168 Prof. Maclaurin on the Influence of Stiffness 



Thus (5) becomes 

 -^i^ r, ■ 6EI 



— + 2 tan -v/r . i/r = 3 cos'^ -^/r [5 COS^ ■\\r — '2'\ sill a/t . -v/r. 



Integrating we get 



, c^ 6EIf .. , ,. , , , ' , J, 



^^ C0s2\/r "" 'vTc' ] ^'^^'' '''(^ ^^ ^ - 2 j smi/r ^^ 



2EI 



^ w^ cos^^(2-3cos2^) 



• cos"' lir ^-^^ cos3 ^^,{2-'^ cos^ »//) 



.'. '\lr =^ — . e I'-'c^' 



^ c 



El 



It will be seen in the sequel that — 3 is an exceedingly 



minute quantity, and neglecting squares and higher powers 

 of this we have, by the exponential theorem, 



f dylr 2Eir.., . , ,, , ., 



.-. s = c\ ^ H 3-1 (o cos" 1/^ — 2) cos Alrrfiir 



' COS' -yjr i^'^ J T ^ T T 



*2ET 



= c tan a/t + -y^ sin i/r cos^ -^/r, (7) 



ICC 



which is the intrinsic equation of the curve. 



If .V be the horizontal distance corresponding to 5, then a? 

 is the quantity to be calculated from an observed value of s. 

 We have : — 



^' =cos^ .-. ^i'=Jcos^^5 = cJ^^ + -^J(3cos^^-2cos->)^,|r 

 Hence 



EI 



= c sinh-i (tan ^) + __^ p^ + i gin 2i/r (1 + 6 cos^ ^^r) ] .(9) 



EI 



The quantity -j-^ [-i/r + i sin 2'\/r (1 + 6 cos-i/r)] is not the 



correction to be applied when the rigidity is taken into account. 

 The value of ^/r is affected by the stiffness, so that c sinh'"^ 

 (tan y^) will be different when the stiffness is regarded than 

 in the case of perfect flexibility. We shall afterwards 



