a Theory of the Collapse of Boiler-flues. 231 



renders it necessary in the present case to have recourse to 

 such arguments. 



12. When a flexural couple is applied to a thin shell, it 

 produces extension of the middle surface and change of curva- 

 ture. Under these circumstances the most natural hypothesis 

 to make is, that this couple is a linear function of the exten- 

 sions and the changes of curvature, together with a constant 

 term depending on the external pressures. When there are 

 no external pressures this hypothesis may be proved to be 

 true by the method of my former paper ; but when the shell 

 is subjected to such pressures, every circle whose plane is 

 perpendicular to the axis of the cylinder is elongated or 

 contracted, independently of the strain produced by the 

 couple. Accordingly we should anticipate that the coefficients 

 of the extensions and the changes of curvature would contain 

 terms depending on the pressures ; and we shall therefore 

 assume that 



8A 3 / mn \ /d 2 w 



„ #k 6 f mn \[<t'w , \ 



4A 3 Cmn(m-n) n \ ( dv , \ /1t . 



+ ^{j^r +/3 j{d4> +w ) +y '- (15) 



where a, /3, 7 are quantities which depend on the pressures. 



When there are no external pressures, a, /3, 7 are each 

 zero, and the above expression for G reduces to that given in 

 my former paper — see Phil. Trans. 1890, p. 441, equations 

 (14) and (16). It will be noticed that this hypothesis is 

 consistent with (13), as it involves nothing more than the 

 assumption of certain definite values for the undetermined 

 quantities A, A x . - 



The preceding argument though plausible is [not [entirely 

 free from danger. When a thin shell is free from surface 

 forces, it might be argued that the effect of any stress is to 

 produce extension, change of curvature, and torsion ; and that 

 consequently the expression for the potential energy must be a 

 quadratic function of the quantities by which these three states 

 are specified. If, however, the expressions for the potential 

 energy of a thin cylindrical or spherical shell be examined — 

 Phil. Trans. 1890, p. 443, equation (24), and p. 467, equation 

 (16) — it will be found that they contain certain terms 

 depending on the differential coefficients of the extensions. 



13. One further difficulty still remains. The equations (6) 

 for determining the small oscillations of the flue involve the 

 two displacements v and w, and the stresses T, N, G. By 

 means of the above value of G these stresses can be eliminated, 



