GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 



143 



In such case, then, the equihbrium or steady state is no longer 

 fixed by (1) as soon as the values of the parameters Pi, Pi, ■ ■ Pj are 

 given, but the equilibrium further requires, for its definition, the 

 statement of the initial values .li, Ai,...A,n of the variables A^, 

 X2, . . . Xm- That is to say, the equations of equilibrium (3) take the 

 form 



•f*! {Ai, A^,- ■ -Am', Am+1- 



= Po (Ai, A2,. . .Am] X 



m+1 ■ 



.Xn) 

 .Xn) 



— Fn (Ai, Ao,. . .Am] X, 



m+1 ■ ■ ■ 



Xn) = 



(21) 



the Xi, Xo, . . . Xm having been eliminated by virtue of the equations 

 of constraint (15), with introduction, in their place, of the initial 

 constants Ai, A^,. . .Am- Equilibrium is now free to be influenced by 

 the arbitrarv selection of the initial values of m of the masses X of the 

 components of the system. This is the case of common interest in 

 physico-chemical transformations. 



It has been shown that, in the case where equations of constraint 

 exist between the variables X, the pohTiomical A (X) contains the 

 factor X, the determinant A (0) vanishing. In fact, it is evident that 

 the diff"erential equation of n^^ order in x which can be derived in 

 well-kno%\Ti manner ^° from the system (7), after rejection of all 

 terms of higher degree than the first, is reducible to an order as much 

 lower than n as there are zero roots for the equation A (X) = 0. 



However, in practice, instead of operating in this manner upon the 

 primitive system (1) or its equivalents (2) or (6), it is preferable to 

 eliminate from the start 7?^ of the variables A' by the aid of the 

 equations of constraint (15). There then remain (?i — m) dependent 

 variables, which may be either certain of the original variables X 



10 The elimination of (?i — 1) of the variables Xi X2. . .Xn in the system (7), 

 discarding the terms of second and higher degree, can be effected at sight ac- 

 cording to the plan 



that is to say 



-^ U 1 ^^ = ' 



an - ^j^, au, 



O21, 



0>i — 



dt' 



ani, 



ani, 



. ai» 



, Unn r; 



at 



Xi 



