a Load at the Centre of an Elastic Chain. 815 



b 



I 



Hence so long as j does not exceed 1/30 we may write 



and 



i. e. 



M 



la 









2e+l b 









2c " I ' 





M 





e h 2e 

 c ~" I 2e + 1 " 



b 



* r 





- 1 



M 



+ l U + ' 1 ' 



From equation (1') we have similarly 

 I 2 



°~26o' 



Hence equation (7) becomes 

 QL + *l)ql I 2 



-2—b"^W Q 



2 \ I 2 I 2 J 



Let <rl = m half the mass of the chain, then 



(M + m) rn _ae 



I bo~W { ° h ' ' * " () 



a cubic equation to determine the depression b if M, m, a, 

 e. /, and b are known. From it we have : — 



M + in in 



e = 9* —r-T 



a b*-b<? W 



Or if ^S\ x and M 2 are any two masses suspended, b l} b 2 , the 

 depressions produced 



M Y + m M 2 + m 

 gl* b t b 2 ( 



€ ~ a b x 2 -b 2 2 ' * * * (iU; 



an equation to determine e from observation of b. 



From equation (10) it is evident that the simple theory 

 which considers the chain to be without mass will give 

 values for e correct to about 1 part in 300 for depressions 

 which do not exceed -^ of the length of the chain, if the 

 masses actually suspended from the centre of the chain be 

 supposed increased by half the mass of the chain itself, and 

 are not too nearly alike to make the difference between the 



values of — , less than one-third that of either. 



