This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
In the above demonstration, it is assumed that the true equilibrium polygon will pass through the center of each abutment, and also through the center of the kevstone; and the test then consists in determining whether the equilibrium polygon which is drawn through these three points wall pass within the middle third at every joint, or at least whether it will pass through the joints in such a way that the maximum intensity of pressure at either edge of the joint shall not be greater than a safe working pressure. With any system of forces acting on an arch, it is possible to draw an infinite number of equilibrium polygons; and then the question arises, which polygon, among the infinite number that can be drawn, represents the true equilibrium polygon and will represent the actual line of pressure passing through the joints. On the general principle that forces always act along the line of least resistance, the pressure acting through any voussoir would tend to pass as nearly as possible through the center of the voussoir; but since the forces of an equilibrium polygon, which represent a combination of lines of pressure, must all act simultaneously, it is evident that the line of pressure will pass through the voussoirs by a course which will make the summation of the intensity of pressures at the various joints a minimum. It is not only possible but probable that the true equilibrium polygon does not pass through the center of the keystone, but at some point a little above or below, through which a polygon may be drawn which will give a less summation of pressures than those for a polygon which does pass through the point a. The value and safety of the method given above, lie in the fact that the true equilibrium polygon always passes through the voussoirs in such a way that the summation of the intensities of the pressures is the least possible combination of pressures; and therefore any polygon which can be drawn through the voussoirs in such a way that the pressures at all the joints are safe, merely indicates that the arch will be safe, since the true combination of pressures is something less than that determined. In other words, the true system of pressures is never greater, and is probably less, than the system as determined by the equilibrium polygon which is assumed to be the true polygon.
When an equilibrium polygon for eccentric loading passes through the arch at some distance from the center of the joint at one part of the arch, and very near the center of the joint in all other sections, it can be safely counted on, that the true polygon passes a little nearer the center at the most unfavorable portion, and a little further away from the center at some other joints where there is a larger margin of safety. For example, the true equilibrium polygon for the third condition of loading (see Fig. 223) probably passes a little nearer the center on the left-hand haunch, and a little farther away from the center on the right-hand haunch, where there is a larger margin; in other words, the whole equilibrium polygon is slightly lowered throughout the arch. No definite reliance should fee placed on this allowance of safety; but it is advantageous to know that the margin exists, even though the margin is very small. The margin, of course, would reduce to zero in case the equilibrium polygon chosen actually represented the true equilibrium polygon. While it would be convenient and very satisfactory to be able to obtain always the true equilibrium polygon, it is sufficient for the purpose to obtain a polygon which indicates a safe condition when we know that the true polygon is still safer.