232 Mr. A. B. Basset on the Difficulties of Constructing 



and a single equation will remain connecting v and w. This 

 is insufficient to determine the period, and the only course 

 open to us is to assume that the middle surface is inextensible, 

 which gives a further relation between v and w, and furnishes 

 a sufficient number of equations. I am inclined to think 

 that the neglect of the extension of the middle surface is not 

 likely to lead to any serious error in the final result ; but at 

 the same time the solution cannot be regarded as perfectly 

 rigorous, for when the surfaces are subjected to uniform 

 pressure (which is the case we are considering), an extension 

 of the middle surface must necessarily take place unless the 

 pressures are specially adjusted. 



14. We are now in a position to obtain a solution of the 

 problem of the collapse of a boiler flue, which though imperfect, 

 is probably substantially correct. 



When the natural form of the cylinder is circular, N is 

 zero in equilibrium ; accordingly we may write N/p = N/a. 

 Also if the extension of the middle surface be neglected, 



ds=a dcj) f 



p~~ a ci \d<f> 2 J> 

 4-«?=0. 



(16) 



p a a \dcj>' 



dv 



d$ 



The equations of motion (6) now become 



^-T+cn^noa-^+njA + ^-nd^+to^^, j>(i7) 



From the first two we get 



? +s +^nO^($ + -)-i-^ + i> as) 



Let D = d/d<l> ; then, since the extension of the middle 

 surface is neglected, we shall have 



G=I(D 2 + 1)™,. ..... (19) 



where 



T 8A 3 / mn \ ,^ v 



