We denote by tm(n) the number of partitions of the positive integer n into non-decreasing parts which are positive or zero powers of a fixed integer m > 1 and we call tm(n) ‘the m-ary partition function’. Mahler(1) obtained an asymptotic formula for tm(n), the first term of which is

Mahler's result was later improved by de Bruijn (2).

Our object here is to refine the theorem proved in (3), and we use the notation of that paper. Let Z1, Z2, …, Zn, where Zr = (Xr, Yr), be independent random variables in two dimensions with zero first-order moments and finite third-order moments; and et the covariance matrix of Zr be

We write

Let (x, y) be the distribution function of the sum and let (x, y) be the normal distribution function having the same first- and second-order moments as (x, y).

In (1) and (2), Gould has discussed some generalized Vandermonde-type convolution identities, for C coefficients satisfying

where

and

Geometric structure of classical field theory in Lagrangian formulation is investigated. Symmetry transformations with generators depending on higher-order derivatives are considered and the corresponding conservation laws are obtained.

When Rossby number is small but Ekman number is very much smaller, study of the flow field far from a body moving at right angles to the axis of rotation of a large body of fluid indicates that the region of influence should not be a Taylor column parallel to the axis, but a trailing Taylor column, bent backwards on both sides of the body at a small angle (proportional to Rossby number) to the axis. The paper reviews the physical significance of, and experimental evidence for, this conclusion.

The title problem is investigated with the view of clarifying the nature of the separation in an adverse pressure gradient when the suction υ0 is very strong. A linear model is investigated, but it is argued that this must contain many of the essential features of the exact (non-linear) problem. With a free-stream U = 1 − x the similarity solution u = (1 − x)f(y) is appropriate when (υ0x − y) ≫ 1, x = O(1), 1−x = O(1). There is an O(1) layer centred on y = υ0x in which more complex x-dependence first appears and then, when (y − υ0x) ≫ 1, the solution assumes its final asymptotic behaviour. The y = υ0x layer is non-uniform in x and exhibits a marked change in behaviour when (1 − x;) = O(l/υ0). The solution in this narrow rectangle is singular like (1 − x)−2 at the trailing edge. It is these singular terms that induce separation and reversed flow in an exponentially small neighbourhood of the trailing-edge.

This paper is concerned with the unsteady hydromagnetic boundary-layer flow past a semi-infinite flat plate when the oncoming free stream is perturbed by an arbitrary function of time and the applied magnetic field is parallel to the plate far away from it. Following Lighthill, the two-dimensional boundary-layer equations are separated into those representing steady and unsteady parts of the flow and they have been solved in sequence. For the unsteady part of the motion two types of solutions are obtained, one for large times and the other for small times. Also the quasi-steady solution is obtained in terms of the steady solution. The skin friction and the tangential magnetic field at the plate are calculated.

The axisymmetric potential problem for a plane circular electrode of radius a in a concentric hole of radius b in a coplanar earthed sheet is formulated in terms of triple integral equations for the Hankel transform of the potential, and reduced to a single Fredholm equation by use of the Erdélyi-Kober fractional operators.

In the limit of small gap width (b − a)/b, the equation takes the form

which is solved by applying the Wiener-Hopf technique to the Mellin transform of f(x). This leads to the asymptotic expression

for the capacity of the disc; for the opposite limit the expression

is derived. Numerical integration of the governing Fredholm equation has been carried out for a range of intermediate values of b/a.

It is shown that the solution of the Rikitake two-discdynamo system may be described by an orbiting point which, for sufficiently large time, lies arbitrarily close to a limit surface of bounded area. Reversal of the field currents of the dynamo coils are shown to occur in the juxtaposed regions of the two sheets of the limit surface. The two singular points of the system are shown, by Liapounov's direct method, to be unstable foci. An asymptotic theory is developed for the case of small difference in dynamo velocities; the theory is applied to cases of small dissipation in which orbits tend to become nearly periodic with a single oscillation between reversals.

The internal labelling problems for the group-subgroup combinations Sp (2n) ⊃ Sp (2n − 2) × Sp (2) and O(n) ⊃ O(n − 1) are solved by the method of elementary multiplets.