150 



LOTKA. 



That is to say, the quantities y are in constant ratio. Furthermore, 

 if iim relates to the substance or substances Sm having the least |x|, 

 all the substances parent to Sm are entirely absent in the last stages 

 of the transformation. 



We recognize here a well-known property of radioactive equilibria, 

 in which the longest-lived substance (e. g. uranium) heads the series, 

 while the other substances are present in constant ratio. However, 

 from the method by which this conclusion has here been reached, it is 

 evident that this property is independent of the particular form of 

 radioactive transformations, but rests on a broader basis and is com- 

 mon to all systems in which is taking place a series of purely suc- 

 cessive transformations according to any law whatever. 



Special Cases. (Continued) 

 Consecutive Reversible ^^ Reactions. 



It may be noted in passing that the conclusions of the last section 

 above are still applicable to a broad class of cases of consecutive 

 reactions, including reversible reactions, provided that at least one 

 among them is irreversible. It will suffice here to point out, by the 

 way of example, one such case, that of five consecutive reactions, 

 of which the second and third only are reversible. It is understood 

 that the reactions considered are otherwise purely consecutive, so 

 that the second, for instance, is wholly independent of the progress 

 of the fourth and fifth. 



The determinant A (X) in this case takes the form : 



A(X) = 



In this case also it will be found that A (X) reduces to its dexter 

 diagonal, all the roots for X being in conseciuence necessarily real. 

 The approach to equilibrium is therefore in this case also asymptotic, 



15 The word "reversible" is here of course not understood in its thermo- 

 dynamical sense. 



