  We show that fiberwise stable vector bundles are preserved by relative
Fourier-Mukai transforms between elliptic threefolds with relative Picard
number one. Using these bundles we define new invariants of elliptic
fibrations, and we relate the invariants of a space with those of a relative
moduli space of stable sheaves on it. As a byproduct, we calculate the
intersection form of a certain new example of an elliptic Calabi-Yau threefold.
