  Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$
denote the set of limit points of the sequence $\{(p_{n+1} - p_n)/\log p_n\}_{n
= 1}^{\infty}$ of normalized differences between consecutive primes. We show
that for $k = 9$ and for any sequence of $k$ nonnegative real numbers $\beta_1
\le \beta_2 \le ... \le \beta_k$, at least one of the numbers $\beta_j -
\beta_i$ ($1 \le i < j \le k$) belongs to $\boldsymbol{L}$. It follows at least
$12.5%$ of all nonnegative real numbers belong to $\boldsymbol{L}$.

Number Theory