  We derive Painlev\'e--type expressions for the distribution of the $m^{th}$
largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the
edge scaling limit. This work generalizes to general $m$ the $m=1$ results of
Tracy and Widom [23]. The results of Johnstone and Soshnikov (see [15], [19])
imply the immediate relevance of our formulas for the $m^{th}$ largest
eigenvalue of the appropriate Wishart distribution.
