1. Introduction. The geometry of slides and turns of oriented lineal elements in the plane was first studied by Kasner [10]. Slides and turns generate whirls, which constitute a three-parameter group Wz. The product of Wz and Mz, the three-parameter group of Euclidean displacements in the plane, yields a sixparameter group of whirl-motions 1 G&. The geometry of turbines 2 , and also of general series of lineal elements, under G 6 was investigated by Kasner in [10] and, in subsequent papers, by

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... asner and DeCicco, particularly in [3], [4], [11], [12]. The author investigated the geometry of series of lineal elements under the seven-parameter group of whirl-similitudes G7 (of which GQ is a subgroup) in [6], [7], [8]. Among other things, the author showed that G7 is isomorphic to the group of collineations of the points in quasi-elliptic three-space, the geometry of which had been previously studied by Blaschke [1], [2] and Griinwald [9] ; he also showed how the geometry of Wz, G%, and £7 can be interpreted kinematically as the displacement of one plane over another. In this paper we investigate the geometry of spherical whirls and whirlrotations of oriented lineal elements on a sphere. Some results in this field have already been obtained by Strubecker [15], who mapped the points of elliptic three-space E z one-to-one upon the oriented lineal elements of a unit sphere. Using synthetic methods, Strubecker deduced, from the geometry of lines in Ez, theorems on spherical turbines and families of curves on a sphere, analogous to others found by Kasner for the plane [10]. We pursue the geometry of whirls and whirl-rotations on a sphere in other directions and by means of other methods. With the aid of quaternions we shall investigate the differential geometry of series of lineal elements on a sphere subject to two groups, SB 3 and @6-analogous respectively to Wz and Ge in the plane-determining their fundamental differential invariants and "Serret-Frenet formulae." Our principal objective is to present a characterization of the geometry of whirls and whirl-rotations on a sphere in terms of the kinematic geometry of continuous