  K$^+$--nucleus quasielastic cross sections measured for a laboratory kaon
beam momentum of 705 MeV/$c$ are presented for 3--momentum transfers of 300 and
500 MeV/$c$. The measured differential cross sections for C, Ca and Pb at 500
MeV/$c$ are used to deduce the effective number of nucleons participating in
the scattering, which are compared with estimates based on the eikonal
approximation. The long mean free path expected for K$^+$ mesons in nuclei is
found. Double differential cross sections for C and Ca are compared to
relativistic nuclear structure calculations.
