Mi; It V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC STABILITY. 211 



We shall consider a tube of indefinite length, of which the inner and outer radii are 



at (so that the thickness is 2t), 



and we shall write 



for the ratio - 

 a 



The com-spoiid'nig stress-system, for the position of equilibrium, is easily obtained.* 

 We have 



TV = 



DO = -v- 



and 



zz = 



& 



. . . (51) 



It can also be shown that e 3 is constant, and equations (46-48) may therefore be 

 taken to express the conditions of neutral equilibrium. The degree of approximation 

 to which these equations have been obtained (p. 20<i) will be maintained for the rest 



>-s 



* 



of this paper, i.e., terms of order -^ '... will be neglected. They may also be written 

 as follows : 



m 



r 



2 



A\ 1 3V /3m-4 A\ 1 dv' 



/ *N r\ A ' I 1 > rt 



rn-2 2/r3r80 \rn-2 2 / r 9 SB 



V = 



I m A)I 

 W-2 2/ r 



-2 4 \" r 8 / ' 4 



A\ 1 3V . /3m 4 A 

 m-2 2 



Bl 



(52) 



m-1 1 3V 



7H-2V 39 s 



m A A 



B 1 1 3V 



- 



and 

 f m 



m-2 4 \ 



4j 82 8r lm-2 4 



3z 



m A/ .a a \ B\l 8V 

 h lm-2 4V rr 4 



_ 

 4\ 1* 



B118V m-1 3V 



_ 



- 



* LOVE, ' Mathematical Theory of Elasticity ' (2nd edition), 100. 



2 E 2 



