:;-jr, MK. L. N. <i. KILON ON TIIK RESISTANCE TO 



8. Methods of Cl<->illi<ni. 



The symmetrical sections selected for numerical treatment are tlmsr for \\liieli 

 ft = 7T/6, 7T/4, ir/3, 7T/2 and a =: WO, 7T/3, ir/2, and 2ir/3, sixteen in all. These 

 sections are shown in figs. 2 to 5. of course is the complement of the half angle 

 between the asymptotes, and all sections having the same /8 have been collected in 

 one figure. 



The numerical calculations were generally based upon the formula? of 4, as they 

 did not require modification for the value 7r/4 of yS. In many cases, however, the 

 alternative series were used also, in order to test the results obtained. 



The calculation of the terms of the series in the expression for the torsion moment 



was carried on until was so great that tanh ~ could be taken sensibly 



-p -p 



equal to unity. The remainder of the series, namely, 



was then obtained by expanding the denominator by the binomial theorem, thus 



The successive terms were easily calculated from the values of 2 - -. given in 







CHRYSTAL'S 'Algebra,' vol. 2, chapter XXX, 15. 



The stresses at A and B were calculated from formulae (18) and (19). If S A , S B 

 denote these stresses, we find easily 



/cosh 2a + cos 2/8\ n i l 00 L'a 



= TC tanh 2 cosh a 4- 8rca I - 



COsh a / ,,,7 7T 2 (2 + I) 2 + 16a s 



- 

 = rc sinh . (sec 2/J - 1) - 8rc/J /^2 a " cos 2^ -j- 



\ cosh / it t 7T 2 (2. + l) s - 16/8* ' 



- sech 



C080 /, 7T S (27l+l) 8 +16 8 



2?t + ITT 



= re tan 2^ cos ,8 - 8rc/3 ^ _ (22) 



- 



, 





The first expressions for S 4 , S B were used in each case as the main basis of t lie- 

 calculation, but the results were partly verified by means of the second expressions. 



