368 J. W. Gibbs — Equilibriuui of Heterogeneous Substances. 



We may write 



Then 



E=z 2' 2 ('^"^ 



(431) 

 (432) 



Also 



- r6 + A'r* — Fr^ + (? = 0. 



, V 



\dx' / ' 



\ Ulx \^ ^ [dx \ 2 /dx dx \ /dx dx \ ) 



^=^ ] ^\d^) WV ~ \d^W \d^dy') S 



~ ( [dx'J \d^') ~ dx' d^' \d^' dy'J S ~ 



, ^\ (dx\^ /dy \^ i^(<^^ \" /dz \^ dx dx dy dy dx dx dz dz \ 

 l\f?a;'/ \dy'] \dx'/ \dy'J dx dy' dx' dy' dx'dy'd^dy'l 



~ \ W/ \dy') "*" W/ W/ ~ d^'dy'da&'di/'S 



^, ^ (<^^ ^y ^^y dx\^ 



\dx' dy' dx' dy'J ' ^ 



This may also be Avritteu 



F= 2' 2 



(434) 



In the reduction of the value of G, it will be convenient to use the 



symbol 2 to denote the sum of the six terms formed by changing 



3+3 

 a;, y, z, into y, z, x; z, x, y\ x, z, y ; y,x,z', and z, y, x; and the 



symbol 2 in the same sense except that the last three terms are to 

 3-3 



be taken negatively ; also to use 2' in a similar sense with respect 



3-3 



to x', y,' z' ; and to use x', y', z' as eqiiivalent to x', y', 2', except that 

 they are not to be affected by the sign of summation. With this 

 understanding we may write 



G= 2' 



3-3 



/dx dx \ /dx dx \ /dx dx\) 



\d^' di) ^ \d^' w ^ W^/ J ■ ^^^^^ 



* The values of F and G given in equations (434) and (438), which are here 

 deduced at length, may be derived from inspection of equation (430) by means of the 

 usual theorems relating to the multiplication of determinants. See Salmon's Lessons 

 Introductory to the Modern Higher Algebra, 2d Ed., Lesson III ; or Baltzer's Theorie und 

 Anwendung der Determinanten, § 5. 



