﻿70 • Mr. B. V. Southwell on the 



of a series of intersecting arcs, of which those shown in 

 the figure (as I poiDted out in the same paper *) are very 

 approximately enveloped by a rectangular hyperbola. 



The curves of which these arcs are fragments are those 

 members of the family represented by the equation 



in which k has positive integral values. h denotes the 

 number of lobes characterizing the cross-section of the tube 

 after collapse, and of the other quantities appearing in (4), 

 besides those which have already been defined — 



E is Young's modulus, and 



— is Poisson's ratio, for the material of the tube; 

 m 



Z is a constant, depending upon the type of the end- 



constraintsf. 



Now it is easy to show that the curve represented by (4) 

 is touched by the rectangular hyperbola 



at a point given by 



1 - d Vq™'" 1 Z d * ,-\ 



l ~ kV / m 2 (tf-1)** 3 ' ' * * ^ 



and the occurrence of k in (5) shows that the family of 

 curves (4) is not exactly enveloped by any one hyperbola. 

 But the hyperbola touching that member of the family for 

 which k = 3 gives values for the collapsing pressure which 

 are in satisfactory agreement with those obtainable from the 

 exact (discontinuous) curve, and which err on the side of 

 safety throughout the practically important range of lengths. 

 We may therefore take the equation to this hyperbola, viz. 



as representing the collapsing pressure of short tubes, and 

 Prof. Bryan's formula ± 



^ = 2 ^T E £ (8) 



* p. 698. 



f Of. Phil. Mag. September 1918, p. 503. 



X Ptil. Mag-. September 1913, p. 504. 



