An account is given of the seiche motion in a curved lake. It is shown that the periods of the first two modes of oscillation are effectively unchanged in comparison with those for a straight lake having a length equal to that of the median line along the curved lake. Near to the nodal positions, the surface elevation is uniform across the lake. In the neighbourhood of the anti-nodes, which includes the ends of the lake, the wave amplitude is shown to be a maximum at the outside of the bend.

Let GF(q) be the finite field containing q = pl elements, where p is a prime and l a positive integer. Let P(x) be a monic polynomial in GF[q, x] of degree m. In this paper we investigate the nature and distribution of monic irreducible polynomials of the following types:

(I) P(xr), where r is a positive integer (r-polynomials).

(II) xm P(x + x−1). (Reciprocal polynomials.) These have the form

(III) xrmP(xr + x−r). (r-reciprocal polynomials.) These have the form Q(xr), where q(x) satisfies (1·1).

Many mathematical problems which do not yield a closed-form solution admit of a solution in the form of a power series; differential equations are an obvious example. The direct use of this power series is limited to the interior of its circle of convergence, and this places a restriction—often a severe restriction—on its usefulness. The method described in this paper enables this restriction to be alleviated in many cases; it also enables the convergence of a power series within its circle of convergence to be improved. The method is based on the Euler transformation.

The linear fluid dynamics is considered when infinite vertical boundaries are set in oscillatory vertical motion. The case of exponential stratification with constant kinematic viscosity is explicitly studied. When the forcing frequency equals the Brunt–Vaisälä frequency for the fluid, the customary boundary layers are absent in the steady-state oscillation, however small be the kinematic viscosity; for a semi-infinite fluid the corresponding horizontal extent of the region influenced by the boundary motion is then of the order of the stratification length. The sign of the phase angle is everywhere dependent on whether the magnitude of the forcing frequency is greater than or less than that of the Brunt–Vaisäla frequency.

In an earner paper ((6), p. 209) the author has given a generating function for the Jacobi polynomials defined by ((5), p. 254)

In the present note we obtain another generating function for these polynomials. The results given earlier by Manocha and Sharma ((4), p. 432) in these proceedings and Brafman in (2) follow as a particular case of our result.

In this paper we have employed Gauss's hypergeometric function, Legendre function and Jacobi polynomial to obtain the solutions of a fundamental differential equation of certain types of forced symmetrical vibrations of a circular elastic plate of finite radius.

The general solution of Chaplygin's equation for the motion of an inviscid compressible fluid is obtained for ‘simple wedge flows’. Some particular examples are given and a discrepancy in earlier Russian papers explained. After a discussion of the properties of the velocity potential and the expression for the physical coordinates in terms of the hodograph variables, a general theorem on sonic jets is proved, namely, that the physical changes due to the presence of solid boundaries in the flow are completed within a finite distance in those directions in which the flow becomes finally a sonic jet.

The paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable system is not identically equal to a unit matrix multiplied by a scalar. The result subsumes that of Lehnigk(1).

The general solution of the Chaplygin equation for ‘simple wedge flows’ is used for the particular case of Réthy flows. The drag coefficient is evaluated and comparisons made with the results obtained from the approximate hodograph equations of Tricomi and of Tomotika and Tamada. A distinction is made between subsonic and transonic régimes and the limitations on the lengths and pressure differences permissible in the various ranges are discussed.

A method is given for deriving refractive index profiles which yield wave functions expressed in terms of the standard transcendental functions for fields of electric type in spherically stratified isotropic media. Some new profiles, as well as all the previously known ones, are derived by applying the method.

Let be a given infinite series with the sequence of partial sums {sn}. Then the sequence-to-sequence Hausdorff transformation of the sequence {sn} is given by

The equations describing transverse propagation of electromagnetic waves in a cylindrically stratified plasma, in which a static magnetic field is imposed parallel to the axis of cylindrical symmetry, are examined. A systematic method developed previously for isotropic media, Heading (3), Westcott (6) is adapted to yield generalized proffles relating to the plasma properties. These proffles are capable of producing exact solutions for the field distribution in terms of standard transcendental functions. Previously known proffles appear as special cases of the analysis.

This paper is concerned with the problem of the flow of an incompressible electrically conducting fluid along a rectangular duet under a transverse magnetic field. The case in which the walls perpendicular to the field are perfectly conducting and those parallel to the field are non-conducting has been considered by Hunt (1), who derives the solution in two ways; as the limiting cases of the flows with (a) non-conducting walls parallel and thin walls of arbitrary conductivity perpendicular to the field, and (b) thin walls of arbitrary conductivity parallel and perfectly conducting walls perpendicular to the field. We show that these two limiting solutions derived by Hunt are in fact equivalent. In addition, we extend the solution of case (b) above by removing the thin wall restriction.

In a recent paper Al-Salam(1) has denned a fractional q-integral operator by the basic integral

(1) Where α ≠ 0, −1, −2, …. Using the series definition of the basic integrals, (1·1) is written as

valid for all α

We denote by b(n) the number of ways of expressing the positive integer n as the sum of powers of 2 and we call b(n) ‘the binary partition function’. This function has been studied by Euler (1), Tanturri (2–4), Mahler (5), de Bruijn(6) and Pennington (7). Euler and Tanturri were primarily concerned with deriving formulae for the precise calculation of b(n), whereas Mahler deduced an asymptotic formula for log b(n) from his analysis of the functions satisfying a certain class of functional equations. De Bruijn and Pennington extended Mahler's work and obtained more precise results.

Riley (1) has considered a magnetohydrodyriamic Stokes flow in which the modifications to the well-known Stokes flow past a fixed sphere were investigated for the case in which there is a magnetic pole situated at its centre. Again, Barthel and Lykoudis (2) and Ludford and Murray (3) have considered the low Reynolds number flow and inviscid flow respectively, past a sphere in the presence of a magnetic field due to a dipole situated at the centre of the sphere and directed parallel to the uniform stream. For the viscous problem, the velocity field and induced magnetic field were calculated for small values of the Hartmann number M and an expression found for the total drag on the sphere. Tamada and Sone (5) have also considered the same problem for a viscous conducting fluid at small Reynolds numbers and calculated the solution for small values of M while an asymptotic solution was found for large values of M. For small values of M they show that, since the Lorentz force retards the fluid motion, the viscous forces are reduced and hence the drag on the sphere is also reduced. However, the magnetic stresses contribute a force on the boundary which actually results in a net increase for the total drag on the sphere.

The NUT metric is interpreted as the field of (a) a mass around the origin of coordinates, and (b) a semi-infinite massless source of angular momentum.

In a recent paper (1) it has been established that a spherically symmetric space-time of class one cannot describe an electromagnetic distribution of energy. In this investigation the inconsistency of class one conditions with Rainich algebraic conditions was shown in case of spherical symmetry. In the present paper it is shown that the result is true for space-time of class one, in general.

New methods are presented for the evaluation of lattice Green's functions for the body-centred cubic lattice.

A systematic analysis is conducted for refractive index profiles capable of yielding exact solutions in terms of hypergeometric or confluent hypergeometric functions for the electromagnetic field of vertically polarized waves propagating in horizontally stratified isotropic media. Previously known profiles emerge as special cases of the analysis. Profiles suitable for use with isotropic lossless ionospheres and of prescribed form n2 = 1 − T(z)/k2, with T(z) independent of k, do not arise in this paper.