We report on what we believe to be the first measurement of the surface elastic constant, KI. This new elastic constant is manifested through the formation of a purely elastic instability. The instability is observed to form in thin smectic films, which are induced at the free surface of a liquid crystal droplet. Under polarising microscope, the instability is viewed as an array of interference fringes, which denote variation of both the orientation and the amplitude of the c director. The instability corresponds to a continuous rotation of the c director coupled to modulations of the molecular tilt angle theta. We propose that this mechanical instability is produced by the surface elastic term, K(1)c(2)divc. This new term to consider in the free energy density is found to be consistent both qualitatively and quantitatively with the experiments.