Notes on obtaining the eigenvalues of Laplace's tidal equation

A review is undertaken of the various forms that have been obtained for the recurrence relation from which the eigenvalues of Laplace's tidal equation may be obtained. Such forms are shown to be analytically consistent and are discussed in relation to their subsequent numerical evaluation. By determining eigenvalues of equivalent depth for a given frequency of oscillation instead of the other way around the problem becomes a straightforward one of matrix diagonalization. If solutions are based on normalized Legendre functions the matrix is symmetric. A method of evaluating the related wind functions is described.