A family of evolutionary cosmological models of considerable generality, chosen spatially flat but otherwise characterized by four physical or kinematic parameters, is formulated and their exact theoretical number counts calculated by computer to determine whether any member of the family will fit the observed number count data of earlier Cambridge surveys and the more recent individual polar counts by Ryle and Neville down to a flux density of S =0.25 × 10−26 w m−3c/s)−1 at 178 Mc/s. For reasons stated in the paper the analysis is confined to the case when radio source density is proportional to galaxy density at all epochs. The four parameters are (i) the index n of the expansion factor R(t), assumed to be of the form R(t) = tn, (ii) the index m determining the evolution of source power P(t), taken to be of the form P(t)∝t−m (m > 0), (iii) the standard deviation σ of the assumed Gaussian dispersion of log P0, P0 being the intrinsic power of a source observed in a given range of high flux density, and (iv) the median power |$\bar P_0$| of the sources so observed. The creation of radio sources is assumed to have started at a certain time t⋆, early in the history of the universe, this time being suggested by the nature of the models. for a spectral index common to all sources of 0.8. In particular, if n = 2/3 then m = 1.5, so that the emission of radiation from space due to the radio sources would vary as t−3.5, which is reasonably close to the time dependence of the total radiative emission from space, varying as t−3, according to the evolutionary theory advanced by one of us earlier (2). For a given n, the value of |$\bar P_0$| is determined by the fact that |$\bar P_0(1-n)^2/n^2$| is constant. It is shown that if n = 2/3 then |$\bar P_0| is equal to 1.5 × 1026 w ster−1 (c/s)−1 at 178 Mc/s.