The principal structural part (or parts) of the warping harness consists of either a pair of crossed beams, as shown in Figure 2, or an x-member, as shown in Figure 3a. It should be capable of providing four equal forces - two pushes and two pulls - equally distributed (90° apart) around the edge of the mirror. The four forces will be equal if, in the crossed beam design, the two beams are pivoted at their midpoints or, in the x-member design, opposite arms are of equal lengths. Preferably, all four arms should be of equal length. Tolerances for lengths of the arms have not been calculated, but it is felt that if reasonable care is exercised in laying out the lengths of the four arms, this specification will have been met satisfactorily.
The force required to warp a glass mirror has not yet been measured or calculated. As a result we can only give a rough value for the strength requirements of the warping harness. The force (at each of the four points on the edge of the mirror) required to bend the mirror is given by the following formula:
where P is the required force, K is a constant which depends primarily on the modulus of elasticity of the glass, t is the thickness of the glass blank, D2 is the calculated illuminated diameter (not the diameter of the glass blank), and wa is the amount of astigmatism which must be produced