We have generalized the classical description of ionization front propagation to the case of a point source in a uniform, cosmologically expanding gas. We present illustrative curves for the comoving radius and peculiar velocity for several turn-on redshifts, z0N, for Ωtot = 1, Ωb = 0.1, h = 1. The quantity RS is the generalized Strömgren radius [RS = RSi (1 + z0N)/(1 + z), RSi = (3Nu/4πnH,i
2 α2)1/3, Nu = photoionizing number flux per source, α2 = recombination rate to n = 2, nH,i = nH
o (1 + z0N)3]. The quantity T0N = 2 (1 + z0N)−3/2/(3H0). We also plot ζ, the value of (2nQ
o) needed to ionize the IGM with overlapping QSO HII regions by redshift z0V for QSO turn-on at various z0N, where Nph, Q = ionizing photon luminosity per QSO, nQ
o = QSO number density (present co-moving value), nH = H density of IGM, and nH/nH
o = nQ/nQ
o = (1 + z)3. From a recent preprint by Koo (1985), we estimate ζ ≲ 1 (for Ωb = 0.1, h = 1) for QSO's with L ∼ 1045 erg s−1. In this case, the observed QSO's cannot be the sole source of the IGM ionization that is implied by the null detection of the Gunn-Peterson effect for QSO's with z > 2.