  Previous results on quasi-classical limit of the KP and Toda hierarchies are
now extended to the BKP hierarchy. Basic tools such as the Lax representation,
the Baker-Akhiezer function and the tau function are reformulated so as to fit
into the analysis of quasi-classical limit. Two subalgebras $\WB_{1+\infty}$
and $\wB_{1+\infty}$ of the W-infinity algebras $W_{1+\infty}$ and
$w_{1+\infty}$ are introduced as fundamental Lie algebras of the BKP hierarchy
and its quasi-classical limit, the dispersionless BKP hierarchy. The quantum
W-infinity algebra $\WB_{1+\infty}$ emerges in symmetries of the BKP hierarchy.
In quasi-classical limit, these $\WB_{1+\infty}$ symmetries are shown to be
contracted into $\wB_{1+\infty}$ symmetries of the dispersionless BKP
hierarchy.
