TORSION OF CKKT.MN FnKMs <>F SHAFTINC 



321 



We find 



M V . JT* V . . . 



= a, sin -|- " sl " +.+ f*l" 



-7 -7 27 



. / i \ + i 47 aiii I 1 -/ . . 167 cos 2-j 



6 = sin 2y 

 -7 



Substituting in (12) and (13) and equating coefficients we obtain 



A 16rV(oo8h2 + coa 27008 2c)( 1)' , 2n + lira 



Vn + VwKZn + !?*>- 167*] 2 7 ' 



BL ""**"" 



whence 



HIT ( W*7T S 47 1 ) 



BO = T<-- sin -Jy sin 2e, 

 4 7 



- 1)" . nir 

 sech 



,, snili 2f sin 2 (w e) 

 = ITC' 



cos 27 



. . mrE HIT (it e) 

 t M= Slnn - cos - 



+ 1 GrcV sin 2e sin 2y -,..+ S(-l)"- 



047^ i 



1 Gnry 2 (cosh 2a+ cos 2y cos 2e) 



2mr(4nV - 167*) cosh 



7 . 



. , 

 sinh 



27 



sin 



-'7 



=o 



(2 + 1) TT [(2 -- 1 )' TT ! - 1(V] cosh 



2n+lir 

 27 



(14). 



It may be noted here that y may vary between and ir/2. It follows that y may 

 have the value ir/2 t and in that case the denominator of the first term under the first 

 2 becomes zero. The same happens to the denominator of the first term of the 

 second 2 when y has the value ir/4. Further, in this latter case the first term in iv 

 also becomes infinite, so that the expression (14) is apparently no longer applicable. 



It is easy to see, however, that the terms, which are apparently infinite in (14), 

 exactly cancel each other. If we write y = w/4 , where is small, simplify and 

 proceed to the limit, we find that when y = 7r/4 the two infinite terms reduce to : 



e" sinh 2f sin 2 (r, - 



- - 4 - - 4 " tanh 2 

 2 IT TT 



- e) + (r; - c) cos 2 (77 - e) sinh 2f ] . 



(15), 



which is finite. 



VOL. CXCIII. A. 



