a side of ΔABC, rays be drawn from two vertices O1, O2, the former meeting AC in b1, b2b3, …, and the latter meeting BC in a1, a2, a3, …, then the lines a1b2, a2b2, … envelope a conic touching the sides AC and BC. This follows since the ranges a1a2a3 …, b1b2b3 …, are homographic.
Transformations of Axes for Whittaker's Solution of Laplace's Equation
Authors: Jeffery G. B.
Published: January 2009

Whittaker has shown that a general solution of Laplace's equation, ∇2V = 0, may be expressed in the form

Since the harmonic property of a function is in no way dependent upon any particular set of axes it follows that the same solution must be capable of being expressed in the form

where X, Y, Z are any second set of rectangular coordinates.
The Effect of a Rise in Prices upon the Amount of Small Money used
Authors: Ford Lester R.
Published: January 2009

At the close of the preceding meeting of the Society a discussion arose concerning the effect of a uniform rise in prices upon the amount of small money necessary for the transaction of business. It was clear that the total amount of money must be increased, but in the case of small money, which is used only when fractional parts of the larger unit are involved, the effect was less obvious. It is the fact, however, that more small money required. The present paper attempts an explanation of that fact.
On certain Integrals
Authors: Tait
Published: January 2009

This paper was based mainly on the results of an investigation which will appear in full in the Transactions of the Royal Society of Edinburgh. Incidentally, however, it led to a discussion of the question:—Find the law of density of a planet's atmosphere, supposing Boyle's law to be true for all pressures, and the temperature to be uniform throughout.
Solution of a Problem proposed by Dr Muir
Authors: Allardice R. E.
Published: January 2009

The problem is to show that the expression

where ω is an imaginary fourth root of unity, is symmetrical with respect to a, b, and c.
The value of cos2π/17 expressed in quadratic radicals.
Authors: Steggall
Published: January 2009

Call 2π/17 α, and let

Then from trigonometry

and

Now let
Amsler's Planimeter
Authors: Steggall
Published: January 2009

There are many proofs of the principle of this planimeter, but all that are accessible to me seem a little beyond the grasp of many students who use the instrument. It seems worth while, therefore, to notice the following proof, which, to the best of my knowledge, is new.
An Elementary Discussion of the Closeness of the Approximation in Stirling's Theorem
Authors: Chrystal
Published: January 2009

[The substance of this paper will appear in the second volume of Professor Chrystal's Algebra.]
The triangle and its escribed parabolas
Authors: Pressland A. J.
Published: January 2009

The problem “to inflect a straight line between two sides of a triangle so that the intercepted portion is equal to the segments cut off” has been discussed in the third volume of the Proceedings.

If we discuss the same analytically; taking CB and CA as axes of x and y (Fig. 1) and calling each segment k, the equation of the line considered is
Note on the preceding Locus
Authors: Pressland A. J.
Published: January 2009

The perpendicular from A on BC = AB·AC/2R, which is constant.
On the Use of Dimensional Equations in Physics
Authors: Peddie William
Published: January 2009

Though every quantity, whatever be its nature, has magnitude, no quantity can be said to be large or small absolutely. When we speak of the size of any body we mean its size relatively to the size of some other body with which we compare it. A yard is large if we compare it with an inch; it is small when compared with a mile. In the former case the number which represents it is more than 60,000 times larger than the number by which it is represented in the latter case. A mere number is therefore useless as regards the statement of magnitude, except when accompanied by a clear indication of what the thing measured is compared with. The quantity in terms of which the comparison is made is called the unit, and the number which tells how often this unit is contained in a given quantity is called the numeric of that quantity.
On a Representation of Elliptic Integrals by Curvilinear Arcs
Authors: M'Cowan John
Published: January 2009

It is well known that the elliptic integral of the second kind may be represented by the arc of an ellipse, and mathematicians have sought with various success to represent similarly by the arc of an algebraic curve the elliptic integral of the first kind. The general solution of the problem has not been obtained, but Serret and Cayley have given solutions of a very general character.
The Wallace line and the Wallace point
Authors: Mackay J. S.
Published: January 2009

In what follows I propose to give the history of two theorenis and to state some of the consequences that have been developed from them.

The first theorem is:

If a triangle be inscribed in a circle, and from any point in the circumference perpendiculars be drawn to the sides, the feet of these perpendiculars lie in a straight line.
The Barycentric Calculus of Möbius
Authors: Allardice R.E.
Published: January 2009

August Ferdinand Möbius was born at Schulpforta, in Saxony, in the year 1790. He studied in the Universities of Leipsic and Göttingen, and, at the age of 2G, was appointed extraordinary Professor of Astronomy and Superintendent of the Observatory at Leipsic. There he remained till his death, in 1868, being appointed ordinary professor in 1844. Between the years 1817 and 1868 Möbius wrote his Barycentric Calculus, a Treatise on Statics, another on the Mechanics of the Heavens, and a large number of papers on Mathematical, Dynamical, and Astronomical questions. Most of these papers were contributed to Crelle's Journal, which was founded in 1826. The works of Mobius have recently been collected under the direction of the Royal Scientific Society of Leipsic, and under the editorship of Klein, Scheibner, and Baltzer.
The Plane Triangle ABC: Intimoscribed Circles, etc.
Authors: Anderson R.E.
Published: January 2009

§ I. On an infinite series of Triad Circles derived from the inscribed circle. Determination of a direct relation between r and the three radii of the nth triad.

