Certain exact tests were developed by Williams (1952) to deal with the goodness of fit of a single hypothetical discriminant function. Bartlett (1951) generalized these results by the use of the geometric method to any number of dependent and independent variables. Bartlett's paper is divided into two parts. The first deals with an approximate factorization of the residual likelihood criterion into an effect due to the difference between the hypothetical and sample functions, and an effect due to non-collinearity. A method is given for constructing confidence intervals from the first factor. The second part of the paper gives two possible exact factorizations of the likelihood criterion, expressing the results in terms of the sample canonical variables. Kshirsagar (1964a) has expressed these results in terms of the original variables and given an analytic proof of the distribution of the factors. Williams (1955, 1961) has outlined a generalization of these results to several discriminant functions and given the result for one of the possible factorizations.

In a previous paper ((l)) the authors gave the solution for the two-dimensional circular inclusion problem in a medium containing a circular cavity. This paper seeks to solve the more general problem of a similar inclusion when the cavity is replaced by an inhomogeneity which could be of a different elastic material. The solution consists in finding three sets of suitable complex potential functions ø(z) and ψ(z) for three regions: the inhomogeneity, the inclusion and the rest of the material. The solution depends upon the evaluation of the complex potentials for a material containing the inhomogeneity when on the former a finite force is acting at some fixed point. It may be noted that two sets of ø(z) and ψ(z) have to be found in this case: one for the inhomogeneity and the other for the rest of the material. This may be taken as an auxiliary problem.

In this note we determine a lower bound for the number of different configurations of monomers and dimers on the hypercubical lattice. This combinatorial problem has important applications in statistical mechanics ((1)) and solid state chemistry.

The possibility of instability arising as overstability in the magnetohydrodynamic Bénard problem is discussed. Results are derived for large Hartmann numbers when the regions outside the fluid layer are conducting or insulating, the interfaces being rigid or free.

If we neglect the viscous dissipation, the temperature distribution in an incompressible fluid flowing in the annulus between two coaxial cylinders with radii r1 and r2 (> r1) is determined by the equation

Initial conditions

Boundary conditions

where ∂Q/∂t is a function of time alone representing the rate of heat generation per unit volume per unit time in the fluid.

Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).

In this paper a method is proposed for solving certain half-plane elliptic boundary-value problems involving mixed boundary conditions. The equation considered is a generalization of the Tricomi equation which contains the space form of the damped wave equation as a special case. Existing methods depend on the use of Fourier integrals and lead to the solution of integral equations. The methods employed here are direct and yield explicit solution formulas without the necessity of solving integral equations and as such avoid the arguments inherent in the use of the Wiener-Hopf technique.

A system of equations is derived for determining the elastic fields in an inclusion and its surrounding finite matrix when the inclusion suffers a physical change and, if not constrained by the matrix, would undergo a deformation . A method for obtaining the exact solution of these equations, when the matrix and inclusion have the same elastic constants, is described and the particular problem of the square inclusion in an infinite matrix solved.

In the usual notation for generalized hypergeometric functions we let

where

and (a) denotes the sequence of parameters

Throughout the present paper we shall suppose that there are A of the a parameters, Bof the b parameters, and so on. Thus ((a))m is to be interpreted as

and similar interpretations hold for ((b))m, etc.

The unsteady two-dimensional compressible boundary layers have been studied by Sarma ((5)) assuming that the Prandtl number is unity and that the wall is in an arbitrary motion, the main stream being steady. In this paper the Prandtl number is taken to be an arbitrary parameter which need not be equal to unity, and it is assumed that the main stream velocity and temperature are perturbing about a steady mean, the wall being in an arbitrary motion. Solutions are obtained in two parts, one when the temperature gradient at the wall is perturbing about a zero mean and the other when the temperature of the wall is perturbing about a steady mean. Thus in this paper the theory given in Sarma ((5)) is made still more general. Following the work of Sarma ((4)–(6)) two types of solutions are developed in each part, one for large times and the other for small times.

Consider the two-dimensional steady flow of an incompressible inviscid fluid under no body forces between two infinite horizontal planes. When an aperture is made in the lower plane, the fluid will issue as a jet bounded by free streamlines along which the pressure is constant. At infinity the flow in the jet will be uniform and parallel.

Recently in these proceedings (see (5), p. 175) by applying Goldstein's form of Parseval's relation ((4), p. 105) to the operational pairs ((3), p. 284)

where R[(b ± c)2] > 0 and ((3), p. 293)

where R(½ − λ + μ) > 0, it was proved that if

then

provided R(½ − λ + μ + ν) > 0 and R[(b ± c)2] > R(a), 0.

A theory is presented for small deformations of a plate which is initially stressed, the state of initial stress not necessarily being obtained by elastic deformation from some other unstressed state. The method used is an asymptotic integration of the full three-dimensional equations. The usual Kirchhoff boundary conditions for stretching and bending are deduced from the analysis.

Complex variable methods are applied to derive exact solutions in closed forms for the small deflexions of uniformly loaded thin isotropic plates bounded by regular curvilinear polygonal contours having n sides. The supported boundary is either clamped or has equal boundary cross-couples. For rectilinear edges the latter conditions agree with those for a simply supported boundary. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the general mapping function

Two different calculations are made of lower bounds for the electronic energy of . In the first the method of truncated Hamiltonians due to Bazley and Fox is adapted in such a way that the nuclear charge rather than the energy becomes the eigenvalue. Lower bounds are calculated for the energies of the six lowest σg and six lowest σu states, as well as of the three lowest of both πg and πu symmetries. This approach gives better convergence than when the energy is used as eigenvalue. In the second calculation the method of Temple and Kato is shown to give a satisfactory value for the energy of the ground state, provided that some necessary knowledge of the energy of the first-excited state is available.

The two-dimensional incompressible fluid flow problem along an infinite flat plate with no heat transfer between the fluid and the plate is discussed when the suction velocity normal to the plate, as well as the external flow velocity, varies periodically with time. Expressions for the velocity and temperature fields in the boundary layer have been obtained in a non-dimensional form. It has been found that the effect of increasing the suction velocity fluctuations is that the amplitude of the skin-friction fluctuations is increased while the phase is decreased and it may be negative for very small values of the frequency. Also the plate temperature decreases with increasing the suction velocity fluctuations for small values of the frequency while there is no effect for very large frequencies.

It has been shown that uncoupled surface waves of SH type can be propagated without any dispersion in an electrically conducting semi-infinite elastic medium provided a uniform magnetic field acts non-aligned to the direction of wave propagation. In general, the velocity of propagation will be slightly greater than that of plane shear waves in the medium.

Formulae for the mean radius and mean square radius of a relativistic Coulomb wave function are obtained. Screening constants for the energy, mean radius and mean square radius are defined relative to non-relativistic wave functions and the results of numerical calculations given. It is shown that if the screening constants so determined are added to the screening constants due to the presence of other electrons as found by the s.c.f. method, good agreement is obtained in a case where both effects have been considered together. The value of solving the relativistic wave equation in a Thomas-Fermi field is also shown.