Two systems of equations of motion, designated as (I) and (II), for thin elastic cylindrical shells are derived which include the effects of both transverse shear deformation and rotatory inertia. The character of the two systems of equations is such that, upon the neglect of transverse shear deformation and rotatory inertia, Eqs. (I) will reduce to those known as Love's first approximation, while (II), which enjoys a considerable mathematical simplicity as compared to (I), will correspond to those given by Donnell. Both systems of Eqs. (I) and (II) are employed in a study for propagation of axisymmetric waves in an infinite cylindrical shell. The agreement between the predictions of the two systems of equations, in all modes of motion, for phase velocities of propagated waves in the complete range of wavelengths is found to be excellent. The results, with reference to the nature of the modes of motion according to both (I) and (II), are further examined and the relative merit of the present paper to the work of other authors is discussed.