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This equation tells us that the amount of flexure should be proportional to the square of the distance from the center of the mirror and in going around the edge of the mirror (say clockwise) the displacement should take the shape of a sine-curve and it should go through two complete cycles (two maxima and two minima) in going once around the mirror.

If we were to balance a mirror blank, face-up, on a small rigid support located under the center of the mirror and then hang two equal weights from the edge of the blank at opposite sides, the glass would bend under the load as a simple beam. The following formula gives the flexure,  h, for a beam so supported and loaded:

where  Kh  is a quantity which is determined by the force applied and other factors,  Z  is the radial distance from the center to the point in question and  r  is one half the diameter of the mirror blank. This equation tells us that in the central part of the mirror the flexure will be proportional to  Z2, but out toward the edge it will flatten out a little compared with this curve. This suggests that as far as the  Z2  feature of the astigmatism formula is concerned, our flexed mirror will follow the desired curve fairly well in its central part, but out toward the edge it may deviate a little from the desired curve.

If we should support the mirror blank at two points at opposite sides at its outer edge instead of at its center, and then hang our two equal weights at points midway between the two points of support (90° each way from the