Thin viscous jets are considered as they slowly fall, in a state of near-neutral buoyancy, through a liquid. An equation is derived which describes the path of the jet. A small perturbation analysis of nearly vertical jets is carried out, and shows that they are necessarily unstable and will eventually deviate significantly from the vertical. Numerical integration of the nonlinear equation describes the nature of this deviation. These results model some experimental observations made by Taylor (1969).