S. N. Roy  obtained in 1943 the distribution of the maximum, minimum and any intermediate one of the roots of certain determinantal equations based on covariance matrices of two samples on the null hypothesis of equal covariance matrices in the two populations. The present paper gives a different method of working out the distribution of any of these roots under the same hypothesis. The distribution of the largest, smallest and any intermediate root when the roots are specified by their position in a monotonic arrangement has been derived for $p = 2, 3, 4,$ and 5 by the new method. The method is applicable for obtaining the distribution of the roots of an equation of any order, when the distributions of the roots of lower order equations have been worked out.
"Distribution of a Root of a Determinantal Equation." Ann. Math. Statist. 19 (1) 47 - 57, March, 1948. https://doi.org/10.1214/aoms/1177730289