﻿Collapse of Tubes by External Pressure. 69 



By calculation, 1 had previously obtained the formula 



V?' 



(2) 



fc being some constant, depending upon the type of the end 

 constraints, of which I have not been able, except in certain 

 ideal cases, to obtain an exact value by analysis*; but, as I 

 have already stated, my definition of the critical length is 

 different from Mr. Cook's. I had concluded, as a result of 

 my analysis, that tubes of length such that the strengthening- 

 effect of the ends is sensible, but small, will collapse under 

 a pressure given by 



®=2[4+4i w 



l being the length of the tube, and a and /3 constants for 

 any given material f. Clearly, as I is increased the col- 

 lapsing pressure given by this equation falls rapidly, and 

 becomes sensibly equal to /3f/d 3 . Hence, adopting a slightly 

 modified form of Professor Love's definition, I took the 

 critical length to be "the least length for which collapse 

 is possible under [a pressure sensibly equal to~] the critical 

 pressure/' L being thus defined, and 8 some small number 

 which we agree to regard as negligible, we have 



or d* A* 



whence equation (2) may be derived, k being equal to 



No hyperbolic relation between collapsing pressure and 

 length occurs in the exact analytical treatment of our 

 problem. But the convenience of a relation of this form, 

 and its satisfactory agreement with experiment, suggest that 

 an hyperbola might with advantage be substituted for the 

 discontinuous curve which represents the exact theoretical 

 expression for the collapsing pressure. A curve of this 

 type is illustrated by the thick lines in fig. 3 (which may be 

 regarded as connecting pressure and length) of my paper in 

 the Philosophical Magazine for May 1913: it is composed 



* Phil. Trans. Roy. Soc. A. vol. ccxiii. p. 227 (1913). 



t Cf. my equation (1), Phil. Mag. September 19 lo, p. oOo. 



