  We extend to the special case of Bhabha scattering the "Z-peak subtracted"
representation previously applied to e+e- to f f(bar) (f different from e).
This allows us to analyze the process at any energy while imposing in an
automatic way the constraints set by high precision measurements at the Z peak.
The procedure turns out to be particularly convenient for the search of a
certain class of new physics effects at variable energy. A few examples are
considered and the information thus obtained is combined with the corresponding
one that can be derived from the other e+e- annihilation processes, both at
present (LEP2) and at future colliders. This shows that the role of Bhabha
scattering in this respect can be quite relevant.
