692 Mr. R. Y. Southwell on the 



We thus obtain the familiar expression 



8mnh* / d 2 iu\ 



™+ 7X2 ; (13) 



3(m + n)a 2 V d$ 2 , 



which may he substituted for G' in (6). The solution is of 

 the form 



ie=W amity, (14) 



where 



n 1 -n 2 = Q , 8my?A ! 3 (P-i). . . . (15) 



This is the required condition for neutral equilibrium. 

 If II 2 is zero, and we give to k the value 2, the condition 

 for stability may be written 



8rf 



1 (m + ra)tt 3 ' v y 



which is Professor Bryan's result. 



The foregoing discussion requires to be read in connexion 

 with Mr. Basset's article, but its purpose maybe summarized 

 in general terms. It is intended to show that the accepted 

 formulae for changes of curvature, in terms of the correspond- 

 ing changes in the stress-couples, need not be restricted to 

 the case of shells whose surfaces are free from stress, — 

 provided only that the changes under consideration are not 

 accompanied by any change in the magnitude of the applied 

 surface-tractions ; and in the boiler-flue problem the latter 

 requirement is satisfied. 



If the problem be approached from considerations of 

 energy, my contention may be expressed as follows : — 

 Although Mr. Basset is right in saying that a complete 

 expression for the potential energy will contain terms depend- 

 ing on II] and II 2 , yet he has not shown that these pressures, 

 when their intensity remains constant, influence to any 

 sensible extent the increase of energy which is involved in a 

 slight displacement from the configuration of equilibrium. 

 The discussion of this paper, and the results which I have 

 obtained independently of the Theory of Thin Shells, seem 

 to show that they will not. 



I have now to show that by the methods described above 

 we may obtain an estimate of the effect of "collapse rings." 

 These rings of course tend to prevent distortion of the tube 

 at its ends, and the consequent strengthening effect must 

 be investigated by the consideration of types of displace- 

 ment in which the departure of the cross-sections from 

 circularity varies in the axial direction. The product of the 

 principal curvatures at a point will now be in general finite 



