The object of this paper is to prove theorems of the following type.

THEOREM 1. If p and q are restricted to odd primes, anddenotes the Legendre symbol, then for x > 2

It is well known that a first approximation to the flow of a viscous liquid in the neighbourhood of a boundary with a corner may be obtained by solving the linearized equations of momentum neglecting the inertia terms. This has been done by Dean and Montagnon [1] for the plane steady flow near a corner formed by two inclined planes. The following is a similar analysis of the modes by which a liquid can form a free surface starting at the edge of a rigid boundary. The formulation differs from that of Dean and Montagnon mainly in that the angle at which the free surface is formed is in the first place unspecified and has to be determined from the analysis.

It was proved by Heilbronn that if ε > 0, N > 1 and ϑ is any real number then there exists an integer n satisfying 1 ≤ n ≤ N such that

where C depends only on ε. Here ║α║ denotes the difference between α and the nearest integer, taken positively. Professor Heilbronn has remarked (in conversation) that the exponent of N cannot be decreased beyond -1, since if p is an odd prime and a is not divisible by p then

for 1 ≤ n ≤ p—1. He has also remarked that if one could improve the exponent of N to -1 + η, say, it would follow that the absolutely least quadratic non-residue (mod p) is less than Cpn. For if a is a quadratic non-residue (mod p) then so is each of the numbers an2 (1 ≤ n ≤ p—1) and ║an2/p║<Cp-1+η implies that an2 is congruent (mod p) to a number of absolute value less than Cpη.

Let a1, …, am and b1, … bm be non-negative real numbers. The well-known inequality of Minkowski states that

if n ≥ 1. If n is a positive integer, this inequality asserts a property of a particular symmetric form (i.e. homogeneous polynomial) in m variables, namely the sum of the n-th powers of the variables. Some time ago, Prof. A. C. Aitken conjectured that similar properties are possessed by certain other symmetric forms. In particular, let E(n)(a) denote the n-th elementary symmetric function of a1, …, am and let C(n)(a) denote the n-th complete symmetric function of a1, …, am, the formal definitions being

Then Prof. Aitken conjectured that

It was proved recently by Roth that if α is any real algebraic number, and κ > 2, then the inequality

has only a finite number of solutions in integers h and q, where q > 0 and (h, q) = 1. This remarkable result answered finally a question which had been only partially answered by the work of Thue and Siegel.

Dr. Erdös has proved the following:

THEOREM. Given any increasing sequence a1, a2, … of positive integers, it is possible to define another increasing sequence, every term of which is representable as ai+aj, and such that none of its terms is divisible by any other.

Let K be a field. We denote by K[t] the integral domain of all polynomials in an indeterminate t with coefficients in K, and by K(t) the quotient field of K[t], i.e. the field of all formal rational functions of t over K. A valuation |f| of the elements f of K(t) can be defined by

for f ≠ 0, and |0| = 0, where e > 1. This valuation is multiplicative, and has the properties

In the solution of many problems in applied mathematics it is often convenient to have expansions for functions, F(r), which satisfy the boundary conditions

In some recent work on hydrodynamic and hydromagnetic stability the author has found certain types of expansions for such functions which have proved very useful and which appear to be novel in this connection. In this note two types of such expansions will be considered and the principles underlying them will be described.

The problem of the isotropic elastic sphere under the application of equal and opposite point couples ±N at its poles has been treated by M. Sadowsky [1] and by A. Huber [2]. The object of this note is to obtain their results by elementary means.

The problem is an example of the torsion of a shaft of varying circular section, the surface of the shaft being obtained by rotating a curve ξ = α about the z-axis.

It was proved by Roth in a recent paper that if α is any real algebraic number, and if K > 2, then the inequality

has only a finite number of solutions in relatively prime integers p, q (q > 0) The object of the present paper is to prove that the lower bound for κ can be reduced if conditions are imposed on p and q. The result obtained is as follows.

It is our purpose here to show that, using results already in the literature, it is easy to prove the following and similar theorems.

For every positive integer d, there exists an integer Ψ (d) such that if K is an algebraic number field of degree d over the field of rational numbers then every cubic form f(x1 x2, …, xn) over K, with n ≥ Ψ(d), has a non-trivial zero in K.

In the present paper a simple technique will be developed for the arithmetical determination of certain class group components and class number factors in finite number fields. This technique is based on classical theories (Hilbert's work on inertia groups, the theory of absolutely Abelian fields as class fields of congruence groups, absolute class fields of number fields). In keeping with the traditional approach to the subject we shall use here the language of ideal theory. The only non-classical concepts to be used (which, however, are of fundamental importance) are those of the inertia groups and the congruence groups associated to p-adic fields. We shall also give some illustrations of the use of our technique in some special cases. Further applications will follow in subsequent papers.

The indentation produced by an axially symmetrical punch bearing on the plane surface of an elastic half-space has been considered by Harding and Sneddon [1], who used Hankel transforms and a well-known pair of dual integral equations, and for the case of a spherical punch they took the indenting surface to be part of the approximating paraboloid of revolution. Chong [2], also using these dual integral equations has treated the case of a symmetrical punch of polynomial form and considers a two-termed expansion for a spherical punch. More recently, Payne [3] has given the exact solution for a spherical punch using either oblate spheroidal coordinates or toroidal coordinates.

Let X be a locally compact space, C(X) the algebra (with point-wise operations) of continuous numerical functions on X. On C(X) we introduce the topology of compact convergence. If f ε C(X), Zf denotes the set of zeros of f; and if I is a subset of C(X), we define

It is well known that an indefinite quadratic form with integral coefficients in 5 or more variables always represents zero properly, and this has raised the problem of proving a similar result for forms of higher degree, namely that such a form, of degree r, represents zero properly if the number of variables exceeds some number depending only on r. For a form of odd degree, no condition corresponding to indefiniteness is needed, but for a form of even degree (4 or more) some even stronger condition must be required.

About twenty years ago, in a note of the same title [2], I obtained the following result.

THEOREM 1. Let u and v be relatively prime integers satisfying u> v ≥ 2 and let ε be an arbitrarily small positive number. Suppose the inequality

is satisfied by an infinite sequence of positive integers n1 n2, … Then

J. R. M. Radok [1] has applied complex variable methods to problems of dynamic plane elasticity. The object of this paper is to show that his results may be obtained in a somewhat simpler way by a more systematic use of complex variable analysis.

In fact it is shown that the problems may be reduced to a form similar to that of the static aelotropic plane strain problems considered by Green and Zerna [2].