IV. THE ARCh HINGED AT CROWN (Fig. 8). This case, though of no practiea.1 importance, is here introduced simply for the sake of completeness. Two conditions must be sought from the theory of elasticity in order to find ff~e reactions. The third is that the moment at the hinge is zero, or that the resultant R, of the loads and reactions on either side of the hinge, must pass through the ~-/+ "-~h hinge. Let us resolve R, the resultant on the right, into its horizontal and vertical' components, V and H. It is clear that if we find the values of V and H we have solved the problem, because then R is known, and we can compound it with the loads on either side of the hinge, and so find either reaction. Let V and H be called positive when they act in the directions given in the figure, that is, when the vertical component of the resultant of the forces on the right of the hinge acts upward, and the horizontal component of this resultant acts toward~the left. For greater clearness, let us denote by subscript r and 1 when the forces R, V, and//, refer to the right-hand or lefthand halves of the arch. Now, if we replace the right-hand half by Rr, the fi>r~s applied to the left-hand half (including R~) will cause a certain horizontal and vertical deflection of the point C,, each of which we can easily calculate, in terms of V and H, which are unknown.
:Mar., 1883.]