On the Equilibrium of Arches. 107 



Let abc, fig. 8. be the curve loaded to the equilibrium ; 

 let fid be a tangent to the carve at a, cd a tariMnt to the 

 cur\'e at c, and he a tangent to the curve at b (the crown). 

 The centre ot gravity 6^ the whole materials is in a vertical 

 line which pafl'es through d; and the centre of gravity of the 

 materials over the arch a,b, is in a vertical line that pafles 

 .through the point e; bccaiife the points J and e are rcfpec- 

 tively the points of Interfeftion of the tangents, drawn from 

 the extreme points of the portions in queftion of the curve. 



26. To find the horizontal diftance 't/' of the centre of gra- 

 vity of the materials contained between the crown and the 

 abutment, from the latter. Drop a vertical line ef, from e, 

 upon the ordinate J', hct af=d, and .v and J' be parallel 

 10, and cotemporary fluxions of, x and j> refpeclively. Then 

 it is evident that x :jf :: .v (= cf) : d ( — qf); 



.TV 



therefore generally d — —.- . 



27. If the curve be the arch of a circle, e,b (fee lafi; figure) 

 Is evidently the tangent of half the arch, which fubtra^ted 

 from half the fpan, leaves d, = fine of the arch — tangent 

 of half the arch. 



38- If the curve be the parabola, 



jf : 2.V : ', j : X = — ^ , by the nature of that fig. 

 therefore d zz ^—^ — = iv. 



29. If the curve be the cquilibrial curve with a horizontal 

 extrados. 



It was {hewn (18) that_^ = ^ar ( — - ' ■ ) = 

 \iax ■\- XX I 



^ X ) ^ zax Jf XX 



therefore d {= ~.--) = sJ — ^^^^ — = V \ — • 



^ X J zax -i- XX 2a + X 



Example. Take the cafe of Dr. Hiitton's mentioned 

 before, wherein r was found = 56.081, rax = I3459> and 

 2.a + X — 52, 



d=s/ -^^^-^ = 16.088. 



XVII. Ob- 



