Consider the Slepian process S defined by S(t) = B(t + 1) − B(t),t ∈ [0, 1] with B(t), t ∈ ℝ a standard Brownian motion. In this contribution we analyze the properties between the maximum ms=max0≤u≤sS(u) and the maximum mt=max0≤u≤tS(u) for 0 ≤ s < t ≤ 1 fixed. Explicit integral expressions are obtained for the joint distribution function between m s and m t and the distribution function of the partial maximum m s . Further, we apply our results for the determination of the moments of m s .