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

Let Q(x1, …, xn) be an indefinite quadratic form in n variables with real coefficients. It is conjectured that, provided n ≥ 5, the inequality

is soluble for every ε > 0 in integers x1, …, xn, not all 0. The first progress towards proving this conjecture was made by Davenport in two recent papers; the result obtained involved, however, a condition on the type of the form as well as on n. We say that a non-singular Q is of type (r, n—r) if, when Q is expressed as a sum of squares of n real linear forms with positive and negative signs, there are r positive signs and n—r negative signs. It was proved that (1) is always soluble provided that

For each index β in a set J let Aβ be a group. The free product

is by definition a group generated by the Aβ in which any two distinct reduced words represent distinct elements. Here a reduced word is either the expression 1 or an expression

with each “syllable” ai ≠ 1 and each βi ≠ βi+1.

In a short and little known paper, Jacobi [1] gives conditions for the cubic residuacity of small primes q = 2, 3, ..., 37 to a prime p in terms of the quadratic partition

in the form L ≡ ±µM (mod q), and LM ≡ 0 (mod q).

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.

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.

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.

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

In all that follows, E denotes a separated locally convex space, E' its topological dual, and <x, x−> the bilinear form expressing the duality. We consider the differentiation of a function f:t → f(t) of a real variable t which takes its value in E; the domain of f will be an open interval which, without loss of generality, may be taken to be the entire real axis. There are various senses in which the derivative may or may not exist, and it is proposed to consider some relations between these senses.

The composite area made up of a semi-infinite strip and a semicircle is represented on a half-plane. Different transformations are in effect used for the two sub-areas and there is accordingly a continuity condition at the line of separation. This condition is satisfied only to an approximation, but a fairly accurate solution is found without much difficulty. The transformation can be used to find the steady two-dimensional flow of perfect fluid in a channel that turns through 180°.

The problem considered is that of determining the stress distribution in an infinite plate which contains a straight crack terminated at one end by a circular hole, and is in a state of plane strain or generalized plane stress under given loads. The problem has some relevance to the engineering practice of “stress relief”, in which holes are made at the ends of a crack with the object of reducing the concentration of stress. When the diameters of the holes are small compared with the length of the crack, the distribution of stress near one hole will not be greatly influenced by the presence of the other, and will then be approximately the same as in the simpler case considered here.

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.