On the EqulUhrlum of Arches. 705 



altitude and half fpan of the arch ; then r may be found as. 

 follows, and thence the ordinate* to any other point. 



Let C = hyp. log. ( ]. 



Then_)' = "^ar X c, 



y- = arc'-^ 



Example. Suppofe the fpan of the arch to be Tco, the 

 altitude of it 40, and the height of the wall above the 

 crown 6 (fame cafe as Dr. Hution's, page 47, Principles of 

 Bridges), 

 a ■= 6, X — 40, y = 50, therefore 



a + * + K' lax -f- .vr ^r. 1 1 11-1 - 1 

 = 15.268, the hyperbolic logarithm 



of which is 2.7257 * — c, c'^ = 7.4297, cc' = 44.578, and 



^— — —^ — = <6.g8i = r = the radius of curvature of 

 ac^ 44.578 ^ 



the arch at the crown. 



19. The WEIGHT, of the incumbent matter on a curve 

 loaded to the equilibrium, contained between the crown and 

 any point B, fig. 3. and which may be expounded by the 

 area AC of the wall above it, has been (hown (14.) to be 

 always = ab X ad — ^li. Let A faid area AC. 



20. If the curve be the arch of a circle, and T = the 

 tabular tangent of the number of degrees contained in the 

 arch, then will / = rT, and confequently A = arT. 



21. If the curve be the parabola, fee fig. 5 ; 



then J/ : 2x : : r : t = ^ 



Therefore t ( = — - = - — ) = y, 



y y 



and y = px = irx 

 ;fore t ( = 

 and A = ya 



ic lo 

 die produa is the rc4uircd hyperbolic lo{^;\iithm. 



i by the nature of the figure. 



• The hyperbolic logarithm of a nvimber may be found by miiliipl\ in<^ 

 the common or Briggs's loj^irithm of the lame number by 1.3025S5, Lc, 



aa. If 



