MR. R. V. sorrmVKI.I. ON T THE GENERAL THEORY OF ELASTIC STABILITY. 



to the new areas of the faces on which they act, they are 



Xi V' , 7' , 

 V/ * y "/' _ ** i 



X _ 

 - 



Y' = 

 1 



(1+^, 



and to the required degree of approximation we may write 



x' x, r i a*/ i a*//] 



" * i \~n \ ^ r "^ ~ ~T 



( 1 + c 2 ) ( 1 + <' 3 ) L 1 +^i oy 1+ 6 vz J 



20 



m2 f/ -,\dit' . dv' , dw'~\ , 



+ 7 : (m-l)-T + -T-+ -T- ..., &c. . . . (11) 



(l-fp 2 )(l+- :! )L 3* 3/ 3 J 



Then if a;, y, z denote the co-ordinates in the final configuration, referred to the 

 original axes, of the point which was originally at (x, y, z), so that 



x = 



t / , ..., &c., 



we may find the stress components in the third configuration, referred to the original 

 axes, and to the strained areas of the faces upon which they act, by the scheme of 

 transformation 



X 



y 



x 



y 1 



z' 



The following expressions are thus obtained (to the required order of approxima- 

 tion) : 



X; = X' X , Y- = Y',, Z; = Z'.,, 



(1 



,) 



, ..., &c. 



(12) 



Now the stress-components (12) must satisfy the ordinary equations of equilibrium 

 which are three of the type 



_ ^ , j 



(13) 



