EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 195 



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

 passing through the axis of X. Then, if we write -* , -T^, ~j~ f r 



the differential coefficients derived from the last equation by treating 

 a, ft, and y as independent variables, 



dS 



, , jfi.j 

 -T- da + -S-Q dB + -7- dy - 0, 

 act a/5 ay 



when 



and a = l, = 0, y = 0. 



mi_ j. ~ j ~ 



That is, -7^ == 0, and -,- = 0, 



when a = l, = 0, y = 0. 



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



Moreover, -^-5 cZ/3 4- -j- dy = 0, 



ctp ay 



when a = 0, da = 0, 



and = 1, y = 0. 



Hence F z = 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, /3, y) are by (379) 



aX x , /3F Y , 7 Z Z . (392) 



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

 surface, 



(1), when the surface is perpendicular to a principal axis of stress ; 



(2), if two of the principal tractions X x , F Y , Z z are equal, when 

 the surface is perpendicular to the plane containing the two corre- 

 sponding axes (in this case the traction across any such surface is 

 equal to the common value of the two principal tractions) ; 



(3), if the principal tractions are all equal, the traction is normal 

 and constant for all surfaces. 



It will be observed that in the second and third cases the positions 

 of the principal axes of stress are partially or wholly indeterminate 

 (so that these cases may be regarded as included in the first), but the 

 values of the principal tractions are always determinate, although not 

 always different. 



If, therefore, a solid which is homogeneous in nature and in state of 

 strain is bounded by six surfaces perpendicular to the principal axes 

 of stress, the mechanical conditions of equilibrium for these surfaces 

 may be satisfied by the contact of fluids having the proper pressures 



