530 Dr. J. Prescott on the Torsion of 



Therefore 



Qf^=4nrA 2 (34) 



The integral involved in this last equation can be obtained 

 either analytically or graphically when the shapes of the 

 boundary curves are known. If t is constant and I is the 

 arithmetic mean of the lengths of the two boundary curves, 

 the torque is given approximately by 



Q= J — (35) 



Equation (34) gives the torque in a thin tube in the 

 general case, and (35) gives the special form when t is 

 constant. It is necessary to recall that A does not mean 

 the area of the section, but means the total area enclosed by 

 the curve passing midway between the two boundaries. 



There is a certain indeterminateness concerning the actual 

 curve which encloses A and along which ds is measured. 

 For a thin tube this indeterminateness does not much matter. 

 If we took the inner boundary instead of the mean curve, 

 equation (34) would give a value for Q below the true value, 

 while if we took the outer boundary that equation would 

 give a result too great. This gives some idea of the error 

 in the formula. 



A Uniform Circular Tube. 



St. Venant's accurate method applied to a circular tube 

 whose inner and outer radii are (r — \t) and (r + ^t) gives 



= 2irnrrt(r 2 + \t 2 ) (36) 



Equation (35) above gives 



= 27rnrrH (37) 



The proportional error in our result in this case is thus 



t 2 

 -r-z. If t is as much as a quarter of the mean radius this 

 4r 2 



proportional error is only 1"5 per cent. 



