  Let \E : 0 --> S --> E --> Q --> 0 be a short exact sequence of hermitian
vector bundles with metrics on S and Q induced from that on E. We compute the
Bott-Chern form of \E corresponding to any characteristic class, assuming E is
projectively flat. The result is used to obtain a new presentation of the
Arakelov Chow ring of the arithmetic grassmannian.
