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In this paper we have established the existence of an arithmetical difference system by a constructive method. Our arithmetical difference systems are a generalization of cyclic difference sets.

Unless stated otherwise, all our matrices (denoted by capital letters) are square matrices of size n × n and composed of real numbers. A' denotes the transpose of A. The characteristic or eigenvectors of matrices are written as column vectors having n coordinates. If ζ is a vector, ζ’ denotes its transpose.

In this paper we establish a connection between rectangular integer matrices and incidence matrices of resolvable balanced incomplete block designs. The definition of these terms has been given in paper I of this series.Our theorem can be stated as follows:THEOREM 2. Let A be a v X b matrix with integer elements such that2.1

In this paper some methods for the construction of balanced incomplete block (b.i.b. for conciseness) designs are given. In the last section it is established that the existence of an affine resolvable b.i.b. design implies the existence of two other b.i.b. designs; §§ 6 and 7 are independent of §§ 3, 4, and 5.

A few years ago, in a short paper (4) Ryser introduced an interesting topic in number theory, viz. the connection between integer matrices (i.e., matrices having only integers as their elements) satisfying certain conditions and 0-1 matrices (i.e., matrices that have no element different from 0 and 1). In this series of papers we shall pursue this topic further.To make the statements of our theorems short we introduce some terminology. We need the definitions of certain 0-1 matrices related to a few well-known combinatorial configurations. By an incidence matrix of a balanced incomplete block (b.i.b. for conciseness) design we mean a 0-1 matrix with v rows and b columns, such that the sum of the elements in each column of A is k, k < v, and the scalar product of any two row vectors of A is λ, λ ≠ 0.

The identity of Abel [1] [2] we deal with here can be stated in the following form: If n is a positive integer,.(In order that all terms be defined we require a ≠ 0, b ≠ n.)This identity and deductions from it have been very useful in many problems, for instance in mathematical statistics [3]. Usually this identity is established by means of the Lagrange - Bűrman theorem [4]. Here we will derive it very simply.

All our matrices are square with real elements. The Schur product of two n × n matrices B = (bij) and C = (cij) (i, j, = 1, 2, …, n), is an n × n matrix A = (aij) with aij = bij cij, (i, j = 1, 2, …, n).A result due to Schur [1] states that if B and C are symmetric positive definite matrices then so is their Schur product A. A question now a rises. Can any symmetric positive definite matrix be expressed as a Schur product of two symmetric positive definite matrices? The answer is in the affirmative as we show in the following theorem.