Abstract: In this paper the stability of linear Rossby waves in a stratified shea:fluid is investigated by using the qualitative theory of ordinary differential equations.It is found that (Ⅰ)if the basic current has no shear at all,the Rossby waves are always stable;(2) if the basic current has horizontal shear only,the unstablecriterion for the waves is sK2+n2f0(f0-∂u/∂y)u/∂y)>0;(4)if the basic current has both horizontal and vertical shears if sK2+n2f0(f0-∂u/∂y)2+n2f0(f0-∂u/∂y)>0,the unstable crtierion is f0(f0-∂u/∂y)n -∂u/∂y)>0