The integral evaluated in this note was suggested by the famous one connected with the Poincaré polynomials of the classical groups (see (1)).

Let X be an n × n matrix whose elements depend on k parameters. Denote by a manifold in Euclidean space of dimension n2, with the property that if X ∈ , then so does XI−i for 1≦i≦n, where I−i is the unit matrix I altered by a minus sign in the (i, i)th place. Suppose further that there exists on a measure which is invariant under the transformation X→XI−i. Such manifolds and measures exist. For example (see (2), § 5), the set of all proper and improper n×n orthogonal matrices H is such a manifold, the H depending on ½n(n−1) parameters because of the orthogonality and normality of the columns of H. Since the set of all H is a compact topological group, an invariant measure exists.