The classic Steinhausen/Groen mathematical description of endolymph flow in a toroidal semicircular canal is extended to the case where the size, shape, and curvature of the canal lumen change continuously through the duct, utricle, and ampulla. The resulting second-order differential equation has three coefficients, unlike the equation of a torsion pendulum, which has only two. The salient anatomical parameters which determine endolymph motion are: the length of the central streamline occupying the center of the canal lumen; the area enclosed by this streamline as projected into the plane of rotation; the average inverse cross-sectional area of the lumen (taken around the central streamline); and the average inverse squared cross-sectional area, weighted by a local wall shape factor. These parameters are evaluated and the average displacement of the face of the cupula is estimated for the human, guinea pig, and rat, based on new anatomical data presented in companion papers. The model predicts that the dynamic range of human average cupula motion lies between 520 A and 10 microns.