Crest lines of the density Σ (blue) and potential Ψ (red, partly concealed by those of the density) for logarithmic m = 2 spirals with a pitch angle α as indicated (top) and an exponentially declining radial density-amplitude (i.e. constant relative amplitude A_m = |Σ_m|/Σ₀), either radially unlimited (top) or with sharp inner and outer edges (bottom). I also plot (dashed) crest lines of the potential obtained via the second-order approximation (24) (to first order, the crest lines of Σ and Ψ are identical).