Figure 3 shows the above relationships applied to the design of 5½' bows. The least curvature is obtained with the light circle having 37" radius. I prefer the ellipse which is shown solid as it gives a slight whipendedness and a tapering limb.
The extreme of whipendedness that is obtainable with an ellipse is shown as the Limit Curve. I have no mathematical reason for using the intermediate curve but wished to stay between the two obvious limits.
and the ellipse
Click for a larger image
The thickness scales are double so that half size reproductions may be scaled directly.
The thickness for various assumptions of R/T ratio are shown in the curve T. The widths are shown in curve W. These proportions are theoretically correct for a rectangular section bow. The weight of the bow will depend only on the elasticity of the wood, which in turn fixes the stress. If these are 1,000,000 and 10,000 respectively, the three bows shown will have weights of 60 lbs., 44 lbs. and 34 lbs.
Inserting a handle affects the proportions only slightly and leaves every inch of the limb almost equally stressed. With such a definite and theoretically correct layout, the bow-builder is concerned only with variations in wood, grain, knots, etc., and with the special forming at handle and tips.
It should be noted, in using such a diagram as Figure 3, that the horizontal scale does not check with linear distances. Those distances are marked along the ellipse for use in laying off dimensions on the stave but thickness and widths should be read directly below them. The solid curve along the bow limbs has semi-axes of 24" and 31¾".
Incidentally, any bow following these proportions may have its pull reduced by any given percentage by reducing the width curve by the same percentage and the stress and curve of the fully drawn bow will remain unchanged.
I suggest the above as the real type of solution of bow design giving equal stress at all points of the limb length; departure from a circle in a desirable direction; a tapered limb thickness and rational looking widths.
Now to tackle the reflex problem. Here's wishing someone luck.