a Load at the Centre of an Elastic Chain. 813 



at ( J , a point at a distance />„ below C. and 



& = e (cosli~-l) (10 



• , I 

 s = f smh- (2') 



Co 



X 



and T = <rge Q cosh— (3') 



c o 



Since the object of the experimental method is to secure a 

 comparatively large depression CCi for a small increase in the 

 length of AC\B. the initial stretching force T should be such as 

 to make the chain nearly straight. The maximum value of 

 T must be within the elastic limit of the material of the chain. 

 In most cases this will mean that the extension cannot exceed 

 1/1000 of the original length. The value of CCi cannot 

 therefore exceed l/ v /lOO0 = l/32 of CB, i.e. the upper limit 

 to tan will be 1/32. 



Hence the upper limit of sinh- and therefore of - = 1/32. 



Since the difference in height of C 2 and B is small we may 



take the stretching force T to be constant throughout the 



chain, and equal to its value T'at C, a point half way between 



B and C, i. e. for which a?=Z/2. 



2e + l 

 Hence T = T' = agc cosh— » — . 



Since there is equilibrium at C/ we must have 

 2T'sin0'=(M-f<7Z>, 



where 0' is the inclination of the curve at C to the axis of X. 



But 



tan 6' = sinh^ — . 



2c 





H^nce 



. a , u 2e + l 



sin o = tann — r 



2c 





Therefore 



2T'tanh-^ = (M + o%, . . . . 



(i) 



and 



r Sinh —rr — = 



2c 2a 



(5) 



When the cl 



lain is unloaded these equations become 







2T'„ tann — —at a, . . . 

 zc *' 



(i') 



and . , I I 



Phil. Mag. S. 6. Vol. 9. Xo. 54. June 1905. 3 [[ 



