Contour integration in the theory of the spherical pendulum and the heavy symmetrical top

Introduction 107 1. History of the method and contents of the present paper. Tables of results ...... 107 2. Basic theory 110 Part I. The spherical pendulum 111 1. Puiseux' inequality by a new method 111 2. An extension of Puiseux' inequality 113 Part II. The heavy symmetrical top . . . 115 1. General . 115 2. a/:3 1. Lower bound .. 121 8. 0 1. Upper bounds .. 122 9. Exact bounds 127 10. Continuity of 4) for a fixed a/: . . . 129 Appendix: The retrogradely precessing heavy symmetrical top and the spherical pendulum 130