  The classical methods of multivariate analysis are based on the eigenvalues
of one or two sample covariance matrices. In many applications of these
methods, for example to high dimensional data, it is natural to consider
alternative hypotheses which are a low rank departure from the null hypothesis.
For rank one alternatives, this note provides a representation for the joint
eigenvalue density in terms of a single contour integral. This will be of use
for deriving approximate distributions for likelihood ratios and linear
statistics used in testing.
