﻿68 Mr. R. V. Southwell on the 



read in conjunction with my theoretical discussions of earlier 

 date*. This relates to the quantity termed the "critical 

 length," of which Mr. Cook's definition differs slightly from 

 my own. 



Both theory and experiment suggest that the length of 

 a tube sensibly affects its resistance to external pressure only 

 in the case of comparatively short tubes, and the earliest de- 

 finitions of the term " critical length," given almost simul- 

 taneously by Profs. A. E. H. Love f — as "the least length 

 for which collapse is possible under the critical pressure " — 

 and A. P. Carman J — as a " minimum length, beyond which 

 the resistance of a tube to collapse is independent of the 

 length," — were in recognition of this fact. Prof. Carman 

 concluded further, from the early experiments of Fairbairn § 

 and from others which he had himself conducted, that " the 

 collapsing pressure varies inversely as the length, for lengths 

 less than the critical length "U* That is to say, the curve 

 suggested by him as expressing the experimental relation 

 between collapsing pressure and length, for a tube of given 

 thickness and diameter, consists of two discontinuous branches 

 — a straight line, representing constant collapsing pressure, 

 for all lengths above the critical length, and a rectangular 

 hyperbola intersecting this line at a point corresponding to 

 the critical length. 



If these views are adopted, the critical length for any 

 definite size of tube may be determined from experiments, 

 by estimating (1) the straight line, parallel to the axis of 

 length, which best represents the collapsing pressure for 

 tubes of considerable length, and (2) the hyperbola which 

 agrees best with the results for the shorter tubes; their point 

 of intersection gives the required value. This is substan- 

 tially the procedure adopted by Mr. Cook, who finds that 

 within the range of his experiments the critical length L, 

 thus defined, is given satisfactorily by the formula 



L=1 - 73 \/?' 



being the thickness and d the diameter of the tube. 



(1) 



* Phil. Trans. Eoy. Soc. A. vol. ccxiii. pp. 187-244 (1913) ; Phil. Mag. 

 September 1913. 



t ' Mathematical Theory of Elasticity,' (2nd edition, 1906) p. 530. 

 t University of Illinois Bulletin, vol. iii. No. 17 (June 1906). 

 § Phil. Trans. Roy. Soc. vol. cxlviii. p. 389 (1858). 

 |l Carman, loc. cit. p. 7. 



