694 



Mr. R. V. Southwell on the 



But in order to find the terms in the expression for 

 (III— n 2 ) which are of order A, we may omit the stress- 

 couples from consideration, since these are of order li z ; and 

 the last two of equations (17) then show that T 1 and T 2 may 

 also be neglected. 



Thus the first three equations become 





3Jj 



dU' + 



-IUH% +J )h^ 



1BP 2=0 



+ n 1 (i + J)- n .(i-J)-o. 



(18) 



J 



(19) 



If we now write P a .... in the forms 



P 1 =[P 1 ]-fP 1 '....etc, . . . 



where [PJ .... are values in the configuration of equi- 

 librium, when the tube is still circular and merely contracted, 



and P x ' are the increments caused by an infinitesimal 



displacement, we have at once 



[P 1 ] = [U 1 ] = [U 2 ]=0; j 



and it is easy to show that, to terms in Ji, 



(20) 



P/ = 2 



iw 



m 2 — 1 



\_0x ma\o<p 



P.' = 2 



m' 



wr 



LV=W 



)] 



1 La \o<p J m ox„ 



(21) 



m+ 1 



V- 



\_a 



30 c^J ' 



E/i - ^ + 



where E is Young's modulus, and — is Poisson's ratio for 

 the material of the tube. m 



Using these relations to simplify (18), and neglecting 

 second order infinitesimals and higher powers of h than the 

 first, we have 



"d 2 u m — 1 1 "d 2 u 

 C^c 2 2 m a 2 



00" \ 2??i Ja'd.v'dcf) \ m) a ^x 



m + 11 B 2 w 



2m a~d<v^<fi 



m — 1 B 2 r 

 2m d#* 



-TT- : ^-, + ^ ^ 4- -,~; -=0, 



1 &* v 



« 2 §^ 2 





1 3 ?« 1 "dv IV \p 



- 5" + - ^ Ta + ■ ■ + [ w ■» 



m o^' <^ 09 « a \ 09", 



; 



0. 



(22) 



