./;• 



Long Beams under Transverse Forces. 301 



and therefore from the second equation 



dx z ft/3 2 7 



Denoting the expression 



N 2 (/3i-/3 2 ) 



(which is of the same dimensions as the moment of inertia of 

 a plane area) by J, there results as the differential equation 

 of the axis of the slightly displaced beam 



J s+^=° w 



This is a well-known transformation of Riccati's equation, 

 and the solution consists of the sum of two BessePs Functions 

 of fractional orders. For the present purpose a more con- 

 venient form of the solution is obtained directly by deter- 

 mination of the coefficients of the series 



d=A 1 x m i + A 2 x m *+ -f A^^-f- 



The result is 



J. 4. 3 ' J 2 . 8. 7.4.3 

 -J».4n(4n-l)(4n-4)(4n — 5) 4.3 



.. A 



b4i- 



}. • (5) 



+ Ba?^l- J7574 + J2. 9.8.5.4" ' * * " 



- J". (4n+ 1) . 4n(4n-3) (4rc— 4) .... 5 . 4 + 



In which A and B are constants to be determined from the 



end conditions assigned in any particular case. The simplest 



case to which the above result is applicable is that of a 



cantilever, rigidly fixed at one end, and supporting a single 



transverse load at the other end, which is free. 



If the origin be taken as the free end, there is no twisting 



dd 

 moment and -j- is zero at that point, hence B = 0, and 



* = A(l-^ + 



=a(i- 



J.4.3 ' J 2 . 8. 7. 4. 3 



- J".4w(4n— 1)....4.3 + 



in which A is the angle through which the free end is 

 turned relatively to its original position. 



