  M. F. Atiyah proved that the index of a transversally elliptic operator
relative to a free action can be computed by using indices of elliptic
operators on the orbit manifold. In this paper, we derive an explicit formula
for the transversal indices on S^1-bundles over complex projective spaces.
Using this explicit formula and the index map, we prove Lefschetz formula for
the n-dimensional complex projective space with the canonical action of T^{n +
1}.
