MR. R. V. SOUTHWELL ON TIIK cl-.NI-.l: \l. TIIKORY OF ELASTIC STABILITY. 225 



are valid. But we may also proceed, aa in the foregoing discussion, by first deter- 

 mining the stress-system for the equilibrium position, and then deriving equations 

 for an infinitesimal displacement. The stress-couples which appear in these equations 

 will be dm* to the ndilitionnl </ rcsses introduced />// tin- distort inn, u,,d since these, to 

 a first approximation >it /<".< ,-nnixli nt the surfaces of t/n tnl,,-. ///// ///// lie. given 

 with sufficient accuraci/ lit/ tlie usual expressions. Moreover, when the distortion is 

 two-dimensional (as in BASSET'S problem), the change in the " hoop " stress-resultant 

 \\ ill be of an order which is negligible, so that the middle surface may be regarded as 

 undergoing no extension relatively to the equilibrium position, even though its area 

 may be sensibly changed in comparison with the unstrained configuration.* 



The method of investigation just described, which follows the actual sequence of 

 occurrences in the material, is suggested as in every way preferable to existing 

 methods, for the investigation of any problem in elastic stability. For the present 

 '\iimple, in particular, it leads to the same results as the more rigorous methods of 

 this paper. 



Comparison uith Existing Formula. 



Previous discussion of the boiler-flue problem by analytical methods have, without 

 exception, dealt with a tube subjected to pressure on one surface only, and almost all 

 of them have been restricted to the case of an indefinitely long flue. Their results 

 have, therefore, to be compared with our equation (80), when $., is zero. It will be 

 found that this equation agrees with the formula obtained by BRYAN! and BASHKT::}: 



FOPPL'S formulat omits the factor ; , which measures the increased resistance to 



m 1 



flexure of a long tube as compared with a circular ring. 



The more general formula may be compared with that of LORKNZ,* if $., IKJ put 

 equal to zero. It will be found that there is a serious want of agreement in regard 

 to both terms in the expression (83). In support of the latter result, it may be 

 urged that LORKNZ' solution gives for the indefinitely long flue a result which does 

 not agree with equation (80) (and, as we have just noticed, this is supported by 

 previous investigations), and which vanishes, not when k = 1, but when k = 0. Now 

 the value 1, in the case of an infinitely long flue, corresponds to translation of the 

 tube as a whole, without distortion, and the value to a change in the diameter 

 of the tube, without any departure from circularity. It is clear that the applied 

 pressures can have no tendency to maintain such a form of distortion, so that LOUDTZ' 

 formula can hardly be correct. 



[* ;t<ttle<l June 8. The arguments of this section are more fully developed in n paper by the author 

 " On the Collapse of Tubes by External Pressure," published in the ' Philosophical Magazine ' for May, 

 1913 (pp. 687-698).] 



t Cf. footnote, p. 209. 



\ Cf. footnote, p. 210. 



VOL. CCX1II. A. 2 O 



