THE EQUILIBRIUM OF BODIES IN CONTACT. 475 



If, moreover, the forces P be supposed to be resistances, the line of 

 resistance will touch the extrados (Fig. 12); 



^ a'2.±Pkeos^ +^ (2Psin4>)^ 

 a2Pcos$ 



« = = ^^^~'^. '-^^^^^ (29). 



Suppose that there is only one force P applied at each extremity ; 

 therefore by equations (28) and (29), 



. tan<1) = ^ ^ (30). 



P sin 4) = ah, a' = ±ak + iPsin $ tan <^ ; 



b 

 i'=«*\/l+i(— ^y' (31). 



^^°^* = i(^ -•(32)- 



This last equation gives that portion of the force P which is resolved 

 in the direction of a horizontal line. If ^ = or the force P be applied . 

 at the angle A of the mass, 



Oft 



tan 4) = -^, P =hy/u^ + \b\ Pcos<t = iZ»^ (33). 



It is worthy of remark, that the last of these expressions is inde- 

 pendent of a, the depth of the voussoirs. 



Let a straight arch be supposed to be supported upon the edges 

 of two vertical piers, (Fig. 12). 



The point of rupture in the extrados of the pier, or its greatest 

 height, so as to stand unsupported, will then be determined by the 

 following equation derived from equation (23), by taking ^ = and 

 writing for <1> its complement, since in the straight arch <J) is measured 

 from the horizontal axis of z, in the pier from the same axis in a 

 vertical position, so that <1> in the one case is the complement of <!> 

 in the other: 



Vol. VI. Part III. 3 P 



