A family of spherical potential-density pairs is presented. The densities are proportional to r^-4^ at large radii and diverge in the centre as r^- γ^ with 0 <= γ <3. The models of Jaffe and Hernquist are included as special cases. The gravitational potential is analytical for all γ. For specific values of γ, most of the intrinsic and projected properties, such as distribution function, surface density and projected velocity dispersion, can be expressed in terms of elementary functions. A comparison to the de Vaucouleurs R^1/4^-profile shows that the model with γ = 3/2 most closely resembles it in both surface density and distribution function. This particular model is completely analytical, and thus it is the best analytical approximation of the R^1/4^-model known so far.