The traditional method of solution to problems in linear viscoelasticity theory involves the direct application of the Laplace transform to the relevant field equations and boundary conditions. If the shape of the body under consideration or the type of boundary condition specified at a point or both vary with time then this method no longer works. In this paper we investigate the applicability of stress function solutions to this situation. It is shown that for time-dependent ablating regions a generalization of the Papkovich Neuber stress function solution of elasticity holds. As an example the stress and displacement fields are calculated for the problem of an infinite viscoelastic body with a spherical ablating stress free cavity and prescribed time-dependent stresses at infinity.

A formal method is developed for deriving a series expansion of the general term in Green-type expansions. The technique is exemplified by detailed calculations for modified Bessel functions of large order.

A necessary and sufficient condition is determined for the modularity of the lattice of congruences on a bisimple inverse semigroup whose semilattice of idempotents is order-anti-isomorphic to the set of natural numbers.

Using a technique due to Macbeath (Jack and Macbeath 1959) this paper gives what the author hopes is a shorter and easier presentation of the evaluation of certain Jacobians of matrix transformations which have occurred in statistics and the theory of quadratic forms.

The generating function for the number of linear partitions was found by Euler, the method being almost trivial. That for plane partitions is due to Macmahon, but, even in a simplified form found by Chaundy, the proof is far from trivial. The number of solid partitions of n, i.e. the number of solutions of

is denoted by r(n). It has often been conjectured that the generating function of r(n) is , but this is now known to be false. We write η(a, b, c) for the generating function of the number of solutions of (1) subject to the additional condition that

Macmahon 1916 found n(a, 1, 1) for general a. Here we find η(a, b, c) for general a, b. c.

The paper describes an investigation of the terminal velocity of uniformly dispersed particles of various shapes, sizes and densities falling through water.

It is concluded that for concentrations above 0–5 per cent by weight, the suspension as a whole behaves as though it were viscous even though the individual particles lie well outside the Stokes range. The shape of the particles has a significant effect only when the concentration is less than 0·5 per cent, and for concentrations between 0·5 and 7·0 per cent, the relative changes in velocity of descent are adequately described for a range of particle shapes from highly angular to spherical and for sizes at least up to 0·65 mm. nominal diameter, by the power series

in which U is the velocity of the suspension, U0 that of a single particle, d the nominal diameter (i.e. that of a sphere having the same volume) and s the mean spacing of the particles.

If the concentration is lower than 4 per cent, the equation may be assumed linear in (d/s) without serious error.

A “roughness parameter”, first used by the author in 1953 (Feather 1953 b) has been re-calculated for 348 points on the mass surface. Systematic features are identified in relation to the variation of this parameter with charge number (Z) and isotopie number (D). In the region of small Z these regularities provide evidence for the persistence of some degree of alpha-unit structure at least as far as Ca. In the region of greater Z (20 < Z < 50) they provide evidence for neutron-proton interactions among the last-added nucleons. Overall, they indicate that the “residuals” characterizing the various semi-empirical mass equations currently in use very probably arise in large part from sub-shell effects which it would be impracticable to attempt to include in the equations.

Until now the chief obstacle to the application of the maximum likelihood method of estimation to factor analysis has been the lack of any really good numerical method of solution. In this paper we give a brief review of recent work which remedies this defect. Two factor analysis models are considered. In each case we derive results which are of use in connection with new methods of solution. Formulae are given for the large-sample variances and covariances of the estimates of parameters in the first model.

Many results concerning real orthogonal matrices have their counterparts in the theory of orthogonal Boolean matrices. In particular, the analogues for Boolean matrices of certain theorems due to Kronecker are established. The structure of the group of orthogonal Boolean matrices of order n is determined in the case where the underlying Boolean algebra is finite.

The problem considered is that of obtaining solutions of the Helmholz equation ∇2V + k2V = 0, suitable for use in connection with paraboloidal co-ordinates. In these co-ordinates the Helmholz equation is separable, and each of the separated equations is reducible to Hill's equation with three terms (the Whittaker-Hill equation). The properties of solutions of this equation are developed sufficiently to make possible the formal solution of simple boundary-value problems for paraboloidal surfaces, principally for the case k2 < 0.

Throughout this paper we shall be concerned with n × n matrices X=[xij] whose elements xij belong to a given Boolean algebra B(≤, ∩, ∪, ′). In tne first part of the paper we show that the set S(λ, x) of matrices with a given eigenvector x and eigenvalue λ is a subgerbier (i.e.), γ-semi-reticulated sub-semigroup of Mn(B), the Boolean algebra of all n × n matrices over B. We also determine the greatest element M(λ, x) of S(λ, x) and consider some of its properties. In the second part of the paper we consider the structure of matrices which possess a given primitive eigenvector x and show in particular that, for any given λ ∈ B, there is a matrix, namely M(λ, x), having x as the maximum primitive eigenvector associated with the eigenvalue λ. We also determine necessary and sufficient conditions under which the matrix M(λ, x) is the same for all λ in any given sublattice of B.

