The higher orders instability of a gas cylinder ambient with an incompressible inviscid liquid endowed with surface tension is analyzed. The perturbation equations up to third order are derived and solved. The surface displacements, the velocity potentials and the dispersion relations are derived for each order of axisymmetric perturbation. It is found that, up to third order, a transition from instability to stability states occurs when the perturbed wavelength equals the circumference of the gas cylinder. The stability discussions for the present ..