{\int _{\alpha }^{\alpha +L}}
{[{\tilde N}(x)-Ax-B]^2}\,dx,$$ where $\tilde N(x)$ is the unfolded number density. In our case there is no need to unfold the spectrum since it is fully contained between 0 and 2$\pi$; $\Delta _3$ is just the least mean square deviation of ${\tilde N}(x)$ from the mean straight line behavior. This statistic is directly proportional to the $<{\Sigma ^2}>$ by $\Delta _3 (L)={2\over {L^4}}
\int _0^L(L^3-sL^2x+x^3)
\Sigma ^2(x)dx$, and thus can be calculated for the Circular Orthogonal Ensembles (COE) as well[@mehta]. The specific theoretical predictions for the averaged ${<{\Delta_3(L)}>}={1\over L}{\sum _{\alpha }
\Delta_3(L,\alpha )}$, are ${{\Delta_3^{(COE)}(L)}}\>={{1\over \pi 
^2}}\ell n\{L\} -0.007,\> \>$ and ${<{\Delta_3^{(Poisson)}(L)}>}\>={L\over 15}\> $. These results are correct in the asymptotic limit valid for $15\leq L$.

In Fig.6 we present our results for $<\Delta_3>$ and $<\Sigma ^2>$. In these figures one clearly sees the transition from Poisson-like (dashes) to COE-like (solid line) behavior as $\epsilon/{{\tilde\hbar}}$ is varied. We note that the $\Delta_3$ statistic does not saturate in the COE limit, even for the maximum interval $L$ that we looked at, as would be expected from semiclassical arguments originally proposed by Berry[@berry]. Furthermore, note that for the largest $L$ considered the Poisson limit does not present the knee 