206 Dr. J. Prescott on the 



This is the general equation when the load is off the 

 centre and there is no couple M at the ends. In the present 

 problem, the origin being taken at the free end, G- = ^wx 2 , 

 and therefore 



Whether p and q are functions of x or constants the 

 solution has the form 



T = a f(x)+a l T(x)+cf>(x), . . . (104) 



where <fi(x) is a particular integral corresponding to — wp. 

 The conditions to be satisfied at the ends are 



-,— = where x = 0, 

 ax 



and t = where x = l. 



These give = a /'(0) + a 1 F(0) + £' (0), 



and = a f{l)+a 1 F(l) + <l>(l). 



Eliminating a^ from these we get 



= a {/'(0)F(Z)-/(/jF'(0)} 



+*(0)F(i)-*(J)F'(0). .... (105) 



Now the analytical condition for instability is that a 

 should be infinite, and this can only occur if the coefficient 

 of a in (105) is zero ; that is, the condition for instability is 



/'(0)F(Z)-/WF'(0) = 0, .... (100) 



which is independent of </> and therefore of p. 



We have now shown, as in the last case, that p has 

 nothing to do with stability. 



Since p does not affect stability we can drop it from our 

 equations. Then, assuming that q is constant, equation (103) 

 becomes, when p is dropped, 



^ = -(m¥ + ^)T, .... (107) 

 where 6 w 1 maw, 



m ~mm> (10b; 



r,_ wq 

 f ~ Kn 



(109 



Now the assumption is being made that q is small, and 



