  We present a formula for computing proper pushforwards of classes in the Chow
ring of a projective bundle under the projection $\pi:\Pbb(\Escr)\rightarrow
B$, for $B$ a non-singular compact complex algebraic variety of any dimension.
Our formula readily produces generalizations of formulas derived by Sethi,Vafa,
and Witten to compute the Euler characteristic of elliptically fibered
Calabi-Yau fourfolds used for F-theory compactifications of string vacua. The
utility of such a formula is illustrated through applications, such as the
ability to compute the Chern numbers of any non-singular complete intersection
in such a projective bundle in terms of the Chern class of a line bundle on
$B$.
