THE CATENARY 83 



Equation (26) is the Cartesian equation of the curve formed by 

 the string; this curve is known as the catenary. 



From equation (23) we obtain the value of s in the form 



= c 2 sinh 2 - > 

 c 



s . , x 



so that - = smh-> 



c c 



/P 5 _* 



where sinh - = \ (e c e c ). 



c 



- -- x 



60. Expanding the exponentials e c , e c , we obtain cosh - in 



the form 



tJC 



So long as x is small, we may neglect all the terms of this series 

 beyond the second. Using the value obtained in this way, we 

 obtain instead of equation (26) 



, a- 2 

 '-'+TS 



showing that so long as x is small the curve coincides very approxi- 

 mately with a parabola of latus rectum 2 c or 2 H/w. 



This parabola, it will be noticed, is one which would be formed by the 

 cable of a suspension bridge of horizontal tension H, w being the weight 

 per unit length of the bridge itself. Indeed, it is clear that when the cable 

 is almost horizontal, it is a matter of indifference whether the cable 

 itself possess weight w per unit length of its arc, or whether a weight w 

 per unit length is hung from it so as to lie horizontally. 



We can also obtain a simple approximation to the shape of the 

 catenary when x is large, i.e. at points far removed from the lowest 



flfi 



point. When x is very large, so that - is very large, the value of 



* -* c 



e c becomes very large, while that of e c becomes very small. Thus 



