where* {e (r), e (0 ) , e(z)} denotes the orthonormal basis associated with the cylindrical coordinate system (r, 0, z), and where a, b, c, g, A, B, D, and K are constants, are the only steady, rotational, universal motions for all incompressible Rivlin-Ericksen fluids in which the proper numbers of the first RivlinEricksen tensor are not all constant. In this paper we show that the motions characterized by (1.1) or by (1.2) are, in fact, dynamically possible without the help of any body force for all incompressible isotropic simple materials, provided that the velocity fields are maintained for all times s ~ ( oe, t], where t denotes some appropriate present time under consideration. As we shall see, velocity fields of the forms (1.1) or (1.2) may be maintained only for some time interval se(-c~,t] but not for all times s e ( o e , + oe) without violating the usual physical restriction that different material points of a body may not occupy the same location in space at the same time. This special feature distinguishes the motions from such other well known universal