  In the Bondi formulation of the axisymmetric vacuum Einstein equations, we
argue that the ``surface area'' coordinate condition determining the ``radial''
coordinate can be considered as part of the initial data and should be chosen
in a way that gives information about the physical problem whose solution is
sought. For the two-body problem, we choose this coordinate by imposing a
condition that allows it to be interpreted, near infinity, as the (inverse of
the) Newtonian potential. In this way, two quantities that specify the problem
-- the separation of the two particles and their mass ratio -- enter the
equations from the very beginning. The asymptotic solution (near infinity) is
obtained and a natural identification of the Bondi "news function" in terms of
the source parameters is suggested, leading to an expression for the radiated
energy that differs from the standard quadrupole formula but agrees with recent
non-linear calculations. When the free function of time describing the
separation of the two particles is chosen so as to make the new expression
agree with the classical result, closed-form analytic expressions are obtained,
the resulting metric approaching the Schwarzschild solution with time. As all
physical quantities are defined with respect to the flat metric at infinity,
the physical interpretation of this solution depends strongly on how these
definitions are extended to the near-zone and, in particular, how the "time"
function in the near-zone is related to Bondi's null coordinate.
