  We study the relation between the periodic Benjamin-Ono equation with
discrete Laplacian and the two dimensional Toda hierarchy. We introduce the
tau-functions tau_pm(z) for the periodic Benjamin-Ono equation, construct two
families of integrals of motion {M_1,M_2,cdots},
{overline{M}_1,overline{M}_2,cdots}, and calculate some examples of the
bilinear equations using the Hamiltonian structure. We confirmed that some of
the low lying bilinear equations agree with the ones obtained from a certain
reduction of the 2D Toda hierarchy.
