208 EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES. 



Also* 

 ir_y/ (?(dx\*/dx\*__y/dx_ dx\ (dx dx\\ 



x dx 



dy f 



dx 



dz^^dx dx dy dy __dx dx dz dz\ 

 dx f \dy') "'dx 7 dy 7 d^ / ~d^^d^'~d^ dot dy'} 



(433) 



dx r 



--^ V 

 / ' 



This may also be written 



dx' dy' 

 dy dy 



(434) 



dx' dy' 



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 



x, 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 Z' in a similar sense with respect 



3-3 



to x f , y', z f ; and to use x', y', zf as equivalent to a? 7 , y', z', except that 

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

 understanding we may write 



Gr= 



3 _ 3 



dx 



,, QfU 

 (4do) 



\dy' dy'J " \dz' dz's 



In expanding the product of the three sums, we may cancel on 

 account of the sign 2' the terms which do not contain all the three 



3-3 



expressions dx, dy, and dz. Hence we may write 



/j__ y/ y (dx dx dy dy dz dz\ 

 "3-33+3 \^ x/ dx' dy' dy' dz dz') 



~(dx dy dz ~,(dx dy dz\\ 

 ~ 3+3 \dx' dy' dz 3 _ 3 \dx' dy' dz')} 



y (dx dy dz\ ~, (dx dy dz\ 



~ z -z\dx' dy f dz') 3 _ 3 \dx' dy' dz'/ 



(436) 



* 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 Intro- 

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

 Anwendung der Determinanten, 5. 



