212 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



where i, e, /, and h denote functions of t. Let us first consider the 

 second of these formulae. Since E, F, and H are symmetrical functions 

 of r ly r z , r 8> if \fa> is any function of t, E, F, H, we must have 



<&"! 



^J2.f. ^72. /. 



(445) 



dr<? 



dr l dr 2 ~ d f r 2 dr s ~~ dr% dr^ 



whenever r 1 = r 2 = r 3 . Now i, e, /, and h may be determined (as 

 functions of t) so as to give to 



their proper values at every temperature for some isotropic state of 

 strain, which may be determined by any desired condition. We 

 shall suppose that they are determined so as to give the proper 

 values to i/r V '> e t c -> when the stresses in the solid vanish. If we 

 denote by r the common value of r lt r 2 , r B which will make the 

 stresses vanish at any given temperature, and imagine the true value 

 of \l^>, and also the value given by equation (444) to be expressed in 

 terms of the ascending powers of 



r i- r o> r 2~n r 3- r o> (446) 



it is evident that the expressions will coincide as far as the terms of 

 the second degree inclusive. That is, the errors of the values of >/>> 

 given by equation (444) are of the same order of magnitude as the 

 cubes of the above differences. The errors of the values of 



dr 1 ' dr 2 ' dr s 



will be of the same order of magnitude as the squares of the same 

 differences. Therefore, since 



d^, dr l 



^. B 



-.dx " dr l -jdx d/r% ..dx dr s .,dx 

 dx' dx' dx' dx' 



whether we regard the true value of \[s v , or the value given by equa- 

 tion (444), and since the error in (444) does not affect the values of 



dr l dr 2 dr, 



3 



..dx' -.dx' -.dx' 

 dx' dx dx' 



which we may regard as determined by equations (431), (432), (434), 

 (437) and (438), the errors in the values of X^, derived from (444) 



