This paper investigates the behavior of vorticity in a polytropic or isothermally stratified fluid under the condition that the density profile remains constant in time. An expression for the velocity field of a general vorticity distribution (analogous to the Biot-Savart law for a uniform fluid) as well as expressions for the far velocity field, vector potential, and kinetic energy, are derived. The particular cases of the infinite straight vortex filament and the vortex ring each having infinitesimal cross-section are presented. They yield the surprising result that vortex filaments in a stratified fluid are propelled transverse to the stratification axis with a velocity given by (K/4πλ)In(b/λ) whereKandb2are the magnitude and cross-sectional area of the filament respectively and λ is the density scale height. For a vortex ring, this transverse motion is manifested as an expansion or contraction of the ring radius, depending on the ring's direction. This effect is unrelated to the curvature of the vorticity; rather, it is a direct result of the expansion/contraction of fluid moving in a density gradient. The implications for the vortex-ring sunspot model and for the formation of hillocks around magnetic flux fibrils are discussed.