1. Let p be a fixed prime number. We denote by k(n) the number of partitions of n and set

A reflection in Euclidean n-dimensional space is a particular type of congruent transformation which is of period two and leaves a prime (i.e., hyperplane) invariant. Groups generated by a number of these reflections have been extensively studied [5, pp. 187-212]. They are of interest since, with very few exceptions, the symmetry groups of uniform polytopes are of this type. Coxeter has also shown [4] that it is possible, by Wythoff's construction, to derive a number of uniform polytopes from any group generated by reflections. His discussion of this construction is elegantly illustrated by the use of a graphical notation [4, p. 328; 5, p. 84] whereby the properties of the polytopes can be read off from a simple graph of nodes, branches, and rings.

Let f(x1, … , xn) be a positive definite quadratic form of determinant Δ; let M be its minimum value for integers x1, … , xn not all zero; and let 2s be the number of times this minimum is attained, i.e., the number of solutions of the Diophantine equation

Methods of mechanical quadrature of functions of more than one variable apparently have received little systematic investigation and the few available results are widely scattered in the literature. In this paper a systematic approach to this problem is given and a number of formulae are derived which may prove to be useful.

A method of proving the impossibility of certain Affine Resolvable Balanced Incomplete Block Designs (A.R.B.I.B.D.) has been given by the author elsewhere [9]. More complete results in the same direction are obtained here using the ideas of a paper by Connor [4].

A set of integers {a0, a1, … , an} is said to be a difference set modulo N if the set of differences {ai — aj (i,j = 0, 1, … , n) contains each non-zero residue mod N exactly once. It follows that N and n are connected by the relation N = n2 + n + 1. If {a0, a1 … an} is a difference set mod N, so is the set {a0 + s, a1 + s, … , an + s} (s = 0, 1, … , N). These difference sets form a finite projective plane of N points, with each difference set constituting a line in the plane.

In recent years the subject of difference sets has attracted a considerable amount of attention in connection with problems in finite geometries [4]. Difference sets arising from higher power residues were first discussed by Chowla [1], who proved that biquadratic residues modulo p form a difference set if (p — l )/4 is an odd square. In this paper we shall prove a similar result for octic residues and develop some necessary conditions which will eliminate all odd power residue difference sets and many others. We also prove that a perfect residue difference set (that is, one in which every difference appears exactly once) contains all the powers of 2 modulo p.

Let/ = /(x1, x2, … , xn) be a polynomial in n > 2 variables with integer coefficients, and let p be a large prime. Very little appears to be known about estimates for the number N of solutions of the congruence

The Waring problem deals with the decomposition of integers into sums of kth powers. Consider

The object of this paper is to consider a problem suggested by Dr. K. F. Roth, on the distribution of integers n that are relatively prime to the integral part of an, a being a fixed real number. He conjectured that the number of positive integers up to x with this property is asymptotic to (or in other words that they have the density ), for irrational a. I prove this and rather more in the following

1. Let a be any irrational real number, and let F(u) denote the number of those positive integers for which (n, [nα]) = 1. Watson proved in the preceding paper that

The independence of postulates for well-known systems is a question of general interest. A closely related question is whether or not, by altering one or more of the postulates in an independent set, the set remains independent. From this standpoint the best sets of postulates are those which involve, first, the fewest postulates and, next, the least number of variables. As a rule progress is made in this direction at the sacrifice of simplicity of postulates. In this paper, in counting postulates, we ignore properties such as closure under the operations and count only identities or those stating that one equation implies another.

In the theory of probability, the conditional can be treated by an operation analogous to division. Many properties of the conditional can best be studied by means of the corresponding multiplication (called the cross-product). An implicative Boolean ring is defined [2] in terms of a cross-product and the usual Boolean operations. The cross-product is the only device yet known in which the events corresponding to conditional probabilities are themselves elements of the Boolean ring. The fact that such advice was not introduced by Boole is probably the reason why Boolean algebra has been very little used in the theory of probability, although probability was one of the principal applications which Boole had in mind.

Cohomology theories of various algebraic structures have been investigated by several authors. The most noteworthy are due to Hochschild, MacLane and Eckmann, Chevalley and Eilenberg, who developed the theory of cohomology groups of associative algebras, abstract groups, and Lie algebras respectively. In this paper we are concerned primarily with a characterization of the third cohomology group of a Lie algebra by its extension properties.

If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of the location of S in G. Thus, for example, famed theorems of Frobenius and Burnside give criteria for the existence of a normal subgroup N of G such that G = NS and 1 = N ⋂ S, and hence in particular for the non-simplicity of G. To aid in locating S in G, and to facilitate exploitation of the transfer, we single out a descending chain of normal subgroups of S. Namely, we introduce the focal series of S in G by means of the recursive formulae

Baer [2] and Neumann [5] have discussed groups in which there is a limitation on the number of conjugates which an element may have. For a given group G, let H1 be the set of all elements of G which have only a finite number of conjugates in G, let H2 be the set of those elements of G, the conjugates of each of which lie in only a finite number of cosets of H1 in G; and in this fashion define H3, H4, …. We shall show that the Hi are strictly characteristic subgroups of G.

The purpose of this note is the classification of thedifferentiable points on curves in the conformai plane. We introduce tangent and osculating circles at such points and study the intersection and support properties of these circles.

Consider any three congruences of an orthogonal ennuple at a point of a Riemannian space. When one congruence is moved by local and a second congruence is moved by parallel displacement in the direction of the third congruence, the rate of change of cosine of the angle between the first two congruences is well known as Ricci's coefficient of rotation and has been extensively studied. It is the purpose of this note to investigate the corresponding rate of change, when the third congruence is replaced by an arbitrary one, in connection with parallelism and equidistance of congruences as studied by Miss Peters [2; 3].

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be written

in a notation explained below.

In this paper it is shown how trigonometric series which are Cesàro-summable to zero may be used to solve differential equations. The explicit solution of the general ordinary linear equation with constant coefficients is found in terms of trigonometric series and special cases are dealt with.