354 J. W. Glbhs — Equilibrium of Ueterogeneotis Substances. 



+ :,/ {a Z^ + /i Zv + ;/ Zz), 

 or, by (375), (376), 



-{-2 a p Xy^2 /3yYz-{- 2 yaZ^. (889) 



We may also choose any convenient directions for the co-ordinate 

 axes. Let us sujjpose that the direction of the axis of ^is so chosen 

 that the value of S for the surface perpendicular to this axis is as 

 great as for any other surface, and that the direction of the axis of Y 

 (supposed at right angles to X) is such that the value of /S" for the 

 surface perpendicular to it is as great as for any other surface 



passing through the axis of X. Then, if we write -:;- , ^— , -r- for 

 ^ ^ '^ ' da' dp' dy 



the differential coefficients derived from the last equation by treating 



a-, (i, and y as independent variables, 



dS J , dS ,,. , dS . 



-r- dot 4 d 6 -f -^— (/]/ 3= (>, 



da ^ .dfi ' dy ^ ' 



when a da -{- fi dfi -(- ;/ dy = 0, 



and « = 1, p =z 0, y =z 0. 



m, . d/S -, djS 



That IS, -J-, == 0, and ~,~- z=z 0, 



dp dy 



when a=. I, // = 0, ;/ = 0. 



Hence, X^ = 0, and Z^ = 0. (390) 



dS - , , diS -, 

 Moreover, ^-- dp -\ — =- dy = 0, 



dp dy 



when a =.0, da- =z 0, 



fJd/J-\- y dy = 0, 



and /^=1, y = 0. 



Hence Yy, = 0. (391) 



Therefore, when the co-ordinate axes have the supposed directions, 

 which, are called the principal axes of stress, the rectangular com- 

 ponents of the traction across any surface (a, /i, y) are by (379) 



«Xx, /3Yy, yZ^. (392) 



Hence, the traction across any surface will be normal to that 

 surface, — 



(1), when the surface is perj^endicular to a principal axis of stress ; 



