Mi:, i: V ^HTTIIWKLL OX THK (IKNKKAL THEORY OF ELASTIC STABILITY. 221 



Combining this result witli (80) \v<> have, us ,,ur final fxpn'ssion for the pressure- 

 'lill'crence which can produce collapse of the flue, 



(83) 



In this equation t/a is the ratio of the thickness to the diameter of the tube, and k 

 is the number of lobes in tin; distorted form of its cross-section. The quantity q is 

 connected with the axial wave-length X of the distortion by the relation 



q\ = 2ira. 



(84) 



We may imagine a flue subjected at its ends to constraints which merely keep the 

 ends circular, without imposing any other restrictions upon the type of distortion.* 

 In this case the end conditions may be written in the form 



when z = 



(85) 



and from (56) it is clear that I, the length of the flue, is equal to 



[* Added June .?. Thin circular discs, inserted into the tube at its ends, but not fixed to it, would 

 approximately realize these conditions.] 



