Mi;, i: V. SOUTHWELL ON Till. CKNKiai. THKORY OF ELASTIC STABILITY. 205 



so that the principal stress.-s in tin- third configuration, referred to the fined areas of 

 the faces on which they act, are 



t " 



I H! + 1 ^ ay i 

 * r a0 a? 



!+<, ~1+J 



20 



I dtl/ cV 1 



!__^__^L 



(l+f 3 )(l +f,) I 1 +e, l+e,J 



2C 

 m-2 



r 



[a' ; IrV I 

 f> r r ?0 

 1- -; n 

 1 + r, !+-, J 



2C 



m-2 r, . v a^ 



+ 71 - rr; - d (i-l)^^ 

 (l+e,)(l+r,)l - . 



Then by the scheme of transformation 



' i a 



+ - ==- 

 r r cflj 



(43) 



tf z' 



1 -m', -n', 



n\ 



n', 



we find for the stress components in the third configuration, at the point which was 

 initially at (r, 6, z), referred to axes in the radial, tangential, and axial directions 

 through the final position of the point, and to the final areas of the faces upon which 

 they act the expressions 



. 



rr = rr', 00 = 00', zz = z/, 



= 07- = 



required approximation), <fec. 



(to the 

 (44) 



