  In an earlier paper, I defined a new winding number of regular closed curves
on complete euclidean/hyperbolic surfaces and showed that this winding number,
together with the free homotopy class, determines the regular homotopy class.
In this paper, I give a Whitney-type formula for the winding number of
non-null-homotopic generic regular closed curves on surfaces with a complete
euclidean or hyperbolic structure, generalizing the formula for curves on a
torus by Tanio and Kobayashi.
