  In the hierarchical theory of the fractional quantum Hall effect, the
low--energy behaviour of a daughter state in the next level of the hierarchy is
described by an interacting system of quasiparticles of the parent state.
Taking the filled lowest Landau level as the parent state, we examine
analytically the quantitative consequences of this approach for electrons
interacting via a pseudopotential interaction. It is shown that the ground
state energy per particle in the daughter state at a filling factor $2/3$ is
exactly equal to that of a system of quasiholes in the parent state with half
filling, precisely as predicted by the hierarchical approach. This is achieved
with {\em only up to two--particle interactions} in the effective Hamiltonian
for the quasiholes. Their single particle energy and two--particle interaction
are derived. The results are generalized to the other filling factors
attainable from the filled Landau level.
