Mi; i;. V. SOITIIU KI.L ON TIIK CKNKKAT, THEORY OF Hl.ASTIC STABILITY. 

 But to a first approximation 



so that we have 



(80) 



which agrees with BRYAN'S result.* 



To complete our discussion of this problem we must consider types of distortion in 

 which the axial wave-length is finite, and thus obtain a theoretical estimate of the 

 strength of short flues with fixed ends. A solution giving A correctly to terms in r a 

 may be derived from the complete fifteen-row determinant ; but we may show that 

 for practical purposes the labour which this evaluation would entail is quite 

 unnecessary. 



We find first of all that those terms in the expression for A which are independent 

 of T contain q* as a factor. Now 2AC being approximately equal to the mean hoop 

 stress in the tube before collapse, it is clear that A must in all cases of practical 

 importance be a very small quantity. It follows that in the expanded equation the 

 terms in A are of primary importance, and A 2 and higher powers may be neglected ; 

 further, since q must also be small, that terms in <f and higher powers of q may 

 be neglected in comparison with terms in g 4 , and that of the terms in r 2 those which 

 involve q are negligible in comparison with the terms already found. 



In accordance with these principles we may derive the terms which are required to 

 complete our solution from a nine-row determinant, obtained by omitting terms in r 2 

 from the general determinant. This simplified determinant is given on pp. 218 

 and 219. Further, we may neglect A 2 in the expansion, and in the coefficient of A 

 retain only those terms which do not involve q ; we thus obtain the equation 



But, by equations (55), 



and therefore, to the approximation of equation (81), 



* Cf. footnote, p. 209. 



