"32 LOTKA. 



It will be seen that in this case the internal parameter H and the 

 •corresponding external parameter He so determine changes in the 

 area a that 



— ^ according as H — He"^ (48) 



that is to say, the parameters H, a and He are related to each other 

 and determine the course of events in a manner analogous to the 

 intensity factor, the capacity factor of an energy, and the "applied 

 force." But it is quite unnecessary to suppose that H and a actually 

 are such factors of an energy in the example cited (population-spread) ; 

 on the contrary, the writer is opposed to this view, which he has taken 

 occasion elsewhere to discuss. ^° For our purposes it is quite immaterial 

 whether P and a are factors of an energy. All we need know is that 

 they enter into the condition (41) as there set forth. 



Condition for Stability toward External Factor. Consider a system 

 for which the condition for equilibrium with the environment is 

 given by 



H = He 



<p (G, H) = constant 



—r^O according as H — Hg^ 

 dt < <• 



Let 



<p (G, H) = const., i.e. H = x (G) (50) 



be plotted as ordinates in a rectangular system in which G is plotted 

 as abscissae. Then it is easily shown that the condition for stability 

 •of equilibrium is that the curve H — x (G) must slope downwards 

 from left to right. 



For, suppose it sloped upwards. Let the system be in equilibrium 

 at a point Ai (Fig. 1), where 



H = Hl= He \ /ri\ 



G=G, j ^^^^ 



-Suppose the system is in any way displaced to the point A^ where 



H2 > H, (52) 



> He (53) 



10 "Economic Conversion Factors of Energy," to appear in a forthcoming 

 issue of Proc. Nat. Ac. 



(49) 



