The asymptotic solution for the Euler-Lagrange equation which describes the surface density profile of liquid metals is derived. The liquid metal is modelled as consisting of a Fermi electron gas and classical one-component-plasma ions. These latter systems are coupled via a first-order electron-ion pseudopotential which is considered to be appropriate for characterizing liquid alkali metals. It is demonstrated here that the density profile can be classified as either a monotonic or an oscillatory type, but our numerical data show that all liquid alkali metals fall into the latter category.
