22 LOTKA. 



changes in x and in one parameter G at a time will be considered, the 

 other paramenters being constant. 



According to (1) a stationary state (which need not be a true 

 equilibrium in the thermodynamic sense) is defined by 



dx 



= ^ = f{xr, G) ' (2) 



where Xi denotes the equilibrium value of x. 



If the parameter G is altered by a small increment 8G, the corre- 

 sponding increment 8xi in the equilibrium value xi of x is, in view of 

 (2), given by 



(3) 

 (4) 



1. Stable State. 



If the stationary state defined by (2) is stable, we must have in the 

 neighborhood of that state ^ 



dx 



We can then distinguish two cases: 



fif 

 a.) 7, > 0. This means that the parameter G is one whose in- 



crease accelerates the transformation the progress of which is meas- 



dxi 

 ured by x. In this case it follows immediately from (4) that tt; > 0. 



oh 



In other words, if the s^\stem is stable in the stationary state defined 

 by (2), then increasing a parameter which accelerates the transforma- 

 tion will shift the position of the stationary state in the direction of 

 increased transformation. From this alone, however, it does not 

 necessarily follow that the new stationary state will actually become 



df 



3 Condition (5) states that the velocity f = — 5x i.s alwavs opposite in 



dx 



sign to the (small) displacement 5x from equilibrium. This is evidently 



necessary for stability of equilibrium. 



