The statistical properties of the eigenvalues of random unitary matrices may be determined from the joint probability density function of the matrix eigenvalues. Earlier theorems have derived the density function for the unitary and symplectic circular ensembles from that for the circular orthogonal ensemble. A method is presented here for successively eliminating variables from the probability density function for the orthogonal circular ensemble; the method generalizes an earlier result, and the resulting function appears to represent the behavior of eigenvalues from a new series of matrix ensembles.