Elastic constants have been calculated for the FCC metals Cu, Al and Ni using the Mobius transform and a short-range three-body potential correction. Unlike a previous calculation based only on pair potentials, the result is in good agreement with experiment since the restriction of the Cauchy relation has been removed. More importantly, it shows the potential application of the Mobius-inversion method for evaluating interatomic potentials from ab initio electronic structure calculations.