  In this paper, we first define the equivariant infinitesimal $\eta$-form,
then we compare it with the equivariant $\eta$-form, modulo exact forms, by a
locally computable form. As a consequence, we obtain the singular behavior of
the equivariant $\eta$-form, modulo exact forms, as a function on the acting
Lie group. This result extends a result of Goette and it plays an important
role in our recent work on the localization of $\eta$-invariants and on the
differential $K$-theory.
