In this paper a few results (of a purely mathematical nature) are obtained, which are useful for studying certain distribution problems in multivariate analysis--e.g., those relating to the characteristic roots of a determinantal equation ([1], [2], [5]). In particular, the results are shown to be readily applicable to the moment problems of the sum of the roots and the distributions of the extreme roots. Most of the results given are in the form of certain recursion formulae for reducing special types of $k$-th order Vandermonde determinants in terms of those of orders $(k - 1)$ and $(k - 2).$ The applications of these results are given by S. N. Roy [6] and the present author [3].