MR. R. V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 237 



ii n lialanced end-thrust, due to the water pressure acting upon the closed ends of the 

 t ul>e. 



It is clear that the expansion of the general fifteen-row determinant will give an 



equation of the form 



... = 0, 



where a, ft, y... depend upon the dimensions of the tube and the type of the 

 distortion. But in any practical case, as we have already observed, A and B must be 

 very small quantities. It follows that an approximate solution may l obtained from 

 the terms 



.......... (107) 



]) .-ui'l S I"- ill'- \.-ilui-s ..r the extent*] pn-^mv .-UK I <>f il ..... ml- thrust, r.-idi <.f 

 which, acting alone, could produce collapse into the assumed type of distortion. Then 

 equation (107) may clearly be written as follows : 



(108) 



where $, and S are the values of the external pressure and end-thrust which can 

 produce collapse when acting in conjunction. 



It may be seen from this equation that $, can have a minimum value for some finite 

 value of the axial wave-length when, and only when, <& exists. If the end-thrust be 

 entirely unl>alaneed, we have 



,, ......... (109) 



and the collapsing pressure may, in this case, be determined from equation (108). 



GENERAL THEORY OF INSTABILITY IN MATERIALS OF FINITE STRENGTH. 

 The Practical Value of a Theory of Instability. 



In the concluding section of this paper an attempt will be made to estimate the 

 practical value of a theory of elastic instability ; to suggest ways in which we may 

 hope to increase this value ; and to indicate the questions to which answers must be 

 found in order that further advance may be possible. 



The first point which must be noticed is the non-realization in practice of our 

 conception of a " critical loading," owing to imperfections which always exist, and 

 which violate our ideal assumptions. In any actual example the displacement of the 

 system increases continuously with the load, and the system collapses at a smaller 

 value of the load than our theory would dictate. It is necessary to inquire whether 

 serious discrepancies are to be expected. 



In some mechanical problems the effects of imperfections may be calculated. We 

 may take, as an example, the system illustrated in fig. 1, and consider any one of the 



