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.
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