  We consider flavor breaking in the the octet and decuplet baryon masses at
second order in large-$N$ chiral perturbation theory, where $N$ is the number
of QCD colors. We assume that $1/N \sim 1/N_F \sim m_s / \Lambda \gg
m_{u,d}/\Lambda, \alpha_{EM}$, where $N_F$ is the number of light quark
flavors, and $m_{u,d,s} / \Lambda$ are the parameters controlling $SU(N_F)$
flavor breaking in chiral perturbation theory. We consistently include
non-analytic contributions to the baryon masses at orders $m_q^{3/2}$, $m_q^2
\ln m_q$, and $(m_q \ln m_q) / N$. The $m_q^{3/2}$ corrections are small for
the relations that follow from $SU(N_F)$ symmetry alone, but the corrections to
the large-$N$ relations are large and have the wrong sign. Chiral
power-counting and large-$N$ consistency allow a 2-loop contribution at order
$m_q^2 \ln m_q$, and a non-trivial explicit calculation is required to show
that this contribution vanishes. At second order in the expansion, there are
eight relations that are non-trivial consequences of the $1/N$ expansion, all
of which are well satisfied within the experimental errors. The average
deviation at this order is $7 \MeV$ for the $\De I = 0$ mass differences and
$0.35 \MeV$ for the $\De I \ne 0$ mass differences, consistent with the
expectation that the error is of order $1/N^2 \sim 10\%$.
