  This paper compares three different methods about computing joint terms in
on-shell action of gravity, which are identifying the joint term by the
variational principle in Dirichlet boundary condition, treating the joint term
as the limit contribution of smooth boundary and finding the joint term by
local SO(1,$d-1$) transformation. In general metric gravitational theory, we
show that the differences between these joint terms are some variational
invariants under fixed boundary condition. We also give an explicit condition
to judge the existence of joint term determined by variational principle and
apply it into general relativity as an example.
