  The mass loss of an isolated gravitating system due to energy carried away by
gravitational waves with a cosmological constant $\Lambda\in\R$ was recently
worked out, using the Newman-Penrose-Unti approach. In that same article, an
expression for the Bondi mass of the isolated system, $M_\Lambda$, for the
$\Lambda>0$ case was proposed. The stipulated mass $M_\Lambda$ would ensure
that in the absence of any incoming gravitational radiation from elsewhere, the
emitted gravitational waves must carry away a positive-definite energy. That
suggested quantity however, introduced a $\Lambda$-correction term to the Bondi
mass $M_B$ (where $M_B$ is the usual Bondi mass for asymptotically flat
spacetimes) which would involve not just information on the state of the system
at that moment, but ostensibly also its past history. In this paper, we derive
the identical mass-loss equation using an integral formula on a hypersurface
formulated by Frauendiener based on the Nester-Witten identity, and argue that
one may adopt a generalisation of the Bondi mass with $\Lambda\in\R$
\emph{without any correction}, viz. $M_\Lambda=M_B$ for any $\Lambda\in\R$.
Furthermore with $M_\Lambda=M_B$, we show that for \emph{purely quadrupole
gravitational waves} given off by the isolated system (i.e. when the "Bondi
news" $\sigma^o$ comprises only the $l=2$ components of the "spherical
harmonics with spin-weight 2"), the energy carried away is \emph{manifestly
positive-definite} for the $\Lambda>0$ case. For a general $\sigma^o$ having
higher multipole moments, this perspicuous property in the $\Lambda>0$ case
still holds if those $l>2$ contributions are weak --- more precisely, if they
satisfy any of the inequalities given in this paper.
