The force polygon _B_, 1, 2, 3, ..., 10, _A_, Fig. 39, is then drawn with the loadings on each ring segment found as already explained. The horizontal forces are taken as represented by the smaller values of _h₁_, _h₂_, _h₃_, _h₄_. Other force polygons with larger values of these horizontal forces were tried and not found satisfactory. Having constructed the force polygon and assumed the trial pole _Pʹ_, the radial lines are drawn from it as shown in Fig. 39. The polygonal frame shown in broken lines in Fig. 38 results from this trial pole. The frame practically passes through _b_ and _c_, but leaves the ring, passing outside of it, above the joint _VU_. The point _q_ in this frame is vertically above _a_. The “three-point” method of finding the frame that will pass through _a_, _b_, and _c_ was then employed. The line _A6_, Fig. 39, was drawn; then _P′D_ was drawn parallel to _qb_, Fig. 38 (not shown); after which _PD_ was drawn parallel to _ab_, until it intercepted the horizontal line _PQ_, the line _PʹQ_ having previously been drawn parallel to _qc_ (not shown). The final pole _P_ was thus found. The polygonal frame shown in full lines in the arch-ring was then drawn with sides parallel to the lines radiating from _P_, all in accordance with the usual methods for such graphic analysis. That polygonal frame lies within the middle third of the arch-ring, although at three points it touches the limit of the middle third. The arch, therefore, is stable. This construction shows that, with the actual loading of the ring, a line of resistance can be found lying within the middle third; its stability under the conditions assumed is, therefore, demonstrated. It does not follow that the line of resistance as determined must necessarily exist, since there may be others located still more favorably for stability. This indetermination results from the fact already observed that the equations of statical equilibrium are not sufficient in number to determine the four unknown quantities (the two horizontal and the two vertical reactions); but the process of demonstrating the stability of the arch-ring is simple and sufficient for all ordinary purposes. The line of resistance found, if not the true one, is so near to it that no sensible waste of material is involved in employing it. This indetermination has prompted some engineers and other analysts to consider all arch-rings as elastic, thus obtaining other equations of condition. While such a procedure may be permissible, it is scarcely necessary, and perhaps not advisable, in view of the fact that many joints of cut-stone arches become slightly open by very small cracks, resulting possibly from unequal settlement, quite harmless in themselves, having practically no effect upon the stability of the structure.