MR. R. V. SOUTHWELL OX THE <:i NKKAL THEORY OF ELASTIC STABILITY. 209 



l-'.c I. F.I;' .-iii'l LM.I: \\.i> ;in<l tin' more |>r;n-t I'M! f-rniul.i ..(' I;\\MM. VQg^el thai 

 this advantage increases without limit as the thickness of the tube is reduced. Such 

 a conclusion is, however, inaccurate, for types of distortion are possible in the case of 

 a tube which do not involve flexure of the axis, and when the tube is thin these 

 types, of which some practical examples are shown in fig. 2, may be maintained by a 

 smaller thrust than would be required to produce failure of the kind discussed by 

 EULER. Moreover, the natural wave-length, for these symmetrical types of distortion, 

 is in general small, so that distortion can occur without hindrance in quite short 

 tubes. Hence, for a considerable range of length the strength of a tube to resist 

 end-thrust is practically constant, and is not given by any of the usual formulae for 

 struts. 



The determination of the strength of tubes to resist these symmetrical types of 

 distortion is obviously a problem of the highest practical importance, and has 

 attracted a great deal of attention in recent years. Illustrations of collapsed tubes, 

 showing symmetrical types of distortion, have been published by A. MALLOCK| and 

 R. LORENZ, and a great deal of experimental work has been carried out by 

 W. E. LILLY. || Theoretical discussions, by approximate methods, have been proposed 

 by A. GROS,1f W. E. LILLY,** S. TiMOSCHENKott and R. LORENZ.}} 



The problem of the boiler flue seems first to have been suggested by the experi- 

 ments commenced by FAIRBAIRN in 1858. These showed that the collapse of tubes 

 under external pressure was in some degree analogous to that of straight columns 

 under end-thrust, and a discussion of the phenomenon, based on EULER'S theory of 

 struts, was given by W. C. UNWIN,|||| who assisted FAIRBAIRN in his research. The 

 similar problem of a circular wire ring subjected to radial pressure has been 

 discussed by M. BRESSE^H and M. LEVY,*** and rational theories of the boiler- 

 flue problem have been given by G. H. BRYAN.ttt A. FoprL.jjJ P. FORCHHKIMER 



* " Sur la force des colonnes," ' Hist. Acad. Berlin,' XIII. (1757), p. 252. 

 t " Sur la figure des colonnes," ' Miscellanea Taurinensia,' V. (1773). 

 J ' Koy. Soc. Proc.,' A, LXXXI. (1908), p. 389. 

 ' Physikalische Zeitschrift,' XII. (1911), p. 241. 



|| ' Proc. Inst. Mech. Eng.,' 1905 ; 'lust. Civ. Eng. Ireland,' 1906 ; ' Engineering,' January 10, 1908. 

 f 'Comptes Rendus,' CXXXIV. (1902), p. 1041. 

 ** 'Inst. Civ. Eng. Ireland,' 1906. 



tt 'Zeitschrift f. Mathematik u. Physik,' LVIII. (1910), p. 337. 



Jt ' Zeitschrift des Vereines Deutscher Ingenieure,' October 24, 1908 ; 'Physikalische Zeitschrift,' XII. 

 (1911), p. 241. 



f ' Phil. Trans. Roy. Soc.,' CXLVIIL, p. 389. 

 HII ' Proc. Inat. C.E.,' XLVI. (1875), p. 225. 

 fH 'Cours de M&anique Appliquee,' 1. Partic, Paris, 1859. 

 *** ' LIOUVILLE'S Journal,' X. (1884), p. 5. 

 ttt 'Proc. Camb. Phil. Soc.,' VI. (1888), p. 287. 

 JJt 'Resistance des Materiuux ' (1901), p. 286. 

 'Zeitschrift des Oesterreichischen Ingenieur- mid Architekten-Vereines,' 1904. 



VOL. COXIII. A. 2 E 



