TOKSloN (IF (T.KTAIX Fo|;Ms OF s||.\FTI\(i. 

 // is tlii-n ilftri-iiiincfl IV tin- Ixxly stress equation 



+-. 



:: 1 .') 



which holds at all {mints of the cross-section, and from the surface-stress equation 



/ {.) + m (.) = o. 



that is. 



da 



-f (ntx ///) T = 



;it tin' iHiundary, where dn is an clement of the outwards normal, and (/, m) are .its 

 i lirect ion-cosines. 



The above conditions allow us to determine w uniquely. They are associated with 

 the condition that the parallel to the generators through the origin remains fixed. 

 If, however, we take any other parallel to the generators through the point (a, b) to 

 remain fixed, then 



= T (y b) z, v = T (x a) 2, 



1/3 dw . . xz dw . , . 



7 = jy + r (* ~ ") 7 = % - T (y - '')< 



and the equation at the boundary becomes 



'I IP . . , > , ,,v 



'' ~~ '.'/) T T (am Hi) = 0. 



Now, if instead of ir we write 



then 



it' = w' + T (<nj !>.r). 



uz (hr : 



= , + rr, - = 



ft d<i ft. 



TV. 



and the equations to determine w are the same which we had before for jr. It follows 

 that the stresses in the cylinder are unaltered, whatever be the parallel to the 

 generators about which it is twisted, the effect of the change being merely to intro- 

 duce a term T (ay bx) into w, which corresponds to a rigid rotation about an axis 

 joining (a, b) to the origin. 



It follows from this that in dealing with stresses due to torsion we may take our 

 origin wherever it is most convenient. 



2 a 2 



