Ort the Equilibrium of Arcles, 1 03 



y = the fluxion of the ordinate to the fame point of the 

 curve = the horizontal diaance of the hnes BC and FI. 



16. Let the curve be the arch of a circle, and let it be re- 

 quired to find ii'. Let r = radius, and y, a, and /. the cft- 

 temporary fluxions of the fine, the arch, and the tangent, 

 (fee fig. 4.) ; alfo, let i = the fecant, and u be drawn p.arall*j'l 



then, ^ : is :: r (= AS) : 5 (=S^) 



ii : ti t: r : s 

 u : i :: r : s 



therefore, y : i : : r"" : s' and ^ — ~^ . 



Subftilute this value for i in the general equation above, 

 and it will be, 



-^^ = <a'y ; therefore 



as^y . , r . ..T 





— being a conflant quantity, w is proportional 



to s^ as found by others. 



17. Let the curve be the parabola, fig. 5. 

 A the vertex or crown. 

 B the point on which lu is required. 

 BK a tangent to the curve at B. 

 X = AP (and by the nature of the parabola 2a* = PK). 

 J/ = I'B ; otlier letters as before. 



y : 2x w'r '. t 

 zxr 



~ y ^ 



'^ ' y^ ' 

 and fubftituting this value for / in the general equation 

 at = u)y)i we have 



By the nature of the figure y^ = px, where p the para- 

 meter = 2r (r being the raclius of curvature at us vcrtiex) 



y^ 



therefore y^ = ant and .v = -— • ; 



alio .T = — — .• 

 r 



G 4 Subftituting 



