26 LOTKA. 



of such constraints we are in no wise assured that the principle holds.* 

 This must be clearly borne in mind in seeking to apply the Le Chate- 

 lier principle, for example, to biological systems. Thus, for instance, 

 the malaria equilibrium under the conditions contemplated by Sir 

 Ronald Ross,^ is independent of the initial amount of malaria in the 

 system (provided only this is not zero). This state of affairs arises 

 out of the fact that there is no equation of constraint of type (S), in 

 this case, connecting the initial amount of malaria with its status at 

 any subsequent epoch. 



Case of more than one variable. A somewhat more complicated case 

 arises if the system under consideration is susceptible of several con- 

 current transformations, so that its state at any instant requires for 

 its definition not one variable x, but a number of such variables. 



It will suffice if we consider here the case for two variables x, y, as, '' 

 for example, the case of a pair of consecutive reversible reactions 



AZi^Z^' (16) 



In this case we have 



dx 



- = J\ (.r, y, G) . (18) 



^ = /, (.T, y, G) (19) 



and equilibrium is defined by 



/i=/2-0 (20) 



Differentiating, in a manner analogous to that followed in the case of 

 a single variable x, we have 



§^«.v, + ^a>/, + f,S6' = (21) 



ax ay ah 



dfo dfo dfo , , 



/^5.r, + f-5.^i + r^6 6' = (22) 



ox dy dG 



4 For there is then no necessary relation between f and A, so that the 



df df 



derivative — r is no longer equal to — , but is indeterminate or meaningless. 



^A OH ^^. 



5 "The Prevention of Malaria," Second English Edition, John Murray, 

 London, 1911, p. 679; Lotka, Nature, Feb. 1912, p. 497. 



