230 MK. K. v. vH-TinvKu. 0* TIIK aiffeAii THEORY OF ELASTIC STABILITY. 



r<..u..I that B' contains terms in 7' and L, * weU as terms independent of q. Thus 

 the complete expression for B is of the form 



f\ m 



and it is clear that B has a minimum value when the axial wave-length has a finite 

 value, given by 



v = 



m 



This minimum value, which alone is of practical importance, is given, to a first 

 approximation in terms of T, by the equation 



(89) 



so that the determination of a and /8 is not required. 

 By expansion of the determinant we find 



m 

 y ~~ 3 m-l' 



and from (55) we deduce, for the minimum thrust required to produce collapse, 



......... (90) 



This expression is correct to terms in t". 



Validity of Investigation by the Theory of Thin Shells. 



A complete investigation of the tubular strut problem must deal with lobed forms 

 of deformation, since it is possible tbat one of these may require a smaller end-pressure 

 for its maintenance than the circular form treated above. We have, therefore, to 

 obtain a general expression for B (when A is zero) in terms both of k and q. 



The derivation of this expression, if we employ the rigorous methods of the present 

 paper, will entail nothing less than the evaluation of the complete fifteen-row 

 determinant ; for the existence of a " favourite type of distortion," of finite axial 

 wave-length, which we have noticed in the particular case (k = 0), is found by 

 practical experiment to be equally a feature of the lobed forms of distortion, and 

 shows that the terms in T Z are important. Now it will be shown that the value of 

 Smh,., when k = 0, may be obtained, correctly to terms in t 2 , by the ordinary theory 

 of thin shells ; and as there is no reason to believe that the latter theory will lead to 



