If the rib has hinged joints at the ends, as in Fig. 34, obviously there can be no bending moment at either of those two points, and hence the two equations of condition which were required in connection with Fig. 33 to determine them will not be needed. There is, therefore, no restriction as to the angle of inclination of the centre line of the rib at those two points. Again, it is obvious that either end _A_ or _B_ may have vertical movement, i.e., deflection in reference to the other, without affecting the condition of stress in any member of the rib; but it is equally obvious that neither _A_ nor _B_ can be moved horizontally, i.e., deflected in reference to the other, without producing bending in the rib and developing stresses in the various members. The unknown quantities in this case are, therefore, only the horizontal thrust _H_ exerted at the two springing points _A_ and _B_, and the two vertical reactions, making a total of three unknown quantities, equations for two of which will be given by equations (36) and (37). The other equation required is the third expression in equation (57), expressing the condition that the horizontal deflection of either of the points _A_ or _B_ in respect to the other is zero, since the span _AB_ is supposed to remain unchanged. By the application of the graphical method to this case, as to the preceding, the employment of equations (36), (37), and (58) will afford an easy and quick determination of the three unknown quantities, whatever may be the curvature of the rib. A ⎲ ⎳ _nMy_ = 0. (58) B If the reactions and horizontal thrust _H_ are found, stresses in every member may readily be computed and the complete design made. If the arch is three-hinged, as in Fig. 35, the condition that the bending moment must be zero at the crown _C_ under all conditions of loading gives a third statical equation independent of the elastic properties of the structure which, in connection with equations (36) and (37), give three equations of condition sufficient to determine the two vertical reactions and the horizontal thrust _H_. In this case, as has already been stated, no elastic equations of condition are required. The determination of the end reactions, bending moments, and horizontal thrust _H_, in these various cases, is all that is necessary in order to compute with ease and immediately the stresses in every member of the rib. These computations are obviously the final numerical work required for the complete design of the structure. These procedures are always followed, and in precisely the manner indicated, in the design of arched ribs by civil engineers, whether the rib be articulated, i.e., with open bracing, or with a solid plate web, like those of the Washington Bridge across the Harlem River. CHAPTER XI.