814 Dr. 0. H. Lees on the Depression due to 



If e is Young's Modulus for the material of the chain, 

 a the total area of the transverse section of the longitudinal 

 part of a link, T ' the initial stretching force, T' that when a 

 mass M is suspended from the centre of the chain, 



s __ s 



1 + - 1 + -" 



ae ae 



where s and ,9 are the half-lengths of the chain in the 



two cases. 



T 

 Since — never exceeds 1/1000 we may write this equation 



Or 



*■-*•'— (i-i) W 



Hence substituting from equations (4), (4'), (2), and (2'), 

 we have 



f • hL \, 2e+l 



(M + <r% ,,2e + l <rla ,, I ) C Smh 2c C0Sh ~2c~ 



- ^— ^coth— s— -~-«f coth^- =ae{ = - - 



2 2c 2 2c J . , / . I 



c sinn -jr— cosh ^~ 

 V. Zc Zc 



Or, expanding the hyperbolic functions, 



^^ 1+ K¥)*-)-*T^(rJ-) 



•+*(&■ 



-1 



Hence, since ( -~ — ) never exceeds 1/1000, 



L -^27T7-^ = 2l(-2Tj--(^ ))' * (7) 



Expanding similarly equations (5) and (1), we have 

 2e + l(, , , /2« + ZV\ M+<rZ . 2e + l { . 1 {2e + iy\ _Mr 



2c 

 and 



i.e. 2* + */^ , 1 (2 g + Q 2 + Z 2 , \b 



2c v +3 4r ' -**•••;- / 



