  We show that positivity of the CM line associated to a family of polarised
varieties is intimately related to the stability of its members. We prove that
the CM line is nef on any curve which meets the stable locus, and that it is
pseudoeffective (i.e. in the closure of the effective cone) as long as there is
at least one stable fibre. We give examples showing that the CM line can be
strictly negative or strictly positive on curves in the unstable locus.
