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1 Paper No. 64, University of California, Graduate School of Tropical Agriculture 
and Citrus Experiment Station, Riverside, California. 
2 The writer wishes to acknowledge his great indebtedness to Dr. G. F. McEwen 
of the Scripps Institution for Biological Research of the University of California for 
valuable assistance in the mathematical work here reported. 
ANALYTICAL NOTE ON CERTAIN RHYTHMIC RELATIONS IN 
ORGANIC SYSTEMS 
By Alfred J. Lotka 
Brooklyn, N. Y. 
Communicated by R. Pearl, May 20, 1920 
Periodic phenomena play an important role in nature, both organic and 
inorganic. 
In chemical reactions rhythmic effects have been observed experi- 
mentally, and have also been shown, by the writer ^ and others, ^ to follow, 
under certain conditions, from the laws of chemical dynamics. 
However, in the cases hitherto considered on the basis of chemical 
dynamics, the oscillations were found to be of the damped kind, and 
therefore, only transitory (unlike certain experimentally observed periodic 
reactions). Furthermore, in a much more general investigation by the 
writer, covering the kinetics not only of chemical but also of biological 
systems, it appeared, from the nature of the solution obtained, improbable^ 
that undamped, permanent oscillations would arise in the absence of 
geometrical, structural causes, in the very comprehensive class of systems 
considered. For it seemed that the occurrence of such permanent oscilla- 
tions, the occurrence of purely imaginary exponents in the exponential 
series solution presented, would demand peculiar and very specific rela- 
tions between the characteristic constants of the systems undergoing 
transformation; whereas in nature these constants would, presumably, 
stand in random relation. 
It was, therefore, with considerable surprise that the writer, on apply- 
ing his method to certain special cases, found these to lead to undamped, 
and hence indefinitely continued, oscillations. 
As the matter presents several features of interest, and illustrates 
certain methods and principles, it appears worth while to set forth the 
argument and conclusions here. 
