The three papers comprise the published and unpublished parts of Zermelo's Habilitation thesis. In Zermelo's words, the thesis 'seeks to explain the flow of an incompressible, frictionless (two-dimensional) fluid in a spherical surface by the use of a theory as systematic as the one which already exists for planar flows. An investigation of this sort is of intrinsic geometric interest in particular because much of what happens in the plane on an infinite scale often defying at least intuition takes place on a sphere on a finite scale. Proceeding along such lines, it should therefore not be impossible to shed light on some of the processes involved both in the propagation of atmospheric cyclones and in the currents of the sea." The published part 1902a begins with a systematic construction of the hydrodynamics of an ideal liquid on an arbitrary two-dimensional surface; s1902b develops the dynamics of point vortices on the sphere; s1902c is devoted to the analysis of the relative motion of vortices and concludes with the absolute motion of three vortices.