Mi;. R. V. SOUTHWELL ON Till! (IF.NKKA!, THKOKY OK HI. ANTIC STAHILITY. 213 



The boundary conditions now require investigation. From the consideration that 

 the cylindrical Ixmndary surfaces of tin- tulx- must continue to be tangent to principal 

 planes of stress, in any possible type of distortion, we deduce the conditions 



m, = 

 n, =0. 



, identically, when r = at. 



(60) 



The other boundary conditions are more complex. Since the pressures acting on 

 the surfaces of the tube are hydrostatic, it is clear that the radial stress, as defined 

 on p. 193, is increased at points on the boundary surfaces of the. tube where the 

 distortion involves positive extension. In the notation employed above, we have 



rr 1 = |Ji, when r = a+t, 

 = $;i when r = at, 

 and from (43) we deduce the following equations, which must be satisfied identically,* 



m : 



u 



v _,_ i at/ _,_ a 

 + + ' wheu 



_, , , 



C [r r26 



r = at, 



.. . . (61) 



Substituting from (56) in the identities (60) and (61), we finally obtain, as the 

 required boundary conditions in U*.,, V M , and W Ai? , 



and 



. (62) 



when r = at. 



* In obtaining these equations it should be noticed that before distortion occurs - 5, and - 1) 3 are the 

 values at the boundary of 



^~s /"^ 



Bid not of rr, if we retain the significance for rr which was assumed on p. 193. The distinction is not 

 really needed for the approximation of the following work, but it may lead to confusion if neglected. 



