  We reexamine the large N limit of matrix integrals over the orthogonal group
O(N) and their relation with those pertaining to the unitary group U(N). We
prove that lim_{N to infty} N^{-2} \int DO exp N tr JO is half the
corresponding function in U(N), and a similar relation for lim_{N to infty}
\int DO exp N tr(A O B O^t), for A and B both symmetric or both skew symmetric.
