172 Influence of Stiffness on a Suspended Wire or Tape, 

 Whence 



</>=39° 19^ = 0-6860 radian; tan0 = O-819O; sec (/)=l-2926 

 -^ _3 X 10^ X 3-1416 X (1-33 x -03937)^ x 39-37 x (-8190)^ 

 ^wcr 16 X -04 X 2-2046 x (3960)^ 



= •0008463 



^ tan</)(2-sec^>) ^ .^.. 0-8190(2- 1-29262) 



* ^^^^^-=^'^^^^ 0^2926? 



= 0-5894. 

 Hence the correction for stiffness in ten chains 



= 2 x 0-0008463 x 0-5894 inch 

 = 0-0009976 in. 

 This is slightly less than eight inches in a thousand miles. 



If we wish to compare the corrections for two chains 

 which differ only in the form of their cross-sections, we notice 

 that if w, s, and T are the same, then ^ and c are the same. 

 The material being the same, E is also the same, so that the 

 corrections are proportional to the moments of inertia of 

 the cross-sections, e. g., if a circular wire were made of the 

 same area of cross-section as the iape considered above, the 

 correction for the stiffness of the wire would be about seven 

 times that for the tape. 



It may be interesting to compare the correction for stiff- 

 ness with that for stretching. If u be the area of' cross- 

 section, ds the unstretched length of an element of the chain, 

 ds' the stretched length, we have by Hooke''s law. 



^ „ ds' — ds ds' ^ T ,T , ,jf 



T = Ea — .-. -T- = 1 + ^ . Also IV ds = to' ds, 



as ds ha. 



The equations of equilibrium are T cos yjr = ivc, and 

 T sin ^Ir = tv's^ = ws ; 



.*. 5 = c tan 'v/r = csinh ^ ; T = ivc cosh 6, 



x= I cos yJr ds' = \ r-— 1 + ^p^- cosh 6 c cosh Q dO 



J ^ Jcosh^L Ea J 



= c6'+^sinh(9. 



