﻿106 Mr. A. E. Young on the Form of a Suspended 

 Moreover, L = ET/p = EI -^, which with (13) gives 



EI§J+U = (14) 



In these equations T is the tension, U the shear, L the 

 bending-moment, s the length of the curve measured along 

 the arc from some fixed point, yjr the angle the tangent makes 

 with the horizontal, l\p the curvature, w the weight of the 

 tape per unit length, E Young's Modulus, and I the moment 

 of inertia of a cross-section of the tape about a line through 

 its centre of gravity perpendicular to the plane in which the 

 tape hangs. 



Eliminating L, IT, and T, the following is the differential 

 equation arrived at for the curve in which the tape hangs: 



oo B +J + 2™*-^[^+*S5tt] • (15) 



EI 



a|t 2 W L' ' ^ 



where the dot • denotes differentiations with respect to s. 



When the tape is perfectly flexible the right-hand side of 

 (15) is zero, and we get 



cos yfr \ - + 2 sin yjr = 0, 



whence %- = — 2 tan yfr ; l/p = yjr= - — £ f 



ip c 



which of course represents the common catenary. 



The flexibility of the surveyor's tape being so nearly 



perfect, Professor Maclaurin proceeds to solve (15) by 



cos ylf 

 approximation, substituting ^ = c Y in the right-hand 



side of (15), and finally arrives at the following for the 

 intrinsic equation of the curve : 



s = c tan yjr +— -, sin ^ cos 2 a/t. . . . (16) 



From this, after further transformations, he finally derives 

 the formula for the stiffness correction previously quoted, 



viz., 



EI r, ,(2-sec 2 tf>)-] 



i ± • -i w 1 i 9_To 2 



where <£ = sm l ^yy and c— ., 



