High resolution elastic constant measurements on a holmium single crystal were made between 80 and 140 K. The results near T//N suggest that in holmium the c-axis stress dependence of T//N is opposite in sign and about three to four times larger than the basal plane stress dependence. The measurements further show that the 95 K anomaly is present only in C//4//4 and that the magnitude of the anomaly diminishes with sample annealing thus supporting a domain origin for the effect.