  In this paper, we obtain the explicit limit value of the Teichm\"{u}ller
distance between two Teichm\"{u}ller geodesic rays which are determined by
Jenkins-Strebel differentials having a common end point on the augmented
Teichm\"{u}ller space. Furthermore, we also obtain a condition under which
these two rays are asymptotic. This is similar to a result of Farb and Masur.
