  The Miller-Morita-Mumford classes associate to an oriented surface bundle
$E\to B$ a class $\kappa_i(E) \in H^{2i}(B;\Z)$. In this note we define for
each prime $p$ and each integer $i\geq 1$ a secondary characteristic class
$\lambda_i(E) \in H^{2i(p-1)-2}(B;\Z)/\Z\kappa_{i(p-1)-1}$. The mod $p$
reduction $\lambda_i(E) \in H^*(B; \F_p)$ has zero indeterminacy and satisfies
$p\lambda_i(E) = \kappa_{i(p-1)-1}(E) \in H^*(B;\Z/p^2)$.
