328 



In like manner if >", = 



Mi; I.. N. <;. 1 1LON ON THE RESISTANCE TO 



1 



If.*', 



ff = tan ' 



ff 



4 



!)TT 



B I sin2tf' 



B, 



sin 40' 



These series converge fairly rapidly if x is at all large, and thus the remainder can 

 l>e obtained. 



Even with the help of all these devices the labour of calculating the moment and 

 stress for the sixteen sections was considerable. 



The values of the hyperbolic functions were taken from GUDERMANN'S Tables 

 (' Theorie der Potenzial oder Cyklisch-hyperbolischen Functionen '), and from 

 GLAISHEK'S and NEUMAN'S Tables of the Exponential Function (' Cambridge Phil. 

 Trans.,' vol. 13). 



9. Values of the Torsional Rigidity. 



The first quantity calculated was the torsion moment. The values found are 

 shown in the table below. 



TABLE of 



It is interesting to compare this table with the table of values of the torsion 

 moment, as given by DE SAINT- VENANT'S empirical formula, viz., 



, . fi 



torsion moment = - - 

 40 I 



where A = area, I = moment of inertia of section about its centroid. 

 Calling M' this value of the torsion moment, we have 



sn 



~-=i v 



' /3 sinh 4a sin 



whence we obtain the following set of values :- 



