  We show that the index of an elliptic Fourier integral operator associated to
a contact diffeomorphism $\phi$ of cosphere bundles of two Riemannian manifolds
X and Y is given by $\int_{B^*X}\hat{A}(T^*X)\exp{\theta} -
\int_{B^*Y}\hat{A}(T^*Y)\exp{\theta}$. Here $B^*$ stands for the unit coball
bundle and $\theta$ is a certain characteristic class depending on the
principal symbol of the Fourier integral operator. In the special case when X=Y
we obtain a different proof of the theorem of Epstein and Melrose.
