The design of the University of California Ten Meter Telescope requires supporting 36 individual hexago-nal mirrors; these 36 mirror segments constitute the primary mirror of the telescope. Each segment is axially supported by a set of whiffletrees, which applies forces at 36 points on the back of the segment. Because these mirror segments are thin (1.8 m diameter, 7.5 em thick), the optimization of the location and magnitude of the supporting forces is a crucial part of the design. We have analyzed this problem by using the finite element method. Careful testing of a finite element model and the associated computer programs is required before the results can be trusted. We began by analyzing the deflection of a thin, flat, circular plate, since this can be calculated independently and then compared with the results of the finite element method. The factors of curvature of the mirror segment, shear, and hexagonal shape were then included one at a time, in order to arrive at the final model. The optimization of the support forces was based on minimizing the r.m.s. surface error, with a design goal of 0.010 Am or less. This is a surface error roughly one hundred times smaller than that resulting from an unoptimized support. The optimization calculations were performed on a large minicomputer. Practical con-siderations required that the physical symmetries of the problem be taken into account, in order to reduce the number of independently optimized variables to a manageable size. The use of the finite element method pro-duced a solution that met the design requirements, and it gave useful information about the sensitivity of mirror deformations to variations in the support structure.