TousloN OF CKIITAIN Fol.vMs OF s|IAFTI\. 



:', I It 



y sinh 4<x sin ly cow 4e sech 'la. {'ly sinli 'la -f- 'la. sin 'Jy cis 'J 



1 1 



s + )siiili )'/ sin Jy ccs -If -J- sinli '2 sin 4y con 4e} 



cos 2c OUB 



'a* sin 'ly (cosh 'la cos 2e -f ctw 2y cos 



1 



1 



Hut tlir scries %- 



, [(L'/J -f l^TT -f lO 



to be 



j + 1 ) J 7r + 

 can IK.- Bummed in finite terms,* and it is (band 



(2a seclr 'Ja tunh - Ja) 



Substituting this value in the expression for the torsion moment we find 



M = - ! (y sinh 4a a sin 4y cos 4e) ( 1 sech-' 2a) 



a sin 4y cos 4< seclr 'la. 'la sech 2a sin 2y cos 2e 



+ sech 2a [sinh 2a sin 2y] (cosh 2a cos 2e -|- cos 'ly cos 4e) 



sin 2y (2a seclr 2a tunh 2*) (cosh 2a cos 2c + cos 2y cos l e ) 



the two series terms. 

 Whence, after some obvious reductions 



M tiinlr L'a , . . 

 - = - (y sinh 4 a sin 4y cos It) 



(1TC 1 



\ sin 2y (cosh 2a cos 2 -f ros 'ly cos 4t) ('lv. sccli' 2a tunh 2a) 



tanh- 



SM 





- 256.* (sin 2y sin ,'e)' 





(ID. 



* SCT t'lii^siAl.'s ' Alge-lira," vol. I' (Ditlereiiliatin^ the renult marke<l (^) on \i. 33). 



