GENERAL KINETICS OF MATERIAL TRANSFORMATIONS. 145 



1. If all the /x's are negative, the series (8) evidently converge for 

 large values of /, and the x's then approach zero as t approaches 

 infinity. In this case, then, the point .vi =^ xo = . = .r„ = corre- 

 sponds to a stable condition. 



2. If, on the other hand, all the m's are positive, then the series (8) 

 converge for large values of — f, and diverge for large values of t. The 

 origin of the .r's in this case corresponds to an unstable condition. 



3. If some of the fx's are positive, others negative, the series (8) 

 evidently in general converge neither for large values of t nor for large 

 values of — /. 



However, in this case "vve can single out a particular solution 



Xi = aiC^'+aii(^^'-\- ... (25) 



in which fi >0; from which it is evident that the state of the system 

 at the origin of the x's is certainly unstable at any rate for certain 

 displacements. Equilibrium at the origin is therefore in this case 

 unstable. 



4. If some of the roots X are pure imaginaries, the presence of even 

 a single positive /x will determine the instability of the equilibrium of 

 the origin. If, on the other hand, those /x's which are not zero, are all 

 negative, then the series may converge for large values of /, and in 

 that case the equilibrium, at the origin of the .r's, would be in a certain 

 sense stable; for, after an arbitrary displacement, though the system 

 would not in general return to the origin, it would ultimately travel 

 in a periodic path not containing the origin. ^^ That is to say, after 

 the lapse of a certain time the point representing the state of the 

 system would thenceforth never be further away from the origin than 

 a certain finite maximum distance M. 



The criterion for the existence of purely imaginary roots is evidently 

 that A (X) and A ( — X) have one or more common factors of the form 

 (X^ 4" P^), where p is a real quantity. 



5. If all the X's are pure imaginaries,^^ (in which case A (X) contains 



12 To be more exact, the point .ri, Xo . . . a'„ in n — dimensional space, which 

 represents the state of the system, describes such a path. 



13 The existence of purely imaginary roots X may perhaps seem of theoretical 

 rather than practical interest, since they could occur only when the coeffi- 

 cients in (7) satisfy very particular conditions, and since it appears improbable 

 that these conditions would be satisfied in nature. 



However, the case, in no way improbable a priori, that the real part n of 

 Xbe very small, in practice will differ but little from the case where m = 0. 

 For, if IX is sufficiently small, any accidental disturbance of the system ( e. g. 

 any accidental change in one of the parameters P) from equilibrium, will set 

 up oscillations with only slight damping, so that the sj-stem will be liable to be 

 kept in more or less constant oscillations. 



