._...,; Mi; I: V ->"! Tl I XVKI.l, ON THE GENERAL THEORY OF ELASTIC STABILITY. 



The " Critical Length." 



A. E. H. LOVK* has investigated the rate at which the strengthening effect of 

 circular ends falls off when the length of a lx>iler flue is increased. His result 

 suggests that at a distance which is great compared with the quantity ^/at the 

 influence of the ends becomes negligible, and the flue collapses under sensibly the 

 same pressure as a tul>e of infinite length ; hence, in order that " collapse rings " may 

 have any appreciable effect, their distance apart must not exceed some experimentally- 

 determined multiple of this quantity. 



The greatest length of tube over which the ends exert any appreciable strengthening 

 influence, or the least length for which collapse is possible under a pressure sensibly 

 equal to the critical pressure, has been called by Prof. Lovst the "critical. length." 

 It is a conception of great importance in experimental work; for, as we have seen,J 

 tests on any length of tube in excess of this limit may be taken to give the strength 

 of an infinite length of the same tube, and their results compared with the theoretical 

 formula (80): but as a basis for the spacing of "collapse rings" it is superseded by 

 the theory of this paper, which yields an expression for the greatest length of tube 

 consistent with stability, when the thickness and diameter of the flue, and also the 

 collapsing pressure, are given ; and Prof. LOVE has suggested to the author that it 

 would be better now to employ the term " critical length " in this more general 

 significance. As we have seen (p. 222), the length of the tube is some multiple of the 

 quantity a/q, and we may therefore obtain from (83) the following formula : 



Critical length = Ma 



where M is a constant, depending upon the type of the collapse ring, and k has 

 that integral value which gives the least value for the right-hand expression of 

 equation (87). 



Before this subject is dismissed, it should be noticed that the theory of this paper 

 does not support Prof. LOVE'S estimate, mentioned above, of the rate of decay of end 

 effects. The term in equation (83) which depends upon the length of the tube may 

 be regarded as negligible, compared with the constant term, when the ratio 





' Proc. Lond. Math. Soc.,' XXIV. (1893), p. 208. 

 t ' Theory of Elasticity ' (2nd edition), 337 (b). 



Page 224. 



f 



i In this sense the term critical length has also been employed by CARMAN, who began his research 

 -mg the strengthening effects of the end plugs with which he sealed his tubes for test. 



