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



increasing as we approach the ends, where it attains a 

 maximum value of 



iv tanh — 



— VT = Jv > P racticall J- 



Since in the example taken a = 3'11 the maximum effect in 

 this case is only 2"'l in a slope of about 9° 9 r . The effect 

 can be also seen by reckoning x from the end instead of the 



middle of the tape and writing x ~ T —a\ so that 



dy _ w 

 tx ~ T 



j sinh 





a G~^) 



2 , al 



a cosh -p- 



= %[^-' v '-^]' verynear] y' 



putting tanh ^ =1. Since <? 3#11 = 22 nearly this shows that 



at only 1 inch from the end the effect of stiffness on the 

 slope is reduced to ^V °f its maximum value, i. e., to about ^ 

 of a second of arc. 



We will now consider the case of ends clamped or con- 

 strained so that the tangent to the tape is horizontal thereat, 

 in which not only is the effect of stiffness greater but the 

 conditions approach more nearly those found in practice, 

 because, in using the long steel tape, the surveyor generally 

 supports it at several points forming several bays in sag, the 

 tangent to the tops at each point of support being horizontal, 

 and a bending moment with curvature concave to the axis of 



a occurring thereat. Putting then -r- =0 where #= ^ in 

 & " dy 2 



equation (3), we get 



/aM w \ . , al wl 

 V I aij 2 



9T 



or 



M = w " 



a *{ X 2sinhf)' 



Substituting this in (3) we get 



_ iv_ Ta 2 ^ 7(coshflM— l) ^j 



dy __ w^ C l sinh ax ^ 



(?) 



