Analytical Note on Certain Rhythmic Relations in Organic Systems

Abstract

Periodic phenomena play an important role in nature, both organic and inorganic. In chemical reactions rhythmic effects have been observed experimentally, and have also been shown, by the writer1 and others,2 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, improbable3 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 oscillations, the occurrence of purely imaginary exponents in the exponential series solution presented, would demand peculiar and very specific relations 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 applying 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. Starting out first from a broad basis, we may consider a system in the process of evolution, such a system comprising a variety of species of matter S 1, S 2…. S n of mass X 1, X 2…. X n . The species of matter S may be defined in any suitable way. Some of …