Long Beams under Transverse Forces. 303 



The first root of this equation is 



R=4-4817.... 



IV. To obtain the condition of critical stability for the 

 case of a beam fixed in azimuth at its ends, it is necessary to 

 modify the equations of equilibrium by inserting in them the 

 couples necessary to maintain the ends in their original 

 positions. 



If at the origin a couple Q in the plane ccy, and a couple S 

 in the plane yz, be applied to the beam, the equations (3) 

 above must be replaced by 



frdV_ ] 



N &»■"*' 



ld±_ ( ly_ % 



Hence 



or 



7 fd_ N(A-&) Q* 

 Ndx 2 ~ && xv /V 



d 2 6 

 J a r - 2 +x 2 + Cx=O, 



dx z ' 



where n AQ 



N(A-A)' 



The solution in this case, 6 being zero when a: is zero, is, in 

 the same manner as before, easily found to be 



J . 3 . 2\ J . 7 . 6 ^ J . 11 . 10 . 7 . 6 ~ 



± J"(4?i + 3)(4n + 2)(4??-l) (4»-2) ....7.6 + ' * ' * J ; 



and 6 being zero when #=Z, the condition of critical stability 

 for a cantilever w 7 hen the free end is prevented from rotating 

 about the axis is 



TJ R 2 Ti n 



7.6 11.10.7.6 " " - (4n + 3) (4n + 2) 7.6 



The first root of this equation is 



R= 101-23.... 



