  This paper is concerned with various aspects of the Slepian process $(B_{t+1}
- B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0
)$. In particular, we offer an analysis of the local structure of the Slepian
zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the
Slepian process for $0 \le t \le 1$. We also establish the existence of a
random time $T$ such that $T$ falls in the the Slepian zero set almost surely
and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.
