302 Dr. J. Prescott on the 



couple N at the fixed end exerting a torsion on the beam, in 

 addition to the couple G whose magnitude is PZ. 

 The bending moment equation at R is 



Ecg = GT-P*T, ...... (14) 



x being measured from A. 



The twisting moment on the section at R of the force P 

 acting at A is 



AF * p =(4l-*) p ' 



The twisting effect at R of the couple G is the component 

 of G about the tangent RF, that is, — G — - nearly. There- 

 fore the equation for the torsion at R is 



*&-**('&')*-*& ■ ■ < 15 > 



Differentiating this last equation we get 



jt d 2 r „ d 2 u ~ d 2 ii 

 dx 1 dx z dx 2 



=( P.,_ G) g. .... (16) 



d 2 y 

 Using the value of -rf 2 from (14) we get, on putting 

 PZforG, dx F * 



^ 2 T P 2 



or 



d# 



., = -„,«(* -«)»T, .... (17) 



where " j4 = eSk ( 18 > 



It is clearly more eonvenient to measure x from the free 



end of the beam. This means using x for (l—x). Then, 



d 2 T 

 since -j-% remains unaltered, 

 dx 2 



S = -'»W (19) 



