  We study the rate of convergence for the largest eigenvalue distributions in
the Gaussian unitary and orthogonal ensembles to their Tracy-Widom limits. We
show that one can achieve an $O(N^{-2/3})$ rate with particular choices of the
centering and scaling constants. The arguments here also shed light on more
complicated cases of Laguerre and Jacobi ensembles, in both unitary and
orthogonal versions. Numerical work shows that the suggested constants yield
reasonable approximations, even for surprisingly small values of N.
