  The correlation energy per electron in the high-density uniform electron gas
can be written as $\Ec(r_s,\zeta) = \lam_0(\zeta) \ln r_s + \eps_0(\zeta) +
\lam_1(\zeta) \,r_s \ln r_s + O(r_s)$, where $r_s$ is the Seitz radius and
$\zeta$ is the relative spin polarization. We derive an expression for
$\lam_1(\zeta)$ which is exact for any $\zeta$, including the paramagnetic and
ferromagnetic limits, $\lam_1(0)$ and $\lam_1(1)$, and discover that the
previously published $\lam_1(1)$ value is incorrect. We trace this error to an
integration and limit that do not commute. The spin-resolution of
$\lam_1(\zeta)$ into contributions of electron pairs is also derived.
