As a continuation of previous work, we determine hadronic-production cross-sectionsσ n (s) satisfying exactly the high-energy empirical laws known for the first, second and third multiplicity moments. The result is obtained in the form of a second-order linear differential equation forσ n (s) which allows one to calculate explicitly all successive moments. In particular, the fourth moment is in excellent agreement with the data. The asymptotic solution of the equation forσ n (s) is given analytically. KNO scaling turns out to be an asymptotic property of our solution. The full solution forσ n (s) is studied numerically and the KNO plot is compared with the data. No free parameters are left to be adjusted except for an overall normalization constant. As expected, KNO scaling sets in rather quickly with increasingn and the agreement with the data is progressively good. This agreement becomes excellent for the whole interval ofn/ for which data exist (0≲4) asn becomes larger, but already forn∼3 the agreement is very good up ton/≅2. It turns out that the asymptotic solution, given in analytic terms, is an excellent approximation to the data and can thus be used for practical purposes instead of the full solution for calculatingσ n (s).