  Laplace's tidal equations govern the angular dependence of oscillations in
stars when uniform rotation is treated within the so-called traditional
approximation. Using a perturbation expansion approach, I derive improved
expressions for the eigenvalue associated with these equations, valid in the
asymptotic limit of large spin parameter $q$. These expressions have a relative
accuracy of order $q^{-3}$ for gravito-inertial modes, and $q^{-1}$ for Rossby
and Kelvin modes; the corresponding absolute accuracy is of order $q^{-1}$ for
all three mode types. I validate my analysis against numerical calculations,
and demonstrate how it can be applied to derive formulae for the periods and
eigenfunctions of Rossby modes.
