with refer e7Ke to Bridges. 231 



referring to some particular work, they may be stated in a 

 few lines. 



Let a = a constant force, estimated in length of the chain, 

 which acts horizontally on A, the apex of the curve: 

 z =: the length of chain or periphery of the curve, between 

 its apex A, and the point of section, by any ordinate 

 EP.: 

 y r: the ordinate : 

 X r=: the absciss. 



Now the curve being sustained in equilibrio by these 

 forces. 

 By the weight of the chain acting perpendicularly 

 downwards ; 



By the force at A acting horizontally : and 

 By the suspension acting in the direction of the 



curve at P. 

 These forces must be represented in magnitude and 

 direction, by the incremental triangle P r p — therefore 



X : y: : z : a, consequently x"^ : y"^ : : z"^ : a' 



a;' +2/' : oJ* :: a'+z": z* 



But a* +2/* = z"^ in all curves ; therefore 



2' : x« : : a'+z'' : 2" And i =: , 



N° 1. x— Va'+z^—a 



Equation A 



N° 2. z = V2aar+x^ 



z'^-x'' 

 N03. a:= -^7- 



Again, 



^1 • • • ax 



X \ y W z : a, consequently ax ~ zy '.' y = 



z 

 substituting from Eq. A N° 2 



ax 



y = 



V^ax-^x"" And 



a +0:4-'^ 2 ax -fa:* a-hx-^z 



y =:• a X A i. ~ = axhL. ~ 



