310 Dr. J. Prescott on the 



Let w be the load per unit length, so that 



W=wl. 



The origin is taken at the end B, x being measured to- 

 wards A, and y upwards in the figure. TR^ is the tangent 

 at R. 



Consider the equilibrium of the portion RB of length x. 



The moment about Rr of the weight RB is %wx 2 , and this 

 has a component about the depth of the twisted section at R 

 of magnitude %wx 2 t. This causes bending in the xy plane. 



Therefore 



Ecg = -i«^r (48) 



Let the coordinates of Q be (#', ?/). Then the twisting 

 couple on the section at R due to the weight wdx' near Q is 



wdx x Qq. 



Since t decreases as x increases it follows that the total 

 twisting couple at R is 



-Kn d ~=\\cQqdx' (49) 



.Differentiating this with respect to the upper limit a, 



-K„£.[„. Q ,]'" + „j;*aw. 



But Qq = y—y' — rg' x tan cf> 



whence £^)=l-J-(^-')3=-(-v)g 



and 



[q?] = 



Therefore 



dx 2 J V J dx 2 



dx' 



d 2 y 



— Oil •■' 



dx 2 



= [*<— t*' f T 



ii/i ^2 «y 



— iiivx 



8 a 



(50) 



