  We consider a relative Fourier-Mukai transform defined on elliptic fibrations
over an arbitrary normal base scheme. This is used to construct relative Atiyah
sheaves and generalize Atiyah's and Tu's results about semistable sheaves over
elliptic curves to the case of elliptic fibrations. Moreover we show that this
transform preserves relative (semi)stability of sheaves of positive relative
degree.
