The author is concerned with the Painlevé equation (1) . The equation (1) has exactly three different types of solutions for : (A) a 2-parameter family of solutions that oscillate infinitely often and slowly approach as ; (B) a 1- parameter family of solutions asymptotic to as ; and (C) a family of solutions, each of which tends to at some finite value of .
It is shown that the solutions of type (B) form a surface separating the solutions of type (A) from these of type (C). Every solution to type (B) must enter the interior of the parabola for the final time through one of the points with positive and uniquely determined slope , where is a real parameter. On the other hand, for each point on there exists a unique positive slope such that the solution of (1) throughout with the slope is asymptotic to as . The function is proved to be continuous and differentiable everywhere except at . Several approximate values of and the graphs of and of the surface are given.