The paper is divided into two parts: A. An elementary derivation of Bartlett's results on the distribution of the canonical correlation coefficients using exterior differential forms. Briefly, our method consists of taking the original multivariate normal distribution, transforming to the canonical correlations and other variables, and then integrating out these extraneous variables. B. A new method of calculating the conditional moments which appear in Bartlett's expansion of this distribution, based on the process of averaging over the orthogonal group. This method allows the calculation of moments of any order.