Buckling of Deep Beams. 207 



the problem is to find t when q is zero and then correct 

 £01* q (or c 2 ). 



When c is zero the solution of (107) that satisfies all the 

 conditions of the problem is 



^4 1 -5T6- + 5T6TTT7I2-}' * (110) 



m being given by the following equation, taken from Case 6 

 in the' first paper, 



m 6 Z 6 =41-30 (Ill) 



The value of t in (110) being denoted by r x the equation 

 we have to solve is approximately 



^+wVt=~cj 2 t 1 , .... (112) 



the term on the right being now regarded as a known 

 function of a. This process amounts to treating c 4 as 

 negligible while c 2 is not negligible. 

 The particular integral of (112) is 



2 , f 1 mV/ 1 1 \ 



Ml 12 * 18 / ' . 1 1 1 \ 



+ 13. UU.2.7.8 + 5. 6. 7. 8 + 5. 6. 11. 12/ 



_ m 1 V 8 / 1 1 



19 ". 20 \1 • 2 . 7 . 8 . 13 . 14 + 5 . 6 . 7 . 8 . 13 . 14 



+ 5 . 6 . 11 . 12 . 13 . 14 + 5. 6. 11. 12. 17. IS/*" ') 

 = -a A 2 F(mV 5 ),say (113) 



Then the complete value of r that satisfies the condition 

 that the torque is zero at the free end where x = is 



T = T 1 -a cVF(mV) 



= a /(mV)-a 6¥F(m 6 ^, . . . (114) 



where /(m 6 ^ 6 ) is the series in the brackets in equation (110). 

 The other condition that has to be satisfied is that r=0 

 when x = L Therefore 



= i\mn G )-c 2 l 2 F(ni 6 l 6 ) (115) 



