THE CATENARY 



85 



61. Sag of a tightly stretched string. A string or wire stretched 

 so as to be nearly horizontal all along its length as for instance 

 a telegraph wire may, as we have seen, be supposed to form a 

 parabola to within good approximation. Thus let A, B be two poles 

 at equal height between 

 which a wire is stretched ; 

 let C be the middle point 

 of AB, and let D be the 

 point of the wire verti- 

 cally below C. Then, 

 from symmetry, D will be the lowest point of the wire and there- 

 fore will be the vertex of the parabola. Thus, from the equation 

 of the parabola, 2 jf 



w 



since, by 60, its latus rectum is 2 H/w. 



Thus if h = AB, the distance between the poles, the " dip " CD is 

 given by 



FIG. 45 



H 



(27) 



To obtain the length of the wire we have to introduce a higher 

 order of small quantities, and so are compelled to return to the 

 equation of the catenary. 



X _X 



We have s = l c (e c e c ) 



laf 



^ec 2 " 1 



The quantity we require is s x, namely DB CB in fig. 45. 

 When the string is tightly stretched c is very great, so that we 

 may neglect the terms in s beyond those written down in the 

 above equation, and obtain 



s x = > approximately, 



6 G 



\ w 2 x 9 

 = 'E'W 



