A new formulation of the single-polynomial analysis is presented, in which the electron density profile is assumed to begin with a flat base at a height equal to the lowest measured virtual height. This formulation gives a more accurately defined profile with only a small number of points and agrees with the assumptions normally made. Near the critical frequency he electron density profile is required to approach a parabolic distribution. Coefficients are given for the rapid analysis of ionograms taken anywhere in the world. The height of the peak, the scale height at the peak, and the subpeak electron content, along with five or six points on the electron density profile, are obtained directly from a small number of measured virtual heights.