  In math.CA/0211148 we observed that $\ln(4/ \pi)$ is an "alternating" analog
of Euler's constant $\gamma$. Here we use the binary expansion of an integer to
give a rational series for $\ln(4/ \pi)$ analogous to Vacca's series for
$\gamma$. Also, using a known generalization of Vacca's series to other bases,
we accelerate Addison's series for $\gamma$. Along the way, we give new proofs
of Vacca's and Addison's formulas.
