"Shorn of technicalities, the problem resolves itself thus: the mirror must rest on a number of self-adjusting supports, geometrically symmetrical with each other in relation to the circular bearing face of the disk, and each taking an equal share of the weight."
Why an equal share of the weight? Why must the supports be self-adjusting? Why is symmetry required?
"Suppose we investigate a three-point support for, say a 10-inch mirror; where would the three supports have to be located? Obviously in a circle of such radius that the mirror is divided into two imaginary portions, an inner circular disk, and an outer annular ring, of equal weight and area."
Why equal weight? Obviously equal weight and equal area are inconsistent because the edge region of the mirror is thicker than the center for most flat-back mirrors. David Chandler's CELL.EXE corrected this contradiction. Note that something as simple as Simpson's Rule integration (perfectly accurate on polynomials up to and including a cubic) works very well on this problem; surely there were people who knew about that!
"Thus a disk 10 inches in diameter has an area of 78.54 square inches. Half of this is 39.27 square inches, equivalent approximately to a 7-inch circle."
Nothing like a little truth to mask the fraud!
"The three supports therefor should be arranged 120 degrees apart at 3 1/2 inches radius, which we shall name the 'radius of equilibrium'. Such supports will be found quite satisfactory for small mirrors, and they may take the form of a raised annular ring of 7 inches mean diameter, not more than one inch broad, at the bottom of the cell, and allow the mirror to select its own three points of support, which it will do in practice."
Thus Hindle proposes a ring support that is inaccurately made. There is no justification for the chosen width of the support ring. We now know, from Selke's formula, that the best radius for uniformly supporting ring is at the .645 zone to minimize maximum deformation, and at the edge to minimize deviation from the parabola of best fit (per Roark's formulas). We now know that for three points of support, the minimum least square deviation from the parabola of best fit is witht he support points at the 38% zone. Of course, Hindle does not propose a uniform pressure support ring...
"It is preferable, however, to glue three strips of cork or felt to the surface of the seating, as shown in Figure 1, to define more accurately the areas of support."
At least he got this part right: there are areas of support, not points. Unfortunately, in a 1935 postscript, he advocates steel ball supports and says, "Cork, felt, or other resilient materials are to be deprecated." Hindle goes on to say that the glass must easily slide on the support points. OK, I will deprecate those materials, and use silicone blobs instead. They provide the desired degree of freedom by their elasticity.
Obviously these are two very different problems. Even in 1933 engineers knew the biharmonic equation for bending of thin plates. They knew that the areas of equal mass would be connected rigidly and that forces and moments would be transmitted across the boundaries. Even in 1933 nobody would have applied that idea to a beam; why would anyone expect it to work on a plate or a solid object? As with a beam, Hindle's flotation might have been best if the mirror were to be cut into separate pieces, but real mirrors are continuous pieces of glass. Some cases of the biharmonic equation could be solved in 1933 by means of fourier series; however, the work of doing this is very great. It was worse, because solving for the deformation was not enough. You would have to find the deviation from the parabola of best fit, and then repeat this for a large number of flotation configurations to determine what is "best". In 1933 a "computer" consisted of an office full of beautiful young women (Admiral Hopper) with mechanical calculators and instructions. It was expensive!
Another fault of Hindle is his concept "radius of equilibrium". This might be a valid concept for a small-angle pie slice of glass that is detached from its neighbors. In that case, the deformation of the glass under its own weight could be found by beam theory. In this case, the center of mass at the 2/3 zone is NOT the support point that yields minimum deformation! Continuous ring concepts cannot be extended to the three-point flotation where the flotation points are 120 degrees apart. The formulas of Roark and Selke, which consider continuous ring supports do not require isolated wedges because they allow for the transmission of moments and forces between regions of the mirror. Well, at least Hindle did not advocate support at the edge as did Couder; that would have been even worse!
Hindle commits a sin of ommission. He overlooks a lot of possible flotation systems because of his ideas of "symmetry". For example, he does not allow for unequal loads on the support points; we now know that in the case of multiring flotations the optimum solutions have unequal loads. He does not allow for unequal angular displacements of the support points in separate rings. We now know that optimum flotations have unequal angular displacements.
Another sin of omission is Hindle's disregard for support configurations other than three, nine, and eighteen points. We now know that configurations with four, five, six, seven, twelve, fifteen, sixteen, nineteen, and other numbers of points are useful and easy to realize in terms of load-spreading levers and triangles. The six point flotation is just like Hindle's eighteen pointer, but with the triangles removed. Why didn't he think of it? The only answer he gives is his own esthetic thoughts of symmetry and balance.
Hindle's mechanical implementation has a number of defects. These are the use of mirror clips, the use of edge supports, the use of anti-rotation pins, and the use of mechanical bearings for the pivots.