Let $\mathbf{S}_i: p \times p(i = 1, 2)$ be independently distributed as Wishart $(n_i, p, \mathbf{\Sigma}_i)$. Let the characteristic roots of $\mathbf{S}_1 \mathbf{S}_2^{-1}$ and $\mathbf{\Sigma}_1 \mathbf{\Sigma}_2^{-1}$ be denoted by $l_i (i = 1,2, \cdots, p)$ and $\lambda_i (i = 1,2, \cdots, p)$ respectively such that $l_1 > l_2 > \cdots > l_p > 0$ and $\lambda_1 > \lambda_2 > \cdots > \lambda_p > 0$. Then the distribution of $l_1, \cdots, l_p$ can be expressed in the form (Khatri [8]) \begin{equation*} \tag{1.1} C|\mathbf{\Lambda}|^{-\frac{1}{2}n_1}|\mathbf{L}|^{\frac{1}{2}(n_1-p- 1)}\{\prod^p_{i