  Given two measured laminations mu and nu in a hyperbolic surface which fill
up the surface, Kerckhoff [Lines of Minima in Teichmueller space, Duke Math J.
65 (1992) 187-213] defines an associated line of minima along which convex
combinations of the length functions of mu and nu are minimised. This is a line
in Teichmueller space which can be thought as analogous to the geodesic in
hyperbolic space determined by two points at infinity. We show that when mu is
uniquely ergodic, this line converges to the projective lamination [mu], but
that when mu is rational, the line converges not to [mu], but rather to the
barycentre of the support of mu. Similar results on the behaviour of
Teichmueller geodesics have been proved by Masur [Two boundaries of
Teichmueller space, Duke Math. J. 49 (1982) 183-190].
