322 MR. L. N. G. FILON ON THE RESISTANCE TO 



In like manner when y = ir/2 the limit of w is easily evaluated. 



For all other values of y, e and a the series in (14) are absolutely convergent as 

 they stand, for all points ( t)) ivithin or on the boundary of the section. The same 

 holds of all the series in the last paragraph for all values of a, y, e. 



The expression for w being given, the shears and torsion moment are obtained as 

 before 



M = ^- fcfy f " d (cosh 4 - cos 4,) + ^ fTT sin ty dr, - ^ f U" siuh 2fd 

 The integrations are all easily effected, and we find 



ILTC^ 



M = -- (y sinh 4<x a sin 4y cos 4e) 



/ire' (a sin 27 + 7 sinh 2a cos 2e) ,_ /^TC' (sinh 4a sin 2y + sinh 2a sin 4y cos 2e) 



4 COS 27 CMS L'y 



n = a& i 



+ 64/*TcV (cosh 2a + cos 2y cor 2e) sinh 2a S * 







2 4 y 4 (cosh 2 a -f- cos 2y cos 2e) 



tanh 



- 256/iToV (sin 2. sin 



J- 



2y)' I S 

 L n=i 



wV - 167 2 ) 2 1024y\ 



. . 7sec7-an7 



Remembering that __-_*- 512y , *, reducing, and re- 



grouping the terms, we find finally 



L - [ = 1 (a sin 4y y sinh 4a) tan'- 2y 



+ ^ (cosh 2a + cos 2y cos 2e) (2y sec" 2y tan 2y) siuh 2a 



+ sin 4y sin'- 2e (2y tan 2y) 



, Zti+l-rra 

 tanh 



- 256y< (cosh 2 + cos 2y cos 2.)' ' 



,130 



, 

 tauh 



. . . . (IG). 



We may test the correctness of the expressions (11) and (Iti) by remembering that 



