In an effort to gain a better understanding of the mechanics involved in the formation ofvortices in buoyancy driven flows, an analysis on the stability of the laminar free convection, due to a line source of heat with ambient shear, was performed by numerical solution of a viscous linear stability equation. The formation yielded a curve of neutral stability, based on plume thickness and ambient shear, delineating stable and unstable domains. It was thereby indicated that increased buoyancy increases stability at a given elevation in the flow field, and that the instability first occurs at a critical elevation. This critical elevation increases with buoyancy strength, but decreases with ambient shear. The solution yielded that the disturbed motion must be a standing-wave phenomenon, of thetype typically developing into vortex flows. For a given disturbance wavelength, the analysis demonstrated a periodic row of vortices, and an optimization analysis of all wavelengths indicated that the flow should break up into a vortex pattern at a most unstable wavelength or vortex spacing upon the attainment of a most unstable plume thickness Reynolds Number. These results have been found to agree closely, in functional form, with experimental results on the formation of multiple fire whirls. In the natural occurrence of vortices, the analysis suggested that the vertical distribution of ambient vorticity is a determinant factor. It was also indicated that a strong enough convection column can overpower the ambient vorticity and thereby prevent or destroy a vortex flow.