This paper deals with the smallest maximizing point τ_mn⁺ of the two-sample empirical process {F_m(x) - G_m(x): x ∈ R} where F_m(x) and G_n(x) are the empirical distribution functions of X₁,..., X_m and Y₁,..., 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 τ_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 τ_mn⁺ has distribution function F(x).