526 Dr. J. Prescott on the Torsion of 



be very small, since f varies from — -JO to JG from one 

 boundary to the other. For such a tube then 



Q = 2nCA (25) 



approximately. It will make no appreciable difference 

 whether we regard A as the arithmetic mean of the areas 

 enclosed by the two boundaries or as the area enclosed by 

 the curve which runs midway between the two boundaries. 



Equation (25) would give an approximate value of the 

 torque in a thin tube if the constant C were known. We 

 shall now look at the problem of the torsion of a thin tube 

 from another point of view and return to the question of the 

 value of C afterwards. 



Shear Lines. — Since the ,t'-axis may have any direction 

 we choose in the plane of a section, it follows from the 

 equation for Sj, namely, 



o at 



that the component shear stress across any element of length 

 ds drawn in a section is S given by 



S=-n|^, (26) 



the relative directions of S and ds being such that S makes 

 a positive right angle with ds, just as Sj makes a positive 

 right angle with dx (see fig. 3). 



Fi£. 3. 



Now along the closed boundaries of the section tlie 

 function f is constant so that, if ds is measured along one of 

 these boundaries, 



M=o, 



which shows that there is no component shear stress per- 

 pendicular to the boundary. Moreover, there are other 

 closed curves inside the boundaries along which f is constant, 



