We have used Schwarzschild's (1979) method to study the variety of self-consistent solutions available to the family of triaxial “perfect ellipsoids” (de Zeeuw 1985). the time-averaged density at 240 points within the mass model is computed for each of 1065 orbits distributed regularly in phase space and covering all four major orbit families. the underdetermined linear system thereby defined is solved in two ways. First, Lucy's (1974) iterative scheme is used to find a “smooth” solution, lying in the interior of the (mathematically-allowed) solution space. Second, linear programming is used, with linear combinations of the x and z components of the total angular momentum as cost functions, to delineate the boundary of the projection of the solution space in the Lx — Lz plane. the above procedure is applied to 21 figures, with axis ratios, b/a and c/a, chosen at equal intervals of 1/8.