Each of the first triad touches two sides of ABC and the inscribed circle : generally, each circle of the mth triad touches two sides of ABC and also touches one of the circles of the (m – 1)th triad.
Note on a property of a quadrilateral
Authors: Steggall J.E.A.
Published: January 2009

The property is that if any quadrilateral, ABCD, skew or otherwise, have its sides AB, DC divided in E, F so that

AE : EB = DF : FC = AD : BO

then the direction of the line EF shall bisect the angle between the directions of BC, AD.

This extension of Euclid's VI. 3 follows immediately from the proposition that if AE : EB = DF : FC, then all such lines as EF are parallel to one plane, namely, the plane parallel to BC, AD; and that they each cut similar lines drawn with reference to BC, AD.
On a Nine-Point Conic, etc.
Authors: Pinkerton P.
Published: January 2009

1. The Pole and Polar Theorem for the Circle may be proved as follows:—

Let P be any point, and C the centre of a circle.

Let AB be the diameter through P, and QQ′ any chord passing through P.

Let AQ, BQ′ meet in M; AQ′, BQ in O; MO, AB in N; and MO, QQ′ in P′.
On the relations of certain conics to a triangle
Authors: Fraser W. G.
Published: January 2009

If we join the angular points ABC of a triangle to any point O the locus of the centres of conies passing through A, B, C, O is a conic bisecting the six joins of the four points and passing through the intersections of OA, BC : OB, AC; and OC, AB. This conic is analogous to the nine-point circle, and at last meeting of the Society Mr Pinkerton showed that its centre lies on the line joining O to the centroid. In what follows an attempt is made still further to generalise this conception.
On the Pascal Hexagram
Authors: Jack J.
Published: January 2009

ABCDEF is any cyclic hexagon;

AB, DE meet in G,

BC, EF „ „ H,

CD, FA „ „ K; then G, H, K are in a straight line.

Draw KLNM parallel to BC and produce DE, EF, AB to meet it in I, N, M. Join DA, DM and let BC, DE meet in P.
Quantitative proofs of certain Algebraic Inequalities
Authors: Dougall John
Published: January 2009

1. By a quantitative proof of an inequality I mean one which exhibits the difference between the two magnitudes compared in a form which shows at a glance whether the difference is positive or negative. Such a proof not merely establishes the existence of the inequality, but also gives a measure of its amount.
A Property of the Harmonic Triangle of the Complete Quadrangle formed by the four points of contact of the four tangents to the Cubic Curve from a point on it
Authors: Milne William P.
Published: January 2009

The following two propositions are necessary for what is to follow, and are very easy to prove.

I. If A, B, C be three points such that B and C are conjugate with respect to the Polar Conic of A, then are C and A conjugate points with respect to the Polar Conic of B, and similarly with regard to the third vertex.

A triangle possessing the property defined above, viz. that each pair of vertices are conjugate points with respect to the Polar Conic of the third vertex, is called an Apolar Triangle.
Note on Professor Allardice's Paper “On the Locus of the Foci of a System of Similar Conics through three Points.” [Proceedings, Vol. XXVII]
Authors: Sanjana K. J.
Published: January 2009

The fundamental equation in this paper,

,

can be derived, easily and without using trilinears, from the following property of conics.
Note on Professor Naraniengar's Paper in the Proceedings of 1909–10
Authors: Teixeira F. Gomes
Published: January 2009

Permit me the liberty of presenting some remarks on an article by Professor Naraniengar inserted in Vol. XXVIII., p. 73, of the Proceedings of the Edinburgh Mathematical Society, 1909–1910.
The Focal and Bi-Tangent Properties of Bi-Circular Quartics
Authors: Milne William P.
Published: January 2009

The following direct method of attack establishes the fundamental properties of the Foci and Bi-Tangents of a Bi-Circular Quartic very simply, using only the properties of a homographic correspondence. The method was suggested to me by my previous paper on the focal properties of Circular Cubics.

I am indebted to Mr Peter Fraser, University of Bristol, for his criticism.
An infinite series of triangles and conics with a common pole and polar
Authors: Donaldson J. A.
Published: January 2009

An infinite family of triangles, having a common pole (determined by three fixed lines through it), and polar (determined by three fixed points on it), and an allied family of conics with imaginary double contact.

Construction for pole of a line with reference to a triangle.