on a 12.5-inch mirror and found to be adequately strong. A value of  K  equal to 14.5x106  will, therefore, give a maximum force that the warping harness need be able to provide. The actual working force needed to produce the required amount of flexure in a pyrex mirror is considerably less than this.

     In order to keep the weight of the warping harness to a minimum, the cross-beams or x-member should be designed to have the material of that member stressed to its yield strength when a load of this magnitude or a little less is applied to the ends of each of its four arms. If the crossed-beams or x-member is made of annealed, or hot-rolled mild steel, this condition will be met if the maximum width (at midsection)  b, and the depth, or thickness,  a, conform to the following equation:

Equation 18

where  D  is the actual diameter of the mirror blank.

     For small telescopes where only a small amount of warpage will be sufficient, a simple design of harness which contacts the glass blank with four small circular shoes should be satisfactory. These shoes should be articulated, or pivoted so that they can seat themselves well on the surface of the glass. In larger telescopes, where larger amounts of astigmatism must be neutralized, the shoe design shown in Figures 2, 3a, and 3b should be employed. These shoes have two contact areas each, separated by a distance which subtends an angle of 30° at the edge of the mirror blank. The force from the crossed-beams or x-member is applied to the midpoint of each shoe which, in turn, divides it equally between its two contact areas. Each shoe is pivoted or articulated so that it