The rib represented by Fig. 33 is supposed to have its ends so fixed that the inclinations of the centre line at _F_ and _D_ will never change whatever may be the loading or the variation of temperature. This requires the application at each of those points of a couple whose moment varies in value, but which is always equal and opposite to the bending moment at the same point produced by the loads imposed on the rib. It is also to be observed that the loads resting upon the rib are not divided between the points of support _F_ and _D_ in accordance with the law of the lever, since the conditions of fixedness at the ends are equivalent to continuity. There are then to be found, as acting external to the rib, the two vertical reactions and the two moments at _F_ and _D_, as well as the horizontal thrust exerted at the ends of the structure, which is sometimes resisted by the tie-rod, making five unknown quantities. Inasmuch as all external loading is supposed to be vertical, equations (36) and (37) are the only statical equations available, and three others, depending upon the elastic properties of the structure, must be supplied in order to obtain the total of five equations of condition to determine the five unknown quantities. Inasmuch as the end inclinations remain unchanged, the total extension or compression of the material at any given constant distance from the axis of the rib taken between the two end sections _F_ and _D_ must be equal to zero. Similarly, whatever may be the amount or condition of loading, the vertical and horizontal deflections of either of the ends _F_ or _D_ in relation to the other must be zero, since no relative motion between these two points can take place. It is not necessary in these lectures to give the demonstration of the equations which express the three preceding elastic conditions, but if _M_ is the general value of the bending moment for any point of the rib, and if _x_ and _y_ are the horizontal and vertical coordinates of the centre line of the rib, taking the central point of the section at either _F_ or _D_ as an origin, those equations, taken in the order in which the elastic conditions have been named, will be the following, in which _n_ represents a short length of rib within which the bending moment _M_ is supposed to remain unchanged. F F F ⎲ ⎲ ⎲ ⎳ _nM_ = 0; ⎳ _nMx_ = 0; ⎳ _nMy_ = 0. (57) D D D The second and third of these equations express the condition that the vertical and horizontal deflections respectively of the two ends in reference to each other shall be zero. The conditions expressed by equation (57) are constantly used in engineering practice to determine the bending moments and stresses which exist in the arched rib with fixed ends. The graphical method is ordinarily used for that purpose, as its employment is a comparatively simple procedure for a rib whose curvature is any whatever.