This paper deals with the smallest maximizing point \tau_mn^+ of the two-sample empirical process \F_m (x) - G_n (x)\colon x \in \open R \ where F_m (x) and G_n (x) are the empirical distribution functions of X_1,\ldots,X_m and Y_1,\ldots,Y_n , respectively, which are two independent samples of i.i.d. random variables with common distribution function F (x) . We determine the distribution function of \tau_mn^+ for finite subsample sizes m and n . It turns out to be a polynomial of degree m + n in the variable F (x) . If m and n are relatively prime then \tau_mn^+ has distribution function F (x) .