Here, 2n + 1 is the degree of precision of the cubature formula, that is, the truncation error indicated by O(h + ) in (2) is zero when F is a polynomial in X and Y with a joint degree smaller than or at most equal to 2/i + l. The N* points Ph at which the function values are sampled, will be called sample points. All cubature formulae considered in the following make full use of the symmetry, which indicates that the functions F(±X, ±Y), F(±Y, ±X) all have the same mean M(F). This symmetry requires that the complete set of sample points can be obtained from the subset in, say, the closed upper half of the fourth quadrant by reflexion in X = O, Y •O, and X = ±Y. The points of this subset will be called the basic points, and the number of basic points will be denoted by N. A basic point together with the points obtained from it by the reflexions mentioned above will be said to form a set of associated points. The considered symmetry further requires that the function values at a set of associated points appear with the same coefficient a; in the cubature formula (2). A square mesh with sides parallel to the boundaries of the domain of integration in (1) provides one way of choosing sample points that is frequently used in the literature. Here, we consider the diagonal square mesh