Previously, a simple yet exact characteristic equation was derived for steady‐state longitudinally uniform positive columns in which the electron densityneis given by the continuity equationD∇2ne+&ngr;ne+kn2e=0 with the coefficients independent of position and withne=0 at the boundaries, which can be of arbitrary cross‐sectional shape. The effective coefficients &ngr; andkmay be positive or negative and can each represent the net effect of various electron production and loss processes. The general characteristic equation contains a dimensionless quantityS(the integral of −∇2ne/neover the cross section) which provides information about the shape of the electron density distribution. TheSintegral has been evaluated numerically for circular and rectangular column cross sections. For the case of a circular cross section the variation of the density profile andShave been determined for all possible relative magnitudes and combinations of signs of &ngr; andk. The results provide a generalized parameter map which indicates steady‐state operating conditions for a large class of discharge columns.Svaries from large positive values for diffuse columns (fork<0, i.e., with electron loss by volume recombination) to large negative values for constricted columns (for &ngr;<0, i.e., with loss by attachment). For &ngr;<0, the results indicate that no steady‐state distributions exist with attachment‐dominated loss at the location of maximumne.