Abstract: Let V_p, n=\left\H: H\right. is an n \times p matrix, \left.n \geqslant p, H^\prime H=I_p\right\ denote Stiefel manifold and U_1 an n \times p random matrix uniformly distributed on V_p, n. Let F_1=\left\X_1 X_n \times p, \Gamma X \stackreld= X, \forall \Gamma\right. \in O(n)\ and F_s=\left\X: X \in F_1\right. and \left.X Q \stackreld= X, \forall Q \in O(p)\right\, where O(n) is the class of orthogonal matrices of order n. In this paper, the characteristic function of U_1 is obtained by using zonal polynomials. The distribution of U_1 can be determined by that of its first row in F_1 or that of any of its element in F_s. Finally, it is pointed out that F_s is a proper subolass of F_7=\left\X: X \in F_1\right. and X l \stackreld= X a for each \left.l \in V_1, p, a \in V_1, p\right\ a guess given by Fang and Ohen.