Vol. 6, 1920 
BIOLOGY: A. J. LOTKA 
415 
and all the higher derivatives of X2 with regard to Xi vanish, as can be 
seen from (29) directly and by successive differentiations. 
Similarly it can be seen that the curve defined by (29) can never cross 
the X2 axis. 
Hence, if any point on any integral curve of (29) lies within the positive 
quadrant, the whole of that curve lies in that quadrant. Thus the 
oscillations can never exceed the limits of positive values Xi, X2. 
We conclude, therefore, that under the conditions of the problem as 
here set forth, neither the species Si nor the species 52 can become extinct 
through severity of the oscillations alone. In practice the eventuality 
might arise, however, that in the course of these oscillations one or the 
other species might be so thinned out as to succumb to any extraneous 
influence that might arise such as has not been taken into account in our 
present considerations. 
We return now briefly to the consideration of the equilibrium defined 
by the equation 
Xi = = 0 (11) 
Applying here the criterion set forth by the author elsewhere,^ it is seen 
that when Ai is positive the determinental equation for X has at this point 
two real roots of opposite sign, which is characteristic of unstable equilib- 
rium. If, on the other hand, Ai is negative in the neighborhood of the 
origin of Xi, X2, then the equilibrium here is found to be stable, the two 
roots for X being both negative. 
In conclusion it may be remarked that a system of equations identical 
in form with (8), (10) is obtained in the discussion of certain consecutive 
autocatalytic chemical reactions. Here, however, the coeflicients A, B 
are constants and the integration can be reduced to a quadrature. Aside 
from a certain number of periodic reactions which have been observed 
more or less as laboratory curiosities, a certain interest is also attached 
to this matter from the fact that rhythmical reactions (e.g., heartbeat, 
which may continue after excision), play an important role in physiology. 
We cannot, of course, say whether in such case geometrical (structural) 
features are the dominating factors. 
1 Lotka, A. J., /. Phys. Chem., 14, 1910 (271-274); Zs. physik. Chem., 72, 1910 (508- 
511); 80, 1912 (159-164) ; P/?};^. Rev., 24, 1912 (235-238) ; Proc. Amer. Acad. Arts Sci., 
55, 1920 (137-153). 
2 Hirniak, J., Zs. physik. Chem., 75, 1910 (675); compare also Lowry and John, J. 
Chem. Soc, 97, 1910 (2634-2645). 
' Lotka, A. J., Proc. Amer. Acad., loc. cit., p. 145, footnote 13. 
^ Lotka, A. J., Science Progress, 14, 1920 (406-417); Proc. Amer. Acad., loc. cit. 
^ Ostwald, Wo., liber die zeitlichen Eigenschaften der Entwickelungsvorgdnge, Leipsic, 
1908, p. 36. 
^ Lotka, A. J., Proc. Amer. Acad., loc. cit., p. 144, et seq. 
