It is known that the moments of several test criteria in multivariate analysis can be expressed in terms of Gamma functions. Various attempts have been made to determine the exact distributions from the known expressions for the moments. Wilks [11] identified such a distribution with that of a product of independent Beta variables, and he gave explicit results in some particular cases. Later, Nair [5], using the theory of Mellin transforms, expressed the distributions as solutions of certain differential equations, which he solved in some special cases. Kabe [3] expressed them in terms of Meijer $G$-functions. Asymptotic expansions of the distribution functions have been given by several authors; notably by Box [2], Tukey and Wilks [8], and Banerjee [1]. In this paper we give exact results in some important cases, in which it is found that the distribution is identical with that of a linear function of Gamma variates.