History of schlicht functions and elementary properties of the nth region Variations of schlicht functions The critical points of the differential equation The} $\Gamma$-structure. Behavior in the large Geodesics. Continuity theorem Functions which are regular in $z < 1$ and satisfy the differential equation The length-area principle. Teichmuller's method Relations between} $P(w)$ and $Q(z)$ Lowner curves Linear forms and the supporting surface The portion of the boundary of} $V_n$ corresponding to single analytic slits Parametrization of the boundary of} $V_n$ The region} $V_3$ A method for investigating the conjecture} $a_4\leq 4$ The region of values of the derivative of a schlicht function (by Arthur Grad).