.u. (x~ = P { ~2 < x }, U (x) = lira U. (x). Let u - n/2 - 1 if n is even, and a - (n - 1)/2, if n is odd. We prove that the distribution function Un(x) is differentiable with respect to x ~ times, but not continuously differentiable ~ + 1 times. In addition, the derivatives of the distribution functions Un(x) as n ~ ~, converge uniformly in x to the corresponding derivative of the limit distribution function U(x). In particular, one has uniform convergence of the densities U~ (x). In this paper we also give asymptotic expansions for the derivatives of the distribution functions Un(x). The estimates of the remainders depend properly on n. The results of the paper generalize and improve the results of Smirnov [I], Anderson and Darling [2], Kandelaki [3], Sazonov [4, 5], Rosenkrantz [6], Kiefer [7], Nikitin [8], Orlov [9], Czorgo [I0], Csorgo and Stacho [II], GStze [12], Borovskikh [13], in which the convergence and rate of convergence of distribution functions Un(x) to U(x) were studied and asymptotic expansions for Un(X ) were also found. We proceed to precise formulations. We denote by C a the class of functions f: R I ~ R I which have = bounded derivatives. THEOREM 1.1. The distribution function Un(x) belongs to the class C ~ but does not belong to the class C a+1, where u - n/2-1 if n is even, and a - (n - 1)/2 if n is odd. Moreover, for any sup ( 1 + x") I U~" (x) - U ~'~ (x) ! < c (s,m)/n.