Introduction. In 1951, Zariski obtained as a special case of his connectedness theorems the fact that the total transform of a normal point by a birational transformation is connected [III]. In a recent paper, [IV], the connectedness of the total transform of a simple point by a birational transformation is proved without using the theory of Fholomorphic functions developed in [III]. In this paper another proof is obtained of this connectedness-theorem for a birational transformation at a simple point. In fact, a somewhat stronger result is proved, namely, the following. Chow has introduced the concept of linear connectedness, a set is said to be linearly connected if every pair of points can be connected by a sequence ofrational curves in that set. In this sense, the total transform of a simple point by a birational transformation is linearly connected.2 Furthermore, some applications are derived for specializations of a complete set of conjugates over a purely transcendental function field. The terminology and notations are from [I]. In conclusion, I want to express my warmest thanks to Professor A. Weil and Professor T. Matsusaka for their advice and interest during the preparation of this paper. I am especially indebted to Professor Weil for suggestions which improve the exposition of the proof of Theorem 1.