  We derive a general formula for the Euler characteristic of a fibration of
projective hypersurfaces in terms of invariants of an arbitrary base variety.
When the general fiber is an elliptic curve, such formulas have appeared in the
physics literature in the context of calculating D-brane charge for M-/F-theory
and type-IIB compactifications of string vacua. While there are various methods
for computing Euler characteristics of algebraic varieties, we prove a
base-independent pushforward formula which reduces the computation of the Euler
characteristic of relative hypersurfaces to simple algebraic manipulations of
rational expressions determined by its divisor class in a projective bundle. We
illustrate our methods by applying them to an explicit family of relative
hypersurfaces whose fibers are of arbitrary dimension and degree.
