  We write the Hamiltonian of the Bose gas with two-body repulsive
$\delta$-function potential in a pseudoparticle operator basis which
diagonalizes the problem via the Bethe ansatz. In this operator basis the
original bosonic interactions are represented by zero-momentum
forward-scattering interactions between Landau-liquid pseudoparticles. We find
that this pseudoparticle operator algebra is complete: {\it all} the
Hamiltonian eigenstates are generated by acting pseudoparticle operators on the
system vacuum. It is shown that one boson of vanishing momentum and energy is a
composite of a one-pseudoparticle excitation and a collective pseudoparticle
excitation. These excitations have finite opposite momenta and cannot be
decomposed. Our formalism enables us to calculate the various quantities which
characterize the static and dynamic behavior of the system at low energies.
