GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 153 



have also been considered. The most general discussion of the sub- 

 ject is that given by Jiittner, but is restricted to a broad treatment of 

 a somewhat general form of the law of mass action. 



In the present paper a very general treatment of the kinetics of 

 material transformations is developed on the sole assumption that the 

 functions F are analytic in the neighborhood of the equilibrium values 

 of the variables X. 



A general solution is given of the system of differential equations 

 (1) for the case that the parameters P are held constant during the 

 transformation. This solution is oscillatory or aperiodic according 

 to the nature (complex or real) of the roots X of a certain determinantal 

 equation A (X) = 



The effect of equations of constraint is discussed. 



Convergence of the series solution is showTi to be related to the 

 stability of the system at the equilibrium point. 



Several special cases are considered. 



1. Single transformation. 



2. Consecutive transformations. It is well kno^^^l that in chains 

 of consecutive radio-active transformations the products are present 

 in constant ratio. This property is now shown to be quite general 

 for the terminal stages of any series of purely consecutive reactions 

 taking place according to any law whatever, provided only that the 

 functions F are analytic as previously assumed. 



3. A similar conclusion applies to all systems in which the process 

 of transformation is aperiodic. 



4. In general, in a complicated system, the process is oscillatory. 

 The entire development set forth presents an analogy to the theory 



of small oscillations according to Lagrange. However, instead of 2n 

 arbitrary constants, corresponding to n co-ordinates, as in Lagrange's 

 theorv, we have here onlv n such constants. This fact stands in close 

 relation to the circumstance that the systems here considered are 

 essentially "inertia-free" or "completely damped," and the trans- 

 formations accordingly are typically irreversible processes, typical 

 cases of evolution (see for example Chwolson, Textbooks of Physics, 

 German Edition 1905, v. 3, p. 499; J. Perrin, Traite de Chimie Phy- 

 sique, 1903, Chapter V). It is on this account, with a view to pre- 

 paring for the treatment of the general problem of evolution that the 

 reflections here set forth were conceived along broad and compre- 

 hensive lines. 



