The dependence of the magnetic surface anisotropy constant K(S) on the magnetic layer thickness 2L has been studied in amorphous Fe70B30/Ag and amorphous C80B20/Ag multilayer films, by magnetometry and ferromagnetic resonance (FMR). Interpretation of the FMR data has required the extension of the theory of FMR in an ultrathin amorphous film so as to include the effect of an in-plane volume anisotropy. A method of determining the value of K(S) from perpendicular magnetization curves has been justified by introducing the surface anisotropy energy through the general exchange boundary conditions rather than as an effective volume anisotropy. The deduced values of K(S) for Fe70B30 and Co80B20 multilayers were found to exhibit a qualitatively similar thickness dependence but to be of opposite sign. A magnetostrictive surface anisotropy mechanism is proposed.