An extensive set of experiments were performed to determine the stability boundary for the longitudinal oscillation of steel spheres settling through Hydroxy Propyl Guar (HPG) contained in long vertical tubes, a phenomenon first reported by Mollinger, et al. (1999). Results are reported in d/D-pH parameter space, where d is the sphere diameter and D is the internal diameter of a tube. Longitudinal oscillations, which owe their existence to the self-healing nature of the gel, occur over the region 0.48 < d/D < 0.86 and 8.31 < pH < 8.82. We discover that the upper pH limit of this stability domain is linked to the HPG gel point. During the course of experimentation, additional instabilities in the form of high-frequency transverse oscillations or helical motions were discovered and documented. The pure transverse oscillations found in the domain 8.035 < pH < 8.31 and 0.9 < d/D < 0.6 result from the elastic nature of the guar in this region. For 6.25 < pH < 8.035 transverse oscillations or stable descents are interspersed between bands of tight helical motion at large d/D and wide helical motion at low d/D. At the lowest value pH = 6.25 the guar is a Newtonian fluid, yet the spiral motion persists in small bands of d/D. We have verified, in the literature and through additional experiments using distilled water as the Newtonian liquid that the spiral motion of spheres falling inside vertical tubes does indeed exist. [ABSTRACT FROM AUTHOR]