Consideration is given to the linear stability of buoyant boundary layers and plumes which belong to the class of flows for which (a) the streamwise velocity vanishes in the free stream and (b) the transverse velocity is inward-directed and has a finite value in the free stream. Disturbance equations for such flows are derived taking account of the fact that the basic flows depend upon the streamwise coordinate. The formulation is specialized to the case of the natural convection plume generated by a horizontal line source of heat. The existence of the so-called bottling effect is demonstrated, wherein the disturbance vorticity and temperature are contained within the boundary layer of the flow. The neutral stability curve exhibits both a minimum Grashof number and a lower branch, in contrast to the neutral curve for the conventional stability analysis, which does not exhibit these features. Consideration is given to the amplification of disturbances and to the frequencies which are the most amplified. Results are also presented for the limiting case of inviscid instability.