In the case of Boolean matrices a given eigenvector may have a variety of eigenvalues. These eigenvalues form a sublattice of the basic Boolean algebra and the structure of this sublattice is investigated. Likewise a given eigenvalue has a variety of eigenvectors which form a module of the Boolean vector space. The structure of this module is examined. It is also shown that if a vector has a unique eigenvalue λ, then λ satisfies the characteristic equation of the matrix.

An investigation has been made of the condensation nuclei created by an electric field in a N2 – H2O mixture. These nuclei are distinguished by the fact that they induce condensation in a vapour which is only 4 per cent supersaturated.

An explanation of these phenomena is found in the presence of nitrogen dioxide vapour, one of the products of reactions induced by the field, exerting a small pressure PNO2<1O−6 mm. Hg.

The observations are consistent with the assumption that the nuclei are created in the reaction, 2NO2 + H2O ⇌ HNO2 + HNO3. It is believed that the reason for the requirement of a more than critically supersaturated vapour is that this must be the condition for the nuclei forming reaction to proceed.

Once the nuclei have been created, any additional quantity of NO2 collected from the vapour forms acid molecules which promote condensation from a vapour which is not necessarily supersaturated.

Drops formed by these nuclei contain a significant quantity of HNO2 + HNO3, so that, unlike drops of pure water, they are stable against reevaporation in a vapour, the relative humidity of which is <100 percent.

The distribution of stress in the neighbourhood of an infinite row of collinear cracks in an elastic body is calculated for general distributions of internal pressure. The case in which each of the cracks is opened out by the same constant pressure is considered in detail; the variation with the ratio of the width of the crack to the distance between two cracks of quantities of physical interest is shown in a series of diagrams.

A multiple integral, whose integrand is an n × n determinant, is evaluated over certain regions of n-dimensional space. Similar integrals are encountered in the theory of Zonal polynomials. In the course of the work a partition problem arises. In the next paper of these Proceedings, Professor Rutherford enumerates these partitions and relates the subject to the theory of the representation of the symmetric group.

Tensor fields and linear connections in an n-dimensional differentiable manifold M can be extended, in a natural way, to the tangent bundle T(M) of M to give tensor fields of the same type and linear connections in T(M) respectively. We call such extensions complete lifts to T(M) of tensor fields and linear connections in M.

On the other hand, when a vector field V is given in M, V determines a cross-section which is an n-dimensional submanifold in the 2n-dimensional tangent bundle T(M).

We study first the behaviour of complete lifts of tensor fields on such a cross-section. The complete lift of an almost complex structure being again an almost complex structure, we study especially properties of the cross-section as a submanifold in an almost complex manifold.

We also study properties of cross-sections with respect to the linear connection which is the complete lift of a linear connection in M and with respect to the linear connection induced by the latter on the cross-section. To quote a typical result: A necessary and sufficient condition for a cross-section to be totally geodesic is that the vector field V in M defining the cross-section in T(M) be an affine Killing vector field in M.

General formulae are derived from first principles for the temporal and spatial autocorrelation functions of stochastic parameters which are defined in terms of superposed, uncorrelated waves with known spectral density distribution. These formulae are first used for obtaining expressions for the autocorrelation functions of the components of the electromagnetic field strength and the electromagnetic energy density in black body radiation fields. The general theory is further applied to compression waves in liquids, and expressions are derived for the temporal and spatial autocorrelation of thermal density fluctuations in liquids, in particular near their critical point. Finally the spectrum of the fluctuations in the total radiation emitted by a thermal source, owing to the fluctuations in the energy supply to the source, is obtained from the appropriate Langevin equation, and the temporal autocorrelation function of the radiation intensity due to this cause is derived from the spectrum.

§ 1. The principal object of this note is to establish formula (16) of the preceding paper by H. Jack (1966). This formula, which was conjectured by Jack, evaluates a certain coefficient which is attached to a symbol {p, q, …, r}. In this symbols, p, q, …, r form a partition of m such that o≤p≤q≤…≤r,p+q+…+r=m. The symbol however vanishes if any two of the integers p, q, … r are equal but non-zero. In the remaining cases we have to show that the coefficient in question has the value

The types of modes which may exist in an infinite parallel-plate waveguide with a centrally-placed unidirectionally conducting screen are studied. The effect of bifurcating an infinite parallel-plate waveguide by such a screen in the region x>O is investigated when a transverse electric mode is incident on the bifurcated region. The problem is solved by the Wiener-Hopf method, and expressions are derived for the amplitudes and phases of the reflected and transmitted modes. It is found that the transmitted field contains a hybrid wave and a slow wave, and the reflected field contains transverse electric and transverse magnetic waves.

A matrix analysis is derived, using the concept of finite elements, for beam-columns of continuously varying cross-section and “small” initial curvature subject to tangential and normal forces acting at discrete points along the centroidal axis.

The relationships between loads applied to the ends of the member and their corresponding deformations are established in the form of stiffness matrices, which are the basis of the Equilibrium Method of Analysis of linear or non-linear elastic plane frames composed of such membeis. In addition, the end loads induced by the tangential and normal forces for various types of boundary conditions are obtained.

Finally, it is shown that the buckling load may be calculated by the determination of the lowest eigenvalue of a certain matrix, provided there is no lateial translation of the ends of the member.