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Introduction J.D.S. Jones and J.H. Rawnsley Hamiltonian circle actions on symplectic manifolds and the signature G M;! M TG G G M G V M G !V;X d X X M G M oeM b M b M oeM M b M j M M oeM jp pjp M p p oeM jp : k b M Hamiltonian momentum map Let beacompact symplectic manifold with a Hamiltonian action of the circle which has isolated fixedpoints. Then where is the signatureof and is the-th Betti number of . A smooth action of a connected Lie group on a symplectic manifold( ) is if there is a map : , where is the dual of the Lie algebra = of , with the following properties. The map is equivariant with respect to the given action of on and the coadjoint action of on . For any let be the vector field on defined by the action of and the element . Then ( )= ( )( ) for anyvector field on . The map is called a for the action of . The purpose of this note is to prove the following theorem. ( )= ( ) ( ) ( ) ( ) The proof of this theorem is not difficult. First, the Atiyah--Bott fixed p