  We present a framework for the analytic calculations of the hierarchical wave
functions and the composite fermion wave functions in the fractional quantum
Hall effect on the sphere by using projective coordinates. Then we calculate
the overlaps between these two wave functions at various fillings and small
numbers of electrons. We find that the overlaps are all most equal to one. This
gives a further evidence that two theories of the fractional quantum Hall
effect, the hierarchical theory and the composite fermion theory, are
physically equivalent.
