Let $F_n(x)$ be the empirical c.d.f. of $n$ independent random variables, each distributed according to the same continuous c.d.f. $F(x)$. The major object of this paper is to obtain in explicit form the probability law of the random variable $$D^+_n(\gamma) = \sup_{-\infty 1\\ 0,\quad\gamma \leqq 1,\end{cases}\end{equation*} for any $n$. This was noted by Daniels [4] and was rediscovered by Robbins [5]. Using (2.3) it is easy to evaluate $\lim_{n\rightarrow\infty} P(F_n(x) \leqq a(n) + \gamma x)$ where $\gamma, (\gamma > 1)$ is fixed and $a(n) = d/n$, where $d$ is fixed. The limiting distribution when $\gamma > 1$ can be used to derive some facts about the Poisson Process which were recently discovered by Baxter and Donsker [1]. The methods used are elementary. To assist the reader, the results are all listed in Section 2 and Section 3 is devoted to giving proofs.