146 LOTKA. 



only even powers of X, with all coefficients of the same sign), then the 

 series (8) may converge for all values of t. In that case the process 

 would be periodic ah initio, and the equilibrium at the origin would be 

 stable in the same sense as in case (4) above. 



To recapitulate, we may enunciate two propositions, of which one 

 is the converse of the other: 



Given that the series (8) are convergent, 



1 . If none of the real parts /x of the roots X of the equation A (X) = 

 is positive, the sj^stem is in stable equilibrium at the origin of the a;'s. 



This stability is absolute if none of the ^t's are zero. 

 It is relative, of the type (4) or (5), if A (X) = has one or more 

 roots in which /x = 0. 



2. If the system is stable (absolutely or relatively) at the origin 

 of the x's, then the equation A (X) = can not have a root X such 

 that /x > 0. 



Special Cases. 



1. Only one dependent variable. 



If ?! — m = 1, i. e., if the number of equations of constraint is only 

 one less than the total number of primitive dependent variables X, 

 then the state of the system at any instant is completely defined by 

 statement of the value of one single variable Y, given the values of 

 the initial constants A] , A^, . . . and of the parameters Pi, P2, ■ The 

 transformation can in this case be regarded as one single reaction. 



The commonest example of this kind is that of one simple chemical 

 reaction, reversible or not, unaccompanied by any simultaneous 

 reaction of any kind (such as side reactions, consecutive reactions, 

 etc.). For the variable Y we may then employ the mass (or con- 

 centration) of an arbitrarily selected component, or any convenient 

 multiple thereof.^* 



In this case, if we omit from the functional parenthesis the para- 

 meters Ai, A2,..., the system of equations (19) reduces to a single 

 equation. 



14 In chemical dynamics it is customary to employ for the variable Y in a 

 single reaction the change in concentration, from the origin of time to the 

 instant /, of one or more, chosen arbitrarily, of the components. The relation 

 between the X's, the A's and the Y's, at constant volume, is then 



Y = A - X. 



A somewhat different convention is followed by Jtittner loc. cit. See also 

 Mellor, Chemical Statics and Dynamics, 1904, pp. 85, 86; 91, 92. 



