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Simplifications in dealing with the equation of the center when theshort-period effects are removed from a zonal Hamiltonian are commonlyattributed to the elimination of parallactic terms. But this interpretation isincorrect, and the simplifications rather stem from the removal of concomitantlong-period terms, an outcome that can also be achieved without need ofeliminating the parallax. To show that, a Lie transforms simplification isinvented that augments the exponents of the inverse of the radius and stillachieves analogous simplifications in handling the equation of the center tothose provided by the classical elimination of the parallax simplification. Theparticular case in which the new transformation does not modify the exponentsof the parallactic terms of the original problem, leads to a new intermediaryof the main problem that, while keeping higher order effects of J2, is formallyanalogous to Cid's first order radial intermediary.

A new radial, natural, higher order intermediary of the main problem four decades after the elimination of the parallax