154 OUTLINES OF NATURAL PHILOSOPHY. 



will represent the two forces which compound the 

 force UH, and each of them will represent the 

 force with which the arch presses on its supports 

 ML and ON. Let RK = HI, and let RK be re- 

 solved into the two forces QK, KP, the one hori- 

 zontal, and the other vertical ; KQ tends to overset 

 the pier, supposed moveable about the point F, and 

 KP adds to its stability. If FC be bisected in W, 

 we may suppose the weight of the pier at W, to act 

 by the lever F W, and the force KP by the lever FC, 

 both tending to keep the pier in its place, while the 

 force KQ acts by the lever KC or F Y to overthrow it. 

 Farther, though the pier is properly only FGAG, 

 yet it is usual to have it loaded with masonry to 

 the height K, and even higher, so that its weight 

 may be taken as proportional to the rectangle FK. 

 This will give an equation between the forces, when 

 their analytical expressions have been obtained. 



Angle UHI = ft, and VHI = 2/3, the area MLOND 



= a?, EC = r ; make HU = , then IH = 



r 



r& 



= KR. 



2 r cos ft 

 Hence QK = . _ L anc [ 



t/J / r/"vo Q \^s o-i r\ /9 * """"" . (71 ^ ' 



Let CK = h, and FC = x , 

 grcosl* 



then the rectangle FK = h x, and the line that re- 

 presents the weight of the pier will be . 



Therefore, 



