EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 207 



fi '/* ft % 



-T-,,. -j, under the most general supposition with respect to the 

 dec dz 



position of the co-ordinate axes. 



For any principal axis of strain we have 



' 



da dp dy 



when a da + /3' d/3' + y dy = 0, 



the differential coefficients in the first of these equations being 

 determined from (420) as before. Therefore, 



a' da' "P d/3' ~y dy' ' 

 From (420) we obtain directly 



Pd(r*) ,y'd(r*)_ 

 "2 'd" ~2~d' 



( ? 



From the two last equations, in virtue of the necessary relation 

 a 2 +/S /2 -hy /2 =l, we obtain 



(428) 



j /- 



or, if we substitute the values of the differential coefficients taken 

 from (420), 



X \A/X \ . i -. / CLX (A/X 



a 



a 



a 



dx 



dx 



Cjl/X CvZ : 



x dx 



dx 



(429) 



If we eliminate a', /^ y' from these equations, we may write the 

 result in the form, 



(dx dx\ 



dx dx 







' 



dx\ 



2 



= 0. 



(430) 



We may write 

 Then 



(431) 

 (432) 



