Collapse of Tubes by External Pressure. 689 



(see fig. 1). Mr. Basset argues, rightly I think, that there 

 are no perfectly general expressions for these quantities, 



Fiff. 1. 



correct to the third power of the thickness, in terms of the 

 final extensions and curvatures of the middle surface, when 

 the surfaces of the shell are subjected to pressure *. But we 

 may proceed by distinguishing two configurations : (1) the 

 position of equilibrium, in which the tube remains circular 

 and merely suffers contraction : this is the configuration of 

 which we are investigating the stability ; and (2) a con- 

 figuration of infinitesimal displacement from this. If a is 

 the radius of the tube in the former position, we may write 



an< 



ds = ad<\> 

 p a a~\ 



tc + 



d 2 w 



y 



(2) 



p a a- \ d(j> 2 J j 



where w is the (infinitesimal) radial displacement at any 

 point, in the outward direction. We may also write T, N 

 and G in the forms 



T=T +TO 



N = N + N', I (3) 



G=G + G-', J 



where T , N , G , are values for the first configuration, and 

 T, N', Gr', are infinitesimal increments due to distortion. 

 It is clear that 



No=0, 



and 



dG 

 d(j> 



= 



1 



a *\ a J \ aj 



Cf. also Lord Rayleigh, Proc. Lond. Math. Soc. vol. xx. p. 379. 



w 



