a Theory of the Collapse of Boiler-flues. 223 



the assumption that R, S, T may be treated as zero gives a 

 perfectly correct expression for the energy to the above order 

 of approximation. 



For a fuller discussion of this assumption and the results 

 to which it leads, the reader is referred to the following 

 papers *. 



4. When the surfaces of the shell are subjected to external 

 pressure, the results furnished by the above hypothesis are no 

 longer correct. This was first pointed out by Lord Rayleigh f, 

 and shortly afterwards by myself. I wrote \ : — ■ 



" The fundamental hypothesis that R, S, T may be treated 

 as zero, is not true when the surfaces of the shell are subjected 

 to external pressures or tangential stresses ; for if the 

 convex and concave surfaces of the shell are subjected 

 to pressures II 1? II 2 , the value of R, as we pass through the 

 substance of the shell from its exterior to its interior sur- 

 face, must vary from — Hi to — H 2 , and consequently (ex- 

 cepting in very special cases) R will contain a term inde- 

 pendent of the thickness. Hence the theory developed in the 

 present paper is not applicable to problems relating to the collapse 

 of boiler-flues. ... In order to obtain a theory which would 

 enable such questions to be mathematically investigated, it 

 would be necessary to find the values of the additional terms 

 in the variational equation of motion which depend upon the 

 external pressures ; and this is a problem which awaits solution^ " 



5. The problem of the collapse of a boiler-flue, in its simplest 

 form, may be stated as follows : — Let the flue be regarded as 

 indefinitely long, and be cylindrical ; let a + h and a — hhe the 

 external and internal radii of its surfaces ; and let the outer 

 and inner surfaces be subjected to pressures n x , II 2 . Then the 

 condition of stability requires that a certain relation should 

 exist between these four quantities, which may be written 



F(a, h, n b II 2 )>0. 



If this condition is not satisfied, the equilibrium will be 

 unstable, and a disturbance will cause the flue to collapse. 



If the flue be regarded as of finite length I, a correction is 

 required whose importance depends upon the magnitude ratio 

 of a to I. 



Taking the simplest case of an indefinitely long tube, the 

 problem of determining F may be attacked in two different 

 ways. In the first place, let the flue be slightly deformed, 

 and let the period of the small oscillations be found ; then 



* Lord Rayleigh, Proc. Lond. Math. Soc. vol. xx. p. 225. Basset, ibid. 

 vol. xxi. pp. 33, 53 ; Phil. Trans. 1890, p. 433. 



t Proc. Lond. Math. Soc. vol. xx. p. 373. 



X Phil. Trans. 1890, p. 437. 



R2 



