Elastic Stability of Long Beams under Transverse Forces. 299 



II. In rectangular coordinates let the axis of the beam 

 before deformation be the axis of x, and the forces be applied 

 in the plane zx. 



Take first a single force N applied at the origin parallel to 

 the axis of z, and suppose that after a small deformation the 

 point of application of N becomes (0, y { , Sj). The bending and 

 twisting moments at the point (x, y, z) will be given by 





(2) 



small quantities obviously of higher order than those retained 

 being neglected. 



Equations (1) and (2) may be combined by using the 

 geometrical relations 



d?z 

 dx 2 



ds ds ' 



<%_ #2 + #l0 



dx' 1 

 d9" 



dx i 



ds ds 



cW 

 ds' 



It may appear at first sight that in the last terms of the first 

 two of these equations the product of small quantities of the 

 order of the strains are being retained, whereas elsewhere all 

 powers above the first have been rejected. This is, however, 

 not really the case, for it is to be observed that 6 is not of 



the same order as -p, but of the order of the product of -r- 

 ds r ds 



and the length of the beam, which is supposed large. 



Substituting in the geometrical relations from equations (1) 

 there result 



dx' 2 

 7 dx ' 



A 



