Consider an m × n rectangular array whose m rows are permutations of 1, 2, …, n. Such an array will be called a constant-sum array if the sum of the elements in each column is the same (and equal to ½m (n+1)). An example of a 3×9 constant-sum array is In contrast to a Latin rectangle, elements in the same column of a constantsum array may be equal. It will be convenient to assume arrays normalised in the sense that the columns are arranged so that, as in (1), the first row is in the standard order 1, 2, …, n.

This is a continuation of [5] and we begin by recalling two definitions and a result of that paper which are needed here. Let be a family of functions with domains contained in a set X and ranges contained in a set Y and let be a function with domain D()= Y and range with the property for each pair of elements ƒ and g of . Since the composition operation is associative, is a semigroup if for ƒ and g in , we define the product ƒg by .

Let E be a vector lattice in the sense of Birkhoff [1]. We use the following notations:

Suppose that the real-valued function ƒ(t) is positive, continuous and monotonic increasing for t ≧ t0. If x = x (t) is a solution of the equation for for t ≧ t0, it is known that the solution x(t) oscillates infinitely often as t → ∞ and that the successive maxima of |x(t)| decrease, with increasing t. In particular x(t) is bounded as t → ∞.

The purpose of this article is to present some results on varieties of metabelian p-groups, nilpotent of class c, with the prime p greater than c. After some preliminary lemmas in § 3, it is established in § 4, Theorem 3, that there is a simple basis for the laws of such a variety, and this basis is explicitly stated. This allows the description of the lattice of such varieties, and in § 5, Theorem 4, it is shown that each such variety has a two-generator member which generates it; this is established by the help of Theorem 5, which states that each critical group is a two-generator group, and Theorem 6, which gives explicitly the varieties generated by the proper subgroups, by the proper quotient groups, and by the proper factor groups of such a critical group.

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

In this paper we investigate finite metabelian p-groups from the point of view of varieties and identical relations. In other words we shall be interested in those properties of such groups which are preserved under the operations of fonning direct products, taking subgroups and taking factor groups. Ideally one would like to characterize such properties in terms of numerical invariants and with the sole restriction that the p-group in question have class less than p we accomplish this.

A formal method is developed for deriving a series expansion of the general term in Green-type expansions. The technique is exemplified by detailed calculations for modified Bessel functions of large order.

Until now the chief obstacle to the application of the maximum likelihood method of estimation to factor analysis has been the lack of any really good numerical method of solution. In this paper we give a brief review of recent work which remedies this defect. Two factor analysis models are considered. In each case we derive results which are of use in connection with new methods of solution. Formulae are given for the large-sample variances and covariances of the estimates of parameters in the first model.

In recent years, a number of special integral equations of the first kind was discussed by several authors (see [l]–[4], [6], [7], [9]–[18]). The kernels of these integral equations are special functions of the hypergeometric family, and it was necessary to restrict the parameters appearing in these functions to secure convergence of the integrals. If these restrictions are removed, the integral fails to converge but it may possess a finite part (in Hadamard's sense), and the question arises whether the methods used in the restricted case will alsoapply in the new situation. Indeed, one could pose the moregeneral problem of Volterra integral equations involving finite parts of divergent integrals [19]

The generating function for the number of linear partitions was found by Euler, the method being almost trivial. That for plane partitions is due to Macmahon, but, even in a simplified form found by Chaundy, the proof is far from trivial. The number of solid partitions of n, i.e. the number of solutions of

is denoted by r(n). It has often been conjectured that the generating function of r(n) is , but this is now known to be false. We write η(a, b, c) for the generating function of the number of solutions of (1) subject to the additional condition that

Macmahon 1916 found n(a, 1, 1) for general a. Here we find η(a, b, c) for general a, b. c.

§ 1. The principal object of this note is to establish formula (16) of the preceding paper by H. Jack (1966). This formula, which was conjectured by Jack, evaluates a certain coefficient which is attached to a symbol {p, q, …, r}. In this symbols, p, q, …, r form a partition of m such that o≤p≤q≤…≤r,p+q+…+r=m. The symbol however vanishes if any two of the integers p, q, … r are equal but non-zero. In the remaining cases we have to show that the coefficient in question has the value

Necessary and sufficient conditions are given for the existence of solutions of equations, in any number of variables, on distributive lattices with least and greatest elements, together with an algorithm for determining a solution when these conditions are satisfied.

A multiple integral, whose integrand is an n × n determinant, is evaluated over certain regions of n-dimensional space. Similar integrals are encountered in the theory of Zonal polynomials. In the course of the work a partition problem arises. In the next paper of these Proceedings, Professor Rutherford enumerates these partitions and relates the subject to the theory of the representation of the symmetric group.

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. Let t be an element of B(X), and let edenote the identity operator on X. Since the earliest days of the theory of Banach algebras, ithas been understood that the natural setting within which to study spectral properties of t is the Banach algebra B(X), or perhaps a closed subalgebra of B(X) containing t and e. The effective application of this method to a given class of operators depends upon first translating the data into terms involving only the Banach algebra structure of B(X) without reference to the underlying space X. In particular, the appropriate topology is the norm topology in B(X) given by the usual operator norm. Theorem 1 carries out this translation for the class of compact operators t. It is proved that if t is compact, then multiplication by t is a compact linear operator on the closed subalgebra of B(X) consisting of operators that commute with t.

The problem considered is that of obtaining solutions of the Helmholz equation ∇2V + k2V = 0, suitable for use in connection with paraboloidal co-ordinates. In these co-ordinates the Helmholz equation is separable, and each of the separated equations is reducible to Hill's equation with three terms (the Whittaker-Hill equation). The properties of solutions of this equation are developed sufficiently to make possible the formal solution of simple boundary-value problems for paraboloidal surfaces, principally for the case k2 < 0.