170 Prot'. Maclaurin on the Influence of Stiffness 



- ) —s'^=c^, and 



they give tanli 9=~ from which 6 is determined when the 



weight of the chain and the tension at the end are known. 

 In the unsymmetrical case when the tensions at the ends 

 of the chain are not equal we have these equations to deal 

 with : — 



T 



-^=ccosh6^i ; 5i = rsinh^^ ; '!'"' =ztanh 6>i. . {a) 



m 



-^=ccosh^2; .^2 = ^'sinh^2 ; ^=tanh6'2, . (/3) 



(s;-.,.,._-g.)--.,, 



'p 2 T 2 



•'•- ^^2 ' =(^l-^2)(^i+^2) (7) 



Either 5i-— 52 or 5^ + 52 is given, and when either is known the 

 other can be calculated from (7). Thus s-^ and 52 can be 

 found, and then 0^ and 6 2 determined by means of (a) and (/5) . 

 The horizontal distance x corresponding to any value of 6 is 

 very readily obtained if a table of hvperbolic functions is 

 available. We have : — 



cosh 6— — =9eG^ + — ^o sin" i/r cos^ i/r 



wc wc^ ^ ^ 



and in the terms of the first order we may put cos '\/r = sech 6 ; 

 sin'>/r = tanh0. 



Thus we set . , ., 2EI sinh^ 6 



to 



cos -v^ = sech 6 4- 



iL'c' ^oih^e 



dx dx ds , , 



do=rs'-dd='''^''''''^^' 



2Elsinh2fy 



c + 



ivc^ cosh'""^ 



.^2Eir sinh20 ,. 



X =c6+ ^o 1 r^MO. 



wc^ } cosh^^ 



Tsin] 



rsinh2^^6^ C z^dz "^ , , ^ 



J cosh^'- = J(i4-^^)^"^^^^^=^^^^^ 



_ 1 r _, z 2r -I 



■. x = c64- 



EI r. ...... sinh6' 2sinh^ 



4:wc^ 



r^ , , . 1 ^, Sinn d' ^smnt^i ,,^, 

 [tan-(smh^)+^-^j^-^^;^^i^] . (10) 



