GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 149 



the approach to it must be ultimately asjTnptotic, the presence of 

 trigonometric terms in (8) being excluded by the reality of all the X's. 



Indeed, the purely successive character of the reactions, as defined 

 above, excludes reversible reactions, so that the final mass of certain 

 of the components, when in equilibrium, is zero. Oscillations beyond 

 the equilibrium point are therefore excluded by the physical nature 

 of the variables A'^ (masses), which do not admit of negative values. 



3. We have seen that the svstem will be stable at the origin of 

 the ?/'s, and at the same time the series (8) will converge, provided 

 that all the ;u's are negative, i. e. in the present case, provided that 



aii<0 (i = 1,2,...!/) (34) 



that is to say, provided that each individual transformation taken by 

 itself tends towards a stable equilibrium (see discussion of equation 



32). 



B. Multiplicative Constants. 



1. The system (33) has the following peculiarity: We may break 

 off at any one of the equations, say the j*^ and leave a self-complete 

 system. 



From this it is easily seen that the series (8) for yj must in this 

 case break off at the terms in Xy = ay,- or its multiples, and can not, 

 for example contain any terms in X (y+i). The coefficients of these 

 further terms must therefore be zero. This also follows directly by 

 (12). 



2. Let — X„i = — amm be the least of the — X's, all the X's being 

 negative. Or, in other words, let | X^ | = I Omm \ be the least of the | X |. 



Then it is evident that for sufficiently large values of t all other terms 

 may be neglected in comparison with those in X^ and of the first 

 degree. In its last phases the process is therefore represented by 



2/1 = 



2/2 = 



2/m-l = 



(36) 



so that in the last phases 



2/1 :2/2 : • ■ • ■-Vm-x :?/,« :?/,«+i :...:?/.: :0 :0: ... :0 : C^ : Ci:. . . :C^ (37) 



