L'l I \1K I: V. SOUTH WKLL ON THE GENERAL THEORY OF ELASTIC STABILITY. 



The differential relations (57-59), with the boundary conditions (62), are 

 theoretically sufficient for an exact solution of our problem : we shall, however, 

 content ourselves with approximate solutions for ($, $ 2 ) and <, correct to terms 

 in r 3 . To obtain these, we assume solutions for U At g , V Ai r W kit , in series of ascending 

 powers of the quantity (ra). Thus we write 



(63) 



\a/ 2!W "J 

 where r = a+h. 



We may now derive, from equations (57-59), any required number of relations 

 between the undetermined coefficients ,...;<,...,..., and the boundary conditions (62) 

 take the form of equations in series of ascending powers of the small quantity T, in 

 which the sums of the odd and of the even powers must vanish separately. If we 

 neglect in these equations terms of order higher than some definite power of T, we 

 may obtain corresponding approximations to the values of A and B, by the elimination 

 of the undetermined coefficients. 



The approximate boundary conditions, correct to terms in r 3 , are* 





(64) 



(65) 



-i:t.fly (67) 



in\ _ 



2 fi-?-|-& = 0,. . . . (68) 



In deriving these boundary conditions it is to be noticed that a- is to a first approximation equa 

 to - 1, 80 that to our approximation - T * may be written for <n-2. 



