  We introduce a deformation of Riemann surfaces and we are interested in the
convergence of this deformation to a point of the Gardiner-masur boundary of
Teichmueller space. This deformation, which we call the horocyclic deformation,
is directed by a projective measured foliation and belongs to a certain
horocycle in a Teichmueller disc. Using works of Marden and Masur and works of
Miyachi, we show that the horocyclic deformation converges if its direction is
given by a simple closed curve or a uniquely ergodic measured foliation.
