& \ell(X^+_B) = 1\\
{7 \over 8} < B < {11 \over 12} & \ell(X^+_B) = {1 \over 2}\\
{11 \over 12} < B < {13 \over 14} & \ell(X^+_B) = 1\\
{13 \over 14} < B < {17 \over 18} & \ell(X^+_B) = {1\over 2}\\
\end{array}$$ These values agree with the predictions of Theorem \[thmposmeas\] that $\ell(X^+_B)\in \frac{1}{4}\{1,2,3\}$. More generally for any $n \in {{\bf N}}$ we observe that $$\begin{array}{ll}
{6n-1 \over 6n} < B < {6n+1 \over 6n+2} & \ell(X^+_B) = {1}\\
     {\rm else} & \ell(X^+_B) = {1 \over 2}.\\
\end{array}$$ Therefore we have $$\begin{aligned}
\ell(X^+) &=& \int_{1/2}^1 \ell(X_B^+) dB \\
&=& \frac{1}{4}+\frac{1}{2}\sum_{n=1}^\infty
\left({6n+1 \over 6n+2} - {6n-1 \over 6n}\right) \\
&=& {1 \over 4} + {1 \over 12}
\sum_{n=1}^\infty {1 \over 3n^2+n} \\
&=& {1 \over 2} - {\sqrt{3}\pi \over 72} - {1 \over 8} \ln 3 \\
&\approx& 0.28709849.\end{aligned}$$ Thus we can obtain a series expression for the measure and in this case we can sum it explicitly. Unfortunately we have not been able to extend this method to one that applies generally.

Irrational parabolic maps
-------------------------

We have not yet obtained any significant results for irrational parabolic area-preserving maps and, in fact, for the all 