MR. R. V. BOOTHWBL ON THE GENERAL THEORY OF ELASTIC STABILITY. 203 



co-ordinates. We Hlmll limit our discussion to stress-systems which produce a 

 displacement symmetrical about an axis, up to the instant at which the equilibrium 

 becomes unstable and distortion occurs : in PKARSON'S notation, the principal stresses 



in the equilibrium configuration an- //, rt#, and zz, and these quantities are functions 

 of r only. 



The new equations are derived by a method very similar to that which has already 

 been explained. The co-ordinates of a point in the unstrained configuration are 



r, e, z; 



in the second configuration (of equilibrium) they are 



r+it, 6, z+ir, 



and in the third configuration (of slight distortion from the position of equilibrium) 

 they are 



r+ ' + "', 





(the radial, tangential, and axial displacements u' t r', iS being ultimately taken as 

 infinitesimal). 



The extension of a line-element joining the point (r, B, z) to the point (r+ir, + 30, 

 z + $z) is 



*> /I I I " ** 



/ r >T0 / ?z 



GIP 

 T I T 1" ^T Till I + 



L cr r < rt \ 

 where 



m = r , and n -^ ; 



cwAl'l "I 



TF/J !' 



and this has a stationary value for a line very slightly inclined to the radius, given 

 by 







where < c,, c 3 are written for dli/2r, w/c, ami < <//< : ivnpectivt'ly. 



2 D 2 



