A Proof of the Puiseux - Halphen Inequality in the Theory of Spherical Pendulum Based on an Interesting Identity of S. Ramanujan

Abstract

The classical problem of finding bounds for the apsidal angle in the case of the spherical pendulum has been considered by Halphen in his classical treatise on 'Fonctions Elliptiques'. His proof is based on certain inequalities among the Elliptic-function constant. We prove these inequalities of Halphen and Puiseux, in a simple way, using an interesting identity of S. Ramanujan; besides, we obtain positive-term series for the quantities involved which may enable one to improve such bounds.