  This Letter is motivated by some recent experiments on pan-cake shaped
nano-samples of metallic glass that indicate a decline in the measured shear
modulus upon decreasing the sample radius. Similar measurements on crystalline
samples of the same dimensions showed a much more modest change. In this Letter
we offer a theory of this phenomenon; we argue that such results are
generically expected for any amorphous solid, with the main effect being
related to the increased contribution of surfaces with respect to bulk when the
samples get smaller. We employ exact relations between the shear modulus and
the eigenvalues of the system's Hessian matrix to explore the role of surface
modes in affecting the elastic moduli.
