150 OUTLINES OF NATURAL PHILOSOPHY. 



the intrados, draw GC, intersecting DH'in'F. 

 Let FL be perpendicular to CE ; join LB, and 

 produce CG, till CK be equal to LB ; K is a point 

 in the extrados; and in the same .way may innu- 

 merable other points be found. 



The line CK is always greater than CF, but ap- 

 proaches to it as the arch AG increases, so that 

 DH is an asymptote to the extrados. 



The equation to the extrados is easily deduced from 

 this construction. Draw KO perpendicular to CE. 

 Let CO = x, OK =#, CA = a, CD = b, the equa- 

 tion to the curve is 



jj.-^ + ^y-y 4 ind 



The extrados, in the case of a circular arch, is, there- 

 fore, a curve of the fourth order, very much resem- 

 bling the Conchoid of NICOMEDES. It has an a- 

 symptote DH, and also a point of contrary flexure, 

 so that it coincides very nearly with the curve jn 

 which a road is usually carried over a bridge. 



Instead of r, it may be convenient to have its value 

 in terms of a, and the depth of key-stone d ; viz. 

 b* = Zad + d?. 



All this holds, whatever portion of a semicircle the 

 .arch be supposed to consist of. 



In 



