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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

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