rebuilding London Bridge. 31 



The height from the bottom of the grating to the Feet > 



vertex of the extrados of the arch, by . . m = 80,75 



45 + 29,5 + 6,25 = 80,75. 



The distance of the extreme point of thrust at the 

 level of the bottom of the grating, from the middle 

 of the arch, by . . . . . . . d = 



The angle of thrust at the extreme point, with a 

 horizontal line at the level of the bottom of the 

 grating, by <p 



In respect of the thickness at the vertex of an arch equally stable 

 in all its parts ; it is shewn in the work referred to*, that the com- 

 pression at the vertex is equal to the weight there multiplied, by 

 the radius of curvature there, and also equal to the horizontal 

 thrust, and thence the force of horizontal thrust = (ng + w) br, 

 and since the area of the vertical section = nb square feet, and 

 the repulsive strength of the material = f\n linear feet; it follows 

 that nbf must represent the force to crush the thickness at the 

 vertex. Whence nbf = (ng + w) br from which is obtained 



n = = = 3 feet 10 inches, supposing the 



Z - 1 1375 _ 1 



r 190 



arch at the vertex uniformly thick, that is, without ribs. 



The formulae given to determine the point and angle of thrust 

 at any level from the vertex to the foundation, extracted from that 

 work, and applied in this case, are as follow : 



d = ( — — + — ) jj v* — 1 in which v is to be obtained from 



the formulce v 3 — ) I + -1 — H — ILv = — an equation of the 

 ) c 2 n { en 



form w s — pv = h falling under the irreducible case wherein 



/ i_ J is greater than ( — J . Therefore by the known for- 



, 3/t /3 , 2d /3 z , 



mulae — / — = cos. z and v = -C / — .. cos. — and a 



'2p V p 3 V p 3 



• Tracts on fault* and Bridges, pages 37 and 67, Tract 2. 



