  In this article, we present the definitive transformation formulae of the
mass aspect and angular momentum aspect under BMS transformations. Two
different approaches that lead to the same formulae are taken. In the first
approach, the formulae are derived by reading off the aspect functions from the
curvature tensor. While in the second and more traditional approach, we read
them off from the metric coefficients. As an application of the angular
momentum aspect transformation formula, we directly verify a relation
concerning the Dray-Streubel angular momentum. It also enables us to
reinterpret our calculations in terms of differential forms on null infinity,
and leads to an exact expression of the Drey-Streubel angular momentum of a
general section. The formulae we obtained played crucial roles in our recent
work on supertranslation invariant charges, and resolved some inconsistencies
in the literature.
