The properties of a formula for single-particle distribution function are investigated in connection with the mechanism of the increase of some spectra with energy. A new form of scaling, which approaches Feynman scaling at asymtotic energies, is proposed. Recent experiments on p+p →(p̅,K- and π-)+anything up to ISR energies are consistent with the new scaling, in which the energy dependence and the longitudinal shape of the spectrum are simultaneously determined by single parameter for each species of the produced particle. The concave rise of the averaged multiplicity up to ISR energies is obtained as a result of the new scaling: above ISR energies, the logarithmic increase is predicted to set in. The transverse shapes of spectra are shown to reveal their common dependence on longitudinal mass, after factorizing out an energy dependent part on the basis of the new scaling.