For any natural $n$, let $U_1,\dots,U_n$ be independent identically distributed
random variables each uniformly distributed on the interval $[0,1]$. As usual, let $U_{n:1}\le\cdots\le U_{n:n}$ denote the corresponding order statistics. Consider

\begin{equation}\label{eq:G_i}
G_i:=U_{n:i}-U_{n:i-1} \quad \text{for}\quad i=1,\dots,n+1,
\end{equation}
where $U_{n:0}:=0$ and $U_{n+1:n+1}:=1$.

One may refer to the $G_i$'s as the gaps or, as it is usually done in the literature, the spacings between the consecutive order statistics. Let now $$ G_{n+1:1}\le\cdots\le G_{n+1:n+1} $$ denote the ordered gaps $G_1,\dots,G_{n+1}$, so that the random sets $\{G_{n+1:1},\dots, G_{n+1:n+1}\}$ and $\{G_1,\dots,G_{n+1}\}$ are the same.

Fisher obtained the distribution of the largest gap, $G_{n+1:n+1}$. Fisher's result was generalized in this note, where the distribution of the $k$th smallest gap, $G_{n+1:k}$, was obtained, for each $k=1,\dots,n+1$.

Does anyone know other references to these or other related results in the literature?