1. In deriving an expression for the number of representations of a sufficiently large integer N in the form

where k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the type

where ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sum

where α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the form

where s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

In (4) Young investigates the representation in terms of his exact seminormal units of substitutional expressions which are unchanged on premultiplication by any permutation of a given set of consecutive letters, or which are changed in sign on premultiplication by an odd permutation of those letters. He illustrates his results by deducing the forms of the matrices representing the positive and negative symmetric groups on a set of consecutive letters.

The generation of acoustic disturbances in a fluid of semi-infinite extent by the motion of a circular piston surrounded by a plane rigid baffle has been studied quite extensively (see (1), (2), (3), (4) and further references given in these papers). Attention has been devoted mainly to the case in which the piston executes a harmonic oscillation of small amplitude, and only comparatively recently has Oberhettinger (2) demonstrated how the time-harmonic solution can be used to solve the more general problem in which the normal velocity of the piston is an arbitrary function of time. The purpose of the present paper is two-fold. Firstly, we point out that for arbitrary normal motion of the piston the " baffled piston problem " can be solved directly, and in a particularly simple manner, by means of a technique involving integral transforms which has been applied by Mitra (5) and Eason (6) to the study of shear wave propagation in an elastic half-space. Secondly, we give a more detailed account than appears to be available in the literature of the structure of the sound pulse generated by the arbitrary normal motion of a baffled piston.

Mohanty (1) and (3) considered the absolute summability of conjugate series and Fourier series by Borel's integral method by proving the following theorems.

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

The non-associative algebras arising in genetics (1), are rather isolated from other branches of non-associative algebra (6). However, in a paper (5), in which he studied these algebras in terms of their transformation algebras, Schafer proved that the gametic and zygotic algebras for a single diploid locus are Jordan algebras.

Barrelled and quasibarrelled spaces form important classes of locally convex spaces. In (2), Husain considered a number of less restrictive notions, including infinitely barrelled spaces (these are the same as barrelled spaces), countably barrelled spaces and countably quasibarrelled spaces. A separated locally convex space E with dual E' is called countably barrelled (countably quasibarrelled) if every weakly bounded (strongly bounded) subset of E' which is the countable union of equicontinuous subsets of E' is itself equicontinuous. It is trivially true that every barrelled (quasibarrelled) space is countably barrelled (countably quasibarrelled) and a countably barrelled space is countably quasibarrelled. In this note we give examples which show that (i) a countably barrelled space need not be barrelled (or even quasibarrelled) and (ii) a countably quasibarrelled space need not be countably barrelled. A third example (iii)shows that the property of being countably barrelled (countably quasibarrelled) does not pass to closed linear subspaces.

A rigid spherical punch vibrates normally on the surface of a semi-infinite isotropic elastic half-space. The essential novelty of this problem, which is treated within the context of classical elasticity, is that of a changing boundary; the radius of the circle of contact on the free surface varies with time. The geometrical co-ordinates are modified to yield a boundary value problem with fixed boundaries. However the governing differential equations become more complicated. These equations are solved by a perturbation procedure for the case where the contact radius a(t) is of the form

where a0 is constant and |ŋ(t)≪1. Finally the normal stress and the total load under the punch are evaluated in the form of series which are valid for sufficiently slowly varying ŋ(t).

The generalised zeta-function ζ(s, α) is defined by

where α>0 and Res>l. Clearly, ζ(s, 1)=, where ζ(s) denotes the Riemann zeta-function. In this paper we consider a general class of Dirichlet series satisfying a functional equation similar to that of ζ(s). If ø(s) is such a series, we analogously define ø(s, α). We shall derive a representation for ø(s, α) which will be valid in the entire complex s-plane. From this representation we determine some simple properties of ø(s, α).

We consider the Dirichlet problem for Laplace's equation in a rectangle with a view toward determining the asymptotic behaviour of the solution for the case in which the width of the rectangle is small in comparison with its length. Although the construction of an explicit representation of the solution is an elementary matter, the resulting formula is inconvenient for present purposes, and we accordingly proceed along different lines.