The distribution of the latent roots depends on a definite integral over the group of orthogonal matrices. This integral defines a function of the latent roots of both the covariance matrix and the estimated covariance matrix. With an integration procedure involving first a substitution and then an expansion of the resulting integrand the first three terms of an expansion for the integral are found. This expansion is given in increasing powers of $n^{-1}$, where $n$ is the sample number less one. A numerical example is given for the distribution of the latent roots using the expansion for the definite integral given in this paper. Improved maximum likelihood estimates for the latent roots are found and the likelihood function is considered in detail.