  Two relations, the virial relation $M_{\rm ADM}=M_{\rm K}$ and the first law
in the form $\delta M_{\rm ADM}=\Omega \delta J$, should be satisfied by a
solution and a sequence of solutions describing binary compact objects in
quasiequilibrium circular orbits. Here, $M_{\rm ADM}$, $M_{\rm K}$, $J$, and
$\Omega$ are the ADM mass, Komar mass, angular momentum, and orbital angular
velocity, respectively. $\delta$ denotes an Eulerian variation. These two
conditions restrict the allowed formulations that we may adopt. First, we
derive relations between $M_{\rm ADM}$ and $M_{\rm K}$ and between $\delta
M_{\rm ADM}$ and $\Omega \delta J$ for general asymptotically flat spacetimes.
Then, to obtain solutions that satisfy the virial relation and sequences of
solutions that satisfy the first law at least approximately, we propose a
formulation for computation of quasiequilibrium binary neutron stars in general
relativity. In contrast to previous approaches in which a part of the Einstein
equation is solved, in the new formulation, the full Einstein equation is
solved with maximal slicing and in a transverse gauge for the conformal
three-metric. Helical symmetry is imposed in the near zone, while in the
distant zone, a waveless condition is assumed. We expect the solutions obtained
in this formulation to be excellent quasiequilibria as well as initial data for
numerical simulations of binary neutron star mergers.
