﻿Collapse of Tubes by External Pressure. 71 



for those cases in which it gives a higher value for ^ than 

 Ave should obtain from (7). 



The theoretical value oE the " critical length," as Prof. 

 Carman and Mr. Cook have defined the term, is then given 

 by the point of intersection of (7) and (8). We find 



T 16 VZ m'-la 6 /n . 



L= WV M~^T¥f' • • * • 00 



which agrees in form with Mr. Cook's equation (1). The 

 latter equation is therefore supported by my analysis, 

 although the " critical length " with which it deals is a 

 quantity differing from that which was considered by me in 

 my earlier papers. 



The significance of the foregoing investigation lies in the 

 fact that the expression for the critical length given by (9) 

 is almost entirely independent of the material composing 

 the tube. In so far as it varies with the elastic constants, 



V m z ' 



Lcc\/l-^, < 10 ) 



and the value of the quantity on the right of (10) ranges 

 from 0*985 in the case of glass (for which l/m = 0*258) to 

 0*951 in the case of copper (for which l/m = 0*428)*. It is 

 highly probable, therefore, that Mr. Cook's equation (1) has 

 an application much udder than the range of his experiments. 



The point is, I think, worth investigation in future ex- 

 perimental work. If Mr. Cook's equation should be found 

 to be thus generally applicable, it will introduce important 

 simplifications into the problem of design. For a knowledge 

 of the critical length and of a formula giving the minimum 

 collapsing pressure for a tube of given thickness, diameter, 

 and material is, as I shall now attempt to show, sufficient for 

 all practical purposes. 



At the present time there is no generally accepted formula 

 for the collapsing pressure of long tubes. The results of 

 theory are naturally unreliable in practice, since the perfectly 

 elastic and homogeneous tube which it presupposes is an ideal 

 not practically realizable : those of experiment have the dis- 

 advantage of restricted scope and, further, give relations 

 between the collapsing pressure and the ratio of thickness to 

 diameter which are not expressible in any simple formula. 

 I have tried to show f that these experimental relations can 



* The authorities for these figures are given by Love. op. eif. p. 103; 

 the figures given above represent an extreme range o( values. 

 t Phil. Mag-. September 1913. 



