104 .On the EquiUbriunt of Arches. 



Subftituting thefe values in the equation juft above, 

 uy-y — zay y — ayy 

 111 — a; fo that the extrados and intrados 

 are precifely fimilar and equal figures, having; their vertical 

 diftance conftantly the fame -: a, the altitude of the wall at 

 the crown. 



18. Let the extrados be a horizontal line DC, fig. 6, re- 

 quired the intrados, a being given, and x, y, tv, as before. 

 By the figure w ^ a + x, and the general equation will ba 



wy = a + X j — ai 'y 



therefore y = . 



Alfo, y : X (fee figure) : : r : t. 

 Therefore y = — = -J 



t a + X 



ati =s arx + rxx, 

 af- — 2arx + rx'-. 



i" = X lax + x^y 



a 



t — — - y^Zax + x^j 



y — — (as above; = 



i_ 



— Tv^ac.v + XX 



— .V 



y ar 





+ C. 



But when x = o,y = o, and C = — s/ar X hyp. log. «. 

 The corre£l fluent is therefore 



y = j^/ar X hyp. log. , 



which is the fame that Dr. Hutton makes it, page 44, new 

 edition of his Principles of ^^ridg^s, wherein Q there = ar, 

 here - the horizontal force. 



If a, X, and y, be given in any cafe, as, for inftance, the 

 altitude of the wall at the crown above the intrados and the 



altitude 



