  Self consistency in the analysis of transmission measurements for K^+ on
several nuclei in the momentum range of 500-700 MeV/c is achieved with a
"t_{eff}(rho)rho" potential and new results are derived for total cross
sections. The imaginary part of the t_{eff} amplitude is found to increase
linearly with the average nuclear density in excess of a threshold value. This
phenomenological density dependence of the K^+ nucleus optical potential also
gives rise to good agreement with recent measurements of differential cross
sections for elastic scattering of 715 MeV/c K^+ by Li^6 and C.
