64 Mr Macdoncdd, On the torsional [May 1, 



2. To obtain the twisting couple, we have 



11 ^ — X ^= ^ sinh -»? cos ^ + ^ cosh « sin P, 

 '^ dx dy drj 8| ^ 



= 2c- I ( ~^ sinh t; cos 1^ + ^ cosh ?; sin M 



(cosh 77 -f cos ^y 



M ^ig-n(a+^) ^/cosh-y3 cosh' a' 



47rT6^c^ 2 -^-T 77t X — '"'TC 



a/ 



1 sinh n {^ — a) Vsinh* fi sinh^ 



Also 

 /"/*.„ „x 7 7 *"■ . , ,,x , /cosh- a cosh^/3\ 



jj(.' + 2,') d^dy = 2 («*-«') + -0' (g^^ - g;^j , 



where c cosech a = a the radius of the circle 7? = a, 



c cosech/3 = a' the radius of the circle r] = ^, 

 and 



c (coth a — coth /S) = 6 the distance between their centres. 



Hence if N is the twisting couple, and n the rigidity of the 

 substance of the prism, 



= -— - (a —a ) — ^inbc 2 



n 2 1 sinh n(^ — a.) ' 



)sech^ 

 + cosech' (3,/3- 2a) + ...] 



= ~ {a' - a") - 27rT6V {cosech'/S + cosech^ (2^ - a) 



= |'K-a'^)-27rr6^(^'^ + (^^^ 





If rt' is small, we have 



N 



n ^ 



and if Tq is the twist per unit length in a solid cylinder of radius a 

 due to the same twisting couple 



4Z>V-\ /, 4h\i 



^>^.(-^^) 



