Kiefer and Wolfowitz (1959) proved that the optimal design for estimating the highest coefficient in polynomial regression is supported by certain Tchebycheff points. Hoel and Levine (1964) showed that the optimal designs for extrapolation in polynomial regression were all supported by the Tchebycheff points. These results were extended by Kiefer and Wolfowitz (1965) to cover nonpolynomial regression problems involving Tchebycheff systems and the large class of designs supported by the Tchebycheff points was characterized. In the present paper it is shown that the optimal design for estimating any specific parameter is supported by one of two sets of points for Tchebycheff systems with certain symmetry properties. Different proofs of the Kiefer-Wolfowitz results are also presented. The author wishes to thank Professor Kiefer for providing one of the counterexamples in Section 6.