TUKSION OF CEUTAIN IOKMS OF SHAFTING. 



323 



\\lu-u \\- make tin- distance c between the foci very great, ami y and a MTV -mall, 

 the section reduces to a rectangle, of which the half sides n and b are given by 



a = ca. cos e, b = <*y cos . 



The first .UK! last terms in (16) are ultimately of negligible order, when multiplied 

 l,v c. 



The second and third reduce to 



f y*a (1 -|- cos 2e) \ y'a (1 - cos 4e) = f y'a (3 + 4 cos 2e + cos 4<) 



The fourth term gives 



'2 11+ \1Tii 





- 1 024 (y cose) 4 2 



Hence 



. 



tonh 



1 = 



\\hich is one of SAINT- VEN ANT'S expressions for the torsion moment of a rectangle 

 of sides 2a, '2b. 



If we treat in a similar manner expression (11), neglecting terms of order greater 

 tlian four in a, y, we get the other expression for the torsion moment of the 

 rectangle. 



G. Recapitulation of Results for the Symmetrical Case. 



By far the most important case we have to deal with is that in which the sections 

 are symmetrical. 



We have then /8' = ft, and therefore e = 0, y = /8. 



liotli solutions then simplify a good deal, and we have the equivalent expressions 



w = \ TV'- sinh 2 sin 2i; sech 2at 

 + IGrcV (cosh 2a + cos 2/8) 

 = ^ re 2 sinh 2f sin 2rj sec 2/8 

 - lGrc ! /8 2 (cosh L'a -f cos 2/3)' 



, 

 ( - 1 )" sinh o 



ir(2n+ l)[(2n 



, 

 - 1)" siuh 



. 2n+lirr) 



o 



. . (17). 



ir (2 + 1 ) [(2 + 1 )V - 1 60 s ] cosh ^ 

 2 T 2 



