The equations of motion of a biased bowl rolling on a plane horizontal grass green are derived by Lagrange's method, using an appropriate representation of the effect of the green. The equations are solved approximately under general initial conditions which include the presence of a perturbing wobble.

An expression is derived for the total angle of precession of the bowl, and the equation of the path is found in parametric form. The theory predicts that (contrary to universal belief) the effect of an initial wobble on the bowl path is negligible under normal playing conditions.

A detailed comparison is made between the predictions of the theory and experimental results obtained on several greens of different speeds.

Much is known about power series with integral exponents, and, as in the case of the Theta-functions, about series whose exponents are the squares of the natural numbers. About other types of power series little is known. I prove here that a certain power series whose exponents are the square roots of the natural numbers is a symmetrical Fourier kernel.

A genetical population in which different genotypes have different viabilities will undergo, by selection, changes in its genetical constitution which can be expressed in a simple mathematical form. We here consider a system of m alleles A1, A2, …, Am segregating at one locus, and we assume that the genotype AiAj has viability aij, where aij = aji ≥ 0. We assume discrete, non-overlapping generations and random mating.

In a previous contribution to these Proceedings (Ramakrishnan(1)) the concept of product density was introduced to describe the statistical distribution of a discrete number of particles in a continuous space E, corresponding to a single point t, where t is the parameter with respect to which the stochastic process evolves. This is extended to densities corresponding to n points on the t axis and correlation problems associated with these density functions are studied with particular reference to electron-photon cascades.

Recently Ragab ((3), p. 538) evaluated an integral involving the product of an associated Legendre function and an E-function. In this note we derive an integral which is more general than the one given by Ragab.

A long column of viscous liquid of radius a and uniform density p is rotating about its axis with angular velocity ω. It is shown that this motion is stable to plane perturbations of wave number 8 provided that the surface tension T satisfies . This critical value is higher than that required for stability of the similar motion of a nonviscous liquid, but is otherwise independent of the coefficient of viscosity. The rate of development of instability when T is less than the critical value is also studied. Some numerical results are given.

The condition has recently been obtained by Hocking (2) for the special cases of very high and very low Reynolds numbers.

The current controversy over the interpretation of radio-stellar counts shows, in my view, that in large-scale physics as well as on the very small scale the idea of size, as applied to distributions of matter, has broken down. I shall expand this view.

A central limit theorem is proved for the estimates of the regression coefficients in a multiple system of regressions when the vector process generating the residuals is a linear process with coefficient matrices the sum of whose norms converges. The regressor variables are assumed to satisfy conditions, due to Grenander, which make their generalized harmonic analysis possible.

Several recent papers (Bastin and Kilmister(1),(2); Newman(3),(4)) have made the role of algebraic structures in physics the subject of detailed discussion. The general purpose of this work has not been to discover new physical laws or predict new results, but rather to give the theoretical physicist some further insight into the significance of familiar concepts. This is, of course, being done in the hope that it will lead to new refinements in our theoretical techniques.

The hypothesis of the existence of abstract structures in physics, which has been made in recent papers, is made precise. Difficulties in previous formulations are removed and insight into Maxwell electrodynamics is obtained. General relativity is related to other field theories which deal with flux from sources, and new field equations for general relativity are proposed.

A method employed by the author for a class of boundary-value problems arising in diffraction theory is applied to the problem of the oscillations of water waves on a sloping beach. The two cases of long and short waves are considered. The approach is valid for all values of the parameters and particular cases do not have to be treated separately. It is shown that the present approach enables certain solutions to be obtained which cannot arise in earlier solutions of the problem.

Formulae for Bessel derivatives with respect to order have generally been obtained by differentiating integral forms of the Bessel functions. In this note an asymptotic expansion will be derived from the differential equation satisfied by the functions, and it will be obtained in a form which terminates when the order is half an odd integer. Previous discussion of these functions has been restricted to the order one half. (See, for example, Ansell and Fisher (1) and Oberhettinger(2).)

In using the Boltzmann equation to calculate the transport properties of thin metallic films, it is usually assumed that a time of relaxation exists for the scattering in the film which is the same as for the bulk metal, although this is only strictly justified in the ideal case of isotropic scattering. The correct Boltzmann equation in the case of elastic scattering, when the cross-section depends on the angle of scatter, is set up and possible methods of solution are discussed.

The equation is solved for a simple anisotropic scattering law and the electrical resistivity and thermo-electric power are found to depend upon two parameters which are measures respectively of the thickness of the film and the anisotropy of the scattering mechanism. The numerical results for the electrical resistivity, however, differ only slightly from those given by the one-parameter formula obtained when a time of relaxation is assumed to exist.

It is well known that a uniform spherical shell produces the same gravitational field outside itself as that of a concentric uniform solid sphere with the same total mass. A given external field may therefore be produced by a variety of mass systems even under the restriction that the bodies of the system should have a prescribed uniform density. It can be asked what further conditions on the system are sufficient to ensure its uniqueness. The problem is of interest in connexion with the geological interpretation of local anomalies in the earth's gravitational field and work has already been done on it by several authors, (3), (6), (7), (8), (9). The results proved so far have all been restricted to systems consisting of a single body and the purpose of the present paper is to extend one of these results to include more general mass systems. The example of the spherical shell, mentioned above, might suggest that the absence of cavities in the bodies is a sufficient additional condition to ensure uniqueness. This is not so however. Celmiņš (3) gives an example of two different mass systems which have the same external field, each system consisting of three bodies of the same uniform density without any cavities in them.

Several writers have dealt with the use of Chebyshev polynomials as aids to interpolation. The present paper considers the theory with associated Legendre polynomials replacing those of Chebyshev. Interpolation formulae of both Bessel and Everett type are derived, and formulae and expansions are given for evaluating the coefficients required in them either directly or in terms of derivatives or central differences. The introduction summarizes a number of formulae for polynomial interpolation and indicates the relationship between the new formulae and the old.

Complex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours with n sides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function . Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.

Copson(1) has shown that the problem of determining the surface density of electric charge induced on a thin circular disk maintained at a constant potential in an external electrostatic field can be solved by two applications of the known solution of Abel's integral equation. This note shows that Copson's method can be extended to determine the surface charge density induced on the disk when situated inside an earthed coaxial infinitely long hollow cylinder. Whilst this problem is not solved in closed form, it is shown to be governed by a Fredholm integral equation of the second kind, which can be solved approximately by iteration when the radius of the cylinder is large compared with that of the disk.

The Jacobi identity for Whitehead products,

where α ɛ πl(X), β ɛ πm(M), γ ɛ πn (X), l, m, n ≥ 2 is now well known. In this note we extend the identity to the case l, m, n > 1; the extension is easy and we are merely filling a small gap in the literature. However, the extension does enable us to improve a result in (2).

The breakdown of stability in a porous homogeneous medium of a fluid subject to a vertical temperature gradient was examined by Lapwood (4). However, in natural conditions the fluids move generally in inhomogeneous media and then it is important to deal with the problem of stability in these media; the purpose of this paper is to consider just such a problem.

In genetics it is of considerable interest to be able to calculate the probabilities that a mutant gene will either be lost to the population or that its allele will be lost, since, in general, if no further mutation occurs, one of these two events must ultimately take place in any finite population. A good deal of research has been done on this problem by Fisher (l) and Kimura (3) and the purpose of this paper is to extend their results to more general situations and to discuss the validity of their approximations.