How much does the rigid body rotate? A Berry's phase from the 18th century

A formula, apparently new, is derived for the spatial rotation of a free rigid body during one period T of its body angular momentum vector. This formula has the structure of recent phase formulas of Berry [Proc. R. Soc. London Ser. A 392, 45-57 (1984); Sci. Am. 259 (6), 46-55 (1988)] and Hannay [J. Phys. A 18, 221-230 (1985)]; namely, it consists of a geometric and a dynamic part. It is derived by applying Stokes' theorem to a certain closed curve in phase space.