We have successfully employed a constrained non-linear least squares analysis of incoherent scatter data from the topside ionosphere over Arecibo, Puerto Rico to simultaneously deduce concentrations of helium ions, protons, and oxygen ions as well as ion and electron temperatures. The constraint encourages smooth temperature profiles, and its use avoids unrealistic parameter values in regions where light ions are more than 80% of the total. Here we report proton concentrations that remain higher than helium ion concentrations, even though the latter reach 30% of the total and exceed the oxygen ion densities at altitudes as low as 700 km at night and 1500 km in the daytime. Before local sunrise the effects of conjugate--point photoelectrons are clearly seen, while after sunrise the transition height between oxygen and helium rises much more rapidly than does the transition from oxygen to hydrogen. ¿ American Geophysical Union 1994