SOO Major-General Sir Howard Douglas 07i 



common two-wheeled carts, by merely laying the ends of the 

 shafts of each cart into the body of that in front of it. 



The learned author next enters into a minute, clear, and full 

 account of all the circumstances attending tlie construction of 

 Suspension-bridges. 



V In a learned note, at pages 307, 308, and 309, the author 

 deduces the equation to the catenary, both in a complete and in 

 an approximate form. 



Let 07 be the depression of the middle of the chain below the 

 horizontal line, called generally by mathematicians the absciss ; 

 9/ half the horizontal distance of the points of suspension, or 

 ordinate; and z half the length of the curve; then, if the 

 weight of a unit of the chain, such as one foot, be w, the 

 weight of half the chain will be w z, and w c and w t will be the 

 tensions at the middle of the chain, and at each point of suspen- 

 sion. From these considerations, and the properties of the sus- 

 pended chain, Sir Howard arrives at his equation (6) ; or, 



and, ^=7/4- • • W 



y 



To these we shall add those given by Professor Sir John Leslie, 

 from which every thing necessary to the construction of suspen- 

 sion-bridges in all ordinary cases may be derived with the great- 

 est simplicity *. Taking the distance between the points of sus- 

 pension equal to d, the depression of the middle of the chain be- 

 low the horizontal line equal to /i, the length of the chain equal to 

 Z, and the strain at the middle of the chain, or parameter of the 

 curve, equal to p ; then, in all practical applications where the 

 depression is small compared with the length of the chain, the 



strain at bottom, or » = -7 -f- ^ /i = — — ^h . . (8) 



Oil D oil 



d^ 7 

 strmns at points of suspension, or 5, = — + ^ A , . (9) 



or, s = ^^^^lh . . (10) 



andZ = d + ^. . . (11) 



• See Elements of Natural Philosophy. 



