MR. R. V. SOUTHWELL OX THE OKNEKAL THEORY OF ELASTIC STABILITY. 2'27 

 has some sufficiently small value ; and -2. being inversely proportional to the length of 







the tube, we deduce for the " critical length," in the original sense of the term, an 

 equation of the form 



where / is constant. Prof. LOVE, as has been said, has obtained an equation of the 

 form 



which is very different ; but he has informed the author that in the light of the above 

 investigation (pp. 210-222) he does not regard his method as adequate.* 



Solution for Tubular Strut : Special Case. 



We may obtain another simplification of the general determinant to ten rows by 

 taking a zero value for k. This corresponds to a type of distortion, possible in the 

 case of a tubular strut, in which the axis remains straight and the cross-sections 

 circular, the diameter varying in a sinusoidal manner. 



The ten-row determinant for this case is given on pp. 228 and 229 ; the factor q* 

 has been cancelled from the sixth column, and terms in A have been omitted, so as to 

 yield a result for tubes collapsed by end pressure alone. The expansion is only 

 correct to terms of order r 2 , and for a first approximation we may also neglect the 

 square and higher powers of B, which must be small in any case of practical 

 importance. Investigating first the terms which are independent of r 3 , we obtain 



We may now employ the substitution 



. . . . (88) 



m q* 



m q 



in the determinant, and expand it from the top row, neglecting terms of higher order 

 than T*. 



A considerable amount of unnecessary lal>our may be avoided by a preliminary 

 examination of the relative importance of the various terms involved. It will be 



[* Added Mail 4- -An argument in favour of the new formula may IKJ drawn from physical considera- 

 tions. The resistance offered by a tube to any given form of distortion is due partly to the extension 

 and partly to the flexure which such distortion entails ; and it is clear that the relative importance of the 

 extcnsional part increases as the thickness is reduced. Hence, other things being equal, the effects of the 

 ends, which necessitate extension of the middle surface, are more important in a thin than in a thick 

 tube ; that is to say, they are sensible over a greater length.] 



2 O 2 



