524 Dr. J. Prescofct on the Torsion of 



Now let x = rcosO 



(18) 



y=r Sill V] 

 and let us regard f as a function of r and #. Then 



= |^(cos 0tfr-r sin 0<J0) + |^(sin 0rf?- + rcos 6W). 

 o<^ oy 



Therefore, keeping 6 constant and varying r, we get 



^C0S6> + " 

 O^' O^ 



# df . # of 







(19) 



r ox r OV 



Whell0e Q=-»jjr|j><*y (20) 



The element of area suitable for polar co-ordinates is the 

 element between two radii and two circles and its magnitude 

 is rdOdr, and this takes the place of dxdy in (21). Therefore 



I rl^rdOdr 



--ff 



i*^-drdd (21) 



The limits for r and in (21) must be such as to include 

 the whole area of the section. Suppose we are dealing with 

 a tubular section as shown in fig. 2. Then the limits for r 



Fig. 2. 



are OK and OH, which we may denote by r and 1\. Now- 

 let the values of f along the outer and inner boundaries of 

 the section be and C. We are at liberty to choose the 

 value of f at any point, since f is indeterminate to the extent 

 of an added constant. The differences of f are, however 



