234 MR. R- V. SOrrmVKLL ON THE GENERAL THEORY OF ELASTIC STABILITY. 

 When k = 0, this becomes 



i a -l 1 ..... (100 



m* q 3 ' 



which shows that q must be great, if * has a value possible in practice. Similarly, 

 when k >0, we see that q must be small, and the approximate expression m this 

 case is 



4,_^!nl __ 2l_ ..... (101) 



m * F(* 3 +l) 



We may now determine sufficiently approximate expressions for the terms in t 2 /a 3 , 

 by treating q as great when k = 0, and as small when k> 1. That is to say, we 

 retain only the highest and the lowest powers of q in the two cases* 



Thus, when k = 0, the important terms are 



and we have 



a > 

 a? 



or 



(102) 



When k > 1, the important terms are 



i 9 4 + i-^ 4 (^- 1 ) 2 = ' ( 103 ) 



771 t* 



whence, to terms in t^/a 3 , 



m'-i 



_m 8 -! q 3 iFjF-l)^ 2 



m 2 F(P+1) *g a F+l a 2 ' 



with sufficient accuracy, when q is small. 

 This leads to the result 



For practical purposes only the stationary values of S are important. It is readily 

 seen that the minimum value obtained from (102) agrees with (90), and is therefore 



* In every case it is legitimate for practical purposes to neglect the term in 2 . 



