152 OUTLINES OF NATURAL PHILOSOPHY. 



a. It can be easily shewn, that if the figure of the 

 chain were reversed, the joints being such, that the 

 force which was a pull in the first situation becomes 

 a thrust in the second, the chain would support itself, 

 and remain in equilibria. 



b. The equation to the catenaria, if x be the abscis- 

 sa, taken from the vertex along the axis or vertical 

 line, and y the corresponding ordinate, is 



:= A + hyp.log. 



c. The constant quantity A is the horizontal tension 

 expressed by the length of a chain of the same sec- 

 tion and the same specific gravity with that of 

 which the catenaria itself is formed. If S be 



half the length of the catenaria, S= 



d Chain Bridges have been proposed, in which the 

 road is to be carried along the catenarian curve sus- 

 pended between two Piers. As in such bridges the 

 curvature must always be small, and as the arch 

 must consist of a small part of the curve near the 

 vertex, an approximation more than sufficiently exact 

 for practice may be obtained in very simple terms. 



e. This approximation gives 



These two formulas are sufficient for the construction 

 of a chain bridge, when y half the span, and x the 

 depression of the middle below the points of suspen- 



sion, are given. 



For 



