185] NON-UNIFORM FLEXURE. 493 



The moments are 



L =j I (yZ 2 -zY 2 }dS = E\z C Cxydxdy - zY = 0, 

 133) M = 



every integral in N vanishing on account of the symmetry. 



We have as a result that to produce this strain we require a 

 transverse force X applied at the free end, together with a couple M 

 about the Y-axis. The transverse sections are buckled, contrary to 

 the old theory and to the case of uniform flexure, while as there 

 the fibres for which x is positive are in longitudinal tension, those 

 for which it is negative in compression. 



It was shown by Clebsch that the integral / / -r^dxdy, which 



occurs in X and M, could be calculated without determining the 

 function V l itself. 



Putting in Green's formula, 137, 55), /"=#, we have, since 



dx - doc 



134 ) 



3V 

 and for this problem, taking the value of -*- from 130) 



135) j I ^ dx dy = j{X cos (nx) + Tcos (ny)} ds 



where 



Converting the line -integral into a surface -integral by the divergence 

 theorem, 



136) 



Inserting this value in 132), 



137) X = pb (2n + 2) I y = 



