196 Dr. J. Prescott on the 



Let G denote the bending moment at D in the vertical 

 plane, G + dG the bending moment at D'. LetT and T + JT 

 denote the torques at D and D' causing the twist in the 

 beam. Representing these couples by vectors perpendicular 

 to their planes we get the system of couples shown in the 

 next figure. 



Fig. 9. 



T+dT 



We denote by </> the angle between the d'-axis and the 

 tangent at D to the central line of the beam, so that d<j) is 

 the change of the angle between D and D'. The vectors 

 for G and (G-f- dQ), being normal to the beam at D and D', 

 contain the angle d<p. 



Resolving the couples on the beam about the tangent 

 .at D' we get, to first order, 



Gd<j> + dT = 0, (71a) 



whence 



dT = _ G d$ 

 dx dx 



But dy -' ■ , , 



■— = tan (p = 9 nearly. 

 clx 



Therefore 



£— g3.- ..... (71) 



dx dx 1 



The moment about the tangent at J)' of the shearing force 

 at D, as well as of whatever load there is on the element 

 DD', nas been neglected in the preceding equations because, 

 assuming that the load is on the central line of the beam, 

 this moment is a quantity of at least the second order. The 

 correct equation when the load is not on the central line is 

 given later (equation 101). 



Now the bending moment G can be resolved again into a 

 pair of components about lines respectively parallel and 



