Mi;, i; V. SOUTHWELL ON THE GENERAL THEORY OF ELASTIC SI ABII.ITY. 195 



Thus the line initially given by the direction-cosines 



1, m,, n } , 



liecumes a direction of principal stress in the final configuration. Its direction-cosines 

 (referred to Ox, O#, and Oz) are then 





1, 



or 



,ai/ 



,<'"' 





1, 



: (6) 



which we shall write as 1, m',, ', ; and its final extension, to terms .if the first order 

 in ?/', /', /'', is 



' ' "' lv\ 



In the same way we find that the other directions of principal strain in the final 

 configuration are given by the direction-cosines 



- 



- 



and 





_ 



- 



(1 +,>,)-(! 



(which we may write as wi',, I, u' 2 , and u',, >*', 1), and that the final extensions 

 in these directions are 



and 



. 



The stresses in these directions, which we shall call the directions of JT', 

 referred to the original areas of the faces on which they act,* are therefore 



, and zf, 



, v ^ ax, s' . ax r ar' . ax, a/ 



. i A r + . -r T - . -T + -r . -r , 

 8<>, ?a: ce, dy oc, dz 



(.0) 



* ('/. the assumption of p. 193. 

 2 O 2 



